ABSTRACT
Title of dissertation: ULTRACOLD BOSONIC ATOMS IN OPTICAL LATTICES
Ana Maria Rey, Doctor of Philosophy, 2004
Dissertation directed by: Prof. Theodore R. Kirkpatrick
Department of Physics
Dr. Charles W. Clark
National Institute of Standards and Technology
This thesis covers most of my work in the field of ultracold atoms loaded in optical lattices. It makes
a route through the physics of cold atoms in periodic potentials starting from the simple noninteracting
system and going into the many-body physics that describes the strongly correlated Mott insulator regime.
Even though this thesis is a theoretical work all the chapters are linked either with experiments already done
or with ongoing experimental efforts.
This thesis can be divided into four different parts. The first part comprises chapters 1 to 3. In these
chapters, after a brief introduction to the field of optical lattices I review the fundamental aspects pertaining
to the physics of systems in periodic potentials.
The second part deals with the superfluid weakly interacting regime where standard mean field
techniques can be applied. This is covered in chapters 4 and 5. Specifically, chapter 4 introduces the
discrete nonlinear Schr?odinger equation (DNLSE) and uses it to model some experiments. In chapter 5 I go
one step further and include the small quantum fluctuations neglected in the DNLSE by studying quadratic
approximations of the Bose-Hubbard Hamiltonian.
Chapters 6 to 8 can be grouped as the third part of the thesis. In them I adopt an effective action
formalism, the so called two particle irreducible effective action (2PI) together with the closed time path
(CTP) formalism to study far-from-equilibrium dynamics. The many-body techniques discussed in these
chapters systematically include higher order quantum corrections, not included in the quadratic approxi-
mations of the Hamiltonian, which we show are crucial for a correct description of the quantum dynamics
outside the very weakly interacting regime.
Finally, chapter 9 to 11 are devoted to study the Mott insulator phase. In these chapters using
perturbation theory I study the Mott insulator ground state and its excitation spectrum, the response of the
system to Bragg spectroscopy, and propose a mechanism to correct for the residual quantum coherences
inherent to the Mott insulator ground state. Even though small these are not ideal for the use of neutral
atoms in optical lattice as a tool for quantum computation.
Ultracold bosonic atoms in optical lattices
by
Ana Maria Rey
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2004
Advisory Committee:
Prof. Theodore R. Kirkpatrick, Chair/Advisor
Dr. Charles W. Clark,
Prof. Bei-Lok Hu,
Prof. Steve Rolston,
Prof. J. Robert Dorfman
c Copyright by
Ana Maria Rey
2004
There are men who fight one day and are good.
There are men who fight one year and are better.
There are some who fight many years and they are better still.
But there are some that fight their whole lives,
these are the ones that are indispensable.
Berthold Brecht
ii
ACKNOWLEDGMENTS
I want to use this lines to thank and acknowledge all the people that made this thesis possible.
First and foremost I would like to thank Charles Clark for giving me the invaluable opportunity to
join the NIST group. Through his intelligent ideas, love for physics and dedication he gave me not only
very good advice and guidance but also all the encouragement I needed to feel the work I was doing was
worthwhile and important. It has been a pleasure to work with and learn from such an extraordinary person.
I also want to thank Prof. Ted Kirkpatrick for serving as my University of Maryland supervisor. He
has always made himself available for help and advice. His ideas and suggestions were really valuable for
the development of this thesis.
I am also grateful to Prof. Bei Lok Hu. He was an invaluable source of knowledge and assistance,
especially with regard to the 2PI-CTP formalism discussed in chapters 6 to 8. There has never been an
occasion when I?ve knocked on his door and he hasn?t given me time. I cannot leave out Esteban Calzetta
and Albert Roura. Discussions with Bei Lok, Esteban and Albert regarding the application of the 2PI-CTP
formalism to the patterned loaded system made possible the publication of the paper.
I would also like to acknowledge help and support from Carl Williams. He was my other advisor at
NIST, willing to help me all the time and providing me with very useful ideas.
I also owe special gratitude to everybody in the Physics Laboratory at NIST and colleagues in the
Radiation Physics Building, in particular Blair Blakie, Zach Dutton, Nicolai Nygaard, Marianna Safranova,
Jamie Williams and Mark Edwards. Besides colleagues they have been wonderful friends and helped me
any time I needed help or advice. Especially I am greatly indebted with Blair who was not only my office
mate when he was at NIST and helped me to sort out all the problems but also for his support and great
ideas even now that he is in New Zealand. I also want to thank Nicolai not only for providing me all the
LaTex style files for writing this thesis but also for all his time reviewing the manuscript. Guido Pupillo and
Gavin Brennen also deserve a special mention. Because of their great ideas and critical thinking the work
iii
described in chapters 9 to 11 was possible. I can not forget to thank Prof. Keith Burnett. I worked with
him during the summer of 2002 when he came to NIST. Besides an incredible person, he is also a bright
physicist. He was the one who gave me the idea of using the superfluid properties to study the approach to
the Mott insulator phase.
I owe my deepest thanks to my family - my mother, my father, my sister and my brother. They have
always stood by me in good and bad times. I really owe all my achievements to them.
Finally, I have no words to express the gratitude I have to my husband Juan Gabriel. He is the most
important person in my life. We came together to pursue the PhD. program and without his love and support
the completion of this work would not have been possible.
iv
TABLE OF CONTENTS
List of Figures ix
1 Introduction 1
2 Optical lattices 10
2.1 Basic theory of optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1 AC Stark Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 Dissipative interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.3 Lattice geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Single particle physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Bloch functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Wannier orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.3 Tight-binding approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.4 Semiclassical dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 The Bose-Hubbard Hamiltonian and the superfluid to Mott insulator transition 28
3.1 Bose-Hubbard Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Superfluid-Mott insulator transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Mean field theory 35
4.1 Discrete nonlinear Schr?odinger equation (DNLSE) . . . . . . . . . . . . . . . . . . . . . 35
4.2 Dragging experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2.2 Linear free particle model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2.3 Effect of the magnetic confinement . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2.4 Interaction effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Dynamics of a period-three pattern-loaded Bose-Einstein condensate in an optical lattice . 46
4.3.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
v
4.3.2 Case of no external potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3.3 Dynamics with a constant external force . . . . . . . . . . . . . . . . . . . . . . . 57
5 Quadratic approximations 67
5.1 The characteristic Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 Bogoliubov-de Gennes (BdG) equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3 Higher order terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.3.1 Hartree-Fock-Bogoliubov equations . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.3.2 HFB-Popov approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.4 The Bose-Hubbard model and superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.5 Expectation values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.6.1 Translationally invariant lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.6.2 One-dimensional harmonic trap plus lattice . . . . . . . . . . . . . . . . . . . . . 87
5.7 Improved HFB-Popov approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.7.1 The two-body and many-body scattering matrices . . . . . . . . . . . . . . . . . 102
5.7.2 The anomalous average and the many-body scattering matrix . . . . . . . . . . . 104
5.7.3 Improved Popov approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6 The two particle irreducible effective action (2PI) and the closed time path (CTP) formalism 112
6.1 2PI effective action ?(z;G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Perturbative expansion of ?2(z;G) and approximation schemes . . . . . . . . . . . . . . 118
6.2.1 The standard approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2.2 Higher order expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.2.3 Zero mode fluctuactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.3 CTP formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.4 HFB approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
vi
6.4.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.4.2 Mode expansion of the HFB equations . . . . . . . . . . . . . . . . . . . . . . . . 128
6.5 Second-order expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.5.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.5.2 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7 Nonequilibrium dynamics of a patterned loaded optical lattice: Beyond the mean field approxima-
tion 139
7.1 Mean field dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.1.1 Dynamical evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.1.2 Comparisons with the exact solution . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.2 2PI-CTP approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.2.1 Initial conditions and parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.2.2 Numerical algorithm for the approximate solution . . . . . . . . . . . . . . . . . . 144
7.2.3 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8 From the 2PI-CTP approximations to kinetic theories and local equilibrium solutions 157
8.1 Rewriting the 2PI-CTP second-order equations . . . . . . . . . . . . . . . . . . . . . . . 157
8.2 Boltzmann equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.3 Equilibrium properties for a homogeneous system . . . . . . . . . . . . . . . . . . . . . . 167
8.3.1 Quasiparticle formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
8.3.2 HFB approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
8.3.3 Second-order and Beliaev approximations . . . . . . . . . . . . . . . . . . . . . . 171
8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
9 Characterizing the Mott Insulator Phase 175
9.1 Commensurate translationally invariant case . . . . . . . . . . . . . . . . . . . . . . . . . 175
vii
9.1.1 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
9.2 Harmonic confinement plus lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
9.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
10 Bragg Spectroscopy 190
10.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
10.2 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
10.3 Linear response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
10.4 Zero-temperature regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
10.4.1 Bogoliubov approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
10.4.2 Inhomogeneous system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
10.5 Mott insulator regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
10.5.1 Commensurate homogeneous system . . . . . . . . . . . . . . . . . . . . . . . . 203
10.5.2 Inhomogeneous system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
10.6 Finite temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
10.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
11 Scalable register initialization for quantum computing in an optical lattice 212
11.1 Homogeneous dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
11.2 Dynamics in presence of the external trap . . . . . . . . . . . . . . . . . . . . . . . . . . 217
11.3 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
11.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
12 Conlusions 226
Bibliography 228
viii
LIST OF FIGURES
2.1 AC Stark shift induced by atom-light interaction. The laser frequency is ! = 2?? which is
detuned from the atomic resonance by ? . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Optical lattice potential.(a)-(e) potentials for different configurations of 3 beams, (f) poten-
tial for the 4 counter-propagating laser beam configuration (The two pair of light fields are
made independent by detuning the common frequency of one pair of beams from the other.
ER is the atomic recoil energy, ER =~2k2=2m. This figure is a courtesy of P. Blair Blakie
[34] ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Band structure of an optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Probability density of the Bloch wave function `(n=1)q=0 for different lattice depths. It can be
observed the localization of the wave function increases with the lattice increases. . . . . 18
2.5 Wannier state log-density distribution corresponding to the lattice configurations shown in
Fig. 2.2. Dotted white lines indicate the potential energy contours. The numbers labels the
near neighbor sites: 1 the nearest neighbors, and so on. This figure is a courtesy of P. Blair
Blakie [34] ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Ground state amplitudes f(0)i as a function of the lattice site i for different values of ? . . 23
2.7 Diffusion of a square wave packet in the presence of a linear potential.The parameters for
the plot are ? = 0:1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.8 Harmonic confinement plus lattice spectrum: The index n labels the eigenvalues in increas-
ing energy order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1 Zero-temperature phase diagram. The vertical lines indicate critical values for different
filling factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1 Condensate density in the x-direction at the observation time tf. Light colors areas indicate
regions of high condensate density. The dashed line indicates the final location of a co-
moving point with the optical lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
ix
4.2 Atom?s average velocity in the lab frame vs. lattice velocity. Line: results from the Bloch
dispersion relation, dots: tight-binding approximation results. vB =~?=am. . . . . . . . 40
4.3 Atom?s average velocity in the lab frame vs. lattice velocity in the presence of a magnetic
trap. vB =~?=am. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4 Velocity of the atoms predicted by the variational model in the case when interactions are
taken into account (dotted line). The single particle case is shown in red. The parameters
used were  (0) = 10;?(0) = 0 and NU = 2ER. . . . . . . . . . . . . . . . . . . . . . . 44
4.5 Phase diagram UN vs vlatt for Vo = 3Er and  o = 10. . . . . . . . . . . . . . . . . . . 45
4.6 Localization of the wave packet in the self-focusing regime vlatt = ?0:7vB. Notice the
change in the vertical scale in the last row . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.7 Evolution of the wave packet in the diffusive regime vlatt = ?0:15vB. . . . . . . . . . . 46
4.8 Patterned loading experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.9 Oscillation period (in units of~=J) as a function of the interaction strength  . . . . . . . . 51
4.10 Minimum value of f2 during an oscillation period as a function of  . As  increases the
population imbalance between wells increases (see text). . . . . . . . . . . . . . . . . . . 52
4.11 Maximum contrast of the Fourier components as a function of  . The maximum contrast is
defined as ?max ? (jc1j2 ?jc0j2)max with the maximum value occurring when f2 is at its
minima. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.12 Comparison between the population evolution of the central three wells for the inhomo-
geneous condensate and the homogeneous model. Inhomogeneous condensate: stars for
the initially populated well and boxes for the initially empty wells. Homogeneous model:
dashed line represents the initially populated wells, and the solid line represented the ini-
tially unpopulated wells. We used  efi as the local mean field energy (see text). The param-
eters used for the simulation were J = 0:075ER and  efi = 2:64. . . . . . . . . . . . . . 55
4.13 Momentum distribution of the inhomogeneous condensate evaluated at various times for
the same parameters as used in Fig.4.12. . . . . . . . . . . . . . . . . . . . . . . . . . . 56
x
4.14 Evolution of momentum peak populations. Upper curves: population of the q = ?2?=3
momentum states. Lower curves: population of the q = 0 momentum state. Inhomoge-
neous condensate (dotted), homogeneous result (solid line), where the comparison is made
by replacing  by an average mean field energy  ave ? ?Pnjznj4 = 1:85. Parameters are
the same as in Fig. 4.12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.15 Evolution of the normalized population for different values of ?. One Bloch period is shown
in the plots except ? = 0 where the period is infinite. Solid line: j~?3nj2, dotted line:
j~?3n+1j2, dash-dot line: j~?3n+2j2. The ?nonclassical? motion can be seen where the 3n+1-
well populations initially increase more rapidly that the populations of the 3n+2-wells. It
can also be seen in the plots that ? = ?resmax=2 and ? = ?resmax are resonant values. . . . . . . 62
4.16 The spectrum of values of external force, ordered in decreasing magnitude, for which a
population resonance occurs i.e. the values of ? for which ~?0 periodically disappears. . . 63
4.17 Effects of interactions on generalized Bloch oscillations for the pattern loaded system. Evo-
lution of j~?nj2 for various interaction strengths. Upper plot: ? = 2?resmax. Lower plot:
? = 2?resmax=7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.18 Comparison between the evolution of a inhomogeneous condensate with the homogeneous
result. Inhomogeneous condensate: j~?0j2 (boxes), j~?1j2 (stars), and j~?2j2 (diamonds).
Homogeneous case: j~?3nj2 (dash-dot line), j~?3n+1j2 (solid line), and j~?3n+2j2 (dotted
line), where we have taken  as the local mean field energy. The parameters used were
J = 0:075ER, ? = 2 and  efi = 1:59 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.19 Evolution of the momentum peak populations. q = 0 (dashed line, squares), q = ?2?=3
(dash-dot line, stars), q = 2?=3 (solid line, diamonds). Inhomogeneous condensate (dotted
curves), homogeneous model (lines) using the same parameters as those in Fig. 4.18, but
replacing  by an average value  ave = 1:11 for the homogeneous model. . . . . . . . . . 66
5.1 Scattering processes included in the quadratic hamiltonian: a) Direct and exchange excita-
tions, b) Pair excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
xi
5.2 Comparisons of the exact solution and BdG (and HFB-Popov) solutions as a function of
Veff = U=J, for a system with M = 3 and filling factors n = 5 and 5:33. Left: number
fluctuations (Exact: solid line, BdG(and HFB-Popov): dotted line), middle: condensate
fraction (Exact: solid line, BdG(and HFB-Popov): dotted line,analytic (5.79:red line ),
right: superfluid fraction fs (Exact: solid line, BdG(and HFB-Popov): dotted line). The
exact second order term (dashed line) of the superfluid fraction, f(2)s is also shown in these
plots. The vertical line shown in the plots is an estimation of V criteff when the system is
commensurate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.3 Comparisons of the exact solution and BdG(and HFB-Popov) solutions as a function of
Veff = U=J, for a system with M = 3 and filling factors n = 50 and 50:33. Left: number
fluctuations (Exact: solid line, BdG(and HFB-Popov): dotted line), middle: condensate
fraction (Exact: solid line, BdG(and HFB-Popov): dotted line, analytic Eq.(5.79): red line
), right: superfluid fraction fs (Exact: solid line, BdG(and HFB-Popov): dotted line). The
exact second order term (dashed line) of the superfluid fraction, f(2)s is also shown in these
plots. In this case the agreement is much better. . . . . . . . . . . . . . . . . . . . . . . . 85
5.4 Comparisons of the condensate fraction given by the BdG solutions and the HFB equations
as a function of Veff = U=J, for a system with M = 3 and filling factor n = 10. . . . . . 87
5.5 Comparisons of the condensate wave function found by numerically solving the BdG equa-
tions with the Thomas-Fermi solution. The parameters used were U=J = 0:2, ?=J =
9:5?10?4 and N = 100. The depletion is also shown in the plot. . . . . . . . . . . . . . 91
5.6 Comparisons of the quasiparticle spectrum found by numerically solving the BdG equa-
tions(BdG) with the Thomas-Fermi solution (TF) and the noninteracting energies (Free).
The parameters used were U=J = 0:2, ?=J = 9:5?10?4 and N = 100. . . . . . . . . . 92
5.7 Low-lying quasiparticle amplitudes found by numerically solving the BdG equations. The
parameters used were U=J = 0:2, ?=J = 9:5?10?4 and N = 100. . . . . . . . . . . . 93
xii
5.8 Comparisons between the product FsnGsn calculated from the Thomas-Fermi approximation
and the exact numerical solution of the BdG equations. The parameters used were U=J =
0:2, ?=J = 9:5?10?4 and N = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.9 Condensate density (triangles), total density (filled boxes) and local depletion (empty
diamonds) as a function of the lattice site i for different values of Veff:.The site indices i
are chosen such that i = 0 corresponds to the center of the trap. Although these quantities
are defined only at the discrete lattice sites we join them to help visualization. The empty
boxes represent the exact solution for the case J=0. . . . . . . . . . . . . . . . . . . . . . 96
5.10 Quasiparticle spectrum predicted by the HFB-Popov theory for different values of Veff:
Empty diamonds (Veff = 0:01), stars (Veff = 0:09), crosses (Veff = 3), filled diamonds
(Veff = 11), empty boxes (Veff = 100) and pentagons (Veff = 312). The letter q labels
the quasiparticle energies in increasing order. The quasiparticle energies are in recoil units. 97
5.11 Number fluctuations in the self consistent HFB-Popov approach as a function of lattice site
for Veff = 0:01 (boxes), Veff = 0:09 (crosses), Veff = 3 (circles),Veff = 11 (triangles),
Veff = 100 (stars) and Veff = 312 (diamonds). The maximum value reached by the
profile decreases as Veff is increased. The empty boxes shown for each of the curves
correspond to the number fluctuations predicted by the homogeneous HFB-Popov model
using a local density approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.12 Quasimomentum distribution as a function of qa, a the lattice spacing, q the quasimomen-
tum, for different values of Veff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.13 First order on-site superfluid fraction as a function of the lattice site i for different values
of Veff..The site indices i are chosen such that i = 0 corresponds to the center of the trap.
Filled boxes: Veff. =0.01, empty boxes: Veff. = 0:09, empty diamonds: Veff. = 3, stars:
Veff. = 11, crosses: Veff. = 100 and triangles: Veff. = 312. . . . . . . . . . . . . . . . . 100
5.14 Diagrammatic representation of the two-body T2b scattering matrix. In this figure jkli
designates the initial states, jiji the final states and jnpi a set of intermediate states . . . . 103
xiii
5.15 Comparisons of the condensate fraction predicted from the exact diagonalization of the
BHH (red), the improved Popov approximation (yellow), the BdG (and HFB-Popov) equa-
tions (blue) and the test field approximation (green) as a function of Veff = U=J, for a
system with M = 3 and filling factors n = 5 and 5:33. . . . . . . . . . . . . . . . . . . . 108
5.16 Comparisons of the condensate fraction predicted from the exact diagonalization of the
BHH (red), the improved Popov approximation (yellow), the BdG (and HFB-Popov) equa-
tions (blue) and the test field approximation (green) as a function of Veff = U=J, for a
system with M = 3 and filling factors n = 50 and 50:33. . . . . . . . . . . . . . . . . . . 109
5.17 Comparisons between no=nTmb = fcTmb and U as a function of U for a system with
M = 3 and filling factors n = 5 and 50. . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.1 Two-loop (upper row) and three-loop diagrams (lower row) contributing to the effective
action. Explicitly, the diagram a) is what we call the double-bubble , b) the setting-sun and
c) the basketball. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.1 Comparisons between the exact and the DNLSE solutions for six atoms and three wells. The
time is given in units of~=J. Top panel, strongly correlated regime ( = 12); middle panel,
intermediate regime ( = 2); bottom panel, weakly interacting regime ( = 0:2). The
solid line is the DNLSE prediction for the population per well: jz0(t)j2 and jz1;2(t)j2, the
triangles are used to represent the exact solution for the population per well calculated using
the Bose Hubbard Hamiltonian: h^ay0^a0i, h^ay1;2^a1;2i. The pentagons show the condensate
population per well calculated from the exact solution: jh^a0ij2 and jh^a1;2ij2. Due to the
symmetry of the initial periodic conditions the curves for the i = 1 and 2 wells are the same
in all depicted curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
xiv
7.2 Comparisons between the exact solution(solid line), the HFB approximation (boxes), the
second-order large N approximation (pentagons) and the full 2PI second-order approxima-
tion(crosses) for the very weak interacting regime. The parameters used were M = 3;N =
6;J = 1 and U=J = 1=30. The time is given in units of ~=J. In the plots where the
population, condensate and depletion per well are depicted the top curves correspond to the
initially populated well solutions and the lower to the initially empty wells. Notice the dif-
ferent scale used in the depletion plot. In the momentum distribution plot the upper curve
corresponds to the k = ?2?=3 intensities and the lower one to the k = 0 quasi-momentum
intensity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.3 Comparisons for the case M = 3;N = 8;J = 1 and U=J = 1=3. The time is given in
units of ~=J. In the plots the abbreviation 1st is used for the initially occupied well and
2nd for the initially empty wells. In the quasimomentum plots k = 2?=a is the reciprocal
lattice vector with a the lattice spacing. . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.4 Comparison between the evolution of the atomic population per well for M = 2;J = 1=2,
NU=J = 4 and N = 20;40 and 80. Time is in units of ~=J. In the plots P1 stands for the
fractional atomic population in the initially populated wells and P2 for the population in the
initially empty wells. The number of atoms N is explicitly shown in each panel. . . . . . 154
7.5 Time evolution of the condensate population per well and the total condensate popula-
tion,for the same parameters as Fig. 7.4. Time is in units of ~=J. In the plots C1 stands
for the fractional condensate population in the initially populated well, C2 for the fractional
condensate population in the initially empty one and CT for the total condensate fraction. 155
7.6 Dynamical evolution of the quasi-momentum intensities. The parameters used were M =
2;J = 1=2, NU=J = 4 and N = 20;40 and 80. Time is in units of ~=J. In the plots
ko denotes the k = 0 quasi-momentum component and k1 the k = ?=a one (a the lattice
spacing). The plots are scaled to set the integrated quasi-momentum density to one for all
N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
xv
9.1 Contour plot of the two dimensional band of the 1-ph excitations to fist order in perturbation
theory. In the plot the brighter the color the higher the energy. The labels are ky = 2?=MR
and kx = 2?=MR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
9.2 Comparisons between the first order corrections to the 1-ph excitations calculated by diag-
onalizing the Bose-Hubbard Hamiltonian inside the 1-ph subspace and the analytic solution
Eq. (9.22). The number of sites used for the plot was M = 11. Energies are in units of J. . 181
9.3 Comparisons between the energy eigenvalues calculated by the exact diagonalization of
the Hamiltonian (red dots), the diagonalization of the Hamiltonian in the restricted 1-ph
subspace (green triangles) and the analytic solution (blue crosses), Eq. (9.22). The index
n labels the eigenvalues in increasing order of energy. The filling factor g, the number of
wells M and the ratio U=J is indicated in each plot. Energies are in units of J. . . . . . . 182
9.4 Quasimomentum distribution as a function of U=J for a unit filled lattice with six wells
(M = 6). The exact solutions are displayed with solid lines and the first order perturbative
results with dotted lines. Due to the lattice symmetry n4?=3 = n2?=3 and n5?=3 = n?=3. . 183
9.5 Comparisons between the exact (red) and the perturbative (blue) spectra for a trapped sys-
tem with N = 9, M = 11 and ?=J = 1:875 deep in the Mott regime. The lines indicate
the energies of the states that are coupled directly to the ground state. . . . . . . . . . . . 188
10.1 Bragg spectroscopic scheme considered in this paper. The lattice potential with atoms
loaded into the ground band is perturbed by a shallow running wave perturbation (the Bragg
potential). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
10.2 Imparted energy: Comparisons between the the exact and Bogoliubov approximation for
N = M = 9. The horizontal axis is in units of ~. For the plot we used J?=~ = 10. See
text Eq. (10.23). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
xvi
10.3 Comparisons of the imparted energy vs. Bragg frequency curves calculated from the exact
diagonalization of the manybody Hamiltonian (red) and the from the HFB-Popov approxi-
mation(blue) for a trapped system with N = 9, M = 11 and ?=J = 1:875. The horizontal
axis is in units of ~. The lines depicted in the plot are located at the different quasiparticle
energies excited by the Bragg perturbation. . . . . . . . . . . . . . . . . . . . . . . . . . 201
10.4 Density profile calculated from the exact diagonalization of the many-body Hamiltonian
(red) and from the HFB-Popov approximation (blue) for a trapped system with N = 9,
M = 11 and ?=J = 1:875. n labels the lattice site. . . . . . . . . . . . . . . . . . . . . . 202
10.5 Dynamical structure factor vs. ~!=J in the Thomas-Fermi (blue) and HFB-Popov (red)
approximations. The horizontal axis is in units of~. The system parameters are U=J = 0:2,
?=J = 9:5?10?4 and N = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
10.6 Eigenvalues of the quasiparticle spectrum plotted in increasing energy order. The vertical
scale is in units of~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
10.7 Transfer energy vs. Bragg frequency for a zero temperature homogeneous system with
M = N = 9 and different transfer momenta q. The horizontal axis is in units of ~. The
perturbation time was set to J?=~= 10. Note the change of the vertical scale between the
two panels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
10.8 Transfer energy vs. Bragg frequency for a zero temperature trapped system with parameters
U=J=45, ?=J = 1:857, M = 11 and N = 9 and different transfer momenta q. The
horizontal axis is in units of ~. The perturbation time was set to J?=~ = 10. Note the
change of the vertical scale between the two panels. . . . . . . . . . . . . . . . . . . . . 205
10.9 Transfer energy as a function of the temperature for different U=J parameters. The plots
are for a homogeneous system with N = M = 9. The horizontal axis in in units of~. . . 210
10.10Transfer energy as a function of the temperature for different U=J parameters. The plots
are for an trapped system with M = 11, N = 9 and ?=J = 1:875. The horizontal axis in
in units of~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
xvii
11.1 Comparisons between the time evolution of the percentage fidelity calculated by diagonal-
izing the Bose-Hubbard Hamiltonian (dots), the perturbative solution inside the one particle
hole subspace (red) and the analytic solution Eq. 11.6 (blue). . . . . . . . . . . . . . . . . 215
11.2 Percentage fidelity as a function of time calculated using the perturbative solution 11.4 (red)
and the analytic solution Eq. 11.6 (blue).The number of sites used for the plot is M = 31 . 216
11.3 Schematic of an inhomogeneous lattice filled with N qubits with an onsite interaction en-
ergy U. An externally applied trapping potential of strength Vj = ?j2, e.g. due to a
magnetic field, acts to fill gaps in the central region of the trap. The center subspace R of
the lattice defines the quantum computer register containing K < N qubits. . . . . . . . . 217
11.4 Comparisons between the time evolution of the percentage fidelity Fcom(K = 5;t) cal-
culated by evolving the initial target state using the Bose-Hubbard Hamiltonian (dots), the
restricted one particle hole basis (red) and the analytic solution Eq. 11.16 (blue). In the plot
we assumed a commensurate unit filled lattice with five sites and infinitely high boundaries. 220
11.5 Percentage fidelity as a function time calculated using the approximated solution 11.16
(blue) and the fidelity found numerically by restricting the dynamics to the one particle
hole subspace. We assumed infinity high boundaries and K = 31. . . . . . . . . . . . . . 221
11.6 Schematic of the relevant couplings in the problem. The unit filled state jTi describing a
target quantum register and the states jS?j i having one doubly occupied lattice site and a
neighboring hole are coupled to first order in J. A catalysis laser resonantly couples the
ground states jS?j i to the excited states jM?j i describing a bound molecule at the dou-
bly occupied site. The bound states quickly decay and give the possibility of monitoring
population in the ?faulty? register states jS?j i. . . . . . . . . . . . . . . . . . . . . . . . . 222
xviii
11.7 Population in the unit filled register state jTiR during continuous measurement of the regis-
ter beginning in the Bose-Hubbard ground state j?gi. The plots show dynamics appropriate
to tunneling in one dimension with U=J = 500. (a) Quantum trajectories corresponding to
a null measurement result for three different register sizes K. The time scale to saturate the
target state is independent of the number of qubits: tsat ? ??1. (b) Long time dynamics
for K = 501, N = 551 and finite detector efficiencies ?. The population in jTiR for ? = 1
is indistinguishable from one. Also shown is the oscillatory dynamics at fundamental fre-
quency U described by Eq. (11.16) if the measurement is turned off after the target state is
reached. The arrow indicates ?T;T(0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
xix
Chapter 1
Introduction
Bose Einstein condensation (BEC) in bosonic gases was predicted by Einstein in 1925 [1] based on the
quantum statistics ideas developed by Bose for photons [2]. The basic idea of BEC is that below a critical
temperature a macroscopic number of particles occupy the lowest energy state: as the temperature, T,
is decreased the de-Broglie wave length, which scales like T?1=2, increases and at the critical point it
becomes comparable to the inter-particle mean separation. At this point the wave functions of the particles
are sufficiently smeared out so that there is always some overlap and a Bose Einstein condensate is formed.
Although Einstein?s prediction applied to a gas of noninteracting atoms, London suggested that, BEC could
be the mechanism underlying the phenomenon of superfluidity in 4He [3], despite the strong interactions
in this system. Further evidence for this point of view came from neutron scattering experiments [4].
Experimental efforts to create a BEC in dilute gases date back to the 1980?s [5]. The first experiments
concentrated on using atomic Hydrogen but mainly the large rates of inelastic collision prevented these ex-
periments from succeeding. It was not until 1995, using the advances made in laser cooling techniques[9],
that BEC in dilute alkali atomic gases was achieved. The first series of experiments were done with Ru-
bidium [6], sodium [7] and lithium [8] vapors. In these experiments atoms are typically collected in a
magneto-optical trap (MOT) and compressed and cooled to micro-Kelvin temperatures using laser cool-
ing techniques. They are then transferred to a magnetic trap where evaporative cooling allows the system
to be cooled to nano-Kelvin temperatures. At a critical phase space density BEC takes place. In such a
condensate a macroscopic number of atoms, generally up to 106, collectively occupy the lowest energy
state.
The experimental realization of BEC in alkali gases opened unique opportunities for exploring quan-
tum phenomena on a macroscopic scale. In contrast to experiments with liquid helium, where the strong
interactions between particles wash out the effects due to the BEC, the relatively weak two-particle interac-
tion in dilute alkali atoms allows these systems to be used as a theoretical and experimental arena to study
coherent matter wave properties. Theoretically, the weakly interacting regime has the advantage that all the
atoms can be described by a single macroscopic wave function. This allows a very intuitive understanding
1
of the system based on the so called Gross-Pitaevskii equation (GPE) [10]. Experimentally, the macro-
scopic wave packet can be probed by interference experiments, where by turning off the trapping fields the
atoms are allowed to expand and the wave packets to interfere with each other.
The GPE assumes that all the atoms are in the condensate and neglects completely quantum corre-
lations. In the weakly interacting regime, a description beyond the simplest mean field theory can be made
by treating the small quantum fluctuations as perturbations. This treatment was proposed by Bogoliubov in
1947 [11]. The fluctuations lead to a small depletion of the condensate mode since other excited states dif-
ferent from the condensate get populated. Since in most of the experiments with dilute gases the condensate
depletion is at most 3%, the GPE together with Bogoliubov analysis have been in general very successful
describing these experiments. Much theoretical and experiment work has been done studying condensate
properties such as condensate collective excitations, phonon modes, sound velocity and superfluid flow
phenomena [12]-[19].
However, weakly interacting dilute gases described by a mean field picture are the simplest many
body systems one can possible found. In order to be in the weakly interacting regime, the ratio between the
interaction energy of uncorrelated atoms at a given density, Eint, and the quantum kinetic energy needed to
correlate particles by localizing them within a distance of order of the mean inter-particle distance, Ekin,
must be small. For three dimensional systems Eint ? n4?~2asm (as is the scattering length, which fully
characterizes the low energy scattering processes, n is the mean particle density and m is the atomic mass)
and Ekin ? ~22mn2=3. Thus, the ratio between these two energies is proportional to n1=3as. In dilute
alkali vapors this ratio is generally of order 0:02. To enter the strongly correlated regime, an obvious way
to proceed is either to raise the density or to raise the scattering length. It is indeed possible to tune the
scattering length to large values by using a Feshbach resonance. This has recently been realized for example
in 85 Rb where the scattering length was tuned over several orders of magnitude [20, 21]. The problem of
this approach is, however, that the lifetime of the condensate strongly decreases due to three-body losses
[22]. An entirely different way to reach the strongly correlated regime is by using optical lattices. By
increasing the depth of the optical lattice the ratio between kinetic energy and potential energy can be
changed without affecting the density or the scattering length. The beauty of this approach is that the lattice
2
depth can be used as an experimental knob to change the kinetic to interaction energy ratio allowing us to
reach different many-body regimes.
? Optical lattices
Optical lattices are periodic Stark shift potentials created by the interference of two or more laser
beams. They have been widely used in atomic physics in the context of atom diffraction [23, 24] with
applications to atom optics and atom interferometry [25]. They have also been used as a way to trap
and cool atoms. The first experiment where atoms were cooled to the micro-kelvin regime in a multi-
dimensional optical lattice was carried out by Hemmerich et al. [26]-[28] followed by Grynberg et al. [29].
There have been various attempts to cool atoms directly in an optical lattice. Among them we can mention
Raman cooling techniques, by which atoms have recently been partially cooled to the ground sate with
filling factors of order one [30, 31, 32]. However, one of the most successful techniques to load ultracold
atoms in the ground state, with almost no discernible thermal component, is by first forming a Bose Einsten
condensate in a weak magnetic trap and then adiabatically turning on the lattice by slowly ramping up the
intensity of the laser beams.
Atom dynamics in optical lattices is closely related to electron dynamics in solid state crystals, but
optical lattice have favorable attributes such as the absence of defects and the high degree of experimental
control [33, 34]. When ultracold bosonic atoms are loaded in shallow lattices, the system is in the weakly
interacting regime and most of the atoms are Bose condensed. Combined with BEC, the ultimate source
of coherent atom , optical lattices provide a way of exploring a quantum system analogous to electrons in
crystals but with complete control over the lattice and the atoms. Beautiful experiments have been done
in this regime and have provided an elegant demonstration of band structure [35, 36, 37], Bragg scattering
and Bloch oscillations [38], as well as coherent matter wave interferometry [39, 40], superfluidity [41] and
quantum chaos [42].
There has also been spectacular recent experimental progress in the strongly correlated regime.
It was first realized by Jaksch et al. [43], that a BEC loaded in a lattice potential is a nearly perfect
experimental realization of the Bose-Hubbard Hamiltonian, which describes bosons with local repulsive
interactions in a periodic potential. M.P.A. Fisher et al. [44] predicted that a system modeled by the Bose-
3
Hubbard Hamiltonian exhibits a quantum phase transition from a superfluid to an insulator state (superfluid-
Mott insulator transition) as the interactions are increased. In fact, the Mott insulator transition in a 3D
lattice starting from a BEC has experimentally observed by M. Greiner and coworkers [46]. Moreover, in
recent years there have been many impressive experiments which have demonstrated the loss of quantum
coherence as the system approaches the strongly correlated regime, for example by measuring number
squeezing [45] or by studying the collapse and revival of coherence in a matter wave field [47].
One of the most important potential applications of the Mott insulator transition is to use it as a
mean to initialize a quantum computer register. Deep in the Mott insulator regime the kinetic energy is very
small with respect to the interaction energy and it is energetically favorable for the atoms to remain local-
ized without tunneling. The negligible number fluctuations makes it possible to prepare the fiducial state
with exactly one atom per site needed to initialize a quantum computer register [43, 48, 50]. Besides the
possibility of high fidelity initialization, the easy scalability, low noise and high experimental control make
ultracold neutral atoms loaded in an optical lattice one of the most attractive candidates for implementations
of quantum computation.
The overall goal of this thesis is to study equilibrium and non equilibrium properties of cold bosons
loaded in optical lattices starting from the superfluid regime, where mean field techniques can be applied,
and going into the rich and complex strongly correlated regime where the standard GPE and Bogoliubov
treatments fail to describe the system and a more general framework is required. Most of the work is done
in the context of ongoing experimental efforts, especially the ones trying to achieve lattice based quantum
information processing.
? Overview
In chapter 2, I start by reviewing the theory of optical potentials, the single particle band structure
and the tight binding approximation. In chapter 3, I go one step further and consider the many-body
properties of the system by introducing the Bose-Hubbard Hamiltonian. I give a review of the generic
issues that characterize the superfluid to Mott insulator transition.
In chapter 4, I describe the equilibrium properties of lattice systems in the superfluid regime, where
a mean field treatment is valid. I study the mean field Discrete Nonlinear Schr?odinger equation (DNLSE),
4
and use it to model two experiments done by the laser-cooling and trapping group at NIST. In the first one,
an optical lattice was moved and the average displacement of the atoms was used as a means to probe the
band structure of the system. In the second experiment,to which I refer as the patterned loading experiment
[33],the atoms were loaded into every third site of an optical lattice, with the aim of having large enough
spatial separation to address individual atoms. This patterned loading method may be a useful technique
for the implementation of lattice based quantum computing proposals.
The DNLSE completely neglects quantum fluctuations. However, if the system is weakly inter-
acting, the small quantum corrections can be included by using the Bogoliubov approximation. In the
Bogoliubov approximation the complicated many-body quartic Hamiltonian is reduced to a quadratic one,
which can be diagonalized exactly. Using the Bogoliubov approximation, in chapter 5 I study different stan-
dard quadratic approximations in two different lattice systems, a translationally invariant one with periodic
boundary conditions which in general allows analytic solutions, and a system closer to real experimental
situations where, besides the lattice potential, there is a superimposed harmonic confinement potential. I
discuss the Bogoliubov de Genes equations (BdG), the Hartree-Fock-Bogoliubov (HFB) approximation and
the HFB-Popov approximation. To test the validity of the different approximations and their departure from
the exact solution as the interactions are increased I compare them with the exact numerical diagonalization
of the Bose-Hubbard Hamiltonian. Because of the exponential scaling of the dimensionality of the Hilbert
space with respect to system size, the exact solution (for a system with N atoms and M wells the number
of states scales like (N + M ?1!)=(N!M!)), the numerical comparisons are restricted to systems with a
moderate number of atoms and wells.
In chapter 5, I also report on our idea of using the superfluid fraction to study the approach to the
Mott insulator transition. By deriving an expression for the superfluid density based on the rigidity of the
system under phase variations we were able to explore the connection between the quantum depletion of
the condensate and the quasi-momentum distribution on one hand and the superfluid fraction on the other.
At the end of the chapter, I present my attempt to approach the strongly correlated regime by using
the improved Popov approximation [52, 53, 55]. The idea presented here is to upgrade the bare potential,
which is the one that explicitly appears in the many-body Hamiltonian, to the many-body scattering matrix.
5
By upgrading the bare potential, we are properly taking into account the effect of the surrounding atoms in
the properties of binary collisions.
In chapter 6 and 7, motivated by the patterned loading experiment, we adopt a functional effective
action approach capable of dealing with non equilibrium situations that require a treatment beyond mean
field theory. Even though a description of the dynamics of the patterned loading system using the DNLSE
was derived in chapter 4, it is shown by comparisons with the exact quantal solution calculated by time
propagating the initial configuration with the Bose-Hubbard Hamiltonian that a mean field solution is valid
only for short times and in the very weakly interacting regime. To deal with the dynamics far from equi-
librium, we adopt a closed time path (CTP) [56] functional-integral formalism together with a two-particle
irreducible (2PI) [57] effective action approach and derive equations of motion. We retain terms of up
to second-order in the interaction strength when solving these equations. Under the 2PI-CTP scheme we
consider three different approximations : a) the time dependent Hartree-Fock-Bogoliubov (HFB) approx-
imation, b) the next-to-leading order 1/N expansion and c) a full second-order perturbative expansion in
the interaction strength. We derive mathematical expressions for the equations of motion in chapter 6 and
apply them to the particular case of the patterned loaded lattice in chapter 7. We use this system to illustrate
many basic issues in nonequilibrium quantum field theory, such as non-local and non-Markovian effects,
pertaining to the dynamics of quantum correlation and fluctuations. We show that because the second-order
2PI approximations include multi-particle scattering in a systematic way, they are able to capture damping
effects exhibited in the exact solution, which a collisionless approach fails to produce. While the second-
order approximations show a clear improvement over the HFB approximation, they fail at late times, when
interaction effects are significant.
The 2PI effective action formalism provides a useful framework where the mean field and the cor-
relation functions are treated on the same self-consistent footing. However, it yields dynamical equation
of motion that are non local in time and hard to estimate analytically. The idea in chapter 8 is to simplify
the 2PI equations and to obtain near equilibrium solutions where, kinetic theories that describe excita-
tions in systems close to thermal equilibrium are valid. In particular, we show in this chapter how the full
second-order 2PI equations are in agreement with current kinetic theories [15],[58]-[62] and reproduce in
6
equilibrium the higher order perturbative corrections well known in the literature since Beliaev?s work [63].
In chapter 9, I jump into the Mott insulator phase and I study the physical properties of the system
deep in the Mott insulator regime, which is the other regime where an analytic treatment based on perturba-
tion theory is possible. A perturbative analysis should be applicable to study current experiments [64] that
reach the strong Mott regime. I derive expressions for the excitation spectrum of the Mott state for both
homogeneous and trapped systems and compare them with the solutions obtained by exact diagonaliza-
tion of the Bose-Hubbard Hamiltonian. The main purpose of this chapter is to understand the many-body
properties of the Mott insulator ground state and the nature of its many-body excitations which will be
crucially important as more elaborate experiments with optical lattices in the strongly correlated regime are
undertaken.
One key piece of evidence for the Mott insulator phase transition is the loss of global phase coher-
ence of the matter wavefunction when the lattice depth increases beyond a critical value [46]. However,
there are many possible sources of phase decoherence in these systems. It is known that substantial decoher-
ence can be induced by quantum or thermal depletion of the condensate during the loading process, so loss
of coherence is not a proof that the system resides in the Mott insulator ground state. Indeed, for this reason,
in the experiments by Greiner et al. [46] a potential gradient was applied to the lattice to show the presence
of a gap in the excitation spectrum. In chapter 10, we show that another common experimental technique,
Bragg spectroscopy [65, 66], not only can identify the excitation gap that opens up in the Mott regime, but
also can be used to map out the excitation spectrum and to determine the temperature of the system when
it is deep in the Mott regime. Specifically, we study the total momentum and total energy deposited in the
system by the Bragg perturbation calculated under a linear response analysis and obtain analytical solutions
in the superfluid and deep Mott insulator regimes. We test the accuracy of the approximations and their
deviation from the full quantal behavior as usual by comparing them with numerical solutions obtained by
diagonalizing the Bose-Hubbard Hamiltonian for a moderate number of atoms and wells.
All of the proposals for quantum computation which utilize a lattice-type architecture have the Mott
insulator transition as the initialization scheme to load exactly one atom per lattice site. Such architecture
requires a lattice commensurately filled with atoms, which does not correspond exactly to the ground Mott
7
insulator state. The ground state has a remaining coherence proportional to the tunneling matrix element.
This degrades the initialization of the quantum computer register and can introduce errors during error
correction. I finish this thesis with chapter 11, where I report on our proposal to solve this problem by using
the spatial inhomogeneity created by a quadratic magnetic trapping potential together with a continuous
measurement procedure which projects out the components of the wave function with more than one atom
in any well.
Publications in the PhD work:
? Chapter 4
Dynamics of a period-three pattern loaded Bose-Einstein condensate in an optical lattice, A. M.
Rey, P. B. Blakie, Charles W. Clark, Phys. Rev. A 67, 053610 (2003).
? Chapter 5
Bogoliubov approach to superfluidity of atoms in an optical lattice, Rey AM, Burnett K, Roth R,
Edwards M, Williams CJ, Clark CW, J. Phys. B: At. Mol. Opt. Phys, 36 , 825 (2003).
? Chapter 6 and 7
Nonequilibrium Dynamics of Optical Lattice - Loaded BEC Atoms: Beyond HFB Approximation,
A. M. Rey, B. L. Hu, E. Calzetta, A. Roura, Charles W. Clark, Phys. Rev A. 69, 033610 (2004).
BEC with fluctuations: beyond the HFB approximation, A. M. Rey, B. L. Hu, E. Calzetta, A. Roura,
Charles W. Clark (Proceedings of the Laser Physics Workshop 2003), Las. Phys., 14, 1, (2004).
8
? Chapter 10
Bragg spectroscopy of ultracold atoms loaded in an optical lattice, Ana Maria Rey, P. Blair Blakie,
Guido Pupillo, Carl J. Williams, Charles W. Clark, submitted to Phys. Rev. Lett. cond-
mat/0406552.
? Chapter 11
Scalable register initialization for quantum computing in an optical lattice, G. K. Brennen, G.
Pupillo, A. M. Rey, C. W. Clark, C. J. Williams, submitted to Phys. Rev. Lett., cond-mat/0312069.
Scalable quantum computation in systems with Bose-Hubbard dynamics, Guido Pupillo, Ana M.
Rey, Gavin Brennen, Carl J. Williams, Charles W. Clark, to appear in J. Mod. Opt, quant-
ph/0403052.
9
Chapter 2
Optical lattices
Optical lattices are periodic potentials created by light-matter interactions. When an atom interacts with an
electromagnetic field, the energy of its internal states depends on the light intensity. Therefore, a spatially
dependent intensity induces a spatially dependent potential energy. If such a modulation is obtained by
the interference of several laser beams, the resultant optical potential felt by the atoms will have different
potential wells separated by a distance of the order of the laser wavelength. The depths of the optical
potential wells that can be obtained in an experiment are in the microKelvin range. Nevertheless, atoms can
be trapped in this potentials when cooled at low temperatures, by laser and evaporative cooling techniques.
Cold atoms interacting with a spatially modulated optical potential resemble in many respects elec-
trons in ion-lattice potential of a solid crystals . However, optical lattices have several advantages with
respect to solid state systems. They can be made to be largely free from defects, such defects for example
prevented the observation of Bloch oscillations in crystalline solids. Optical lattices also can be controlled
very easily by changing the laser field properties. For example the lattice depth can be changed by modi-
fying the laser intensity, the lattice can be moved by changing the polarization of the light or chirping the
laser frequency and the lattice geometry can be modified by changing the laser configuration. Moreover, in
contrast to solids, where the lattice spacings are generally of order of Angstrom units, the lattice constants
in optical lattices are typically three order of magnitude larger.
The idea of this chapter is to introduce the basic theory of optical lattices and to review the single
particle properties of atoms loaded in such periodic potentials.
2.1 Basic theory of optical lattices
Neutral atoms interact with light in both dissipative and conservative ways. The conservative interaction
comes from the interaction of the light field with the induced dipole moment of the atom which causes
a shift in the potential energy called ac-Stark shift. On the other hand, the dissipation comes due to the
absorption of photons followed by spontaneous emission. Because of conservation of momentum, the net
effect is a dissipative force on the atoms caused by the momentum transfer to the atom by the absorbed and
10
spontaneously emitted photons. Laser cooling techniques make use of this light forces.
For large detunings spontaneous emission processes can be neglected and the energy shift can be
used to create a conservative trapping potential. This is the physics that describes optical lattices.
2.1.1 AC Stark Shift
Consider a two level atom, with internal ground state jgi and excited state jei and energy difference ~!o
in a lossless cavity of volume V , interacting with a monochromatic electromagnetic field with frequency
! = 2?? as schematically shown in Fig.2.1. Assume also that the experiment is performed within a time
smaller than the spontaneous emission rate so that spontaneous emission can be neglected.
Figure 2.1: AC Stark shift induced by atom-light interaction. The laser frequency is ! = 2?? which is
detuned from the atomic resonance by ?
The uncoupled Hamiltonian describing the atoms and the electromagnetic field is given by
^Ho =~!ojeihej+~!(^ay^a+1=2); (2.1)
where ^a is the photon annihilation operator. If the detuning of the laser from the atomic transition, ? =
! ? !o, is small j?j ? !o, then the state with the atom in the ground state and N photons in the field,
j0i ? jg;Ni has similar energy to the state with the atom in an excited state and N ? 1 photons j1i ?
je;N ?1i;E1 ? E0 = ?~?. The effect of the interactions is to couple these states. Under the dipole
11
approximation which assumes that the spatial variation of the electromagnetic field is small compared with
the atomic wave function, the coupling Hamiltonian denoted as ^HI in the interaction picture is given by:
^HI = ?~d? ~E (2.2)
= ?jeihgjei!ot +jgiheje?i!ot?
 ~??(~x)
2 ^a
yei!t + ~?(~x)
2 ^ae
?i!t
?
:
Here ?(x) is the Rabi frequency given by~?(~x) = ?2
q
~!hNi
22V u(~x)~"?~d, with ~" the unit polarization vector
of the field, ~d the dipole moment of the atom and u(~x) the field mode evaluated at the atomic position x,(for
plane waves, for example u(~x) = e?i~k?~x). In the rotating wave approximation, valid in the limit j?j? !o;
the type of processes with a rapidly oscillating phase, exp(?i(!o + !)t), are neglected and only the near
resonant frequency processes are considered. The interaction Hamiltonian is then reduced to
^HI ?
 ~?(~x)
2 jeihgj^ae
i?t + ~??(~x)
2 jeihgj^a
ye?i?t
?
: (2.3)
Physically, the resonant process correspond to either the excitation of the atom along with the emission of
a photon or the relaxation of the atom with the absorbtion of a photon. In the previous line we also assume
a large number of photons and neglect the variation in the coupling constant due to ?N; i.e N ? N ?1:
If the detuning is large compared to the Rabi frequency, j?j  ?, the effect of the interactions on
the states, j0i and j1i, can be determined with second order perturbation theory. In this case, the energy
shift E(2)0;1 is given by
E(2)0;1 = ?jh1j
^Hintj0ij2
~? = ?~
j?(~x)j2
4? ; (2.4)
with the plus and minus sign for the j0i and j1i states respectively. This energy shift is the so called ac-
Stark shift. Since the atoms are practically always in the ground state, the energy of the atoms is changed
according to the stark shift~j?(~x)j24? ; which defines the optical potential.
Furthermore, if instead of interacting with a monochromatic electromagnetic field, the atoms are
illuminated with superimposed counter-propagating laser beams, the beams interfere and the interference
pattern results in a periodic landscape potential or optical lattice.
12
2.1.2 Dissipative interaction
In the above discussion we implicitly assumed that the excited state has an infinite lifetime. However, in
reality it will decay by spontaneous emission of photons. This effect can be taken into account phenomeno-
logically by attributing to the excited state an energy with both real an imaginary parts. If the excited
state has a life time 1=?e corresponding to a e-folding time for the occupation probability of the state, the
corresponding life time for the amplitude will be twice this time. The energy of the perturbed ground state
becomes a complex quantity which we can write as
E(2)0 = ~4 j?(~x)j
2
??i?e=2 = V(~x)+i sc(~x); (2.5)
V(~x) = ~j?(~x)j
2?
4?2 ??2e ?~
j?(~x)j2
4? ; (2.6)
 sc(~x) = ~2 j?(~x)j
2?e
4?2 ??2e ?~
j?(~x)j2?e
8?2 : (2.7)
The real part of the energy corresponds to the optical potential whereas the imaginary part represents the
the rate of loss of atoms from the ground state. The sign of the optical potential seen by the atoms depends
on the sign of the detuning. For blue detuning , ? > 0, the sign is positive resulting in a repulsive
potential, and the potential minima correspond to the points with zero light intensity. On the other hand, in
a red detuned light field, ? < 0, the potential is attractive and the minima correspond to the places with
maximum light intensity. Because the effective spontaneous emission rate of the atoms increases with the
light intensity, the spontaneous emission in a red detuned optical lattice will always be more significant
than in a blue detuned one.
The proper detuning for an optical lattice depends on the available laser power I (j?j2 / I) and the
maximum dissipative scattering rate that can be tolerated. On one hand, with small detuning it is possible
to create larger trap depths for a given laser intensity since the optical potential scales as V ? I=?: On the
other, the inelastic scattering rate is inversely proportional to the detuning squared and scales like ?e=? V .
Therefore, the laser detuning should be chosen as large as possible within the available laser power in order
to minimize inelastic scattering processes and create a conservative potential.
13
2.1.3 Lattice geometry
The simplest possible lattice is a one dimensional(1D) lattice lattice. It can be created by retroreflecting
a laser beam, such that a standing wave interference pattern is created. This results in a Rabi frequency
?(x) = 2?o sin(kx) which yields a periodic trapping potential given by
Vlat(x) = Vo sin2(kx) = ~?
2o
? sin
2(kx); (2.8)
where k = 2?=? is the absolute value of the wave vector of the laser light and Vo is four times times the
depth of a single laser beam without retro-reflection, due to the constructive interference of the lasers.
Periodic potentials in higher dimensions can be created by superimposing more laser beams. To
create a two dimensional lattice potential for example, two orthogonal sets of counter propagating laser
beams can be used. In this case the lattice potential has the form
Vlat(x;y) = Vo?cos2(ky)+cos2(kx)+2"1 ?"2 cos`cos(ky)cos(kx)?: (2.9)
Here k is the magnitude of the wave vector of the lattice light, "1 and "2 are polarization vectors of the
counter propagating set and ` is relative phase between them. If the polarization vectors are not orthogonal
and the laser frequencies are the same, they interfere and the potential is changed depending on the relative
phase of the two beams. This leads to a variation of the geometry of the lattice in a chequerboard like
pattern. A simple square lattice with one atomic basis can be created by choosing orthogonal polarizations
between the standing waves. In this case the interference term vanishes and the resulting potential is just
the sum of two superimposed 1D lattice potentials. Even if the polarization of the two pair of beams is the
same, they can be made independent by detuning the common frequency of one pair of beams from that
the other. Typically a negligible frequency difference compared with the optical frequency is required to
achieve independence, thus even in this case to a good approximation the wave vectors can be considered
equal.
A more general class of two 2D lattices can be created from the interference of three laser beams
[34, 33] which in general yields non separable lattices. Such lattices can provide tighter on-site confine-
ment, better control over the number of nearest neighbors and significantly reduced tunneling between sites
14
compared with the counter propagating four beam square lattice. In Fig. 3 we show a variety of possible
2D optical lattice geometries that can be made by three and four interfering laser beams. Similarly a 3D
lattice can be created by the interference of at least 6 orthogonal sets of counter propagating laser beams.
y  [
?]
(a)
a1
a2
?1
0
1 (b)
a1
a2 (c)
a1
a2
x  [?]
y  [
?]
(d)
a1
a2
?1 0 1
?1
0
1
x  [?]
(e)
a1
a2
?1 0 1 x  [?]
(f)
a1
a2
?1 0 1
VLatt(r) [ER] 
?10 ?5 0 5
Figure 2.2: Optical lattice potential.(a)-(e) potentials for different configurations of 3 beams, (f) potential
for the 4 counter-propagating laser beam configuration (The two pair of light fields are made independent
by detuning the common frequency of one pair of beams from the other. ER is the atomic recoil energy,
ER =~2k2=2m. This figure is a courtesy of P. Blair Blakie [34] )
2.2 Single particle physics
In this section for simplicity we are going to restrict the analysis to a one dimensional lattice. Generalization
to higher dimensions can be done straightforwardly, especially if the lattice geometry is separable. The main
purpose of this section is to review the basic aspects that describe the behavior of noninteracting particle
subject to a periodic potential.
15
2.2.1 Bloch functions
One of the most important characteristics of a periodic potential is the emergence of a band structure.
Consider a one dimensional particle described by the Hamiltonian H = ^p22m + Vlat(x), where Vlatx) =
Vlat(x+a). Bloch?s theorem [67, 68] states that the eigenstates `(n)q (x) can be chosen to have the form of
a plane wave times a function with the periodicity of the potential:
`(n)q (x) = eiqxu(n)q (x); (2.10)
u(n)q (x+a) = u(n)q (x): (2.11)
Using this ansatz into the Schr?odinger equation, H`(n)q (x) = E(n)q `(n)q (x), yields an equation for u(n)q (x)
given by:
"
(^p+~q)2
2m +Vlat(x)
#
u(n)q (x) = E(n)q u(n)q (x) (2.12)
Bloch?s theorem introduces a wave vector q. The quantity q should be viewed as a quantum number charac-
teristic of the translational symmetry of the periodic potential, just as the momentum is a quantum number
characteristic of the full translational symmetry of the free space. Even though it is not the same, it turns out
that~q plays the same fundamental role in the dynamics in a periodic potential as the momentum does in the
absence of the lattice. To emphasize this similarity~q is called the quasimomentum or crystal momentum .
In general the wave vector q is confined to the first Brilloiun zone, i.e. ??=a < q ? ?=a.
The index n appears in Bloch?s theorem because for a given q there are many solutions to the
Schr?odinger equation. Eq. (2.12) can be seen as a set of eigenvalue problems in a fixed interval, 0 < x < a,
one eigenvalue problem for each q. Therefore, each of them, on general grounds has an infinite family of
solutions with a discretely spaced spectrum of modes labelled by the band index n. On the other hand,
because the wave vector q appears only as a parameter in Eq. (2.12), for an infinite lattice, the energy levels
for a fixed n has to vary continuously as q varies. The description of energy levels in a periodic potentials
in terms of a family of continuous functions E(n)q each with the periodicity of a reciprocal lattice vector,
2?=a; is referred to as the band structure.
16
In the simple case of a sinusoidal potential, which is the one used in experiments, Vlat(x) =
V0 sin2(kx), the band structure can be solved analytically. In this case the Schr?odinger Equation is given
by
? d
2
dy2`
(n)
q (y)+
V0
4ER (2?2cos(2y))`
(n)
q (y) =
E(n)q
ER `
(n)
q (y) (2.13)
where ER = ~2k2=2m is the atomic recoil energy and y = kx. For convenience, lattice depths are
generally specified in recoil units. Equation (2.13) is just the Mathieu equation ([69])
d2y
dx2 +(a+2scos(2x))y = 0: (2.14)
.
Solutions of the Mathieu equation are generally written in the Floquet form ei?xP(x) where ? is known as
the characteristic exponent and a = a(?;s) is the characteristic parameter which is a complicated function
of s and ?. In the lattice language ? correspond to the quasimomentum , s = V0=4ER and a = E(n)q =ER?
V0=2ER.
Minus1 Minus0.5 0 0.5 1qaSlash1?0
24
68
ESlash1E
R
VoEqual4ER
Minus1 Minus0.5 0 0.5 1qaSlash1?0
5
10
15
ESlash1E
R
VoEqual20ER
Minus1 Minus0.5 0 0.5 1qaSlash1?0
24
68
ESlash1E
R
VoEqual0ER
Minus1 Minus0.5 0 0.5 1qaSlash1?0
24
68
ESlash1E
R
VoEqual0.5ER
Figure 2.3: Band structure of an optical lattice
Fig. 2.3 shows the band structure of a sinusoidal potential for different potential depths. For V0 = 0,
the particles are free so the spectrum is quadratic in q. As the potential is increased the band structure
appears. For small V0 the discontinuity occurs only at the edge of the first Brillouin zone qa ?? and the
17
gap is proportional to V0=2. As the depth increases, the band gap increases and the band width decreases.
For very deep lattices the spectrum is almost degenerate in q and exhibits a dependence on n similar to the
one of a particle in a fixed finite interval(determined by the period of the potential).
In the absence of a lattice the eigenfunctions of the free system are plane waves. As the lattice depth
is increased the barrier height between adjacent lattices sites increases and the eigenstates of the system
tend to get localized at each lattice site (regions around the potential minima). In Fig. 2.4 the probability
density of the Bloch wave function `(n=0)q=0 is plotted for different lattice depths. It can be observed how the
amplitude of the wave function in between adjacent lattice sites decreases with increasinglattice depth.
0 1 2 3 4
xSlash1a
Density
VertBar1?0LParen11RParen1LParen1xRParen1VertBar12
0 1 2 3 4
xSlash1a
Density
VertBar1?0LParen11RParen1LParen1xRParen1VertBar12
0 1 2 3 4
xSlash1a
Density
VertBar1?0LParen11RParen1LParen1xRParen1VertBar12
0 1 2 3 4
xSlash1a
Density
VertBar1?0LParen11RParen1LParen1xRParen1VertBar12
Figure 2.4: Probability density of the Bloch wave function `(n=1)q=0 for different lattice depths. It can be
observed the localization of the wave function increases with the lattice increases.
2.2.2 Wannier orbitals
Wannier orbitals are a set of orthonormal wave functions that fully describe particles in a band and are
localized at the lattice sites. They are defined as:
wn(x?xi) = 1pM
X
q
e?iqxi`(n)q (x); (2.15)
18
where the sum is over the first Brillouin zone, M is the total number of lattice sites and xi is the position
of the ith lattice site. Wannier orbitals are thus a unitary transformation of the Bloch functions and are
formally an equivalent representation to describe the periodic system. They constitute a more appropriate
representation as the lattice depth is increased and particles get localized at individual lattices sites. The
actual form of a Wannier function may be seen if we assume the periodic function u(n)q (x) in equation
2.10 to be approximately the same for all Bloch states in a band. Under this approximation the Wannier
function centered at the origin can be shown to be
wn(x) ? u(n)(x)sin(kx)kx : (2.16)
y  [
?]
(a)
1
1
11
1
1
2
2
2
2
2
2
3
3
3
3
33
?1
0
1 (b)
1
1 1
1
2
2
33
3
3 3
3
(c)
11 22 3
3
x  [?]
y  [
?]
(d)
1
1 2
2 3
3
?1 0 1
?1
0
1
x  [?]
(e)
1
1
2
2
2
2
3
3
?1 0 1 x  [?]
(f)
11
1
1
2
2
22 33
3
3?1 0 1
log10|w0(r)|2   [??2]
?15 ?10 ?5 0
Figure 2.5: Wannier state log-density distribution corresponding to the lattice configurations shown in Fig.
2.2. Dotted white lines indicate the potential energy contours. The numbers labels the near neighbor sites:
1 the nearest neighbors, and so on. This figure is a courtesy of P. Blair Blakie [34] )
This looks like u(n)(x) at the site center, but spread out with oscillations of gradually decreasing amplitude.
The oscillations are needed to ensure orhogonality between Wannier functions. In Fig. 2.5 we show the
Wannier orbital centered at the origin site for each of the lattice configurations shown in Fig.2.2. The lattice
depth in this pictures in deep enough that the Wannier orbitals are well localized. To observe the small
oscillatory tails at the neighboring sites, the plots use a log-scale for the density color map.
19
2.2.3 Tight-binding approximation
The tight-binding approximation deals with the case in which the overlap between Wannier orbitals at dif-
ferent sites is enough to require corrections to the picture of isolated particles but not too much as to render
the picture of localized wave functions completely irrelevant. Sometimes a very good approximation is
only to take into account overlap between nearest neighbor orbitals. This tight-binding model is commonly
used to solve the problem of a particle in a periodic potential when also an external potential is applied and
it is going to be fundamental when considering particle interactions.
By expanding the wave function in Wannier orbitals,
?(x;t) =
X
n;i
z(n)i (t)wn(x?xi); (2.17)
and using it in the Schr?odinger equation that describes a particle moving in the potential of a 1D lattice plus
a perturbative external potential V(x), we get the following equations of motion
?i~@@tz(n)i =
X
jn0
?J(n0)ij ?nn0z(n0)j (t)+V (n;n0)ij z(n0)j (t); (2.18)
with
J(n)ij = ?
Z
dxw?n(x?xi)Hown(x?xj)dx; (2.19)
V (n;n0)ij = ?
Z
dxw?n(x?xi)U(x)wn0(x?xj)dx: (2.20)
If we assume that the external perturbation is not strong enough or sharp enough to induce interband
transitions, we may represent the moving particle quite satisfactory by using Wannier functions of only
the first band. Moreover, if the lattice is deep enough such that tunneling to next to nearest neighbors can
be ignored, and the external perturbation is a slowly varying function such that it can be assumed constant
inside each individual lattice site, i.e. V (n;n0)ij = V(xi)?ji?n;n0, the equations of motion reduce to
?i~@@tzi(t) = ?J (zi+1(t)+zi?1(t))+V(xi)zi(t)+"ozi(t); (2.21)
with
20
J = ?
Z
dxw?0(xi)How0(x?xi+1)dx; (2.22)
"o =
Z
dxw?0(x)How0(x)dx (2.23)
where J is the tunneling matrix element between nearest neighboring lattice sites, "o is the unperturbed on
site energy shift and and zi the first band coefficients zi(t) = z(0)i (t): Eq. (2.21) is known as the discrete
Schr?odinger equation (DSE) or tight-binding Schr?odinger equation.
For deep lattices the localized Wannier orbitals can be approximated by a Gaussian function. How-
ever, because the Gaussian ansatz neglects the small oscillations characteristic in the tail of the Wannier
function, the tunneling matrix element J is then underestimated by almost an order of magnitude. For
the special case of a sinusoidal potential Vlat(x) = Vo sin2(kx) for which Mathieu functions are the exact
solutions an analytic expression for J can be obtained by using tabulated Mathieu functions in Eq. (2.22)
and fitting the data. The best fit we get is:
J=ER = fi
?~
Vo
? 
e?fl
p~
Vo (2.24)
with fi =1.39666, fl =1.051 and  =2.12104 and ~Vo = Vo=ER.
2.2.4 Semiclassical dynamics
Formalism
Solving Eq. (2.21) is presumably not possible for arbitrary V(xi). However, we get general insight to the
nature of solutions by an application of the correspondence principle. It is well known that wave-packet
solutions of the Schr?odinger equation behave like classical particles obeying the equations of motion
derived from the same classical Hamiltonian. The classical Hamilton equations are
_x = @H@p ; _p = ?@H@x : (2.25)
Using the correspondence principle and the resemblance of the crystal momentum or quasimomentum to
real momentum, the semiclassical equations of a wave packet in the first band of a lattice can be written as
([67], [68])
21
_x = v(0)(q) = 1~dE
(0)
q
dq ; ~_q = ?
dV(x)
dx : (2.26)
The semiclassical equations of motion describe how the position and wave vector of a particle evolve
in the presence of an external potential entirely in terms of the band structure of the lattice. If we compare
the acceleration predicted by the model with the conventional newtonian equation, m?x = ?dV(x)=dx, we
can associate an effective mass induced by the presence of the lattice, m?. This is given by
1
m? =
1
~2
d2
dq2E
(0)
q : (2.27)
Translationally invariant lattice
If no external potential is applied, V(x) = 0, plane waves are the solutions of DSE: zj(t) = f(q)j e?it(Eq+"o)=~,
f(q)j = 1pMeiqaj with M is the total number of lattice sites. If also periodic boundary conditions are
assumed, the quasimomentum q is restricted to be an integer multiple of 2?Ma. The lowest energy band
dispersion relation in this case is given by
Eq = ?2J cos(qa): (2.28)
From the above equation it is possible to see the connection between J and the band width
J = ?Eq=?a ?Eq=0?=4: (2.29)
Linear external potential and Bloch oscillations
If V(x) is assumed to be a linear potential, V(x) = J?a x, it can be shown (see for example [70],[71],[72]),
that in this case the energy spectrum of Eq. (2.21) is discrete and evenly spaced
(Es ?E0) = J?s s = 0;1;2;3:::: (2.30)
The corresponding eigenfunctions are spatially localized and the amplitudes zj(t) given by :
zj(t) = f(s)j e?iEst=~; (2.31)
f(s)j = Js?j(?2=?) s = 0;1;2;3;:::: (2.32)
22
With Js the sth order Bessel function of the first kind. Because the quantities f(s)j are just displaced Bessel
functions, the sth eigenstate of the system tends to be exponentially localized around the sth site as ? is
increased. The localization is almost complete when ? = 4, i.e. when the potential energy drop over a
lattice period is equal to the zero field band width. The localization can be observed in Fig.2.6 where the
ground state amplitudes f(0)i are plotted for different values of ?.
Minus30Minus20Minus10 0 10 20 30isiteMinus0.6
Minus0.4Minus0.2
00.2
0.40.6
f iLParen10RParen1
?Equal1
Minus30Minus20Minus10 0 10 20 30isiteMinus0.2
00.2
0.40.6
0.8
f iLParen10RParen1
?Equal4
Minus30Minus20Minus10 0 10 20 30isiteMinus0.2
Minus0.1
0
0.1
0.2
f iLParen10RParen1
?Equal0.1
Minus30Minus20Minus10 0 10 20 30isiteMinus0.4
Minus0.2
0
0.2
0.4
f iLParen10RParen1
?Equal0.5
Figure 2.6: Ground state amplitudes f(0)i as a function of the lattice site i for different values of ?
An insight on the dynamics of a particle in this linear potential is obtained by calculating the time
dependent probability of finding it at the site j at time t , jzj(t)j2, if it was in the state localized at the origin
at time zero zj(0) = ?0j. After some algebra it can be shown to be given by
Pj = jzj(t)j2 = J2j
 4sin(??=2)
?
?
; (2.33)
with ? = tJ=~. The dynamical evolution of Pj6=0, for a given j consists of an initial period of growth until
a maximum is reached followed by an oscillatory decay. The sequence is repeated again backwards until a
time ?= 2?=? at which Pj6=0 vanishes and a new cycle starts.
In Fig. 2.7, the diffusion of a square wave packet occupying at t = 0 the central 20 sites is plotted.
It can be seen that the center of mass of the wave function accelerates and then returns to its initial position
after a period ?= 2?=?.
A simplified picture of the dynamics of a wave packet in this linear potential can also be obtained
23
Minus40Minus20 0 20 40isite
VertBar1? iVertBar12
?LParen12?Slash1?RParen1Equal0.8
Minus40Minus20 0 20 40isite
VertBar1? iVertBar12
?LParen12?Slash1?RParen1Equal1.
Minus40Minus20 0 20 40isite
VertBar1? iVertBar12
?LParen12?Slash1?RParen1Equal0.4
Minus40Minus20 0 20 40isite
VertBar1? iVertBar12
?LParen12?Slash1?RParen1Equal0.6
Minus40Minus20 0 20 40isite
VertBar1? iVertBar12
?LParen12?Slash1?RParen1Equal0.
Minus40Minus20 0 20 40isite
VertBar1? iVertBar12
?LParen12?Slash1?RParen1Equal0.2
Figure 2.7: Diffusion of a square wave packet in the presence of a linear potential.The parameters for the
plot are ? = 0:1.
if we use the semiclassical model. If we assume a packet initially centered at x(t = 0) = 0, with average
initial quasimomentum q(t = 0) = qo; the semiclassical model predicts periodic oscillations of the center
of mass with frequency ? and amplitude 2=?.
x(t) = 2?(1?cos(?? +qoa)): (2.34)
This periodic motion, known as Bloch oscillations, has been observed in systems of cold atoms [73] and
Bose-Einstein condensates in optical lattices [38, 36]. On the other hand, if the external potential is strong
enough to cause interband transitions, the tight-binding model does not apply anymore. In this case the
atoms may gain enough energy from the external field to tunnel out through the energy gap into a second
band when they reach the Brillioun zone boundary. This phenomenon is known as of Landau-Zener
tunneling and it also has been observed in experiments [40].
24
Harmonic potential
A case of experimental interest is when, besides the periodic potential, the system is subject to a harmonic
confinement, V(x) = 12m!2Ta2(x=a)2 ? ?(x=a)2, with !T the trapping frequency (note that ? is unre-
lated to the Rabi frequency discussed in section 2.1). The energy spectrum in this case is determined by the
equation
Esf(s)i = ?J(f(s)i+1 +f(s)i?1)+?i2f(s)i : (2.35)
It is possible to show that the solutions f(s)i of Eq. 2.35 are the Fourier coefficients of the periodic
Mathiue function with period ? and the eigenvalues are the characteristic value of these periodic Mathiue
functions [69]. The symmetric and antisymmetric solutions and eigenvalues are given respectively by
f(s=2r)i = 1?
Z 2?
0
ce2r
 
x; 4J?
?
cos(2ix)dx r = 0;1;2:::; (2.36)
f(s=2r+1)i = 1?
Z 2?
0
se2r
 
x; 4J?
?
sin(2ix)dx;
Es=2r = ?4 a2r
 4J
?
?
Es=2r+1 = ?4b2r
 4J
?
?
; (2.37)
with ce2r(x;4J=?) and se2r(x;4J=?) the even and odd periodic solution (with period ?) of the Mathieu
equation with characteristic parameter a2r(4J=?) and b2r(4J=?) respectively. The eigenvalues Es are
complicated functions of 4J? and therefore we are only going to consider expressions for certain limits:
? Case J  ?: This is the case where most experiments have been developed, for example for a 5 ER
lattice the parameter q = 4J? varies between 1.7x104 for a magnetic trap of 9 Hz to 35 for a trap of
200 Hz. In this regime the excitation spectrum is given by:
Es = ?2J +p4?J(s+ 12)??
 (2s+1)2 +1
32
?
(2.38)
??
r
?
J
 (2s+1)3 +3(2s+1)
210
?
+:::
which to a first approximation is just the harmonic potential found if no lattice were present but with
25
an effective frequency !? and effective mass m? given by
~!? = p4?J =~!t
r m
m?; (2.39)
m? = ~
2
2Ja2; (2.40)
It is worth it to emphasize that this result was derived under the tight-binding approximation and
therefore it is not valid for very shallow lattices. The harmonic character of the spectrum can be
easily checked if we take the continuous limit of Eq. (2.35). The differential equation obtained is just
the harmonic oscillator equation,
(Es +2J)f(s)(x) = ? ~
2
2m?
@f(s)(x)
@x2 +
1
2m
?!?2x2f(s)(x): (2.41)
The average width of the wave function is just aho=
q
}
m?!? .
? Case J ? ?: The single particle spectrum in this case is given by
Es=2n t Es=2n?1
= ?4
?
(2n)2 +
 4J
?
?2 1
((2n)2 ?1) +O
 4J
?
?4
+::
!
n ? 3;
E4 t E3 = ?4
?
16+
 4J
?
?2 1
30 +::
!
;
E2 = ?4
?
4+
 4J
?
?2 5
12 +::
!
; (2.42)
E1 = ?4
?
4?
 4J
?
?2 1
12 +::
!
;
E0 = ?4
?
?
 4J
?
?2 1
2 +:::
!
:
It can be appreciated that the spectrum is nearly twofold degenerate and it goes like ?n2 to first order in
(J=?) . In this regime then, the trapping potential is what mostly determines the energy spacing. In Fig.
2.8 the spectrum is shown for different values of 4J? :
26
5 10 15 20nth4
162
321
EnSlash1J
4JSlash1CapOmegaEqual1
5 10 15 20nth40
1624
3208
EnSlash1J
4JSlash1CapOmegaEqual0.1
5 10 15 20nth
2
4EnSlash1J
4JSlash1CapOmegaEqual100
5 10 15 20nth1
17
33
EnSlash1J
4JSlash1CapOmegaEqual10
Figure 2.8: Harmonic confinement plus lattice spectrum: The index n labels the eigenvalues in increasing
energy order.
27
Chapter 3
The Bose-Hubbard Hamiltonian and the superfluid to Mott insulator transition
In chapter 2 we studied the physics of non interacting particles in a periodic system. In this chapter we go
one step ahead and describe the basic properties of a bosonic many body system with repulsive interactions
in a periodic lattice potential.
The simplest non trivial model that describes interacting bosons in a periodic potential is the Bose
Hubbard Hamiltonian. It includes the main physics that describe strongly interacting bosons, which is the
competition between kinetic and interaction energy. This model has been used to describe many different
systems in solid state physics, like short correlation length superconductors, Josephson arrays, critical be-
havior of 4 He and, recently, cold atoms in optical lattices [43]. It has been studied analytically with many
different techniques such as mean field approximations [74, 75, 77], renormalization group theories [44]
and strong coupling expansions [78]. Numerically most of the studies are based on quantum Monte Carlo
methods and density matrix renormalization techniques [79]. The Bose Hubbard Hamiltonian predicts a
quantum phase transition from a superfluid to a Mott insulator state. This transition has been observed
experimentally in atoms confined in a 3D optical lattice [46]. In this chapter we start by reviewing the basic
properties of the Bose Hubbard Hamiltonian and outline the most important aspects that characterize the
superfluid to Mott insulator transition.
3.1 Bose-Hubbard Hamiltonian
We begin the analysis by considering the second quantized Hamiltonian that describes interacting bosonic
atoms in an external trapping potential plus lattice. In the grand canonical ensemble it is given by:
^H =
Z
dx^'y(x)
 
?~
2
2mr
2 +Vlat(x)
?
^'(x)+(V(x)??)^'y(x)^'(x) (3.1)
+12
Z
dxdx0^'y(x)^'y(x0)Vat(x ? x0)^'(x)^'(x0);
where ^'y(x) is the bosonic field operator which creates an atom at the position x, Vlat(x) is the periodic
lattice potential, V(x) denotes any additional slowly- varying external potential that might be present (such
as a magnetic trap), ? is the chemical potential and acts as a lagrange multiplier to fix the mean number of
28
atoms in the grand canonical ensemble and Vat(x) is the interatomic scattering potential, which is in general
a complicated function. However, since for cold gases the thermal de Broglie wavelength is much larger
than the effective extension of the interaction potential, the only relevant scattering process is the s-wave
scattering, and the actual inter-particle potential plays a minor role. Therefore, to a good approximation,
the interatomic potential can be replaced by an effective contact interaction:
Vat(x) ? 4?as~
2
m ?(x); (3.2)
with as is the scattering length and m the mass of an atom. In this thesis we are going to assume that the
scattering length is positive so the interactions are repulsive.
Similar to the noninteracting situation, where we used Wannier orbitals to expand the single particle
wave function, it is convenient to expand the field operator in a Wannier orbital basis. If the chemical
potential of the system is less than the single particle excitation energy to the second band, only the first
band has to be considered and the field operator can be written as:
^'(x) = X
n
^anw0(x ? xn); (3.3)
where w0(x) is the Wannier orbital of the lowest vibrational band localized at the origin, and ^an is the
annihilation operator at site n which obeys bosonic canonical commutation relations. The sum is taken
over the total number of lattice sites. If Eq. (3.3) is inserted in ^H and only tunneling between nearest
neighbors is considered, we obtain the Bose Hubbard Hamiltonian
^HBH = ?J X
<n;m>
^ayn^am + 12U
X
n
^ayn^aynanan +
X
n
(Vn ??)^ayn^an; (3.4)
with
J = ?
Z
dxw?0(x ? xn)
?
^p2
2m +Vlat(x)
!
w0(x + xn+1); (3.5)
U = 4?as~
2
m
Z
dxjw0(x)j4 ; (3.6)
Vn = V(xn): (3.7)
29
The first term in the Hamiltonian proportional to J is a measure of the kinetic energy of the system.
J is the hopping matrix element between nearest neighbors (see also 2.24). Here the notation < n;m >
restricts the sum to nearest-neighbors sites. Next-to-nearest neighbor tunneling amplitudes are typically
two orders of magnitude smaller than nearest-neighbor tunneling amplitudes, and to a good approximation
they can be neglected.
The second term determines the interaction energy of the system. The parameter U measures the
strength of the repulsion of two atoms at lattice site n.The integral (3.6) is not as sensitive as Eq. (3.5)
to the oscillatory tails characteristic of the Wannier orbitals, and a Gaussian approximation can be used
to estimate it [43]. Under the Gaussian approximation, the ground state wave function centered at the
origin of a separable sinusoidal lattice, with lattice depth Vj0 in the jth direction, has the form w0(0) ?
Q
j exp?
? x
jp
2ahoj
?2
, with ahoj =
q
~
m!hoj and !hoj =
q
4ErVj0
~2 . Using this ansatz in the integral (3.6)
yields an onsite interaction given by
U = ~asp2? !hoa
ho
; (3.8)
with the bar indicating the geometric mean. To guarantee the validity of the one band approximation,
the mean interaction energy per particle UN must be smaller than the energy gap to the first vibrational
excitation, i.e. UN <~!ho. N is the mean number of atoms. This inequality is readily satisfied in practice.
The third term in the Hamiltonian takes into account the energy offset at site n due to a slowly
varying external potential V(x).
The realization of the Bose-Hubbard Hamiltonian using optical lattices has the advantage that the
interaction matrix element U and the tunneling matrix element J can be controlled by adjusting the intensity
of the laser beams. As the intensity of the lattice is increased the tunneling rate decreases exponentially and
the onside interaction increases as a power law, V d=40 , (d is the dimensionality of the lattice).
3.2 Superfluid-Mott insulator transition
At zero temperature the physics described by the Bose-Hubbard Hamiltonian can be divided into two dif-
ferent regimes. One is the interaction dominated regime when J is much smaller than U, and the system is
30
in the Mott insulator phase. The other is the kinetic energy dominated regime, where tunneling overwhelms
the repulsion and the system exhibits superfluid properties. The onset of superfluidity is a consequence
of the competition between the kinetic energy, which tries to delocalize the particles, and the interaction
energy, which tries to localize them and make the number fluctuations small.
In the superfluid regime, the kinetic energy term dominates the Hamiltonian. In this regime, quantum
correlations can be neglected and the system can be described by a macroscopic wave function since the
many body state is almost a product over identical single particle wave functions. There is a macroscopic
well-defined phase and the system is a superfluid. Because atoms are delocalized over the lattice with equal
relative phases between adjacent sites, they exhibit an interference pattern when the lattice is turned off, as
expected from an array of phase coherent matter wave sources. As interaction increases the average kinetic
energy required for an atom to hop from one site to the next becomes insufficient to overcome the potential
energy cost. Atoms tend to get localized at individual lattice sites and number fluctuations are reduced. In
the Mott insulator phase the ground state of the system instead consists of localized atomic wave functions
with a fixed number of atoms per site. The lowest lying excitations that conserve particle number are
particle-hole excitations (adding plus removing a particle from the system). This phase is characterized by
the existence of an energy gap. The gap is determined by the energy necessary to create one particle-hole
pair.
The phase diagram [44],[74]-[76] that describes the Mott insulator transition of a translational in-
variant system with Vn = 0 exhibits lobe-like Mott insulating phases in the J-? plane, see Fig. 3.1. Each
Mott lobe is characterized by having a fixed integer density. Inside these lobes the compressibility, @?=@?
with ? the average density of the system, vanishes. The physics behind this diagram can be understood as
follows: If we start at some point in the Mott insulating phase and increase ? keeping J fixed, there is going
to be a point at which the kinetic energy of adding an extra particle and letting it hop around will balance the
interaction energy cost. With an extra particle free to move around the lattice phase coherence is recovered
and the system enters the superfluid regime. Similarly, by decreasing ? from a point in the Mott phase, at
some point eventually it will be energetically favorable to remove one particle from the system. The extra
mobile hole will induce also phase coherence and the system will condense in a superfluid state. Since
31
the kinetic energy of the system increases with J the width of the lobes decreases with J. The distance in
the ? direction at fixed J between the upper and lower part of the lobe is the energy gap. At J = 0 the
gap is just equal to U. Also at J = 0 the intersection points between the lobes with density ? = n and
the ones with density ? = n + 1 are degenerate. Because there is no energy barrier to the addition of an
extra particle at these degenerated points the system remain superfluid. Mott insulator phases occur only at
integer densities; non-integer density contours lie entirely in the superfluid phase because there is always
an extra particle that can hop without energy cost.
The phase diagram includes two different types of phase transition. One type takes place at any
generic point of the phase boundary, and it is driven by the energy cost to add or subtract small numbers
of particles to the incompressible Mott state as explained above. On the other hand, the other type only
occurs at fixed integer density and takes place at the tip of the lobes. This transition is driven at fixed
density by decreasing U=J and enabling the bosons to overcome the on site repulsion. The two kinds of
phase transition belong to different universality classes. In the generic one, the parameter equivalent to
the reduced temperature ? = T ?Tc for finite temperature transitions that measures the distance from the
transition is ? ? ???c, with ?c the chemical potential at the phase boundary. For the special fixed density
on the other hand one must take ? ? (J=U) ? (J=U)c. As shown in ([44]) provided ? ? ? ? ?c, the
compressibility has a singular part scaling near the transition as ??fi, fi > 0. This scaling does not apply
at the special fixed density Mott-Superfluid transition for which ? ? (J=U)?(J=U)c and differentiation
with respect to delta becomes inequivalent to differentiation with respect to the chemical potential. Most of
the features of the phase diagram discussed above can be verified by simple calculations done using a mean
field approximation. The solution of the mean field model [74]- [76] predicts that the critical value at the
tip of the Mott lobe depends on the density and dimensionality of the lattice as
(U=J)c = t
?
2n+1+
p
(2n+1)2 ?1
?
; (3.9)
where n is the integer density of the lobe, t the number of nearest neighbors, t = 2d, with d the dimen-
sionality of the system. The parabolic phase boundary of the lobes in the plane ? vs (J=U) vs is given
by
32
?=J = 12
?
(2n?1)?Jt=U ?+
p
(Jt=U)2 ?2(Jt=U)(2n+1)+1
?
: (3.10)
LParen1JSlash1URParen1c
JSlash1U
?Slash1
U
Phase Diagram
nEqual1
nEqual2
nEqual3
nEqual4
Figure 3.1: Zero-temperature phase diagram. The vertical lines indicate critical values for different filling
factors
Up to this point we have discussed the basic aspects of the superfluid-Mott insulator transition in a
translationally invariant system. The situation is fundamentally different for a inhomogeneous system with
a fixed total number of atoms and external confinement. This is the case realized in experiments, where
besides the lattice there is a harmonic trap that collects the atoms at the center. In this case the density of
atoms is not fixed since the atoms can redistribute over the lattice and change the local filling factor.
To deal with the inhomogeneous case, it is possible to define an effective local chemical potential,
?n = ??Vn, at each lattice site n [43]. If the change in the mean number of atoms between neighboring
sites is small, the system can be treated locally as an homogeneous system. Because in the inhomogeneous
case, the local chemical potential is fixed by the density, as the ratio U=J is changed the system can locally
cross the boundary between the superfluid and Mott insulator phases. Therefore even for the situation
when the local density was not commensurate at the beginning, as the ratio U=J is changed a local phase
transition can take place. For the inhomogeneous case, the gradient in the local chemical potential leads to
a shell structure with Mott insulator regions and superfluid regions in between. In [79], the authors used
33
quantum Monte Carlo simulations to study the ground state of the one dimensional Bose Hubbard model in
a trap. They found that Mott phases exist in extended domains above a threshold interaction strength, even
without the commensurate filling required in the translationally invariant case.
34
Chapter 4
Mean field theory
In this chapter we introduce the mean field Discrete Nonlinear Schr?odinger equations and use it first to
investigate the behavior of a BEC adiabatically loaded into an optical lattice moving at constant velocity,
and second to model the dynamical evolution of a BEC initially loaded into every third site of an optical
lattice.
4.1 Discrete nonlinear Schr?odinger equation (DNLSE)
In the weakly interacting regime, quantum fluctuations can be neglected to a good approximation. In this
regime most of the atoms occupy the same condensate wave function. The macroscopic occupation of a
single mode implies that the commutator of the annihilation and creation operators that create and destroy
a particle in the condensate mode can be neglected with respect to the total number of atoms in the mode.
Therefore, to a good approximation the field operator can be replaced by a c-number,
^'(x) ! ?(x): (4.1)
This procedure can be interpreted as giving to the field operator a non zero average, h^'(x)i = ?(x), and
thus a well defined phase to system. Because the original Hamiltonian is invariant under global phase
transformation, this definition of BEC corresponds to an spontaneous symmetry breaking [11, 12, 80].
The function ?(x) is a classical field having the meaning of an order parameter and is often called the
?condensate wave function?. It characterizes the off-diagonal long range order present in the one particle
density matrix [81].
lim
jx0?xj!1
h^'y(x0)^'(x)i = ??(x0)?(x): (4.2)
Strictly speaking, neither the concept of broken symmetry, nor the one of off-diagonal long range order
can be applied to finite size systems with well defined number of particles. However the condensate wave
function can still be determined by diagonalization of the one particle density matrix. The eigenstate with
larger eigenvalue corresponds to ?(x).
35
If we use the approximation (4.1) in the many-body Hamiltonian (3.1) we get a classical Hamiltonian
given by:
H =
Z
dx?y(x)
 
?~
2
2mr
2 +Vlat(x)
?
?(x)+(V(x)??)?y(x)?(x) (4.3)
+12 4?as~
2
m
Z
dx?y(x)?y(x)?(x)?(x);
The time dependent equations of motion for the condensate wave function can be obtained from the varia-
tional principle by minimizing the quantity i~@@t ? ^H. The minimization yields
i~@?(x)@t =
 
?~
2
2mr
2 +Vlat(x)+V(x)+ 4?as~2
m j?(x)j
2
?
?(x) (4.4)
This equation is known as the Gross-Pitaevskii equation (GPE). It was derived by independently by Gross
and Pitaevskii [10].
If the chemical potential is small compared to the vibrational level spacing and the lattice depth
is deep enough that a tight binding picture is valid, the condensate order parameter can be expanded in a
Wannier basis keeping only the lowest band orbitals:
?(x;t) =
X
n
zn(t)w0(x ? xn): (4.5)
Using this ansatz in Eq. (4.4), the Gross-Pitaevskii equation reduces to the discrete nonlinear
Schroedinger equation (DNLSE), where the zn amplitudes satisfy:
i~@zn@t = ?J
X
<n;m>
zm +(Vn +UNjznj2)zn; (4.6)
where < n;m > restricts the sum over nearest neighbors, the parameters J and U are the ones defined in
Eq. (3.5) and (3.6) and N is the total number of particles. The normalization condition for the amplitudes
is given by:
X
n
jznj2 = 1: (4.7)
36
Notice the DNLSE can also be derived by assuming the field operators present in the Bose-Hubbard Hamil-
tonian to be c-numbers,i.e. ^an = zn(t). This assumption leads to a tight-binding classical Hamiltonian,
which can be minimized using the variational principle.
H = ?J
X
<n;m>
z?nzm +
X
n
 
Vnjznj2 + U2 jznj4
?
: (4.8)
The DNLSE is similar to its linear counterpart, the DSE or tight-binding Schr?odinger equation (see
2.21), but with an extra cubic term, which takes into account mean field interaction effects. To see the
modification that the nonlinear term introduces in the dispersion relation, we consider a one dimensional
optical lattice with M sites and periodic boundary conditions.
In the translational invariant case with no other external potential present, Vn = 0, the solutions of
the DNLSE that are similar to those of the noninteracting case are, zn = pno exp(?inq)exp(?iEqt=~),
with no = N=M the condensate density. Using this plane wave ansatz in the DNLSE we get a dispersion
relation given by:
Eq = ?2J cos(qa)+Uno (4.9)
In the tight-binding approximation, mean field interactions do not modify the shape of the lowest band: it
has still a cosine dependence with bandwith equal to 4J. The only difference with respect to the single
particle case is an energy shift in the dispersion relation which takes into account the average mean field
energy Uno.
It is important to mention though, that outside the range of validity of a tight-binding model, nonlin-
ear effects does not necessarily change the dispersion relation in a simple way, especially at the edge of the
Brillouin zone. It has been shown in the literature [82, 83] that if the chemical potential is bigger than the
gap vibrational excitation energy to higher bands, the lowest band becomes triple valued near the Brilloiun
zone edge and a loop appears at this point. This loop reflects the existence of new solutions, which only
exist when the tight-binding approximation is not valid. These new solutions have a non zero velocity at the
edge of the Brillouin zone, as opposed to the linear case. The nonzero velocity carried by the Bloch waves
is a manifestation of the superfluidity of the system. For free particles the flow is stopped completely by
Bragg scattering from the periodic potential, but if interactions are strong enough the superfluid flow can
no longer be stopped.
37
4.2 Dragging experiment
In this section we investigate an experiment done at NIST where a Bose-Einstein condensate was prepared
in a harmonic trap and adiabatically loaded into a one dimensional optical lattice moving at constant velocity
Ref. [37]. We first study the dynamics in a translationally invariant lattice using a simple single particle
picture, and show how the center-of-mass motion of the condensate can be used as a probe of the lattice
band structure. Then we discuss the effects of the trapping potential and interaction effects on the dynamics.
When mean field effects are included, they cause the effective mass of the Bloch state to depend on time,
influencing the condensate dynamics in the lattice. As a consequence, characteristic nonlinear effects such
as solitons, self focusing effects and dynamical stabilities might be observed.
4.2.1 Experiment
In the experiment, a condensate prepared in the ground state of a harmonic trap is loaded in an optical lattice,
which is moved at constant velocity. The loading of the condensate in the optical lattice is done by turning
on the lattice, linearly in time, up to some maximum height. The important part of this turning on is that it
is done sufficiently slow to be adiabatic with respect to band excitations. After the maximum is reached, at
t = tr, the lattice depth is held constant for a fixed period of time and the system is allowed to expand in the
lattice. Finally at t = tf the condensate density distribution along the lattice axis is imaged. The temporal
dependence of the optical lattice depth is shown in Fig. 4.1. In the figure, experimental results obtained for
the final condensate density for different lattice depths are also shown. The general observation is that the
deeper the lattice, the greater the response of the condensate to the lattice motion. In this experiment, the
condensate has a very narrow momentum spread compared to the lattice recoil momentum.
4.2.2 Linear free particle model
In this section to a first approximation we are going to neglect interactions and apply a single particle picture
to understand the dynamics.
Because the momentum spread of the condensate is very narrow compared with the lattice recoil
momentum at t = 0, the initial wave function can be approximated by a plane wave with zero momentum.
38
Figure 4.1: Condensate density in the x-direction at the observation time tf. Light colors areas indicate
regions of high condensate density. The dashed line indicates the final location of a co-moving point with
the optical lattice.
This wave function, in the lattice frame, which is moving with velocity vlatt with respect to the lab frame,
has the form ?(x0) = e?ikox0, with vlatt = ~ko=m (we used primes to denote the coordinates and wave
functions in the moving frame). As t increases, the optical potential is ramped up, the wave function
evolves, but due to the fact that the Hamiltonian is invariant under translation, the quasimomentum is
conserved (at least in this linear model with no other potential than the lattice) during the ramping time and
the average quasimomentum at the end of the loading should remain ?ko. From tr to tf the lattice depth is
held constant. During this period of time the average velocity of the atom, as seen in the lattice frame, is:
v = 1~dE
(n)
q
dq
flfl
fl
q=?ko
; (4.10)
with E(n)q the nth band Bloch-like atomic dispersion relation evaluated at the final lattice depth. Trans-
39
forming back to the lab frame, Eq. (4.10) implies that in the lab frame, during tr to tf, the average velocity
of the atoms is given by:
vatom = ~kom + 1~dE
(n)
q
dq
flfl
fl
q=?ko
: (4.11)
Because x(tf) is the observable of the experiment, we can calculate it as
?x = hx(tf)i?hx(tr)i? vatom(tf ?tr): (4.12)
Therefore, by measuring the average position of the atoms for a fixed tf ?ti the experiment measures the
atom?s velocity, which is a determined by the band structure of the lattice.
In Fig. 4.2 we plot the atom?s average velocity in the lab frame vs. the lattice velocity. The velocity
in Eq.(4.11) is calculated using the Bloch dispersion relation for a three recoil lattice. The tight-binding
results (dotted) are also shown. Even though the tight-binding velocities do not exactly match the velocities
calculated using the Bloch dispersion relation, we can say that for a three recoil lattice the tight-binding
approximation is fair.
0 0.5 1 1.5
vlattSlash1vB
0
0.5
1
v atom
Slash1v B
VoEqual3Er
Bloch
TBA
Figure 4.2: Atom?s average velocity in the lab frame vs. lattice velocity. Line: results from the Bloch
dispersion relation, dots: tight-binding approximation results. vB =~?=am.
We emphasize the following points:
? Because the initial quasimomentum is determined by the lattice velocity, vlat =~ko=m, to move the
lattice at different speeds corresponds to loading the condensate at a different Bloch state. How much
40
the atoms are dragged depends on the derivative of the dispersion relation evaluated at the loaded
quasimonentum.
? At zero lattice depth the dispersion relation is quadratic, 1~dE
(n)
q
dq
flfl
fl
q=?ko
=~ko=m and therefore Eq.
(4.11) predicts no dragging. As the lattice depth increases the dispersion relation desviates more from
free-particle behavior and dragging ensues. This is consistent with the experimental results shown in
Fig. 4.1.
? For small lattice velocities (jkoaj? 1, a the lattice constant) the atoms are almost stationary because
then the dispersion relation is more free-particle like. On the other hand, as ko approaches the k of
the Brillouin zone edge, ko = ?qB, the atoms are dragged the most. At this point, as opposed to the
free free system, Bloch waves carry zero velocity, i.e dE(n)q =dqjq=?qB = 0.
4.2.3 Effect of the magnetic confinement
In the experiment a harmonic confinement was present in addition to the lattice potential. To model the
effect of the magnetic trap on the atoms dynamics we used a semiclassical model (see chapter 2). The
semiclassical model is a very good approximation whenever it is unnecessary to specify the position of the
atom on a scale comparable with the spread of the wave packet. The validity of the semiclassical model
requires that the external potential varies slowly over the dimension of the wave packet and ignores the
possibility of inter band transitions. In the experiment, the strength of the harmonic trap is less than one
percent of the potential depth and we expect the semiclassical model to be valid. Following this model, in
the lattice frame, the time evolution of the average position and wave vector of the atoms are given by Eq.
(2.26):
_x = v(n)(q) = 1~dE
(n)
q
dq ; ~_q = ?m!
2(x+vlatt); (4.13)
with ! the magnetic trapping frequency. The initial conditions are k(0) = ?ko and x(0) = 0. We solved
these equations of motion, using the analytic dispersion relation for a sinusoidal lattice in terms of Mathieu
functions. We used the parameters V0 = 3ER and 1=2m!2=ER = 0:000025. The results are shown in Fig.
4.3.
41
0 0.5 1 1.5v
lattSlash1vB
0
0.5
1
v atom
Slash1v B
SemiclassicalmodelLParen1VoEqual3ErRParen1
notrap
trap
Figure 4.3: Atom?s average velocity in the lab frame vs. lattice velocity in the presence of a magnetic trap.
vB =~?=am.
We observe in the plot that when atoms are loaded in the first band, for a range of lattice velocities
below half the Brillouin velocity (vlatt . vB=2 with vB = ~?=am), the motion of the atoms with and
without trap is almost the same. For larger velocities, as long as the atoms remain in the first band, trapped
atoms move faster than the un-trapped ones. However, if the lattice velocities exceeds the Brillouin velocity,
vB, the opposite behavior is observed and for an interval of velocities, vB < vlatt . 3vB=2, trapped atoms
slow down compared with the ones in the homogeneous lattice.
This behavior can be understood as follows: Because of the trap, the atoms feel a force opposite to
the direction of motion. The force not only increases with time but also with the lattice velocity (see Eq.
(4.13)). The average quasimomentum of the system evolves in the presence of the force, and if at time t = 0
the atoms are loaded with quasimomentum q(0) = ?ko, then at tf the final quasimomentum is q(tf) =
?ko+?(ko;tf). Here ?(ko;tf) is a negative quantity because the force is a restoring force. Therefore, the
final velocity of the atoms in the trap , is less or greater than the velocity of the atoms without the external
trap depending on the sign of the effective mass (sign of the curvature of the dispersion relation) evaluated at
the final momentum. If the effective mass is positive, d2Eqdq2 > 0, (0 < q < ?=2a, ?=a < q < 3?=2a,...), the
group velocity is an increasing function of q and the velocity of the trapped atoms is less than the untrapped
atoms? velocity. On the other hand, when the effective mass is negative,(?=2a < q < ?=a,...), the group
42
velocity is a decreasing function of q and the trapped atoms speed up with respect to the untrapped ones.
The more pronounced difference between trapped and un-trapped atom velocities observed in the second
band with respect to the first one, is not only because ? increases with the lattice velocity but also because
of the discontinuous change of the band structure when crossing the Brillouin zone edge. Besides the sign
difference in the curvature the second band has a larger band width than the first one.
4.2.4 Interaction effects
To model the effect of the interatomic interactions on the dynamics we used the tight-binding variational
method proposed in Ref. ([84, 85]). In these reference the authors assume a Gaussian profile wave packet
zn(t) = 4
r 2
 2?e
[?(n??)2= 2+ik(n??)+i?(n??)2=2]; (4.14)
and used ?; ;k;? as variational parameters. The equations of motion they found are:
_k = ?@V
@? ; (4.15)
_? = 2J sin(k)e??; (4.16)
_? = 2J cos(k)e??
 4
 4 ??
2
?
+ 2NUp? 3 ? 4 @V@ ; (4.17)
_ = 2J cos(k)e?? ?; (4.18)
with ? =
?
1
2 2 ?
?2 2
8
?
, V = p2= 2?R dxV(x)exp[?2(x??)2= 2] and the time scaled as t ! tEr=~.
Notice that the variable k is just the average quasimomentum of the wave packet. We can also associate an
inverse effective mass given by m??1 = 2J cos(k)e??.
We numerically integrated the equations of motion assuming no other external trapping potential
besides the lattice, V = 0. In this case the Hamiltonian is invariant under translations and we expect
conservation of the quasimomentum. This is seen in the above equations, because if V = 0 then dk=dt = 0.
The parameters were chosen according to the experiment:  (0) = 10, ?(0) = 0 and NU = 2ER. The
bandwidth J, was found using Mathieu functions assuming a 3ER lattice. We plot Fig. 4.4 vatom vs vlatt
evaluated at t = 50. The curve found is plotted in
When mean field effects are included, the dynamics of the system becomes very rich. Depending
on the the initial quasimomentum and width of the packet, a variety of dynamical regimes can be observed.
43
0 0.2 0.4 0.6 0.8 1 1.2 1.4
vlattSlash1vB
0
0.2
0.4
0.6
0.8
1
v atom
Slash1v B
vatomEqualvlatt
linear
nonlinear
Figure 4.4: Velocity of the atoms predicted by the variational model in the case when interactions are
taken into account (dotted line). The single particle case is shown in red. The parameters used were
 (0) = 10;?(0) = 0 and NU = 2ER.
Physically, the different regimes can be understood by realizing that the nonlinear term makes the effective
mass time dependent. The effective mass, m? = e?=(2J cosk), not only depends on the initial quasi-
momentum but also on the width of the wave packet, and when the nonlinearity is present ? is no longer
constant. The different dynamical behaviors include effects such as self-trapping, diffusion, and for special
cases solitons or breathers [84].
The self-focusing is a genuine nonlinear effect, characterized by a diverging effective mass. In
particular the self-trapped wave packet cannot translate along the array. The final value for  f, for a packet
in the self-focusing regime predicted by the variational model is,
 f = UN oUN ?4Jp?cos(k) 
o
; (4.19)
With  o =  (0) the initial width of the packet, which is assumed big compared to one.
The diffusive regime, on the other hand, is characterized by the unbounded growth of the width of
the packet. This is the behavior expected in regular linear quantum mechanics. The model predicts that the
transition between the two regimes occurs, again assuming  o  1, at
44
(UN)c = 4Jp?cos(k) o jkj < ?=2
(UN)c = 2J
p?jcos(k)j
 o jkj > ?=2 (4.20)
Using a potential depth of 3ER and  o = 10, the predicted dynamical phase diagram as a function of UN
vs vlat is shown in Fig. 4.5. It can be seen that for UN = 2ER, the self-focusing regime is expected for
lattice velocities greater than vlat = 0:4vB. At vlat = 0:5vB the noninteracting effective mass becomes
negative.
0 0.2 0.4 0.6 0.8 1
vlat Slash1vB
0
2
4
6
8
UN
Phase Diagram VoEqual 3 Er
Diffusion Self MinusFocusing
Figure 4.5: Phase diagram UN vs vlatt for Vo = 3Er and  o = 10.
Consistently, in Fig. 4.4 we see that for lattice velocities approximately greater than vlat = 0:4vB,
the velocity of the interacting system becomes greater than the noninteracting one, and approximately at
vlat = 0:7vB the velocity of the atoms is just the velocity of the lattice so, the atoms stop moving in
the lattice frame. This regime is a self-trapping regime and nonlinear effects tend to localize the atoms
in the lattice frame. On the other hand, for vlatt . 0:4vB the dynamics of the system is similar to the
non interacting case. To appreciate the different behaviors exhibited by the system below and above vB=2
we plot in Figs. 4.6 and 4.7, the evolution of the wave packet, for two different initial velocities. For
vlatt = ?0:7vB the wave packet localizes, whereas for vlatt = ?0:15vB the wave packet diffuses.
45
Minus30 0 30n0
0.5
1
VertBar1z nLParen1
tRParen1VertBar12
tEqual30
Minus30 0 30n0
0.51
1.5
VertBar1z nLParen1
tRParen1VertBar12
tEqual40
Minus30 0 30n0
0.51
1.52
VertBar1z nLParen1
tRParen1VertBar12
tEqual50
Minus30 0 30n0
0.5
VertBar1z nLParen1
tRParen1VertBar12
tEqual0
Minus30 0 30n0
0.5
VertBar1z nLParen1
tRParen1VertBar12
tEqual10
Minus30 0 30n0
0.5VertBar1z nLParen1tRParen1VertBar12
tEqual20
Figure 4.6: Localization of the wave packet in the self-focusing regime vlatt = ?0:7vB. Notice the change
in the vertical scale in the last row
Minus60Minus300 30n0
0.5
VertBar1z nLParen1
tRParen1VertBar12
tEqual30
Minus60Minus300 30n0
0.5
VertBar1z nLParen1
tRParen1VertBar12
tEqual40
Minus60Minus300 30n0
0.5
VertBar1z nLParen1
tRParen1VertBar12
tEqual50
Minus60Minus300 30n0
0.5
VertBar1z nLParen1
tRParen1VertBar12
tEqual0
Minus60Minus300 30n0
0.5
VertBar1z nLParen1
tRParen1VertBar12
tEqual10
Minus60Minus300 30n0
0.5
VertBar1z nLParen1
tRParen1VertBar12
tEqual20
Figure 4.7: Evolution of the wave packet in the diffusive regime vlatt = ?0:15vB.
4.3 Dynamics of a period-three pattern-loaded Bose-Einstein condensate in an optical lattice
In this section we discuss the dynamics of a Bose-Einstein condensate initially loaded into every third site
of a one dimensional optical lattice, motivated by the recent experimental realization of this system by
the NIST group [33]. A condensate loaded in this way is not an eigenstate of the final period-a lattice,
and the condensate will continue to evolve in the final system. Outside the strongly correlated regime,
the GPE equation is expected to give a good description of the condensate dynamics. In our system we
assume the lattice is sufficiently deep that a tight-binding description is applicable, and the DNLSE is
valid. For the periodic initial condition of equal wavefunction amplitude at every third site and zero at
46
all others, symmetry arguments can be used to reduce the wavefunction evolution to a two mode problem
(analogous to a double well system with an energy offset) for which an analytic solution of the dynamics
can be given. We show that for large ratios of the interatomic interaction strength to tunneling energy the
condensate evolves with self-maintained population imbalance, whereby the condensate population tends
to remain localized in the initially occupied lattice sites. A similar phenomenon has been studied in double-
well systems [87, 88, 89, 90, 91]. We show that the momentum distribution of an interacting condensate
changes in time in a manner which can be related to the spatial tunneling of condensate, and would be a
suitable experimental observable.
Under the influence of an external force a Bloch state will exhibit Bloch oscillations, as described
in chapter 2. To illustrate how a linear potential affects the motion of a pattern loaded condensate we find
analytic solutions for the noninteracting case with periodic initial conditions and show how the dynamics
for this system can be interpreted in terms of the interference of three Bloch states of the lowest band
undergoing Bloch in unison. We present numerical results for more general (non-periodic) initial conditions
and consider characteristic properties of the momentum distribution.
4.3.1 Experiment
In the experiment, a combination of two independently controlled lattices was used to load the condensate
into every third site of a single lattice. Briefly the procedure consists, as shown in Fig. 4.8, of loading a
condensate into the ground band of lattice with periodicity 3a, so that the condensate is well localized in
the potential minima of this lattice. A second lattice of periodicity a which is parallel to the first lattice, is
then ramped up so that the superimposed light potentials form a super-lattice of period 3a. For the ideal
case both lattice potentials are inphase, and the addition of the second lattice will not shift the locations of
the potential minima from those of the first lattice alone, and the condensate will remain localized at these
positions. Finally, by removing the first period-3a lattice on a time scale long compared to band excitations,
but short compared to the characteristic time of transport within the lattice, the condensate will be left in
every third site of the period-a lattice.
47
Figure 4.8: Patterned loading experiment.
4.3.2 Case of no external potential
The major interest of this section is in the tunneling properties of the condensate in the lattice, and in the
noninteracting case the time scale for tunneling is determined by the hopping matrix element J. For this
reason it is convenient to define a new dimensionless scale of time ? = Jt=~, and dimensionless energy
En = Vn=J and coupling constant ? = NU=J. In terms of these new variables the tight-binding evolution
equation (4.6) takes the form
i@zn@? = ?(zn?1 +zn+1)+(En +?jznj2)zn; (4.21)
? The case of periodic initial conditions: reduction to a two mode system
We treat first a model case in which no external potential is present, (En = 0), and in which the
initial occupancies of each third site are the same, and in which the condensate initially has a uniform
48
phase. At ? = 0, the amplitudes zn(?) are given by
z3n(0) = p?; (4.22)
z3n+1(0) = z3n+2(0) = 0; (4.23)
where ? = 3=M with M the total number of lattice sites. N? represents the initial number of atoms
per occupied site. This initial condition is homogeneous in the sense that each occupied site has the
same amplitude and phase along the length of the lattice. For an infinite lattice, or one with periodic
boundary conditions, the amplitudes for all initially occupied sites (z3n) evolve identically in time,
and the amplitudes for the initially unoccupied sites satisfy z3n+1(?) = z3n+2(?) for all ? and all n.
This allows us to reduce the full set of equations (4.21) to a set of two coupled equations
i@z0@? = ?2z1 +?jz0j2z0; (4.24)
i@z1@? = ?(z1 +z0)+?jz1j2z1; (4.25)
where z3n ? z0 and z3n+1 = z3n+2 ? z1 for all n. The normalization condition is
jz0j2 +2jz1j2 = ?: (4.26)
The Hamiltonian function, H, of this system is (from Eq. 4.8)
H = J?
?
?2(z?0z1 +z0z?1)?2jz1j2 + ?2jz0j4
+?jz1j4
?
(4.27)
= J??2 :
By writing z0=?0p? and z1=?1p?=2, we can transform Eqs. (4.24)-(4.25) to the form
i @@?
0
BB
@
?0
?1
1
CC
A =
0
BB
@
 j?0j2 ?p2
?p2  j?1j2p2 ?1
1
CC
A
0
BB
@
?0
?1
1
CC
A; (4.28)
where  = ?? is the ratio of on-site repulsion to tunneling energies, and the normalization condition
(4.26) is now j?0j2 + j?1j2 = 1. The factor of p2 difference in the definition of ?0 and ?1 arises
because ?1 represents the amplitude of the two initially unoccupied sites. With this factor incorpo-
rated, the matrix appearing in Eq. (4.28) is explicitly Hermitian. We note that this equation of motion
is identical to that for a condensate in a double well trap in the two mode approximation [92, 93].
49
We note in passing that a similar reduction, to a system of [m=2] + 1 equations, exists for lattice
systems that are loaded such that only every m-th site is initially occupied.
? Solution of the equations of motion
It is convenient to write the lattice site amplitudes appearing in Eqs. (4.24) and (4.25) as z0=fei 0p?
and z1= gei 1p?, where f;g, and  are real. By introducing the phase difference ` =  0 ? 1, Eqs.
(4.24), (4.25) and (4.27) can be recast as
:f = 2gsin`; (4.29)
:g = ?f sin`; (4.30)
 
2 = ?4fgcos`+
 
2
?f4 +2g4??2g2; (4.31)
The analytic solutions of Eqs. (4.29)-(4.31), found using a procedure similar to that presented by
Raghavan et al. [87], can be expressed in terms of Weierstrassian elliptic functions }(?;g2;g3) [69].
The analytic solutions are
f(?) =
s
1? 2412}(?;g
2;g3)+(9+2 + 2)
; (4.32)
g(?) =
r
1?f(t)2
2 ; (4.33)
where the parameters g2 and g3 are given by
g2 = ?81?14 2 +4 3 + 4?=12; (4.34)
g3 = ?729+243 2 ?46 3 ?15 4 + 6?=216: (4.35)
The solutions f(?) and g(?) are oscillatory functions whose amplitudes and common period, T( ),
are determined by the parameter  (see Figs. 4.9 and 4.10). It is useful to qualitatively divide
this behavior into two regimes, separated by  = 2. Analysis of Eqs. (4.29) - (4.31) shows that
f(?0) = g(?0) for some value of ?0 when  ? 2, and for  > 2, f(?) > g(?) for all ?.
Tunneling dominated regime
For  . 2, we find that the oscillation period is essentially constant (see Fig. (4.9).
50
? (J)
T (
h/J
)
Figure 4.9: Oscillation period (in units of~=J) as a function of the interaction strength  .
In this case the role of interactions is relatively small, and the behavior can be approximately under-
stood by taking  = 0, in which case the matrix of Eq. (4.28) is constant in time. The equations of
motion in this case are equivalent to those of a two-state Rabi problem [94], where the two levels are
coupled by a Rabi frequency of strength p2, which is detuned from resonance by ?1. This system
will undergo Rabi oscillations whereby atoms periodically tunnel from the initially occupied site into
the two neighboring sites. Because the coupling is detuned from resonance the transfer of populations
between wells is incomplete, with j?1j2 attaining a maximum value of 8=9. The Rabi model predicts
that the cycling frequency between the levels is equal to the difference between the eigenvalues of
the matrix of matrix of Eq. (4.28), which gives the period of oscillation as TRabi = 2?=3 in units of
~=J (see Fig. 4.9).
Interaction dominated regime
The effect of interactions on the mean-field dynamics is to cause the energies of the initially occupied
sites to shift relative to those of the unoccupied sites. As  increases and this energy shift increases
relative to the strength of coupling between sites, the tunneling between sites occurs at a higher
frequency, but with reduced amplitude. The population of the initially occupied sites becomes self
trapped by the purely repulsive pair interaction, which in the context of a double well system has been
called ?macroscopic quantum self trapping? [93]. This is demonstrated quantitatively in Fig. 4.10
where we plot the minimum value of f2 occurring during the oscillation as a function of  . In con-
51
trast to the tunneling dominated regime, where tunneling periodically populates all sites equally, the
condensate tends to be localized on the initially occupied sites in the interaction-dominated regime.
(  )
? (J)
Figure 4.10: Minimum value of f2 during an oscillation period as a function of  . As  increases the
population imbalance between wells increases (see text).
? Momentum space dynamics
Typically the spacing between individual wells in an optical lattice is too small to resolve the localized
density distributions of atoms in neighboring sites using standard imaging techniques.
The momentum distribution is a more convenient observable which approximately corresponds to the
expanded spatial distribution of the released condensate. Here we calculate the momentum dynamics
of the condensate loaded into every third site of an optical lattice, and show how this relates to the
evolution of the spatial amplitudes given in Eqs. (4.32) and (4.33).
In the tight-binding approximation the condensate order parameter ?(x;?) (4.4) is expressed as a sum
over the lattice sites . Because of the periodicity of the system, the momentum space wavefunction,
which we denote as ~?(k;?), is expressible as the Fourier series
~?(k;?) =
X
m
zm(?)e?ikam?(k); (4.36)
where
?(k) = 1p2?
Z 1
?1
e?ikxw0(x)dx: (4.37)
To compute the momentum distribution, we invoke the identity PM?1n=0 eikna = M?k;2?m=a, where
52
M is the number of lattice sites, and m an integer. Since there are only two independent amplitudes
in the set fzmg, we find that
~?(k;?) =
p
3=?cm(?)?m?k;qm=3;
?m = ?(qm=3); (4.38)
cm(?) =
r 1
3?
?
z0 +z1e?iqm=3 +z2e?i2qm=3
?
;
where q is the reciprocal lattice vector q = 2?=a. The momentum distribution of the system has
very sharp peaks of relative amplitude jcmj2 at momentum k = qm=3, arising from the 3-lattice
site spatial periodicity of the condensate wavefunction. In addition, ?m describes a slowly varying
envelope determined by the localization of the Wannier states at each lattice site.
Using the analytic solutions for f and g, and Eq. (4.31), we obtain only two independent Fourier
amplitudes
jc3n(?)j2 = 13
?
1+  4(3f2 +1)(f2 ?1)
?
; (4.39)
jc3n+1(?)j2 = 13
?
1?  8(3f2 +1)(f2 ?1)
?
(4.40)
= jc3n+2(?)j2: (4.41)
In the reduced zone scheme, where we only consider momenta in the range k 2 [?q=2;q=2], the
momentum wavefunction then consists of three peaks corresponding to Bloch states of quasimomenta
0;?q=3. The identical behavior of jc3n+1j2 and jc3n+2j2 means that the ?q=3 peaks always have the
same intensity. If interactions between the atoms are ignored (i.e.  = 0), the momentum components
are constant in time (see Eqs. (4.39) and (4.40)), even though tunneling occurs between the lattice
sites. However, when interactions are considered, the momentum intensities explicitly depend on the
occupations of each site and will vary in time when tunneling occurs. The magnitude of the time
variation of the jcnj2 is proportional to  , but will reduce for sufficiently large values of  , where
the self-trapping effect causes the tunneling between lattice sites to stop (i.e. f2 ? 1 at all times).
In Fig. 4.11 we show the maximum contrast between the intensity of the Fourier peaks, defined as
?max ? (jc1j2 ?jc0j2)max, where the value is maximized by evaluating the jcnj2 at the time when
53
f takes its minimum value (we note that f(?) is given by Eq. (4.32)). We note that the maximum
contrast occurs for  ? 3:9; in this case, the zero quasimomentum component c0(?) vanishes once
during each period of oscillation T( ). For values of  greater than 3:9 the contrast between the
intensities starts to decrease due to the reduction in tunneling caused by the nonlinearity- induced
self-trapping.
0 5 10 15 20
0
0.1
0.2
0.3
0.4
0.5
|? max
|2
? (J)Figure 4.11: Maximum contrast of the Fourier components as a function of  . The maximum contrast is
defined as ?max ? (jc1j2 ?jc0j2)max with the maximum value occurring when f2 is at its minima.
? Application to an inhomogeneous condensate
Here we wish to consider the dynamics for an inhomogeneous condensate, applicable to a condensate
initially prepared in a harmonic trap. For the pattern loaded condensate, we use inhomogeneous to
refer to the overall spatial envelope of the period three initial condition. The previous homogeneous
theory we have presented is expected to accurately describe inhomogeneous cases when the initial
pattern of population in every third site extends over many lattice sites i.e. M  1 so that mean-
field energy associated with each triplet of sites U(n) = ?2 P3i=1jz3n+ij4 varies slowly across the
system. Taking a particular example we choose a Gaussian envelope to the periodic arrangement of
atoms into every third site, so that the initial state is
z3n+1(0) = z3n+2(0) = 0; (4.42)
z3n(0) = 3
? 2
?M2
?1=4
exp(?(9n=M)2): (4.43)
For the simulations shown here, the parameters used were N = 105, M = 76, U = 2:11?10?5ER
54
and J = 0:075 ER; these correspond to a condensate of 105 atoms of 87Rb produced in a magnetic
trap with axial and radial frequencies of 9 and 12 Hz respectively, and loaded into a lattice with a
depth of 4.5ER, with ER = 2:2kHz. These parameters are typical of the experimental regime, but
also lie in a range in which the homogeneous model is expected to give a fair description. The total
number of occupied wells and the strength of the on site interatomic interaction were calculated by
preserving the value of the chemical potential of the system upon reduction to one spatial dimension
and by assuming that each of the localized orbitals in the tight-binding description are Gaussian. The
hopping rate J was estimated by using Mathieu functions. In Fig. 4.12 we show the evolution of the
population of the central wells (normalized to one) compared with the homogeneous model with  
taken to be the local mean field energy  efi = ?
q
2
? (9=M).
Fig. 4.12 shows the results of numerical integration of the equations of motion and the approximate
analytical solution given by the quasi-homogeneous model described above. We see that the numer-
ical and analytical results agree well at short times, but differ more as time progresses due to the
different mean-field seen by different wells.
Figure 4.12: Comparison between the population evolution of the central three wells for the inhomogeneous
condensate and the homogeneous model. Inhomogeneous condensate: stars for the initially populated well
and boxes for the initially empty wells. Homogeneous model: dashed line represents the initially populated
wells, and the solid line represented the initially unpopulated wells. We used  efi as the local mean field
energy (see text). The parameters used for the simulation were J = 0:075ER and  efi = 2:64.
To understand the disagreement as time evolves, we show in Fig. 4.13 the numerical Fourier spectrum
55
for the inhomogeneous case evaluated at several different times. The variation of the intensities of the
peaks, which, as shown below, is related to the spatial tunneling between lattice sites in the presence
of the mean field, can be seen in the plot. Initially all occupied sites are in phase and the three
distinctive momentum peaks have a narrow width determined by the intrinsic momentum uncertainty
of the condensate envelope. That is the reason why the homogeneous model fits very well. However,
as time progresses the mean field variation across the lattice causes the tunneling rates to vary with
position and leads to momentum peak broadening. This effect eventually causes the homogeneous
model to become an inaccurate description of the inhomogeneous system.
t = 1.7ms t = 2.6ms
t = 0ms t = 0.8ms
momentum momentum
density
 (arb. units)
0 q/3 2q/3 0 q/3 2q/3
density
 (arb. units)
momentum momentum
0 q/3 2q/3 0 q/3 2q/3
Figure 4.13: Momentum distribution of the inhomogeneous condensate evaluated at various times for the
same parameters as used in Fig.4.12.
We note that momentum space signature for spatially tunneling in the interacting system is still
present in the inhomogeneous case. This is shown in Fig. 4.14, where we plot the evolution of
the quantities jc3n(?)j2, jc3n+1(?)j2, jc3n+2(?)j2 (calculated from the numerical simulation by parti-
tioning the numerical Fourier spectrum in three equal non overlapping sections, each centered around
56
the respective peak and adding the square of the norm of the Fourier components within each section)
vs. the ones calculated with the homogeneous model, but using an averaged value  ave ? ?Pnjznj4
instead of  = ??. It can be observed that the predictions of the simple model are in very good agree-
ment with the numerical results when the three peaks of the spectrum are well defined. For longer
times, the width of the Fourier peaks increases, until a point when they split. At this point the quan-
tities jc3n(?)j2, jc3n+1(?)j2, jc3n+2(?)j2 are not meaningful anymore. Because the parameters used
for the numerical calculations were chosen to be experimentally achievable, and as shown in the
plots the model predictions are fair at least for one period of oscillation, we conclude that the Fourier
distribution can be used as a signature of the mean-field quantum tunneling inhibition.
Figure 4.14: Evolution of momentum peak populations. Upper curves: population of the q = ?2?=3
momentum states. Lower curves: population of the q = 0 momentum state. Inhomogeneous condensate
(dotted), homogeneous result (solid line), where the comparison is made by replacing  by an average mean
field energy  ave ? ?Pnjznj4 = 1:85. Parameters are the same as in Fig. 4.12.
4.3.3 Dynamics with a constant external force
? Homogeneous three state model
In this section we consider the dynamics of a periodically loaded condensate in the presence of a
linear external potential, corresponding to a uniform force parallel to the lattice. In what follows we
assume that the force is sufficiently weak that band excitations due to Landau-Zener tunneling are
negligible, so that a tight-binding picture of the lowest band is sufficient to describe the dynamics. In
57
this case the evolution equation differs from what we considered in the previous section by the term
En in Eq. (4.21) taking the form En = n?, where ? is the potential difference between lattice sites (in
units of the hopping matrix element J). Taking the initial conditions (4.22)-(4.23), and transforming
the Wannier amplitudes as z3n+j(?) = ~?3n+j(?)e?i3n?? (j = 0;1;2), we obtain the evolution
equations
i@
~?3n
@? = ?(
~?3n?1ei3?? + ~?3n+1)
+?j~?3nj2~?3n; (4.44)
i@
~?3n+1
@? = ?(
~?3n+2 + ~?3n)+?~?3n+1
+?j~?3n+1j2~?3n+1; (4.45)
i@
~?3n+2
@? = ?(
~?3n+1 + ~?3n+3e?i3??)+2?~?3n+1
+?j~?3n+1j2~?3n+1: (4.46)
Assuming periodic boundary conditions, the periodicity of the initial conditions and equations of
evolution allow considerable simplification from the full set of M coupled equations. In particular,
these assumptions mean that every third Wannier amplitude evolves identically (i.e. ~?n = ~?n+3)
and so the evolution of the system can hence be reduced to the three independent equations
i@
~?0
@? = ?(
~?2ei3?? + ~?1)+?j~?0j2~?0; (4.47)
i@
~?1
@? = ?(
~?2 + ~?0)+?~?1 +?j~?1j2~?1; (4.48)
i@
~?2
@? = ?(
~?1 + ~?0e?i3??)+2?~?2
+?j~?2j2~?2; (4.49)
where the new amplitudes map onto the original set according to ~?0 $ f~?3ng, ~?1 $ f~?3n+1g,
and ~?2 $f~?3n+2g, and obey the normalization condition
2X
j=0
j~?jj2 = 3M: (4.50)
The equations of motion (4.47)-(4.49) are more difficult to treat analytically than the case considered
in the last section due to the presence of a linear potential. In this section we derive an analytic solu-
tion for a noninteracting condensate (i.e. ?=0), which provides valuable insight into the complicated
58
tunneling dynamics the system exhibits in the absence of nonlinearity, yet should furnish a good de-
scription for dilute condensates satisfying  ? 1. For the nonlinear regime we present numerical
results to illustrate the typical behavior.
? Analytic solution for linear dynamics
Defining the vector x(t) = (~?0(?); ~?1(?); ~?2(?)), and using the transformation y(?) = P(?)x(?),
where P(?) is the unitary matrix
P(?) =
0
BB
BB
BB
@
1 1 1
e?i?? e?i(???2?3 ) e?i(??+2?3 )
e?i2?? e?i(2??+2?3 ) e?i(2???2?3 )
1
CC
CC
CC
A
; (4.51)
the linear version of Eqs. (4.47)-(4.49) can be decoupled, directly yielding the solutions
~?n(?) = e?in??p
3M
?
e?i?0(?) +ei(2n?3 ??1(?))
+e?i(2n?3 +?2(?))
?
; (4.52)
where we have defined the phase terms
?n(?) = ?2?
?
sin(?? ? 2n?3 )+sin(2n?3 )
?
; (4.53)
for n = 0;1;2.
These solutions for the spatial amplitudes ~?i(?) can be most easily understood by considering a
Bloch state decomposition of the condensate wavefunction. The nature of our system allows us to
construct an analytic form for the initial wavefunction. Because the system has a three lattice site
period and is assumed to be in the lowest band, the wavefunction can be expressed as a superposition
of three Bloch waves (of the lowest band) which are symmetrically spaced in quasimomentum. As-
suming the condensate initially has a total crystal momentum of zero, at this time the wavefunction
must be of the form
?(x;0) = fi0`0(x)+fi+`q=3(x)+fi?`?q=3(x); (4.54)
59
where `k(x) is a Bloch state with quasimomentum k, and the fi are complex constants determined
by the lattice depth, with jfi+j = jfi?j.
The action of an external force on a Bloch state causes it to linearly change its quasimomentum in
time according to
k(?) = ??a? +k(0): (4.55)
The periodicity of the Bloch dispersion relation in k, and hence of the group velocity of the Bloch
wave, gives rise to the well-known phenomenon of Bloch oscillations (see chapter 2). For the case
we are considering here, the system consists of three Bloch states whose quasimomenta will translate
in unison under the action of the external force. During this evolution each state accumulates phase
at a rate determined by the instantaneous Bloch energy, i.e.
?n(?) =
Z ?
0
E(kn(s))ds; (4.56)
where kn(?) = ???=a+kn(0) is the quasimomentum of Bloch state n at time t. In the tight-binding
approximation the dispersion relation for the Bloch states has the analytic form
E(k) = ?2cos(ka); (4.57)
for which ?n(?) can be evaluated, and yields the results given in Eqs. (4.53)). The wavefunction
evolution in the Bloch basis is
?(x;?) = fi0`??
at
(x)e?i?0(?) +fi+`??
at+
q
3
(x)e?i?1(?)
+fi?`??
at?
q
3
(x)e?i?2(?): (4.58)
From this solution we can obtain solutions for the evolution of the spatial amplitudes Eqs. (4.52). To
do this we expand the Bloch states in terms of Wannier functions according to, `k(x) = Pn eiknaw0(x?
na) (see chapter 2) and make use of Eq. (4.5). Note: we take all fi = ?3 as determined by the initial
conditions, Eqs.(4.22) and (4.23).
Bloch Oscillations
The evolution of the spatial amplitudes, and in particular the population in each well is then
determined by the interference of the Bloch phases ?n. These functions are all periodic in time
60
with period ?B = 2?=? (in units of ? = tJ=~). This is the normal Bloch oscillation period, and
gives the time scale over which the quasimomenta of the Bloch states increases by exactly one
reciprocal lattice vector.
Small ? solution - Non classical transport
To understand the dynamics, we first start by considering the case when ? is small. For this
case, the population in the wells is given by
j~?0(?)j2 = 13M
?
5+4cos(3?)+?2h(?)
?
(4.59)
j~?1(?)j2 = 13M
?
(2+3?)[1?cos(3?)? ?
2
2 h(?)]
?
(4.60)
j~?2(?)j2 = 13M
?
(2?3?)[1?cos(3?)? ?
2
2 h(?)]
?
(4.61)
where
h(?) = ?
3
2 (6? +3? cos(3?)?4sin(3?)): (4.62)
The above solution shows that when the force is applied, the degeneracy in the population of
the wells represented by ~?1 and ~?2 is lifted. For ? > 0, atoms in ~?0 start to tunnel to ~?1
more rapidly than to ~?2. This should be compared with the results in the absence of the force,
where ~?1 and ~?2 behave identically. Thus, in this weak limit, the effect of a linear potential
is to enhance the tunneling from the initial populated 3n wells to their 3n + 1 neighbors ones,
making the system closer to resonance, in the sense of Eq. (4.28), where the resonance condition
results in the initially populated wells becoming empty at some later time. It is interesting to
note that the system exhibits ?nonclassical? dynamics whereby the atoms start to tunnel in the
direction opposite the direction of the force (this statement applies even when the external field
is not weak).
Resonances
In Fig. 4.15 we show the temporal evolution of the spatial amplitudes ~?i for a range of values of
?. For certain choices of ? the initially occupied ~?0 amplitude periodically disappears - we refer
to these as resonances. By requiring ~?0 = 0 in Eq. (4.52) we obtain the following conditions
61
0
?= 514 ? ?=3?
?= 27 ? ?= ?
?
0
?=0
0
?= 12 ?
 (h/J) ?  (h/J)
?  (h/J) ?  (h/J)
?  (h/J) ?  (h/J)
Figure 4.15: Evolution of the normalized population for different values of ?. One Bloch period is shown
in the plots except ? = 0 where the period is infinite. Solid line: j~?3nj2, dotted line: j~?3n+1j2, dash-dot
line: j~?3n+2j2. The ?nonclassical? motion can be seen where the 3n+1-well populations initially increase
more rapidly that the populations of the 3n+2-wells. It can also be seen in the plots that ? = ?resmax=2 and
? = ?resmax are resonant values.
on the phases for these resonances
?1(?)??0(?) = ?
??
3 +2?s
?
; (4.63)
?2(?)??1(?) = ?
??
3 +2?m
?
; (4.64)
where s and m are integers and the same sign choice on the right hand side must be made for
both equations. When these conditions are satisfied the population is not shared between the
adjacent sites, but preferentially tunnels to one of the neighbors. In particular, the +sign choice
in Eqs. (4.63) and (4.64) gives the phase condition for complete tunneling into ~?1. Similarly
the ?sign case gives the condition for complete tunneling into ~?2. Solving for the values of ?
62
0 20 40 60 80 100
n
0.1
0.2
0.3
0.4
0.5
Figure 4.16: The spectrum of values of external force, ordered in decreasing magnitude, for which a popu-
lation resonance occurs i.e. the values of ? for which ~?0 periodically disappears.
for which Eqs. (4.63) and (4.64) hold, one obtains the spectrum
?resn=(s;m) = ?6
p3
?
(1+3s)
(1+3s)2 +3(s+2m+1)2; (4.65)
which is shown in Fig. 4.16. For large jsj or jmj the resonant values of ? are close together, and
become spaced further apart as the magnitude of s decreases.
In the absence of an applied force (i.e. ? = 0) the phase terms are time-independent with
?0 = ?2, ?1 = ?1 and ?2 = ?1 so that the resonance conditions can never be achieved.
When the external force is applied the phases oscillate at a rate that increases with ? and an
amplitude that decreases with ?. The largest value of ? for which a resonance can be found is
?resmax ? 6p3=2?, since for values of ? greater than this the amplitudes of the phase oscillations
?i are so small that they cannot satisfy the condition for population resonance. In the regime
? > ?resmax, j~?0j exhibits only one (nonzero) minima per Bloch period and the dynamics of the
system is dominated by the Bloch oscillations.
Large Force Limit
For ? > ?resmax the system exhibits a population imbalance as the Bloch oscillation suppresses
the ability of the system to tunnel between sites. In the limit ?  ?resmax the population in the
63
wells is described by rapid, small amplitude oscillations around its initial value,
j~?0(?)j2 ?! 3M
 
1? 8?2 sin2
 ??
2
??
; (4.66)
j~?1(?)j2 ?! 12M 1?2 sin2
 ??
2
?
; (4.67)
j~?2(?)j2 ?! 12M 1?2 sin2
 ??
2
?
: (4.68)
The case of ? = 3?resmax depicted in Fig.4.15 shows an approach to this behavior.
? Numerical results for the non-linear case
 (h/J)
 (h/J)
Figure 4.17: Effects of interactions on generalized Bloch oscillations for the pattern loaded system. Evo-
lution of j~?nj2 for various interaction strengths. Upper plot: ? = 2?resmax. Lower plot: ? = 2?resmax=7.
64
Because of the difficulty of solving analytically the equations of motion the nonlinear equations were
solved numerically. Although the dynamics in this situation is considerably more complicated than in
the linear case, the resonance picture still gives us useful guidance concerning the expected behavior.
In general, there exist critical values of ? and ? above which no resonances occur. Furthermore, non-
linearity tends to destroy the periodicity of the Bloch oscillations that is present in the noninteracting
case. For example, in Fig. 4.17, we see that for ? = 2?resmax, the introduction of interactions brings the
system into resonance, but that for large interaction strength, nonresonant behavior is restored. For
? = 2?resmax=7, on the other hand, the introduction of interactions eventually draws the system out of
resonance.
Concerning the momentum distribution, similarly to the untilted case, interatomic interactions induce
time variation of the momentum intensities. The contrast between momentum components vanishes
at ? = 0. As ? increases, the contrast increases to a maximum value, then eventually decreases
towards zero when macroscopic imbalance is achieved, analogous to what is seen in Fig. 4.11 for the
untilted case. Because the external field together with the nonlinearity breaks down the periodicity
of time evolution, the dynamics of the momentum distribution is quite complex.
To compare the predictions of the homogeneous model in the presence of a linear external potential
to a more realistic case, we again solved numerically the discrete nonlinear Schr?odinger equation for
a condensate loaded into every third lattice site but, instead of being homogeneous, initially with a
Gaussian profile. Very good agreement between the model and the numerical results was found for
short times and modest mean-field energies, if, as in the untilted case, we use an effective mean field
energy. In Figs. 4.18 and 4.19 we present a comparison between the evolution of the normalized
population at the central wells found numerically and the prediction of the model for the parameters:
NU = 4:8ER, M = 290, J = 0:075 ER and ? = 2. For longer times, the model predictions start
to disagree with numerical simulation due to the spread of the three Fourier peaks. We observe that
the dynamics are much more sensitive to inhomogeneous effects in the presence of an external force,
and for the more inhomogeneous initial state used in Figs. (4.12 - 4.14) (which extends over M = 76
sites), the inhomogeneous result much more rapidly departs from the homogeneous prediction than
65
the example presented here.
Figure 4.18: Comparison between the evolution of a inhomogeneous condensate with the homogeneous
result. Inhomogeneous condensate: j~?0j2 (boxes), j~?1j2 (stars), and j~?2j2 (diamonds). Homogeneous
case: j~?3nj2 (dash-dot line), j~?3n+1j2 (solid line), and j~?3n+2j2 (dotted line), where we have taken  as
the local mean field energy. The parameters used were J = 0:075ER, ? = 2 and  efi = 1:59
Figure 4.19: Evolution of the momentum peak populations. q = 0 (dashed line, squares), q = ?2?=3 (dash-
dot line, stars), q = 2?=3 (solid line, diamonds). Inhomogeneous condensate (dotted curves), homogeneous
model (lines) using the same parameters as those in Fig. 4.18, but replacing  by an average value  ave =
1:11 for the homogeneous model.
66
Chapter 5
Quadratic approximations
As discussed in chapter 3, the Bose-Hubbard Hamiltonian is an appropriate model, when the loading pro-
cess produces atoms in the lowest vibrational state of each well, with a chemical potential smaller than
the energy gap to the first vibrationally excited state. The quartic form of the Hamiltonian makes it very
difficult to deal with it in all the different regimes. The aim of this chapter is to rewrite it so that a sys-
tematic approach can be developed in the weakly interacting regime, when a condensate is present. The
basic idea, which was first proposed by Bogoliubov in 1947 [11], is to treat the field operator as a c-number
plus a fluctuating term. The c-number describes the condensate or the coherent part of the matter field.
In the weakly interacting regime, quantum fluctuations are small, and therefore the dominant terms in the
Hamiltonian are quadratic in the fluctuating field. The non-quadratic terms should be of higher order and
can be treated perturbatively.
In this chapter, we apply the Bogoliubov approximation to the Bose- Hubbard Hamiltonian and
derive the correspondent Bogoliubov-de Genes(BdG) equations as developed in Refs. [77, 95]. We also
deal with the higher order terms by considering different approximations. Among them we discuss the
Hartree-Fock-Bogoliubov (HFB) approximation, the HFB-Popov approximation and the improved Popov
approximation where the bare potential is upgraded to the many-body scattering matrix. All of these ap-
proximate approaches are valid only in the weakly interacting regime when quantum fluctuations are small.
They will fail to give a good description as interactions become important and the system approaches the
Mott insulator transition. To test the validity of the approximations we compared them with numerical
solutions obtained by diagonalizing the Bose-Hubbard Hamiltonian.
In this chapter we also derive an explicit expression for the superfluid density based on the rigidity
of the system under phase variations. We show how the superfluid fraction can be thought of as a natural
order parameter to describe the superfluid to Mott insulator transition.
Because we are interested in the quantum effects caused by interactions that drive the quantum
phase transition instead of thermal quantum fluctuations, the analysis in the present chapter is going to be
restricted to the zero temperature case.
67
5.1 The characteristic Hamiltonian
In the very weakly interacting regime, to a good approximation, the creation and annihilation operators on
each site can be replaced by a c-number(see chapter 4). To include the small quantum fluctuations in the
description of the system we assume that the field operator can be written in terms of a c-number plus a
fluctuation operator:
^an = zn + ^?n: (5.1)
Replacing this expression for ^an in the Bose-Hubbard Hamiltonian Eq. (3.4) leads to :
^H = H0 + ^H1 + ^H2 + ^H3 + ^H4; (5.2)
with
H0 = ?J
X
<n;m>
z?nzm +
X
n
?
(Vn ??)jznj2 + U2 jznj4
?
; (5.3)
^H1 = ?J X
<n;m>
^?nz?m +
X
n
?V
n ??+Ujznj2
?z?
n ^?n +h:c:; (5.4)
^H2 = ?J X
<n;m>
^?yn ^?m +
X
n
(Vn ??)^?yn ^?n +
U
2
X
n
?^?2y
n z
2
n + ^?
2
nz
2?
n +(^?
y
n ^?n + ^?n ^?
y
n)jznj
2?; (5.5)
^H3 = U X
n
^?yn ^?yn ^?nzn +h:c:; (5.6)
^H4 = U
2
X
n
^?yn ^?yn ^?n ^?n: (5.7)
Where < n;m > restricts the sum over m to nearest neighbors and h:c: stands for the hermitian conjugate.
The terms of the Hamiltonian have been grouped in equations according to the number of non condensate
operators which they contain.
5.2 Bogoliubov-de Gennes (BdG) equations
In this section we neglect higher order terms and focus only on the quadratic Hamiltonian. The first step in
the diagonalization is the minimization of the energy functional H0 . This requires the condensate ampli-
tudes zn to be a stationary solution of the DNLSE (4.6), which in turn means that the linear Hamiltonian
68
^H1 vanishes. For the time independent situation the DNLSE can be written as
?zn = ?J
X
<m;n>
zm +(Vn +Ujznj2)zn (5.8)
The Hamiltonian H0 gives a description which includes only the contribution from the condensate. The
quadratic Hamiltonian H2 allows the leading order effects of the non condensate to be taken into account.
By including second order terms in the Hamiltonian two classes of interactions are included besides the
condensate-condensate ones: a) Interactions between one excited atom and one condensate atom leading to
transitions of the form j0ji ! j0ii (direct and exchange excitations)and b) interactions between two con-
densate atoms which cause the atoms to be excited to non condensate states j00i!jiji (pair excitations).
Schematically they are shown in fig.5.1.
Figure 5.1: Scattering processes included in the quadratic hamiltonian: a) Direct and exchange excitations,
b) Pair excitations
The quadratic Hamiltonian can be diagonalized by finding a basis of so-called quasiparticle states
which do not interact which each other. Mathematically this involves a linear canonical transformation of
the single-particle creation and annihilation operators ^ayn and ^an into quasiparticle operators ^fiys, ^fis: The
transformation is known as Bogoliubov transformation and is given by
^?n =
X
s6=0
?us
n^fis ?v
?s
n ^fi
y
s
?: (5.9)
69
In general the spectrum of fluctuations includes a zero mode. This mode is the Goldstone boson associated
with the breaking of global phase invariance by the condensate. The zero mode is essentially non perturba-
tive and it introduces an artificial infrared divergence in low dimensional models. For this reason quadratic
approximations are actually improved if the contribution from this mode is neglected all together [96]. A
different way to deal with the zero mode has been proposed by Castin and Dum [98], Gardiner [97] and
Morgan [16]. Here the theory is written in terms of operators which exchange particles between zero and
nonzero modes, conserving the total particle number, and one further operator which changes total particle
number. As shown by [16] both the latter and the former approaches lead to the same physical predictions,
so for simplicity we are going to ignore zero mode fluctuations and restrict the fluctuation operators to act
only on the excited states.
The Bogoliubov transformation Eq. (5.9) is required to be canonical, which means that it preserves
the commutation relations and leads to bosonic quasiparticles. To satisfy this, the amplitudes fusn;vsng are
constrained by the conditions:
X
n
(u?sn us0n ?v?sn vs0n) = ?ss0; (5.10)
X
n
(usnvs0n ?v?sn u?s0n ) = 0: (5.11)
The necessary and sufficient conditions that the quasiparticle amplitudes have to fulfill to diagonalize the
Hamiltonian are provided by the so called Bogoliubov-de Gennes (BdG) equations
0
BB
@
L M
M? L
1
CC
A
0
BB
@
us
vs
1
CC
A = !Bs
0
BB
@
us
?vs
1
CC
A+cs
0
BB
@
z
?z?
1
CC
A; (5.12)
with us = (us1;us2;:::) , vs = (vs1;vs2;:::) and z = (z1;z2;:::). The matrices L and M are given by
Lnm = ?J
X
<n;l>
?nl?lm +?nm(2Ujznj2 +Vn ??) (5.13)
Mnm = ?Uz2i?nm: (5.14)
with ?nl the Kronecker delta which is one if n = l and zero otherwise. The parameters cs ensure that the
solutions with !Bs 6= 0 are orthogonal to the condensate [16]. These parameters are given by
70
cs = U
P
njznj
2(z?nusn ?znvsn)
P
njznj2
(5.15)
The BdG equations without the cs parameters have the same quasiparticle energies as Eq.(5.12), neverthe-
less the solutions are orthogonal to the condensate only in the general sense, i.e. Pn(z?nusn ?znvsn) = 0.
Each of the terms, Pn z?nusnand Pn znvsn are not necessarily zero. To obtain the desired orthogonal modes
one way to proceed is to solve the BdG equations setting the cs to zero and then to remove from the so-
lutions their projection onto the condensate. If the BdG equations are satisfied, the quadratic Hamiltonian
takes the form
^H2 = X
s6=0
?
!Bs ^fiys^fis + 12 ?!Bs ?Lss?
?
: (5.16)
If the Hamiltonian ^H2, is positive definite, which is the case when the condensate amplitudes correspond
to a stable state of the system, the eigenvalues of the system are real. The solutions !Bs come in pairs with
positive and negative energies, if !Bs is a solution for the amplitudes (us;vs) then ?!Bs is also a solution
for the amplitudes (vs?;us?). There is always a solution with !Bs = 0 and in this case the amplitudes
must be proportional to the condensate (v0;u0) / (z;z). We explicitly exclude the zero mode solution to
guarantee that the excitations are orthogonal to the condensate.
To have a complete description of the quadratic Hamiltonian, we define the one body density fluc-
tuation matrix, ?nm and the anomalous average matrix, mnm as:
?nm = h^ayn^ami?h^aynih^ami = h^?yn ^?mi; (5.17)
mnm = h^an^ami?h^anih^ami = h^?n ^?mi: (5.18)
The averages denotes an ordinary quantum expectation value. At zero temperature, the average is over the
ground quasiparticle state given by ^fisj0i = 0 and the ?nm and mnm can be written in terms of quasiparticle
amplitudes as:
71
?nm =
X
s6=0
vs?n vsm; (5.19)
mnm = ?
X
s6=0
usnvs?m: (5.20)
Physically, the quantity ?nn represents the non -condensate population (or depletion) at position n or de-
pletion at position n. If the average number of atoms is N, the depletion and condensate atoms are related
by the equation:
N =
X
n
?^ay
n^an
fi = X
n
?jz
nj2 +?nn
?: (5.21)
The interpretation to the physical meaning of the anomalous term is postponed to section 5.7.2.
5.3 Higher order terms
In the preceding section we described the diagonalization of the Hamiltonian containing only terms up
to quadratic order in the field operators. We now want to go beyond this approximation and consider the
corrections introduced by the cubic and quartic terms. Taking into account higher order terms we are mainly
introducing many-body effects on the scattering.
5.3.1 Hartree-Fock-Bogoliubov equations
Third and quartic terms in the Hamiltonian can be included by treating them in a self-consistent mean
field approximation, which relies upon the factorization approximation: products of many operators are
approximated by paring the operators in all possible ways and replacing these pairs by their expectation
value [15]. Using the factorization approximation, one can reduce third and quartic products of fluctuation
field operators to:
?yn?n?n ! 2?nn?n +2mnn?yn; (5.22)
?yn?yn?n?n ! 4?nn?yn?n +mnn?yn?yn +m?nn?n?n ?(2?2nn +jmnnj2): (5.23)
Eq.(5.23) is justified by Wick?s theorem [51], which gives h?yn?yn?n?ni = 2?2nn+jmnnj2, while Eq.(5.22)
is justified by analogy [16].
72
Therefore, using Eq.(5.23) and Eq.(5.22) in ^H3 and ^H4 ( Eqs. (5.6) and (5.7)) one can reduce them to
quadratic forms in terms of the functions ?nm and mnm. The c-number term in Eq.(5.23) introduces also a
energy shift in H0 :
? ^H0 = U2 (2?2nn +jmnnj2) (5.24)
^H3 = U X
n
(2?nn ^?n +mnn ^?yn)zn +h:c:; (5.25)
^H4 = U
2
X
n
(2?nn ^?yn ^?n +mnn ^?n ^?n)+h:c:: (5.26)
The corrections from higher order terms yield a modified quadratic Hamiltonian which can be also diago-
nalized. The diagonalization leads to the following equations
?HFBzn = ?J
X
<m;n>
zm +(Vn +Ujznj2 +2U?nn +Umnn)zn; (5.27)
0
BB
@
LHFB MHFB
M?HFB L?HFB
1
CC
A
0
BB
@
us
vs
1
CC
A = !HFBs
0
BB
@
us
?vs
1
CC
A+
0
BB
@
cHFB?s z
?cHFB+s z?
1
CC
A; (5.28)
LHFBnm = ?J
X
<n;l>
?nl?lm +?nm(2Ujznj2 +Vn ??HFB +2U?nn); (5.29)
MHFBnm = ?(Uz2i +Umnn)?nm; (5.30)
cHFBcurrency1s = U
P
njznj
2(z?nusn ?znvsn)currency1U P
n (m
?nnznusn ?mnnz?nvsn)
P
njznj2
: (5.31)
The parameters cHFBcurrency1s are there again to enforce the orthogonality of the quasiparticle modes with the
condensate [16].
The above set of equations (5.27) to (5.30) are known as Hartree-Fock-Bogoliubov (HFB) equa-
tions. Even though they take into account higher order corrections, they have the problem that violate the
Hugenholtz and Pines theorem [99] which states that the energy spectrum of a Bose gas is gapless, i.e, there
exists an excitation with an energy which tends to zero as the momentum tends to zero.
73
5.3.2 HFB-Popov approximation
One way to solve the gap problem is to set the anomalous term emnn to zero in HFB equations. This
procedure is known as HFB-Popov approximation. The HFB-Popov equations were first introduced by
Popov [52], and they are considered to be a better approximation than the HFB equations because they
yield a gapless spectrum.
The HFB-Popov equations have the form:
?Pzn = ?J
X
<m;n>
zm +(Vn +Ujznj2 +2U?nn)zn; (5.32)
0
BB
@
LP MP
M?P L?P
1
CC
A
0
BB
@
us
vs
1
CC
A = !Ps
0
BB
@
us
?vs
1
CC
A+cs
0
BB
@
z
?z?
1
CC
A; (5.33)
LPnm = ?J
X
<n;l>
?nl?lm +?nm(2Ujznj2 +Vn ??P +2U?nn); (5.34)
MPnm = ?(Uz2i )?nm; (5.35)
cs = U
P
njznj
2(z?nusn ?znvsn)
P
njznj2
: (5.36)
5.4 The Bose-Hubbard model and superfluidity
The concept of superfluidity is closely related to the existence of a condensate in the interacting many-
body system. Formally, the one-body density matrix ?(1) (~x;~x0) has to have exactly one macroscopic
eigenvalue which defines the number of particles in the condensate; the corresponding eigenvector describes
the condensate wave function `0 (~x) = ei?(~x)j`0 (~x)j. A spatially varying condensate phase, ?(~x), is
associated with a velocity field for the condensate by
~v0 (~x) = ~m~r?(~x): (5.37)
This irrotational velocity field is identified with the velocity of the superfluid flow,~vs (~x) ?~v0 (~x) ([100],[101])
and enables us to derive an expression for the superfluid fraction, fs. Consider a system with a finite linear
74
dimension, L, in the ~e1?direction and a ground?state energy, E0, calculated with periodic boundary condi-
tions. Now we impose a linear phase variation, ?(~x) =  x1=L with a total twist angle  over the length of
the system in the ~e1?direction. The resulting ground?state energy, E will depend on the phase twist. For
very small twist angles,  ? ?, the energy difference, E ?E0, can be attributed to the kinetic energy, Ts,
of the superflow generated by the phase gradient. Thus,
E ?E0 = Ts = 12mNfs~v2s; (5.38)
where m is the mass of a single particle and N is the total number of particles so that mNfs is the total
mass of the superfluid component. Replacing the superfluid velocity, ~vs with the phase gradient according
to Eq. (5.37) leads to a fundamental relation for the superfluid fraction.
fs = 2m~2 L
2
N
E ?E0
 2 =
1
N
E ?E0
J (? )2 : (5.39)
where the second equality applies to a one dimensional lattice system on which a linear phase variation has
been imposed. Here the distance between sites is a, the phase variation over this distance is ? , and the
number of sites is M. In this case, J ?~2=(2ma2).
Technically the phase variation can be imposed through so-called twisted boundary conditions [102].
In the context of the discrete Bose-Hubbard model it is, however, more convenient to map the phase varia-
tion by means of a unitary transformation onto the Hamiltonian [103]. For simplicity we are going to focus
in a one dimensional lattice. The resulting ?twisted? Hamiltonian in this case
^H =
MX
n=1
^nnVn ?J
MX
n=1
(e?i? ^ayn+1^an +ei? ^ayn^an+1)+ U2
MX
n=1
^nn(^nn ?1) : (5.40)
exhibits additional phase factors e?i? ? the so-called Peierls phase factors ? in the hopping term [104,
105]. These phase factors show that the twist is equivalent to the imposition of an acceleration on the lattice
for a finite time. It is interesting to note that the present experiments enable us to make a specific connection
between the formal and operational aspects of the system.
We calculate the change in energy E ?E0 under the assumption that the phase change ? is small
so that we can write:
e?i? ? 1?i? ? 12(? )2 : (5.41)
75
Using this expansion the twisted Hamiltonian (5.40) takes the following form:
^H ? ^H0 +? ^J ? 1
2(? )
2 ^T = ^H0 + ^Hpert ; (5.42)
where we retain terms up to second order in ? . The current operator ^J (Note that the physical current is
given by this expression multiplied by 1~) and the hopping operator ^T are given by:
^J = iJ
MX
n=1
(^ayn+1^an ?^ayn^an+1) (5.43)
^T = ?J
MX
n=1
(^ayn+1^an + ^ayn^an+1) : (5.44)
The change in the energy E ? E0 due to the imposed phase twist can now be evaluated in second order
perturbation theory
E ?E0 = ?E(1) +?E(2) : (5.45)
The first order contribution to the energy change is proportional to the expectation value of the hopping
operator
?E(1) = h?0j ^Hpertj?0i = ?12(? )2h?0j^Tj?0i : (5.46)
Here j?0i is the ground state of the original Bose-Hubbard Hamiltonian (4.27). The second order term is
related to the matrix elements of the current operator involving the excited states j??i (? = 1;2;:::) of the
original Hamiltonian
?E(2) = ?
X
?6=0
jh??j ^Hpertj?0ij2
E? ?E0 = ?(? )
2X
?6=0
jh??j^Jj?0ij2
E? ?E0 : (5.47)
Thus we obtain for the energy change up to second order in ? 
E ?E0 = (? )2
?
? 12h?0j^Tj?0i?
X
?6=0
jh??j^Jj?0ij2
E? ?E0
?
= M(? )2D;
D ? 1M
?
? 12h?0j^Tj?0i?
X
?6=0
jh??j^Jj?0ij2
E? ?E0
?
: (5.48)
The quantity D, defined above, is formally equivalent to the Drude weight used to specify the DC conduc-
tivity of charged fermionic systems [106]. The superfluid fraction is then given by the contribution of both
76
the first and second order term:
fs = f(1)s ?f(2)s ;
f(1)s ? ? 12NJ
?
h?0j^Tj?0i
?
; (5.49)
f(2)s ? 1NJ
?X
?6=0
jh??j^Jj?0ij2
E? ?E0
?
:
Here N is the number of atoms in the lattice. In general both the first and the second order term contribute.
For a translationally invariant lattice the second term vanishes (as is going to be shown latter) if one uses a
quadratic approximation. However, in exact calculations the second order term plays a role.
We can further understand this approach to the superfluid density by calculating the flow that is
produced by the application of the phase twist. To do this we work out the expectation value of the current
operator expressed in terms of the twisted variables:
^J = iJ
MX
n=1
(e?i? ^ayn+1^an ?ei? ^ayn^an+1) : (5.50)
We expand this to find the lowest order contributions, i.e.:
^J ? ^J +J? 
MX
n=1
(^ayn+1^an + ^aym^an+1) = ^J ? ^T? : (5.51)
We use first order perturbation theory on the wave function to obtain the following expression:
h?(? )j^J j?(? )i = 2? 
?
? 12h?0j^Tj?0i?
X
?6=0
jh??j^Jj?0ij2
E? ?E0
?
(5.52)
= 2NJfs? : (5.53)
Thus, the physical current, Js, Eq. (5.53) multiplied by 1~, can be expressed as:
Js = 1~h?(? )j^J j?(? )i = Nfs? ~m?a2 : (5.54)
This is the total flux, js, and we need to divide by M to get the flux density, i.e.
js = 1~M h?(? )j ^J j?(? )i =
 ~? 
m?a
? Nf
s
aM
?
= vsns : (5.55)
So we see that the Drude formulation of the superfluid fraction (5.49) gives an intuitively satisfying expres-
sion for the amount of flowing superfluid.
77
5.5 Expectation values
Using the quadratic approximations we can evaluate expectation values of meaningful physical quantities.
Useful quantities as the system approach the Mott insulator state are the number fluctuations and the quasi-
momentum distribution.
The number fluctuations at the site xi are given in the Bogoliubov approximation by
?nn =
q
h^ayn^an^ayn^ani?h^ayn^ani2
= jznj2
X
s
jusn ?vs?n j2: (5.56)
The quasimomentum distribution of the atoms released from the lattice is important because it is
one of the most easily accessible quantities in the experiments. The quasimomentum distribution function
nq is defined as [103]
nq = 1M
X
n;m
eiq?(xn?xm)haynami
= 1M
0
@
flfl
flfl
fl
X
n
zneiq?xn
flfl
flfl
fl
2
+
X
n;m
?nmeiq?(xn?xm)
1
A; (5.57)
where the quasimomentum q can assume discrete values which are integer multiples of 2?aiMi , where ai and
Mi are the lattice spacing and number of lattice sites in the i direction and M = Qi Mi is the total number
of lattice sites.
Neglecting interaction effects during the expansion, the quasimomentum distribution represents the
Fourier transform of the original spatial distribution in the lattice. In the superfluid regime because of the
phase coherence of the system, when it is released from the lattice the gas shows a nice interference pattern.
As the interactions are increased phase coherence is lost, and instead of interference peaks a incoherent
background is observed.
For a one dimensional optical lattice, the superfluid fraction in terms of quasiparticles amplitudes
78
and energies can be written as:
fs = f(1)s ?f(2)s ; (5.58)
f(1)s =
MX
n=1
f(1)sn = 12N
MX
n=1
h
(zn+1z?n +z?n+1zn)+
X
s
(vsnvs?n+1 +vs?n vsn+1)
i
; (5.59)
f(2)s = JN
?X
s
flflP
n
?us
n +vsn
?(z
n+1 ?zn?1)
flfl2
!s
!
+ (5.60)
J
N
0
@X
s;s0
flflP
n
?us
n+1vs
0
n ?usnvs
0
n+1
?flfl2
!s +!s0 +?ss0
flflP
n(u
sn+1vsn ?usnvsn+1)flfl2
2!s
1
A:
5.6 Applications
5.6.1 Translationally invariant lattice
To understand many-body effects included in the quadratic approximations we start by studying the case
where no external confinement is present, Vn = 0. We assume a d dimensional separable square optical
lattice with equal tunneling matrix element J in all directions and periodic boundary conditions. M is the
total number of wells.
Due to the translational symmetry of the system the condensate amplitudes are constant over the
lattice, zn = pno. Also, the quasiparticle modes have a plane wave character and therefore can be related
to quasimomentum modes :
uqn = 1pMeiq?xnuq; vqn = 1pMeiq?xnuq: (5.61)
Here the vector q denotes the quasimomentum whose components assume discrete values which are integer
multiples of 2?a dpM with a is the lattice spacing. The amplitudes uq and uq must satisfy the condition
juqj2 ?jvqj2 = 1 and can all be chosen to be real and to depend only on the modulus of the wave vector
(uq = u?q, vq = v?q).
The translationally invariant case has the advantage that the quasi-particle modes are always orthogonal to
the condensate and therefore the parameters cs are always zero.
? Bogoliubov-de Gennes (BdG) equations and HFB- Popov approximation
79
In the simplest quadratic approximation the DNLSE reduces to
? = ?tJ +noU; (5.62)
where t is the number of nearest neighbors t = 2d.
The BdG equations in the translationally invariant case become the following 2?2 eigenvalue prob-
lem
0
BB
@
Lqq ?Mq?q
Mq?q ?Lqq
1
CC
A
0
BB
@
uq
vq
1
CC
A = !q
0
BB
@
uq
vq
1
CC
A; (5.63)
with
Lqq = ?q +noU; Mq?q = noU: (5.64)
Here we have introduced the definition ?k = 4JPdi=1 sin2(kia2 ):
The quasiparticle energies !q and modes are found by diagonalizing Eq.(5.64):
!2q = L2qq ?M2q?q = ?q2 +2Uno?q; (5.65)
u2q = Lqq +!q2!
q
= ?q +noU +!q2!
q
; (5.66)
v2q = Lqq ?!q2!
q
= ?q +noU ?!q2!
q
; (5.67)
uqvq = ?Mq?q2!
q
= noU2!
q
(5.68)
and
n = no + 1M
X
q6=0
v2q; (5.69)
with n the total density, n = N=M. The constrain that fixes the number of particles can be written as
no = n? 1M
X
q6=0
 ?
q +Uno
2!q ?
1
2
?
: (5.70)
80
In the homogeneous case, due to the cancellation of the ?nn: terms in the quasiparticle equations
(5.34), the HFB-Popov approximation leads to the same quasiparticle energy spectrum and ampli-
tudes than the BdG equations. The only difference is in the chemical potential which has an extra
term 2U?nn
?(P) = ?tJ +Uno +2U?nn: (5.71)
As opposed to the free particle system, where the single particle energy dominates at high momenta
(it grows as q2), the single particle excitations in the presence of the lattice are always bounded by
4Jd. Therefore, in the regime Uno=J > 1 the interaction term dominates for all quasimomentum
and so !q ? p2Uno?q. On the other hand, in the weakly interacting regime, the most important
contribution to the depletion comes from the low-lying modes. For these modes is also true that
!q ?p2Uno?q. Thus, to a good approximation the condensate fraction can be written as:
no ? g ?
r
Uno
J fi; (5.72)
with
fi = fi(d;M) ? 1M
X
q6=0
pJ
2p2?q; (5.73)
g =
8>
><
>>:
n+ M?12M Un=J & 1
n Un=J ? 1
: (5.74)
In Eq.(5.74) the term M?12M is a finite size effect term which is important to keep for low density
systems. fi is a dimensionless quantity which depends only on the dimensionality of the system. In
81
the thermodynamics limit, M !1 the sum can be replaced by an integral and we find
fi(1;1) ! 12p2? ln(cot(qo))
flfl
fl
qo!0
; (5.75)
fi(2;1) = 0:227293; (5.76)
fi(3;1) = 0:160287: (5.77)
(5.78)
The infrared divergence in the one dimensional thermodynamic limit is a consequence of the im-
portance of long wave length correlations in low dimensional systems. For finite one dimensional
systems however fi(1;M) has a finite value.
Because Eqs.(5.65)-(5.66) are completely determined if no is known, by solving Eq.(5.72) we obtain
all necessary information. The solution of the algebraic equation is:
no ? g + fi
2U
2J ?
r
gfi
2U
J +
fi4U2
4J2 : (5.79)
Eq.(5.79) tells us that for very weak interactions almost all the atoms are in the condensate. As
interaction increases the condensate fraction decreases, but under the BdG (and HFB-Popov) approx-
imation it only vanishes when U=J ! 1. The BdG (and HFB-Popov) equations therefore do not
predict any superfluid to Mott insulator phase transition.
The calculated quasiparticle amplitudes can be used to get analytic expressions for the number fluc-
tuations, the momentum distribution and superfluid fraction. If we restrict our attention to the one
dimensional system we have
82
?nn = no
X
q
?q
!q????!M!1
2
?no arctan(
2J
noU); (5.80)
nq = no?q;0 + 1Mjvqj2?q;k; (5.81)
= no?q;0 + 1M
 ?
q +Uno
2!k ?
1
2
?
?q;k; (5.82)
fs = f(1)s ?f(2)s ; (5.83)
f(1)s = MN
h
no + 1M
X
q
jvqj2 cos(qa)
i
; (5.84)
f(2)s = 0: (5.85)
Eq. (5.80) has been studied in Ref. [95] to produce results for squeezing. There the authors show
that Eq. (5.80) is consistent with those of other approaches previously reported in the literature
[107, 108, 109, 110].
It is important to emphasize that due to the translational invariance of the lattice [see Eq. (5.60)], the
second order term vanishes in the Bogoliubov limit.
The expression for the superfluid fraction gives a direct insight into the behavior of the system as
atoms are pushed out of the condensate due to interactions. In Eq. (5.84) the sum involving the
Bogoliubov amplitudes vq characterizes the difference between the condensate fraction, which is
given by the first term, and the superfluid fraction. For weak interactions and small depletion, the
depletion of the condensate has initially little effect on superfluidity. As interactions are increased
the depleted population spreads into the central part of the band, (where the cos(qa) term has a
negative sign) and the superfluid fraction is reduced. Finally for even larger interaction strengths, the
population in the upper quarter of the band again produces a positive contribution to the superflow and
the contribution from the sum decreases. In a sense the interactions are playing a role akin to Fermi
exclusion ?pressure? in the case of electron flow in a band. This, however can lead to perfect filling
and cancellation of the flow. In the case of our Bogoliubov description we can only see reduction of
the flow, not a perfect switching off of the superfluid. This happens in the Mott insulator state, which
cannot be described by the Bogoliubov approximation.
83
5 1015202530
Veff
0.51
1.52CapDeltan nEqual5.33
5 1015202530
Veff
0.20.4
0.60.8f c nEqual5.33
5 1015202530
Veff
0.20.4
0.60.8fs,fs2 nEqual5.33
10203040
Veff
0.51
1.52CapDeltan nEqual5
10203040
Veff
0.20.4
0.60.8f c nEqual5
10203040
Veff
0.20.4
0.60.8fs,fs2 nEqual5
Figure 5.2: Comparisons of the exact solution and BdG (and HFB-Popov) solutions as a function of Veff =
U=J, for a system with M = 3 and filling factors n = 5 and 5:33. Left: number fluctuations (Exact:
solid line, BdG(and HFB-Popov): dotted line), middle: condensate fraction (Exact: solid line, BdG(and
HFB-Popov): dotted line,analytic (5.79:red line ), right: superfluid fraction fs (Exact: solid line, BdG(and
HFB-Popov): dotted line). The exact second order term (dashed line) of the superfluid fraction, f(2)s is
also shown in these plots. The vertical line shown in the plots is an estimation of V criteff when the system is
commensurate
In Figs. 5.2 and 5.3 we compare the number fluctuations per lattice site, the condensate fraction and
the total and second order superfluid fraction determined from the exact solution of the Bose-Hubbard
Hamiltonian to the solution obtained from the BdG (and HFB-Popov) equations as a function of the
ratio Veff = U=J. The systems used for the comparisons are one dimensional lattices with three
wells, M = 3, and commensurate filling factors n = 5 and 5:33 and n = 50 and 50:33. We were
restricted to consider only three wells due to computational limitations. The size of the matrix needed
in the exact solution for N atoms and M wells scales as (N+M?1)!N!(M?1)! . However, if the approximate
approach works well for these small systems we expect it to provide a good description of the larger
systems prepared in the laboratory.
Because the second order term of the superfluid fraction (second term of Eq.(5.49) vanishes in the
quadratic approximations (see Eq. (5.60)), we only expect them to give a good description of the
superfluid fraction in the region where the second order term is extremely small, provided it predicts
accurately the first order term. This is exactly what is observed in the plots. When the second order
84
50100150200
Veff
1
3
5
CapDeltan
nEqual50.33
50100150200
Veff
0.20.4
0.60.8f c nEqual50.33
50100150200
Veff
0.20.4
0.60.8fs,fs2 nEqual50.33
50100150200
Veff
1
3
5
CapDeltan
nEqual50
50100150200
Veff
0.20.4
0.60.8f c nEqual50
50100150200
Veff
0.20.4
0.60.8fs,fs2 nEqual50
Figure 5.3: Comparisons of the exact solution and BdG(and HFB-Popov) solutions as a function of Veff =
U=J, for a system with M = 3 and filling factors n = 50 and 50:33. Left: number fluctuations (Exact:
solid line, BdG(and HFB-Popov): dotted line), middle: condensate fraction (Exact: solid line, BdG(and
HFB-Popov): dotted line, analytic Eq.(5.79): red line ), right: superfluid fraction fs (Exact: solid line,
BdG(and HFB-Popov): dotted line). The exact second order term (dashed line) of the superfluid fraction,
f(2)s is also shown in these plots. In this case the agreement is much better.
term starts to grow, typically above 0:5V criteff ( with V criteff = (U=J)c, see Eq.(3.9)), the BdG (and
HFB-Popov) equations starts to fail. An estimate of V criteff is shown by a vertical line in some of
the figures. With increasing filling factor the critical value is shifted towards larger values of the
interaction strength, and the region in which the BdG (and HFB-Popov) equations are accurate gets
larger. It is interesting to note that the number fluctuations predicted by the theory are accurate in a
greater range than the other physical quantities shown. Its predictions of squeezing agree very well
with the exact solutions right up to the point where the number fluctuations become less than unity.
In the plots for the condensate fraction we also show the analytic approximation given by Eq.(5.79).
It can be observed that the analytic solution agrees well with the numerical solution of the BdG( and
HFB-Popov) equations.
For the cases with non-commensurate fillings the agreement is significantly better for all quantities.
This is not surprising because when the filling is not commensurate there is always a superfluid
fraction present and the Mott transition doesn?t occur. As can be seen in the plots for these cases the
85
second order term is always very small.
? HFB equantions
In the translationally invariant case the HFB Hamiltonian leads to the following equations:
0
BB
@
Lqq ?Mq?q
Mq?q ?Lqq
1
CC
A
0
BB
@
uq
vq
1
CC
A = !q
0
BB
@
uq
vq
1
CC
A; (5.86)
with
Lqq = ?q +noU ?Uem; Mq?q = noU +Uem: (5.87)
The diagonalization of them yields
? = ?tJ +Uno +2Uen+Uem; (5.88)
vq2 = uq2 ?1 = Lqq ?2!q2!
q
= ?q +noU ?Uem?!q2!
q
; (5.89)
uqvq = Mq?q2!
q
= noU +Uem2!
q
; (5.90)
!q2 = ?q2 +2U(no + em)?q ?4U2noem; (5.91)
with
en ? 1M
X
q6=0
v2q; (5.92)
em ? ? 1M
X
q6=0
vquq: (5.93)
From the above equation it can be seen that the quasiparticle energies predicted under the HFB
approximation don?t approach zero as q goes to zero. As discussed in the previous section, the most
important contribution to the depletion of the condensate in the BdG equations comes from the low
lying modes, because of the !q?1 dependence. The finite value of !q has in the HFB approximation
as q goes to zero explains why the HFB equations are always going to predict smaller condensate
depletion than the gapless approximation.
In Fig. 5.4 we compare the condensate fraction predicted by the HFB equations with the condensate
fraction obtained from the BdG equations, for a system with M = 3 and filling factor n = 10. It can
86
be seen that the condensate fraction is always higher in the HFB solutions and remains constant at
higher values of U=J which is not physically correct.
0 10 20 30 40
Veff
0.8
0.85
0.9
0.95
1
f c
nEqual10, MEqual3
HFB
BdG
Figure 5.4: Comparisons of the condensate fraction given by the BdG solutions and the HFB equations as
a function of Veff = U=J, for a system with M = 3 and filling factor n = 10.
5.6.2 One-dimensional harmonic trap plus lattice
In this section we consider the experimentally relevant case when there is an external harmonic magnetic
confinement in addition to the lattice potential. For simplicity we focus our attention on the one dimensional
case. We consider the two most relevant approximations which are the BdG equations and the HFB-Popov
approximation. Firstly we study the BdG equations in the weakly interacting regime and get some insight
on the solutions by deriving analytic results using the so called Thomas-Fermi approximation [12]. Sec-
ondly we study the HFB-Popov approximation and use it to explore the limits of validity of the quadratic
approximations as the interactions are increased.
? BdG equations
In the presence of the harmonic trap the BdG equations take the form:
87
!Bs usn +cszn = ?J(usn+1 +usn?1)+(2Ujznj2 ??+?n2)usn
?Uz2nvsn; (5.94)
?!Bs vsn ?csz?n = ?J(vsn+1 +vsn?1)+(2Ujznj2 ??+?n2)vsn
?Uz?2n usn; (5.95)
?zn = ?J(zn+1 +zn?1)+(Ujznj2 +?n2)zn; (5.96)
N =
X
n
(jznj2 + ~nn); (5.97)
~nn =
X
s
jvsnj2; (5.98)
cs = U
P
njznj
2(z?nusn ?znvsn)
P
njznj2
: (5.99)
In the parameter regime where U=J ? 1, but the number of atoms is large enough such that
NU=Jaho  1, where aho=
q
}
m?!? (See Eq. (2.39)) the Bogoliubov approximation takes a rather
simple analytic form. The effect of increasing the ratio NU=Jaho is to push the atoms outwards,
flattening the central density and increasing the width of the condensate wavefunction. The quantum
pressure, which is proportional to the kinetic energy, takes a significant contribution only near the
edges of the wavefunction, and to a good approximation it can be neglected. This is the so-called
Thomas-Fermi approximation (TF). This approximation has been very useful to derive analytic ex-
pressions in magnetic confined condensates without the lattice and proved to be in agreement with
experimental measurements [12, 18, 19].
Under the Thomas-Fermi approximation one gets a condensate density profile of the form of an
inverted parabola which vanishes at the turning points RTF = p?=?:
jznj2 =
n ???n2U n < RTF
0 n > RTF
(5.100)
The chemical potential is determined by fixing the total number of particles. To first order it is
possible to neglect the depletion of the condensate and assume that Pjznj2 = N. Changing the sum
to an integral it can be solved for ? to get
88
? = ?R2TF;
RTF ?
 3
4
NU
?
?1=3
: (5.101)
To solve for the excitation spectrum we use the Thomas-Fermi condensate amplitude, Eq.(5.100) into
Eqs.(5.94)and (5.95) [18, 111] and use the parameter J as an expansion parameter in these equations.
We define the variables Fqn = uqn + vqn and Gsn = usn ?vsn. For simplicity we are going to set the
constants cs to zero and ignore the orthogonality constraint at the beginning. We correct this at the
end by removing from the modes their projection onto the condensate. To first order in J we get the
following set of equations:
2R2TF?
 
1? n
2
R2TF
?
Gsn = !Bs Fsn; (5.102)
?J
8<
:F
s
n+1 +F
s
n?1 ?F
s
n
0
@
q
1? (n+1)2R2
TF
+
q
1? (n?1)2R2
TFq
1? n2R2
TF
1
A
9=
;
= !Bs Gsn: (5.103)
These two equations can be combined to get a simple equation of the form
(s
1? (n+1)
2
R2TF
s
1? n
2
R2TF
?e
Fsn+1 ? eFsn
?
(5.104)
?
s
1? (n?1)
2
R2TF
s
1? n
2
R2TF
?e
Fsn?1 ? eFsn
?)
= 2
 !B
s
!?
?2
eFsn;
with eFsn = Fsn=
q
1? n2R2
TF
.
The above equation is just the discretized form of the Legendre equation, @@x ?(1?x2) @@xPn? +
n(n+1)Pn = 0. Therefore the solutions of Eq. (5.104) are approximately given by :
89
!Bs = ~!?
r
s(s+1)
2 ; (5.105)
Fsn =
s
?RTF(2s+1)
!Bs
s
1? n
2
R2TF Ps
 n
RTF
?
; (5.106)
Gsn =
s
!Bs (2s+1)
4?R3TF
1q
1? n2R2
TF
Ps
 n
RTF
?
: (5.107)
The larger the size of the condensate the closer are the approximate solutions, Eq.(5.107), to the exact
solutions of Eqs. (5.104).
Finally, to make the solutions orthogonal to the condensate we have to remove their projection onto
the condensate mode:
usn ! usn ?cszn=!Bs ; (5.108)
vsn ! vsn ?csz?n=!Bs : (5.109)
If we calculate the constants cs using Eq. (5.99), and orthogonality properties of the Legendre poly-
nomial we find that the only mode that needs to be corrected is the quadrupole mode, s = 2:
cs =
r
4~!?R5?2
15U ?s;2: (5.110)
From Eqs. (5.110) it follows that the desired F and G amplitudes orthogonal to the condensate are
thus given by
Fsn ! Fsn ?2cszn=!Bs ; (5.111)
Gsn ! Gsn: (5.112)
Once solved for the quasiparticle amplitudes, it is possible to calculate the depletion of the conden-
sate, eN = Pn;sjvsnj2. It is important to point out that the term
?
1? n2R2
TF
??1=2
in the G amplitudes
makes the sum divergent. The divergence however is nonphysical and only indicates the failure of
90
the TF approximation near RTF . To perform the sum, we have to exclude the contribution from the
boundary layer of thickness d =
? a4
ho
2RTF
?1=3
where the TF approximation starts failing [111]. Fi-
nally, to fix the number of particles to N it is necessary to renormalize the condensate wave function
and the chemical potential. This is done by renormalizing the Thomas Fermi radius and replacing N
by N ? eN in Eq.(5.101).
In Fig. 5.5 we plot the condensate density profile found by numerically solving the BdG equations
and compare it with the TF condensate solution. The TF approximation reproduces the numerical
solution very accurately except at the edges where the kinetic energy can not be ignored with respect
to the potential energy. In the same plot we also show the condensate depletion which is small
for the chosen parameters. A small condensate depletion guarantees the validity of the quadratic
approximation to describe the many-body system.
Minus30 Minus20 Minus10 0 10 20 30
n
0
0.5
1
1.5
2
2.5
#atoms
TF
depletion
Condensate
Figure 5.5: Comparisons of the condensate wave function found by numerically solving the BdG equations
with the Thomas-Fermi solution. The parameters used were U=J = 0:2, ?=J = 9:5?10?4 and N = 100.
The depletion is also shown in the plot.
In Fig. 5.6 we show comparisons between the quasiparticle energies found numerically, the TF
quasiparticle and excitation spectra of the noninteracting system (see Eq. (2.37)). It can be seen
91
2 4 6 8 10 12 14 16 18 20
sth eigenvalue
0.05
0.1
0.15
0.2
0.25
?sLParen1
E RRParen1
USlash1JEqual0.2CapOmegaSlash1JEqual9.5x10Minus4 NEqual100
Free
TF
BdG
Figure 5.6: Comparisons of the quasiparticle spectrum found by numerically solving the BdG equa-
tions(BdG) with the Thomas-Fermi solution (TF) and the noninteracting energies (Free). The parameters
used were U=J = 0:2, ?=J = 9:5?10?4 and N = 100.
in the plot that only the lowest lying modes are well described by the TF approximation. Higher
excitation energies are closer to the noninteracting ones. In the regime where ? < J and the system?s
size is large enough that discretization effects, neglected in Eqs. (5.105), are not important, the
first quasiparticle excitation energy coincide with the first noninteracting excitation energy . This
first excitation is known as the dipole mode and describes the oscillatory motion of the center of
mass when the system is displaced. The dipole quasiparticle amplitudes (us=1n ;vs=1n ), calculated by
numerically solving the BdG equations are shown in Fig. 5.7. In the same graphic the quasiparticle
amplitudes for the s = 2;3;4 excitation modes are also depicted. The s = 2 mode is known as the
quadrupole mode and is related to the breathing of the condensate when shaken. Higher order modes
describe more complicated collective excitations. In general, quasiparticle excitations other than the
dipole are affected by interactions and they have different energy than the noninteracting excitations.
As shown in Fig. 5.6 TF quasiparticle energies lie below their noninteracting conterparts.
To test the accuracy with which the TF approximation describes the quasiparticle amplitudes, in Fig.
92
5.8 we compare the product FsnGsn as a function of the lattice site n using the TF approximation
results (Eqs. (5.112) and (5.111)) to the product found by numerically solving the BdG equations.
We observe good agreement between the two solutions except near the TF radius. This behavior is
consistent with the fact that near the edge of the atomic cloud, the kinetic energy becomes comparable
to the potential energy and therefore the TF approximation breaks down.
Minus30Minus20Minus10 0 10 20 30
n
0u ns LParen1v ns RParen1
sEqual3
Minus30Minus20Minus10 0 10 20 30
n
0u ns LParen1v ns
RParen1
sEqual4
Minus30Minus20Minus100 10 20 30
n
0u ns LParen1v ns RParen1
sEqual1
unsvn
s
Minus30Minus20Minus10 0 10 20 30
n
0
u ns LParen1v
ns RParen1
sEqual2
Figure 5.7: Low-lying quasiparticle amplitudes found by numerically solving the BdG equations. The
parameters used were U=J = 0:2, ?=J = 9:5?10?4 and N = 100.
? HFB-Popov approximation
In the translationally invariant system, both the BdG and the HFB-Popov approximations give the
same quasiparticle amplitudes and energies. However, when the harmonic confinement is present this
is no longer the case. The HFB-Popov approximation shifts the quasiparticle energies depending on
the spatial variation of the non condensate density in the region of the condensate. In this section we
investigate the limits of validity of the HFB-Popov approximation as the parameter U=J is increased.
We use the HFB-Popov approximation instead of the BdG approximation because of the relevant
role of the noncondensate atoms into the condensate as the system is driven to the Mott insulator
transition.
93
Minus20 100 20n0
F ns G
ns
sEqual3
TFBdG
Minus20 100 20n0
F ns G
ns
sEqual4
TFBdG
Minus20 100 20n0
F ns G
ns
sEqual1
TFBdG
Minus20 100 20n0
F ns G
ns
sEqual2
TFBdG
Figure 5.8: Comparisons between the product FsnGsn calculated from the Thomas-Fermi approximation
and the exact numerical solution of the BdG equations. The parameters used were U=J = 0:2, ?=J =
9:5?10?4 and N = 100.
In the presence of an harmonic confinement the HFB-Popov equations take the form:
!susn +cszn = ?J(usn+1 +usn?1)+(2U(jznj2 + ~nn)??+?n2)usn
?Uz2nvsn; (5.113)
?!svsn ?csz?n = ?J(vsn+1 +vsn?1)+(2U(jznj2 + ~nn)??+?n2)vsn
?Uz?2n usn; (5.114)
?zn = ?J(zn+1 + zn?1)+?U(jznj2 +2~nn)+?n2?zn; (5.115)
~nn =
X
s
jvsnj2; (5.116)
N =
X
n
(jznj2 + ~nn); (5.117)
cs = U
P
njznj
2(z?nusn ?znvsn)
P
njznj2
: (5.118)
The HFB-Popov are nonlinear equations and therefore it is more complicated to get analytic approxi-
mations. Instead, we solved the HFB-Popov equations numerically by an iterative procedure, similar
to the one followed in Ref. [19]. Each cycle of the iteration consists of two steps. In the first step we
94
solve Eq. (5.115) subject to the constraint Eq. (5.117) by using the ~nn obtained in the previous cycle.
This generates new values for the zn. In the second step we solve for fusn;vsng in Eqs. (5.113) using
the ~nn from the previous cycle and the newly generated zn. The fusn;vsng are used then to update ~nn.
Because the HFB-Popov is gapless, it is possible to keep the orthogonality of the excitations to the
condensate by solving Eqs. (5.113) with the cs set to zero but removing in each cycle the projection
of the calculated fusn;vsng amplitudes onto the condensate. Convergence is reached when the change
in Pnj~nnj2 from one cycle to the next is smaller than a specified tolerance.
The parameters chosen for the numerical calculations were ? = 0:0015ER, with ER the one photon
recoil energy, which for the case of a rubidium condensate corresponds to a trap frequency of approx-
imately 90 Hz. We used a total number of 1000 atoms, N = 1000; and set UN = 1:0ER. J was
varied to achieve a range of Veff = U=J between 0.01 and 312. The range was chosen based on
a local mean field approach [77], which for our parameters estimates the transition region between
Veff ? 640 (at the center where the local filling factor is approximately 80) and Veff ? 12 (at the
wings).
The results of the numerical calculations are summarized in Figs. (5.9-5.13). In Fig. 5.9 we plot
the evolution of the density profile (black boxes), the condensate population (triangles) and the on-
site depletion (empty diamonds) as Veff is increased. In the plots we also show, for comparison
purposes, the ground state density profile for J = 0 (empty boxes). This has the advantage that it can
be calculated exactly from the Hamiltonian. In general we observe the reduction of the condensate
population and thus the increment of the depletion with increasing interaction strength. When the
system is in the superfluid regime, most of the atoms are in the condensate, but as J is decreased the
depletion of the condensate becomes very important.
For the chosen parameters, the density profile has a parabolic shape reflecting the confining potential.
By comparing the evolution of the density as J is decreased with the exact solution at J = 0,
we can crudely estimate the validity of the HFB-Popov calculations. The density evolves from a
Gaussian type (see plots for Veff = 0:01 and 0:09) with smooth edges towards a Thomas-Fermi
95
Veff=100
-10 -5 0 5 10i
20
40
60
80 Veff=312
-10 -5 0 5 10i
20
40
60
80
Veff=3
20
40
60
80 Veff=11
20
40
60
80
Veff=0.01
20
40
60
80 Veff=0.09
20
40
60
80Figure 5.9: Condensate density (triangles), total density (filled boxes) and local depletion (empty dia-
monds) as a function of the lattice site i for different values of Veff:.The site indices i are chosen such that
i = 0 corresponds to the center of the trap. Although these quantities are defined only at the discrete lattice
sites we join them to help visualization. The empty boxes represent the exact solution for the case J=0.
profile with sharp edges adjusting its shape to the J = 0 profile. We can appreciate that around
Veff = 3 both profiles are almost equal. For lower values of J the HFB-Popov density starts to
differ from the J = 0 limit, even though the system is closer to the J = 0 limit. We can say
that beyond this point higher order correlations, neglected by the theory, begin to be important. The
departure of the HFB-Popov density profile from the J = 0 one as J is decreased begins at the edges
(see the panels corresponding to Veff = 11 and 100). This is expected if we consider the on-site
depletion. For such values of Veff the local depletion in the wings corresponds to a considerable
percentage of the condensate populations, and thus the validity of the HFB-Popov assumptions starts
to be dubious. The homogeneous results shown in the previous sections corroborate our present
statements for the confined system. For the smallest filling factor (see Fig. 5.2) the differences
between the homogeneous HFB-Popov calculations and the exact solutions become important for
96
values of Veff greater than 20. For higher values of Veff, see plot for Veff = 312, the HFB-Popov
density predictions differs from the J = 0 solution even at the central wells. At this point the failure
of the method is clear and a fully quantal method is required.
0 2 4 6
q
0
0.2
0.4
0.6
?q
Figure 5.10: Quasiparticle spectrum predicted by the HFB-Popov theory for different values of Veff: Empty
diamonds (Veff = 0:01), stars (Veff = 0:09), crosses (Veff = 3), filled diamonds (Veff = 11), empty
boxes (Veff = 100) and pentagons (Veff = 312). The letter q labels the quasiparticle energies in increas-
ing order. The quasiparticle energies are in recoil units.
The HFB-Popov quasiparticle spectrum is shown in Fig. 5.10. It can be observed how the lower
energy eigenvalues evolve from a linear non degenerated spectrum to an almost degenerated one as
J is decreased. It is worth mentioning that the small energy difference between the ground and first
excited states for high values of Veff makes the numerical solution very unstable in the sense
that it is very easy to jump to an excited state when solving for the condensate wave function. The
decrement in the energy spacing predicted by the HFB-Popov theory as the system approaches the
transition is very useful to keep in mind for the experimental realization of the Mott transition. As
the optical lattice depth is ramped up the adiabaticity criteria is harder to fulfill.
In Fig. 5.11 we plot the results for the number fluctuations found numerically using the inhomoge-
97
-10 -5 0 5 10
i
0
2
4
6
?n i
Figure 5.11: Number fluctuations in the self consistent HFB-Popov approach as a function of lattice site
for Veff = 0:01 (boxes), Veff = 0:09 (crosses), Veff = 3 (circles),Veff = 11 (triangles), Veff = 100
(stars) and Veff = 312 (diamonds). The maximum value reached by the profile decreases as Veff is
increased. The empty boxes shown for each of the curves correspond to the number fluctuations predicted
by the homogeneous HFB-Popov model using a local density approximation.
neous HFB-Popov approach. The number fluctuations profile reflects the condensate profile. We also
show the number fluctuations evaluated by using a local density approximation (empty boxes). The
latter was calculated by substituting in the number fluctuations expression (Eq. (5.56)) the fuq;vqg
amplitudes found for the homogeneous system (Eqs. (5.66) and (5.67)), but replacing the condensate
density in each lattice site by the one found numerically for the trapped system (see Fig. 5.9). The
complete agreement between the two approaches justifies the validity of the local density approxi-
mation for the estimations of local quantities in confined systems. Based on this agreement and the
results for the homogeneous system shown in the previous section, we expect that the inhomogeneous
HFB-Popov results for squeezing also agree with the exact solutions right up to the transition.
In Fig. 5.12 we present the quasimomentum distribution for the same parameters used in the previous
plots. The distribution for the two lowest values of Veff corresponds to the one that characterizes
an uncorrelated superfluid phase with a narrow peak at small quasimomenta. As the hopping rate
98
Veff=100
0 1 2 3aq0
100
200
n q
Veff=312
0 1 2 3aq
Veff=3
0
100
200
300
n q Veff=11
Veff=0.01
0
100
200
300
400
n q Veff=0.09
Figure 5.12: Quasimomentum distribution as a function of qa, a the lattice spacing, q the quasimomentum,
for different values of Veff.
is decreased we observe that the sharpness of the central peak decreases and the distribution ex-
tends towards large quasi-momenta. It is interesting to note the appearance of a small peak between
q = 0:5 and 1 which is most noticeable for the Veff = 3 case. This agrees with numerical solutions
of the Bose-Hubbard Hamiltonian using Monte Carlo simulations [112]. We attribute the origin of
the small peak to the depletion of the condensate at the wings. For the parameters when the small
peak is present, the most important contribution to the quasimomentum distribution still comes from
the condensate atoms. The step function like shape of the condensate profile causes an oscillatory
jsin(x)=xj shape of the quasimomentum distribution. As the lattice depth is increased the hopping
becomes energetically costly, the long-range order starts to decrease and the Fourier spectrum be-
comes broader.
In Fig. 5.13 we plot the first order on site superfluid fraction f(1)sn which was defined in Eq.(5.60).
99
-10 -5 0 5 10
i
0
0.02
0.04
0.06
0.08
0.1
f sc(1
)
Figure 5.13: First order on-site superfluid fraction as a function of the lattice site i for different values of
Veff..The site indices i are chosen such that i = 0 corresponds to the center of the trap. Filled boxes: Veff.
=0.01, empty boxes: Veff. = 0:09, empty diamonds: Veff. = 3, stars: Veff. = 11, crosses: Veff. = 100
and triangles: Veff. = 312.
The curves corresponding to Veff = 0:01 ? 11, which are in the regime where the HFB-Popov is
expected to be valid, depict how as Veff is increased the superfluid profile decreases faster at the
wings and at the center but no major change is observed in the middle section. The evolution of the
on-site superfluidity as the interaction strength is increased, exhibiting a domain localized decrement
instead of a global one, is in agreement with the development of uncompressible regions surrounded
by superfluid rings predicted for trapped systems [79] as the transition is approached.
The Mott transition is a quantum phase transition and as for all critical phenomena, its behavior
depends strongly on the dimensionality of the system. In the present analysis, due to computational
limitations, we considered one dimensional systems. Experimentally, the Mott transition has been
achieved [46] in a 3 dimensional lattice with filling factors between 1 and 3. Even though the HFB-
Popov approach fails to describe the strong coupling regime for the one dimensional systems we
considered, we showed how the method is incredibly powerful in describing most of its characteristic
features as they are driven from the superfluid regime towards the transition. We expect the HFB-
100
Popov method to give a better description of the transition as the dimensionality of the system is
increased and therefore to be a good model in an experimental situation.
As shown in previous studies [44], [79] the Mott transition in a d-dimensional homogeneous system
has two different critical behaviors: one (d+1) XY- like, for systems with fixed integer density as
the interaction strength is changed, and one mean field-like exhibited when the transition is induced
by changing the density. Different from the homogeneous case where the Mott transition is charac-
terized by the global offset of the superfluidity, for confined systems, commensuration is only well
defined locally. The inhomogeneity introduced by the confining potential allows the existence of
extended Mott domains (above a critical interaction strength) surrounded by superfluid ones [79],
thus the total superfluid fraction doesn?t vanishes in the Mott regime. This issue, together with the
fact that the finite length scale introduced by the trap suppresses the long wave fluctuations which
are responsible for destroying the mean field [16]1, make us believe the critical behavior in confined
systems to be more mean-field like. Because the critical dimension for the latter type of transition is
two [44], [79], we expect that for trapped systems in d = 3, the range of validity of the HFB-Popov
extends closer to the transition.
5.7 Improved HFB-Popov approximation
The corrections to the quadratic Hamiltonian from higher order terms have essentially two major effects.
Firstly, they include the effect of the interactions between condensate and noncondensate atoms on the
behavior of the condensate. Secondly, they take into account the effect of the surrounding atoms on the
nature of the interatomic collisions. We mentioned previously that although the HFB treatment should
represent an advance over the BdG theory it is not useful in practice because it is not gapless. In the first
part of this section we show, how, by taking into account the anomalous average em (Eq. (5.18)) we actually
upgrade the bare interaction potential U to the many body scattering matrix which gives a better description
of interparticle collisions. The problem with the HFB equations is that not all the interactions are upgraded
1One obvious consequence of this is that BEC is possible in one and two dimensions in a trap whereas in the homogeneous,
thermodynamic limit it can not occur in fewer than three dimensions
101
and the diagonal elements still contain bare interaction terms. The different treatment of the off diagonal
and diagonal terms is caused by the factorization approximation on which the HFB approximation is based.
The factorization is appropriate for the quartic term in the Hamiltonian (as it is justified by Wick?s theorem
[16]) but not for the cubic terms.
To solve the problem the first approximation that one might think of is to upgrade by hand the bare
interactions in the diagonal terms and replace them by the many-body scattering matrix. This procedure
is known as the improved Popov approximation [53, 16]. In this section, we discuss the improved Popov
approximation and apply it to a translationally invariant system. We explicitly show that the improved
Popov approximation gives a better description of the many-body physics as the Mott insulator transition is
approached in comparison with the regular HFB-Popov approximation.
5.7.1 The two-body and many-body scattering matrices
The two-body scattering matrix, T2b, describes the scattering of two particles in vacuum. It is defined as a
function of a complex parameter z by the Lippmann-Schwinger equation:[51]
T2b(z) = V +V 1z ? ^HspT2b(z); (5.119)
Here V is the interatomic potential and ^Hsp is the single particle Hamiltonian. Although T2b is defined
for a general complex parameter z, this is physically interpreted as the energy of the scattering process.
Inserting a complete set of eigenstates in Eq. (5.119) one obtains:
T2b(z) = V +
X
pq
Vjpqi 1z ?("
p +"q)
hpqjT2b(z); (5.120)
where "p and "q are single particle energies (eigenvalues of ^Hsp), and the kets jpqi are the corresponding
two particle eigenstates and describe the intermediate states in the collision of two atoms. A diagrammatic
representation of T2b is shown in Fig. 5.14. Because of the ladder shape of the diagrams included in the
scattering matrix, they are usually referred as ladder diagrams.
T2b describes the interaction between two particles in vacuum. In the interacting Bose system,
binary collisions do not occur in vacuum but in the presence of other atoms. To describe their influence
102
Figure 5.14: Diagrammatic representation of the two-body T2b scattering matrix. In this figure jkli desig-
nates the initial states, jiji the final states and jnpi a set of intermediate states
on the scattering the concept of the many-body scattering matrix is introduced. The many-body scattering
matrix, Tmb, plays the same role as its two-body counterpart, but describes scattering occurring in many-
body systems. For a bosonic gas there are two major effects which need to be accounted for as compared
to scattering in vacuum. First of all, the relevant states which enter in the matrix elements should be
many-body states rather than single particle ones. Second, Tmb should account for Bose enhancement of a
scattering process. Various forms for the Tmb have been been proposed in the literature [16, 53, 54, 113]
depending upon the approximations made. The one that we are going to use is a simplest generalization of
the two-body matrix given by [16, 113]:
Tmb(z) = V +
X
pq
Vjpqi 1+np +nqz ?(!
p +!q)
hpqjTmb(z); (5.121)
where z is again a complex parameter, !p and !q are quasiparticle energies, nq and np are thermal quasi-
particle occupation factors which vanish at T = 0, and jpqi correspond to single-particle wave functions.
The full quasiparticle character of the intermediate states is not taken into account in this simple definition.
A more general many-body scattering matrix which includes the full quasiparticle wave functions has been
discussed by Bijlsma and Stoof [53]. However this simpler Tmb matrix is the one that naturally appears in
the many-body theory, at least at lowest order.
103
At this point it is important to highlight that the zero energy of the many-body scattering matrix is
not the same as the zero energy in its two-body counterpart. Because the two-body matrix is written in
terms of single particle energies, in this case the z is measured relative to the energy of a stationary particle.
In the many-body case Tmb is defined in terms of quasiparticle energies and therefore z is measured with
respect to the condensate chemical potential. In the dilute gas limit, where the inter-particle distance is
large compared with the s-wave scattering length, the ladder diagrams included in the Tmb give the largest
contribution to the modification of the bare potential due to interactions.
5.7.2 The anomalous average and the many-body scattering matrix
In this section we are going to follow [16, 53] to show how by including the anomalous average the many-
body scattering matrix is introduced into the theory. For interpretation purposes we are going to restrict the
analysis to the translationally invariant system.
The anomalous average em is defined in terms of the quasiparticle amplitudes uq and vq according to
Eq.(5.93). The product uqvq, on the other hand, is related to the off diagonal matrix elements, see Eq.(5.90).
If we combine the two equations we get an expression for the off diagonal matrix element Mq?q given by
Mq?q
no = U +
U
M
X
p
1
?(!p +!?p)
Mq?q
no : (5.122)
To make the connection with the many-body scattering matrix, let us take the inner product of Eq. (5.121),
with the states hq?qj and j00i and evaluate it at z = 0. At zero temperature, the thermal occupation
factors vanish, nq = 0 and we get:
hq?qjTmb(0)j00i = hq?qjVj00i+
X
pk
hq?qjVjpki 1?(!
p +!k)
hpkjTmb(0)j00i: (5.123)
The quantity hk1k2jV jk3k4i is the momentum representation of the two-body interaction potential
V . In the Bose-Hubbard Hamiltonian we assume a contact potential with amplitude U which yields a
potential in the momentum representation given by
hk1k2jV jk3k4i = UM?k1+k2;k3+k4: (5.124)
104
Using Eq. (5.124) into (Eq. 5.123) we obtain
Tmb = U + 1M
X
p
U 1?(!
p +!?p)
Tmb; (5.125)
with
hp?pjTmb(0)j00i ? 1MTmb: (5.126)
Eqs. (5.125) and (5.122) imply that Mq?q = noTmb, and therefore that by including the anomalous
average in the equations, the off diagonal matrix element given by Uno in the BdG equations is upgraded
to Tmbno in the HFB equations.
The many-body scattering matrix also appears in the DNLSE since Eq. (5.89) contains em:
? = ?tJ +Uno +2Uen+Uem = ?tJ +Tmbno +2Uen: (5.127)
Finally, to conclude this section we just rewrite the HFB equations in terms of Tmb obtaining:
0
BB
@
?q +no(2U ?Tmb) ?noTmb
noTmb ??q ?no(2U ?Tmb)
1
CC
A
0
BB
@
uq
vq
1
CC
A = !q
0
BB
@
uq
vq
1
CC
A: (5.128)
This allow us to explicitly see the different treatment that the theory gives to the diagonal and off diagonal
terms: The bare potential is completely upgraded to Tmb in the off diagonal terms but is not in the diagonal
ones, and in the chemical potential.
5.7.3 Improved Popov approximation
The appearance of a gap in the HFB equations is because of the different treatment of the off diagonal
and diagonal terms. We will show in chapter 8 that as we go beyond the quadratic approximation of the
Hamiltonian and include higher order corrections we start to incorporate the many-body scattering in the
diagonal terms. However, a naive way to correct for the gap problem in the HFB equations is simple to
upgrade by hand the interactions in the diagonal terms and replace them by the many-body scattering
105
matrix. This way to proceed is known as the improved Popov approximation [16, 53]. The matrix we have
to diagonalize under the improved Popov approximation is:
0
BB
@
?q +noTmb ?noTmb
noTmb ?(?q +noTmb)
1
CC
A
0
BB
@
uq
vq
1
CC
A = !q
0
BB
@
uq
vq
1
CC
A (5.129)
? = ?tJ +Tmbno +2eTmben (5.130)
Eqs. (5.129) and (5.130) together with Eq. (5.125) and Eq. (5.69) form a closed set of equations. They
are exactly the same than the HFB-Popov equations if U is replaced by Tmb. The reason why there are
two different coupling constant in the equation for the chemical potential Tmb and eTmb, is because they
describe two different scattering processes. The second term in the right hand side of Eq. (5.130) describes
the scattering between two atoms in the condensate colliding at zero momentum, hq?qjTmb(0)j00i, and
that is why the coupling constant is Tmb (see Eq. 5.126). On the other hand, the third term describes the
collision between one condensate atom and one atom out of the condensate and the coupling constant in
this case should be evaluated at different energy eTmb /Pphp0jTmb(!p)j0pi.
The extra difficulty that the improved Popov equations have compare with the HFB-Popov equations
is that Tmb depends on the quasiparticle energies and all equations must be solved in a self consistent way.
To get simple analytic expressions we start by analyzing limiting regimes.
? Case noTmb > J
In this regime the quasiparticle energy can be approximated by !q v p2noTmb?q. Using this
approximation in the expression for the many-body scattering matrix we obtain
Tmb = U
1+
q
U2fi2
noJTmb
; (5.131)
with fi defined in Eq. (5.73). Solving the algebraic equation we get two possible solutions
Tmb = U ? U
2
2Jno
?
fi2 +
r
fi4 + 4Jnofi
2
U
!
: (5.132)
106
Because we want the root which increases as U increases we choose the solution with positive sign.
? Case noTmb < J
In general one can show that the condition noTmb < J is only satisfied if the quantity Tmb=J is
small. This also implies that the parameter U=J must be small. In this regime, therefore, we can
perturbativelly expand Eq.( 5.131 ) in powers of U=J to get:
Tmb v U
 
1+O
 U
J
??
: (5.133)
The fact that Eq. (5.134) has the correct asymptotic behavior in the limit noTmb < J (it approaches to U
as U goes to zero) even though it was derived under the assumption noTmb > J, allow us to extend the
validity of Eq. (5.132) to all regimes. Therefore, to a good approximation, we get a general formula for
Tmb given by:
Tmb = U + U
2
2Jno
?
fi2 +
r
fi4 + 4Jnofi
2
U
!
: (5.134)
We have checked numerically that in fact, Eq. (5.134) is a very good approximation of Tmb in all regimes.
Having an analytic expression for Tmb, we can solve for the condensate density to get:
no = n? 1M
X
q6=0
?q +Uno
2!q +
1
2;
? g ?fi
r
Tmb
J no: (5.135)
In the above equation we used the same approximations that lead to Eq.(5.72), which was explicitly showed
to be good in Figs. 5.2 and 5.3 and the definition of g given in Eq. (5.74).
If we solve Eq. (5.135), we finally obtain an analytic expression for no
no ? g ?fi
r
Ug
J : (5.136)
Surprisingly, the expression for the condensate density that we get after including the many-body scattering
matrix is exactly the same, as the one we get if in the BdG equations for the condensate, Eq. (5.72), we
replace no by g in the right hand side. Except for the term M?12M , this corresponds to the lowest order
107
solution we can get of Eq. (5.72), where instead of solving self consistently the algebraic equation we
replace no by n. The term M?12M is very small at high densities when n+1=2 ? n, regime where in fact we
only expect a mean field treatment to be valid and it might be interpreted as a finite size effect. Hereafter we
are going to refer to the non self consistent solution of the BdG equations as the test field approximation.
Finally, if we use the expression for the condensate density Eq. ( 5.136) in Eq. (5.134), and assume
again that n+1=2 ? n, after some algebraic manipulations the final result we get is
noTmb = Un: (5.137)
For us this a striking result. The net effect of including the many-body scattering matrix in the theory
reduces to replacing Uno in the BdG equations with Un. This result is hard to understand because what
it implies is that the lowest order approximation beyond the simple DNLSE, is the one that reproduces the
best the exact solution as the interactions in the system are increased.
0 5 10 15 20 25 30
USlash1J
0.7
0.8
0.9
1
fc
nEqual5 MEqual3
testPopov
ManyBExact
0 5 10 15 20 25 30
USlash1J
0.7
0.8
0.9
1
fc
nEqual5.33 MEqual3
testPopov
ManyBExact
Figure 5.15: Comparisons of the condensate fraction predicted from the exact diagonalization of the BHH
(red), the improved Popov approximation (yellow), the BdG (and HFB-Popov) equations (blue) and the test
field approximation (green) as a function of Veff = U=J, for a system with M = 3 and filling factors
n = 5 and 5:33.
To check the improvement that we get by including the many-body scattering matrix in the theory,
in Figs 5.15 and 5.16 we show comparisons of the condensate fraction, fc = no=n, as a function of
U=J calculated from the exact diagonalization of the many-body Hamiltonian (red) with the improved
Popov(yellow) and the regular BdG (and HFB-Popov)(blue) approximations. We also plot the test field
108
0 50 100 150 200 250
USlash1J
0.7
0.8
0.9
1
fc
nEqual50. MEqual3
testPopov
ManyBExact
0 50 100 150 200 250
USlash1J
0.7
0.8
0.9
1
fc
nEqual50.3 MEqual3
testPopov
ManyBExact
Figure 5.16: Comparisons of the condensate fraction predicted from the exact diagonalization of the BHH
(red), the improved Popov approximation (yellow), the BdG (and HFB-Popov) equations (blue) and the test
field approximation (green) as a function of Veff = U=J, for a system with M = 3 and filling factors
n = 50 and 50:33.
approximation results (green). In the plots we used a one dimensional lattice with M = 3 and densities
n = 5 and n = 5:33 in Fig. 5.15 and n = 50 and n = 50:33 in Fig. 5.16.
It is clear in the plots that the improved Popov is a better approximation than the regular BdG
(and HFB-Popov) approximation. In particular in the non commensurate, high density case the agreement
between the improved Popov and the exact solution is very good. It also can be seen in the plots how
the improved Popov reduces to the simple test field approximation. The small differences seen in the low
density case disappear as the density is increased. To corroborate the validity of Eq. (5.137), we plot in Fig.
5.17 Tmbfc, calculated from Eqs. (5.136) and (5.134), vs U. The curves just overlap in the high density
regime n = 50 and in the low density case n = 5 the disagreement is very small.
Phase Transition
One of the signatures of the entrance to the Mott insulator phase is a zero condensate density. As we
mentioned below, the BdG (and HFB-Popov) approximation does not predict the Mott insulator transition
because the condensate density only vanishes in the limit when U=J ! 1. On the other hand, the im-
proved Popov approximation, does predict the existence of a critical value , V ceff = (U=J)c, at which the
condensate density vanishes:
109
0 5 10 15 20 25 30
U
05
1015
2025
30
Tmb
fc
nEqual5
Tmbfc U
0 50 100 150 200 250
U
0
50
100
150
200
250
Tmb
fc
nEqual50
Tmbfc U
Figure 5.17: Comparisons between no=nTmb = fcTmb and U as a function of U for a system with M = 3
and filling factors n = 5 and 50.
V ceff = n+
M?1
2M
fi2 ??!n 1
n
fi2: (5.138)
If we compare the above expression with the equation of the critical point calculated from a mean field
model starting in the Mott insulator phase (see chapter 3) given by, V ceff = t
?
(2n+1)+
?p
(2n+1)2 ?1
??
,
we see that in the high density limit, the only regime where mean field theories are expected to be valid,
both equations describe a critical point which scales linear with n and with a proportionality constant which
depends only on the dimensionality of the system. In the improved Popov approximation the proportional-
ity constant is 1=fi2, in the mean field theory starting from the Mott phase it is 8t (remember t is the number
of nearest neighbors). For a one dimensional lattice in the thermodynamic limit the improved Popov ap-
proximation has the infrared divergency problem, fi !1 and the critical point approaches zero. However,
this is consistent with the calculations done in Ref. [44] where the authors showed using renormalization
group techniques that the upper critical dimension for the transition at constant integer density is dc = 2.
Nevertheless, the phase transition predicted by the improved Popov approximation does not have a
clear connection to the real Mott insulator phase transition. A specific problem is the fact that the improved
Popov approximation does not distinguish between commensurate and incommensurate fillings. This dis-
tinction is crucial to reproduce the characteristic superfluid to Mott insulator phase diagram. Furthermore,
there is not clear signature that the system becomes incompressible in the Mott phase, and no gap in the
excitation spectrum opens up as the transition is reached.
110
5.8 Conclusions
In summary, in this chapter we have developed quadratic approximations for describing the approach of
a superfluid system towards the Mott insulator transition. We have shown that this method can be used
to predict the relevant physical quantities over a useful range. However, the quadratic approximations are
developed assuming small fluctuations and therefore as soon as quantum correlations become important it
is clear that a fully quantal method is required.
111
Chapter 6
The two particle irreducible effective action (2PI) and the closed time path (CTP) formalism
In the last section we focused our attention on quadratic approximations of the many body Hamiltonian and
we used them to describe equilibrium properties of ultra cold atoms loaded in optical lattices. In this chapter
extend our analysis to non equilibrium systems. The description of the evolution of condensates far from
equilibrium has gained considerable importance in matter-wave physics, motivated by recent experimental
achievements such as the colliding and collapsing condensates [20, 21, 115, 114], collapses and revivals of
the coherent matter field [47] and the NIST patterned loading experiment described in chapter 4. To date
most theoretical descriptions of nonequilibrium dynamics of BEC?s have been based on the time dependent
Gross-Pitaevskii equation, coupled with extended kinetic theories that describe excitations in dilute weakly
interacting systems close to thermal equilibrium ([15],[58]-[62],[116]). However, experiments such as
those mentioned above have been able to achieve regimes where the standard mean field description is
inapplicable, so new methods are required.
To treat far-from-equilibrium dynamics, in this chapter we adopt a closed time path (CTP) [56]
functional-integral formalism together with a two-particle irreducible (2PI) [57] effective action approach.
We retain terms of up to second-order in the interaction strength when solving these equations. This method
has been generalized for and applied to the establishment of a quantum kinetic field theory [117, 118, 119]
with applications to problems in gravitation and cosmology [120, 121], particles and fields [122, 123],
BEC [54, 125] and condensed matter systems [124] as well as addressing the issues of thermalization and
quantum phase transitions [126, 127, 128].
In this chapter we consider under the 2PI-CTP scheme different approximations: the Hartree-Fock-
Bogoliubov (HFB) approximation, the next-to-leading order 1/N expansion of the 2PI effective action up
to second-order in the interaction strength and a second-order perturbative expansion in the interaction
strength. In chapter 7 we apply the 2PI-CTP approximations derived here to describe the the patterned
loading experiment [33] previously discussed in chapter 4.
112
Figure 6.1: Two-loop (upper row) and three-loop diagrams (lower row) contributing to the effective action.
Explicitly, the diagram a) is what we call the double-bubble , b) the setting-sun and c) the basketball.
6.1 2PI effective action ?(z;G)
The first requirement for the study of nonequilibrium processes is a general initial-value formulation depict-
ing the dynamics of interacting quantum fields. The CTP or Schwinger-Keldysh effective action formalism
[56] serves this purpose. The second requirement is to describe the evolution of the correlation functions
and the mean field on an equal footing. The two particle irreducible (2PI) formalism [57] where the cor-
relation functions appear also as independent variables, serves this purpose. By requiring the generalized
(master) CTP effective action [118] to be stationary with respect to variations of the correlation functions
we obtain an infinite set of coupled (Schwinger-Dyson) equations for the correlation functions which is a
quantum analog of the BBGKY hierarchy. The 2PI effective action produces two such functions in this
hierarchy. In this section we shall focus on the 2PI formalism, and then upgrade it to the CTP version in the
next section.
Our starting point is the one dimensional Bose-Hubbard Hamiltonian (see chapter 3).
^H = ?JX
i
(^ayi ^ai+1 + ^ayi+1 ^ai )+ 12U
X
i
^ayi ^ayi ^ai ^ai +
X
i
Vi^ayi^ai (6.1)
where ^ai and ^ayi are the bosonic operators that annihilate and create an atom on the site i. Here, the
parameter U denotes the strength of the on-site repulsion of two atoms on the site i; the parameter Vi
113
denotes the energy offset of each lattice site due to an additional slowly- varying external potential that
might be present (such as a magnetic trap), and J denotes the hopping rate between adjacent sites. Because
the next-to-nearest neighbor amplitudes are typically two orders of magnitude smaller, tunneling to next-to-
nearest neighbor sites can be neglected. The Bose -Hubbard Hamiltonian should be an appropriate model
when the loading process produces atoms in the lowest vibrational state of each well, with a chemical
potential smaller than the distance of the first vibrationally excited state.
Hereafter we consider for simplicity only a homogeneous lattice with periodic boundary conditions
and no other external potential (Vi = 0). Once the equations of motion are derived, it is straightforward to
generalize them to higher dimensions or to include additional external potentials. As usual, we denote the
total number of atoms by N and the number of lattice sites by M.
The classical action associated with the Bose-Hubbard Hamiltonian (6.1), is given in terms of the
complex fields ai and a?i by
S[a?i;ai] =
Z
dt
X
i
i~a?i(t)@tai(t)
+
Z
dt
X
i
J?a?i(t)ai+1(t)+ai(t)a?i+1(t)? (6.2)
?
Z
dt
X
i
U
2 a
?
i(t)a
?
i(t)ai(t)ai(t):
To simplify our notation we introduce aai(a = 1;2) defined by
ai = a1i; a?i = a2i: (6.3)
In terms of these fields the classical action takes the form
S[a] =
Z
dt
X
i
1
2haba
a
i(t)~@ta
b
i(t (6.4)
+
Z
dt
X
i
 
J abaai+1(t)abi(t)? U4N ( abaai(t)abi(t))2
?
;
where N is the number of fields,which is two in this case, and summation over repeated field indices
a;b = (1;2) is implied and hab and  ab are matrices defined as
114
hab = i
0
BB
@
0 ?1
1 0
1
CC
A;  ab =
0
BB
@
0 1
1 0
1
CC
A: (6.5)
In terms of the familiar Pauli matrices,  ab =  x and hab = ? y.
After second quantization the fields aai are promoted to operators. We denote the expectation value
of the field operator, i.e. the mean field, by zai (t) and the expectation value of the composite field by
Gabij (t;t0). Physically, jzai (t)j2 is the condensate population and the composite fields determine the fluctua-
tions around the mean field:
zai (t) = haai(t)i; (6.6)
Gabij (t;t0) = ?TCaai(t)abi(t0)fi?haai(t)i?abi(t0)fi: (6.7)
The brackets denote the expectation value with respect to the density matrix and TC denotes time ordering
along a contour C in the complex plane.
All correlation functions of the quantum theory can be obtained from the effective action ?[z;G],
the two particle irreducible generating functional for Green?s functions parameterized by zai (t) and the
composite field Gabij (t;t0). To get an expression for the effective action we first define the functional
Z[J;K] [57] as
Z[J;K] = eiW[J;K]=~ (6.8)
=
Y
a
Z
Daa exp
8<
:
i
~
0
@S[a]+
Z
dt
X
i
Jia(t)aai(t)+ 12
Z
dtdt0
X
ij
aai(t)abj(t0)Kijab(t;t0)
1
A
9=
;;
where we have introduced the following index lowering convention
Xa =  abXb: (6.9)
The functional integral (6.8) is a sum over classical histories of the field aai in the presence of the local
source Jia and the non local source Kijab: The coherent state measure is included in Da. The addition of
the two-particle source term is what characterizes the 2PI formalism.
115
We define ?[z;G] as the double Legendre transform of W[J;K] such that
?W[J;K]
?Jia(t) = z
a
i (t); (6.10)
?W[J;K]
?Kijab(t;t0) =
1
2[z
a
i (t)z
b
i(t
0)+Gab
ij (t;t
0)]: (6.11)
Expressing J and K in terms of z and G yields
?[z;G] = W[J;K]?
Z
dt
X
i
Jia(t)zai (t)? 12
Z
dtdt0
X
ij
zai (t)zbj(t0)Kijab(t;t0)
?12
Z
dtdt0
X
ij
Gabij (t;t0)Kijab(t;t0): (6.12)
From this equation the following identity can be derived:
??[z;G]
?zai (t) = ?Jia(t)?
Z
dt0
X
j
(Kijad(t;t0))zdj(t0); (6.13)
??[z;G]
?Gabij (t;t0) = ?
1
2Kijab(t;t
0): (6.14)
In order to get an expression for ?[z;G] notice that by using Eq.(6.8) for W[J;K] and placing it in Eq.(6.12)
for ?[z;G], it can be written as
exp
 i
~?[z;G]
?
=
Y
a
Z
Daa exp
?i
~
 
S[a]+
Z
dti Jia(t)[aai(t)?zai (t)] (6.15)
+12
Z
dtidt0j ?aai(t)Kijab(t;t0)abj(t0)?zai (t)Kijab(t;t0)zbj(t0)?? 12TrGK
? 
=
Y
a
Z
Daa exp
?i
~
 
S[a]?
Z
dti ??[z;G]?za
i (t)
[aai(t)?zai (t)]
?
Z
dtidt0j [aai(t)?zai (t)] ??[z;G]?Gab
ij (t;t0)
?ab
i(t
0)?zb
i(t
0)?+TrG??[z;G]
?G
!)
;
where Tr means taking the trace. For simplicity we have denotedR dtPi byR dti. Defining the fluctuation
field, ?ai = aai ?zai , we have
116
?[z;G]?TrG??[z;G]?G = ?i~ln
Y
a
Z
D?a exp
 i
~S[z;G;?]
?
; (6.16)
S[z;G;?] = S[z +?]?
Z
dti ??[z;G]?za
i (t)
?ai(t)?
Z
dtidt0j ?ai(t) ??[z;G]?Gab
ij (t;t0)
?bi(t0): (6.17)
By introducing the classical inverse propagator iD?1(z) given by
iDijab(t;t0) ?1 = ?S[z]?za
i (t)?zbj(t0)
(6.18)
= (?ijhab@t +J(?i+1j +?i?1j) ab)?(t?t0)
?UN (2zia(t)zib(t)+ abzci(t)zic(t))?ij?(t?t0);
the solution of the functional integro-differential equation (6.16) can be expressed as
?[z;G] = S[z]+ i2TrlnG?1 + i2TrD?1(z)G+?2[z;G]+const: (6.19)
The quantity ?2[z;G] is conveniently described in terms of the diagrams generated by the interac-
tion terms in S[z +?] which are of cubic and higher orders in ?
Sint[z +?] = ? U4N
Z
dti(?ib(t)?bi(t))2
?UN
Z
dti?ai(t)zia(t)?bi(t)?ib(t): (6.20)
It consists of all two-particle irreducible vacuum graphs (the diagrams representing these interactions do
not become disconnected by cutting two propagator lines) in the theory with propagators set equal to G and
vertices determined by the interaction terms in S[z +?] .
Since physical processes correspond to vanishing sources J and K, the dynamical equations of
motion for the mean field and the propagators are found by using the expression (6.19) in equations (6.13)
and (6.14) , and setting the right hand side equal to zero. This procedure leads to the following equations:
117
hab~@tzbi(t) = ?J(zi+1a(t)+zi?1a(t)) (6.21)
+UN (zid(t)zdi (t)+G cii c(t;t))zia(t)
+UN ((Giiad(t;t)+Giida(t;t))zdi (t)
???2[z;G]?za
i (t)
;
and
G?1ijab(t;t0) = Dijab(t;t0)?1 ??ijab(t;t0); (6.22)
?ijab(t;t0) ? 2i ??2[z;G]?Gab
ij (t;t0)
: (6.23)
Equation (6.22) can be rewritten as a partial differential equation suitable for initial value problems by
convolution with G. This differential equation reads explicitly
hac~@tG cbij (t;t0) = ?J(Gabi+1j(t;t0)+Gabi?1j(t;t0))+ UN (zid(t)zdi (t))Gabij (t;t0)+ (6.24)
2U
N z
a
i (t)G
cb
ij(t;t
0)zic(t)+i
Z
dt00k? aikc(t;t00)Gcbkj(t00;t0)+i?ab?ij?(t?t0):
The evolution of za and Gab is determined by Eqs. (6.22) and (6.24) once ?2[z;G] is specified.
6.2 Perturbative expansion of ?2(z;G) and approximation schemes
The diagrammatic expansion of ?2 is illustrated in Fig. 6.1, where two and three-loop vacuum diagrams
are shown. The dots where four lines meet represent interaction vertices. The expression corresponding
to each vacuum diagram should be multiplied by a factor (?i)l(i)s?2 where l is the number of solid lines
and s the number of loops the diagram contains.
The action ? including the full diagrammatic series for ?2 gives the full dynamics. It is of course
not feasible to obtain an exact expression for ?2 in a closed form. Various approximations for the full 2PI
effective action can be obtained by truncating the diagrammatic expansion for ?2. Which approximation
is most appropriate depends on the physical problem under consideration.
118
6.2.1 The standard approaches
1. Mean-field approximation:
If, in Eq. (6.19), we discard all terms to the right of S[z], we recover the DNLSE. This gives us the
usual mean-field description, in which the system remains a pure condensate.
2. One-loop Approximation:
The next approximation consists of discarding ?2 altogether. This yields the so-called one-loop ap-
proximation. The one-loop approximation has an equation for the fluctuations identical to the BdG
equations (See chapter 5), however the equation of motion for the condensate does include the deple-
tion and anomalous terms. The presence of this terms is necessary for the time dependent evolution
because they guarantee the conservation of particle number and energy. The unequal treatment of
the condensate and fluctuations present in this approximation introduces limitations and makes this
approach not very attractive for studying the non equilibrium dynamics.
3. Time-dependent Hartree-Fock-Bogoliubov (HFB) approximation:
A truncation of ?2 retaining only the first order diagram in U, i.e., keeping only the double-bubble
diagram, Fig. 6.1, yields equations of motion for z and G which correspond to the time dependent
Hartree-Fock-Bogoliubov (HFB) approximation. This approximation violates Goldstone?s theorem,
but conserves energy and particle number [132, 15, 133]. It is important to point out that between
all the quadratic approximations discussed in chapter 5 only the HFB approximation is suitable for
describing non equilibrium dynamics because it is the only one that obeys conservation laws. The
HFB equations can also be obtained by using cumulant expansions up to the second-order [136] in
which all cumulants containing three or four field operators are neglected. The HFB approximation
neglects multiple scattering. It can be interpreted as an expansion in terms of Ut=J; (where t is
the time of evolution) and is good for the description of short time dynamics or weak interaction
strengths. It will be described in more detail in section 6.4.
119
6.2.2 Higher order expansions
We make a few remarks on the general properties of higher order expansions and then specialize to two
approximations.
? 2PI and Ladder Diagrams
Since the work of Beliaev [63] and Popov [52] it is well known in the literature that including higher
order terms in a diagrammatic expansion corresponds to renormalizing the bare interaction potential
to the four-point vertex (see for example Ref. [53, 54]), thus accounting for the repeated scattering
of the bosons. In particular, in chapter 5 we explicitly showed how by including the anomalous
average in the equations the bare potential was upgraded in the off diagonal elements to the many
body scattering matrix. In chapter 8 we will also show that by taking into account the two-loop
contribution of the 2PI effective action, the bare potential in the diagonal terms is also upgraded to the
many body scattering matrix up to second-order in the ladder expansion. The two-loop contribution
includes diagrams topologically identical to those found by Beliaev, but with the exact propagator
instead of the one-loop propagator. In the dilute gas limit, where the inter-particle distance is large
compared with the s-wave scattering length, the ladder diagrams give the largest contribution to the
four-point vertex. To lowest order in the diluteness parameter, the T-matrix can be approximated by
a constant proportional to the scattering length (the pseudopotential approximation). However this
approximation is only valid in the weak interaction limit and neglects all momentum dependencies
which appears in the problem as higher order terms. In that sense the 2PI effective action approach
allows us to go beyond the weakly interacting limit in a systematic way and to treat collisions more
accurately.
? Nonlocal Dissipation and Non-Markovian Dynamics
Higher order terms lead to nonlocal equations and dissipation. The presence of nonlocal terms in the
equations of motion is a consequence of the fact that the 2PI effective action really corresponds to
a further approximation of the master effective equation [118]. Though strict dissipation can never
be observed in an energy conserving closed system, characteristic features of dissipative systems
120
like exponential damping of correlations can be exhibited once interactions are properly taken into
account.
Non-Markovian dynamics is a generic feature of the 2PI formalism which yields integro-differential
equations of motion. This makes numerical solution difficult, but is a necessary price to pay for a
fuller account of the quantum dynamics. Many well acknowledged approaches to the quantum ki-
netics of such systems adopt either explicitly or implicitly a Markovian approximation [62]. The
Markovian approximation assumes that only the current configuration of the system, but not its his-
tory, determines its future evolution. Markovian approximations are made if one assumes instanta-
neous interactions, or in the kinetic theory context that the time scales between the duration of binary
collisions ?0 and the inverse collision rate ?c are well-separated. In the low kinetic energy, weakly
interacting regime the time between collisions (or mean free path) is long compared to the reaction
time (or scattering length): ?c >> ?0. The long separation between collisions and the presence of in-
termediate weak fluctuations, allow for a rapid decay of the temporal and spatial correlations created
between collision partners, which one can use to justify the Markovian approximation. However,non-
Markovian dynamics needs to be confronted squarely in systems such as the patterned loaded lattice,
in which the lattice which confines the atoms to the bottom of the wells with enhanced interaction
effects, accompanied by the low dimensionality of the system and far-from-equilibrium initial condi-
tions. That is the rationale for our adoption of the CTP 2PI scheme. Now, for the specifics:
1. Second-order expansion:
A truncation retaining diagrams of second-order in U consists of the double-bubble, the setting-sun
and the basketball (see Fig.6.1). By including the setting-sun and the basketball in the approximations
we are taking into account two particle scattering processes [121, 119]. Second-order terms lead to
integro- differential equations which depend on the time history of the system.
2. Large-N approximation
The 1/N expansion is a controlled non-perturbative approximation scheme which can be used to
study non-equilibrium quantum field dynamics in the regime of strong interactions[127, 126]. In
121
the large N approach the field is modelled by N fields and the quantum field generating functional
is expanded in powers of 1/N. In this sense the method is a controlled expansion in a small pa-
rameter, but unlike perturbation theory in the coupling constant, which corresponds to an expansion
around the vacuum, the large N expansion corresponds to an expansion of the theory about a strong
quasiclassical field.
6.2.3 Zero mode fluctuactions
In chapter 5 we neglected zero mode fluctuations due to the non perturbative character of the zero mode.
In the linearized theory, this approach introduces an artificial infrared divergence in low dimensional mod-
els thus the theory is actually improved if the contribution from this mode is neglected all together [96].
However, from a physical point of view the zero mode exists and is quantum in nature. There are both
fundamental and practical reasons why isolating and subtracting the zero mode is not as compelling in the
dynamical evolution we want to describe in this chapter. Firstly, because the 2PI formalism goes beyond the
linearized approximation, the zero mode does not have the impact it has in the linearized formalism and it
is not clear that subtracting it necessarily leads to a better approximation. Secondly, the initial state that we
are going to assume for our analysis is a coherent state rather than a proper state of the total particle number
as will be described in chapter 7. Moreover, as the total particle number is not very high, quantum fluctua-
tions in the total particle number are real, and non-negligible. Discarding these fluctuations would spoil the
integrity of the formalism. Therefore, in this chapter we shall not attempt to isolate the contributions from
the zero mode. A full non-perturbative treatment in the future is certainly desirable.
6.3 CTP formalism
In order to describe the nonequilibrium dynamics we will now specify the contour of integration in Eqs.
(6.22) and (6.24) to be the Schwinger-Keldysh contour [56] along the real-time axis or closed time path
(CTP) contour. The Schwinger-Keldysh formalism is a powerful method for deriving real and causal
evolution equations for the expectation values of quantum operators for nonequilibrium fields. The basic
idea of the CTP formalism relies on the fact that a diagonal matrix element of a system at a given time,
122
t = 0, can be expressed as a product of transition matrix elements from t = 0 to t0 and the time-reverse
(complex conjugate) matrix element from t0 to 0 by inserting a complete set of states into this matrix
element at the later time t0. Since each term in the product is a transition matrix element of the usual or
time reversed kind, the standard path integral representation for each one can be introduced. However, to
get the generating functional we seek, we have to require that the forward time evolution takes place in the
presence of a source J+ but the reversed time evolution takes place in the presence of a different source
J?, otherwise all the dependence on the source drops out.
The doubling of sources, the fields and integration contours suggest introducing a 2 x 2 matrix
notation. This notation has been discussed extensively in the literature [117, 119]. However we are going
to follow Refs. [126] and [127] and introduce the CTP formalism in our equation of motion by using
the composition rule for transition amplitudes along the time contour in the complex plane. This way is
cleaner notationally and has the advantage that all the functional formalism of the previous section may be
taken with the only difference of path ordering according to the complex time contour Cctp in the time
integrations.
The two-point functions are decomposed as
Gabij (t;t0) =  ctp(t;t0)Gab>ij (t;t0)+ ctp(t0;t)Gab<ij (t;t0); (6.25)
where
~Gab>ij (t;t0) = ??ai(t)?bj(t0)fi; (6.26)
~Gab<ij (t;t0) = ??bi(t0)?aj(t)fi; (6.27)
with ?i being the fluctuation field defined prior Eq. (17) and  ctp(t?t0) being the CTP complex contour
ordered theta function defined by
123
 ctp(t;t0) =
8>
>>>>
>>>
>><
>>>>
>>>
>>>:
 (t;t0) for t and t0 both on C+
 (t0;t) for t and t0 both on C?
1 for t on C? and t0 on C+
0 for t on C+ and t0 on C?
: (6.28)
With these definitions the matrix indices are not required. When integrating over the second half C?, we
have to multiply by a negative sign to take into account the opposite direction of integration.
To show explicitly that the prescription for the CTP integration explained above does lead to a
well-posed initial value problem with causal equations, let us explicitly consider the integral in Eq. (6.24).
The integrand has the CTP ordered form
?(t;t00)G(t00;t0) = (6.29)
 ctp(t;t00) ctp(t00;t0)?>(t;t00)G>(t00;t0)+ ctp(t;t00) ctp(t0;t00)?>(t;t00)G<(t00;t0)
 ctp(t00;t) ctp(t00;t0)?<(t;t00)G>(t00;t0)+ ctp(t00;t) ctp(t0;t00)?<(t;t00)G<(t00;t0);
where we have omitted the indices because they are not relevant for the discussion. Using the rule for CTP
contour integration we get
Z
dt00?(t;t00)G(t00;t0) =
Z t
0
dt00? (t00;t0)?>(t;t00)G>(t00;t0)+ (t0;t00)?>(t;t00)G<(t00;t0)?
+
Z 1
t
dt00? (t00;t0)?<(t;t00)G>(t00;t0)+ (t0;t00)?<(t;t00)G<(t00;t0)?
?
Z 1
0
dt00?<(t;t00)G>(t00;t0): (6.30)
If t > t0, we have
Z
dt00?(t;t00)G(t00;t0) =
Z t
0
dt00(?>(t;t00)??<(t;t00))G>(t00;t0)
?
Z t0
0
dt00?>(t;t00)(G>(t00;t0)?G<(t00;t0)): (6.31)
On the other hand, if t < t0
124
Z
dt00?(t;t00)G(t00;t0) =
Z t
0
dt00(?>(t;t00)??<(t;t00))G<(t00;t0)
?
Z t0
0
dt00?<(t;t00)(G>(t00;t0)?G<(t00;t0)): (6.32)
The above equations are explicitly causal.
It is convenient to express the evolution equations in terms of two independent two-point functions which
can be associated to the expectation values of the commutator and the anti-commutator of the fields. We
define, following Ref. [127] the functions
G(F)abij (t;t0) = 12 ?Gab>ij (t;t0)+Gab<ij (t;t0)? (6.33)
G(?)abij (t;t0) = i?Gab>ij (t;t0)?Gab<ij (t;t0)?; (6.34)
where the (F) functions are usually called statistical propagators and the (?) are called spectral functions
[137]. With these definitions Eq.(6.24) can be rewritten as:
hac~@tG (F)cbij (t;t0) = ?J
?
G(F)abi+1j (t;t0)+G(F)abi?1j (t;t0)
?
+ UN
?
zic(t)zci(t))G(F)abij (t;t0)
?
+
2U
N
?
zai (t)G(F)cbij (t;t0)zic(t)
?
+
Z t
0
dt00k?(?)acik (t;t00)G(F) bkj c (t00;t0)
?
Z t0
0
dt00k?(F)acik (t;t00)G(?)bkjc (t00;t0); (6.35)
hac~@tG (?)cbij (t;t0) = ?J
?
G(?)abi+1j (t;t0)+G(?)abi?1j (t;t0)
?
+ UN
?
zic(t)zci(t))G(?)abij (t;t0
?
+
2U
N
?
zai (t)G(?)cbij (t;t0)zic(t)
?
+
Z t
t0
dt00k?(?)acik (t;t00)G(?)bkjc (t00;t0): (6.36)
In particular, we define the normal, ?, and anomalous, m, spectral and statistical functions as
G21(F)ij (t;t0) ? ?(F)ij (t;t0) = 12
D
?yi(t)?j(t0)+?j(t0)?yi(t)
E
; (6.37)
G21(?)ij (t;t0) ? ?(?)ij (t;t0) = i
D
?yi(t)?j(t0)??j(t0)?yi(t)
E
; (6.38)
G11(F)ij (t;t0) ? m(F)ij (t;t0) = 12 h?i(t)?j(t0)+?j(t0)?i(t)i; (6.39)
G11(?)ij (t;t0) ? m(?)ij (t;t0) = ih?i(t)?j(t0)??j(t0)?i(t)i: (6.40)
125
With these relations in place, we now proceed to derive the time evolution equations for the mean
field and the two-point functions from the 2PI-CTP effective action for the Bose-Hubbard model under
the three approximations described before.
6.4 HFB approximation
6.4.1 Equations of motion
The HFB equations include the leading order contribution of ?2. They describe the coupled dynamics of
condensate and non-condensate atoms which arise from the most important scattering processes which are
direct, exchange and pair excitations. The basic damping mechanisms present in the HFB approximation are
Landau and Beliaev damping associated with the decay of an elementary excitation into a pair of excitations
in the presence of condensate atoms[133, 134]. However, these kinds of damping 1 found in the HFB
approximation (due to phase mixing, as in the Vlasov equation citeBalescu) are different in nature from the
collisional dissipation (as in the Boltzmann equation) responsible for thermalization processes. Multiple
scattering processes are neglected in this approximation. We expect the HFB equations to give a good
description of the dynamics in the collisionless regime when interparticle collisions play a minor role.
The leading order contribution of ?2 is represented by the double-bubble diagram. Its contribution
to ?2 is z independent and has an analytic expression of the form
?(1)2 [G] = (6.41)
? U4N
Z
dti ?G aiia(t;t)G biib(t;t)+2Giiab(t;t)Gabii (t;t)?;
the factor of two arises because the direct and exchange terms are identical.
Using the first order expression for ?2 in Eqs. (6:22) and (6.24) yields the following equations of
motion.
1we make a distinction between the meaning of the words ?damping? and ?dissipation?, the former referring simply to the phe-
nomenological decay of some function, the latter with theoretical meaning, e.g., in the Boltzmann sense.
126
hab~@tzbi(t) = ?zHFB; (6.42)
?zHFB ? ?J?zai+1(t)+zai?1(t)?+ UN ?zid(t)zdi (t)+G dii d(t;t)?zai (t)
+2UN ?zib(t)Gabii (t;t)?; (6.43)
hac~@tGcbij (t;t0) = ?GHFB; (6.44)
?GHFB ? ?J(Gabi+1j(t;t0)+Gabi?1j(t;t0))+ UN (zid(t)zdi (t)+G dii d(t;t))Gabij (t;t0)
+2UN (zai (t)zic(t)Gcbij (t;t0)+G aii d(t;t)Gdbij(t;t0))+i?ab?ij?C(t?t0):
In terms of the spectral and statistical functions, Eqs. (6.37) to (6.40), and setting N = 2, the above
equations take the form
i~@tzi(t) = ?J(zi+1(t)+zi?1(t))+U(jzi(t)j2 +2?(F)ii (t;t))zi(t)
+Um(F)ii (t;t)z?i (t); (6.45)
?i~@@t?(F)ij (t;t0) = Lik(t)?(F)kj (t;t0)+M?ik(t)m(F)kj (t;t0); (6.46)
?i~@@t?(?)ij (t;t0) = Lik(t)?(?)kj (t;t0)+M?ik(t)m(?)kj (t;t0); (6.47)
i~@@tm(F)ij (t;t0) = Lik(t)m(F)kj (t;t0)+Mik(t)?(F)kj (t;t0); (6.48)
i~@@tm(?)ij (t;t0) = Lik(t)m(?)kj (t;t0)+Mik(t)?(?)kj (t;t0); (6.49)
with
Lij(t) = ?J(?i+1j +?i?1j)+2U?ij
?
jzi(t)j2 +?(F)ii (t;t)
?
; (6.50)
Mij(t) = U?ij
?
zi(t)2 +m(F)ii (t;t)
?
: (6.51)
The time dependent HFB equations are a closed set of self-consistent equations that describe the
coupled dynamics of the condensate and non-condensate components of a Bose gas. It can be checked that
127
they preserve important conservation laws such as the number of particles and energy. The conservation
properties of the HFB equations can also be understood by the fact that these equations can also be derived
using Gaussian variational methods [15]. These methods always yield a classical Hamiltonian dynamics
which guarantees probability conservation. Because they are local in time they can be decoupled by a mode
decomposition.
6.4.2 Mode expansion of the HFB equations
To decouple the HFB equations we apply the well known Bogoliubov transformation to the fluctuation field
(see chapter 5)
?j(t) =
X
q
?uq
i(t)^fiq ?v
?q
i (t)^fi
y
q
?; (6.52)
where (^fiq; ^fiyq) are time independent creation and annihilation quasiparticle operators and all the time de-
pendence is absorbed in the amplitudes fuqi(t);v?qi (t)g: To ensure that the quasiparticle transformation is
canonical, the amplitudes must fulfill the relations
X
i
?uq
i(t)u
?k
i (t)?v
q
i(t)v
?k
i (t)
? = ?
qk; (6.53)
X
i
?uq
i(t)v
k
i (t)?v
q
i(t)u
k
i (t)
? = 0: (6.54)
In the zero temperature limit, where the quasiparticle occupation number vanishes, ?^fiyq^fikfi = 0, the statis-
tical and spectral functions take the form
?(F)ij (t;t0) = 12
X
q
?vq
i(t)v
?q
j (t
0)+uq
j(t
0)u?q
i (t)
?; (6.55)
?(?)ij (t;t0) = i
X
q
?vq
i(t)v
?q
j (t
0)?uq
j(t
0)u?q
i (t)
?; (6.56)
m(F)ij (t;t0) = ?12
X
q
?uq
i(t)v
?q
j (t
0)+uq
j(t
0)v?q
i (t)
?; (6.57)
m(?)ij (t;t0) = ?i
X
q
?uq
i(t)v
?q
j (t
0)?uq
j(t
0)v?q
i (t)
?: (6.58)
Notice that at equal time and position due to Eqs.(6.53) and (6.54), ?(F)ii (t;t) and m(F)ii (t;t) satisfy
128
?(F)ii (t;t) =
X
q
jvqi(t)j2 + 12 (6.59)
m(F)ii (t;t) = ?
X
q
uqi(t)v?qi (t): (6.60)
Replacing Eqs. (6.59)-(6.60) into Eqs. (6.45)-(6.49) and using the constraints (6.53) -(6.54) we
recover the standard time dependent equations for the quasiparticle amplitudes
i~@@zi(t) (6.61)
= ?J(zi+1(t)+zi?1(t))+U(jzi(t)j2 +2?(F)ii (t;t))zi(t)+Um(F)ii (t;t)z?i (t);
i~@@tuqi(t) (6.62)
= ?J(uqi+1(t)+uqi?1(t))+2U(jzi(t)j2 +?(F)ii (t;t))uqi(t)?U(m(F)ii (t;t)+zi(t)2)vqi(t);
?i~@@tvqi(t) = (6.63)
?J(vqi+1(t)+vqi?1(t)))+2U(jzi(t)j2 +?(F)ii (t;t))vqi(t)?U(m?(F)ii (t;t)+zi(t)?2)uqi(t):
If we compare Eqs. (6.61)-(6.63) with their time independent version, Eqs. (5.27) to (5.87), we notice
that the equations are identical as they should be if we replace i~@@tuqi ! !HFBuqi; i~@@tvqi ! !HFBvqi
and i~@tzi ! ?HFBzi. The factor of 1=2 difference between ?(F)ii and ?ii (see Eq. (5.19)) just leads to
a trivial renormalization of the chemical potential. In the time dependent equations we have included the
zero mode fluctuations, which were neglected in Eqs. (5.27) to (5.87). Hence we do not need the cHFB?s
variables, which simply keep the orthogonality between of the condensate and the excitations. Eqs. (6.61)
-(6.63) correspond to a set of M(2M + 1) coupled ordinary differential equations, where M is the total
number of lattice sites. They can be solved using standard time propagation algorithms. Once the time
dependent quasiparticle amplitudes are calculated we can derive the dynamics of physical observables
constructed from them as a function of time, such as the average number of particles in a well ni(t), etc.
129
6.5 Second-order expansion
6.5.1 Equations of motion
Full second-order
The second-order contribution to ?2 is described in terms of the setting-sun Fig. 6.1b and the basketball
Fig. 6.1c diagrams. The basketball diagram is independent of the mean-field and is constructed with
only quartic vertices. The setting-sun diagram depends on z and contains only three-point vertices. The
second-order ?(2)2 effective action can be written as
?(2)2 [z;G] = (6.64)
i
 U
N
?2Z
dtidtjzib(t)zjb0(t0)?
?
Gbb0ij (t;t0)Gijdd0(t;t0)Gdd0ij (t;t0)+2Gbd0ij (t;t0)Gijdd0(t;t0)Gdb0ij (t;t0)
?
+
i
 U
2N
?2Z
dtidt0j
?
Gijbb0(t;t0)Gbb0ij (t;t0)Gijdd0(t;t0)Gdd0ij (t;t0)+
2Gijbb0(t;t0)Gbd0ij (t;t0)Gijdd0(t;t0)Gdb0ij (t;t0)
?
:
To simplify the notation, let us introduce the following definitions [127]:
?ij(t;t0) = ?12Gijab(t;t0)Gabij (t;t0); (6.65)
?ijab(t;t0) = ?D(t;t0)Gijab(t;t0); (6.66)
D(t;t0) = zib(t)zja(t0)Gbaij (t;t0)??ij(t;t0); (6.67)
? bij a(t;t0) = ?Gcbij(t;t0)Gijca(t;t0); (6.68)
? bij a(t;t0) = ?G bcij (t;t0)Gijac(t;t0); (6.69)
?acij (t;t0) = ?(zid(t)zjb(t0)+Gijdb(t;t0))Gabij (t;t0)Gdcij (t;t0)+?acij (t;t0): (6.70)
With the above definitions we find from Eqs. (6.22) and (6.24) the following equations of motion:
130
hab~@tzbi(t) = ?zHFB + (6.71)
i
 2U
N
?2Z
dt0j zjb(t0)
?
?ij(t;t0)Gbaji (t0;t)+? bij c(t;t0)Gacij (t;t0)
?
;
hac~@tGcbij (t;t0) = ?GHFB + (6.72)
i
 2U
N
?2
zai (t)
Z
dt00kzkc(t00)
?
?ik(t;t00)Gcbkj(t00;t0)+? cik d(t;t00)Gdbkj(t00;t0)
?
+i
 2U
N
?2Z
dt00(? aik d(t;t00)+?caik (t;t00)zic(t)zkd(t00))Gdbkj (t00;t0);
where ?zHFB and ?GHFB are defined in Eqs. (6.43) and (6.45).
To get explicit expressions of the equations of motion in terms of ?(F;?) and m(F;?) we introduce
the functions
?(F)ij [f;g] = f(F)ij (ti;tj)g(F)ij (ti;tj)? 14
?
f(?)ij (ti;tj)g(?)ij (ti;tj)
?
; (6.73)
?(?)ij [f;g] = f(F)ij (ti;tj)g(?)ij (ti;tj)+f(?)ij (t;t0)g(F)ij (ti;tj): (6.74)
Using the spectral and statistical functions and setting the number of fields N equal to 2, the equations of
motion derived under the full second-order approximation can be written as:
i~@tizi = ?J(zi+1(ti)+zi?1(ti))+U(jzij2 +2?(F)ii )zi +Um(F)ii z?i (6.75)
?2U2
X
k
Z ti
0
dtk
?
zk?(?)ik [?;??]+zk?(?)ik [m;m?]+z?k?(?)ik [m;?]
?
?(F)ki
?2U2
X
k
Z ti
0
?
z?k?(?)ik [?;??]+z?k?(?)ik [m;m?]+zk?(?)ik [m?;??]
?
m(F)ki
+2U2
X
k
Z ti
0
dtk
?
zk?(F)ik [?;??]+zk?(F)ik [m;m?]+z?k?(F)ik [m;?]
?
?(?)ki
+2U2
X
k
Z ti
0
?
z?k?(F)ik [?;??]+z?k?(F)ik [m;m?]+zk?(F)ik [m?;??]
?
m(?)ki ;
131
?i~@ti?(F)ij = (6.76)
?J(?(F)i+1j(ti;tj)+?(F)i?1j(ti;tj))+2U(jzij2 +?(F)ii )?(F)ij +U(m?(F)ii +z?2i )m(F)ij
?2U2
X
k
Z ti
0
dtk
?
ziz?k?(?)ik [?;?]+2zizk?(?)ik [?;m?]+2z?i z?k?(?)ik [m;?]
?
?(F)kj
?2U2
X
k
Z ti
0
dtk
?
?(?)ik [?;?]+2z?i zk
?
?(?)ik [?;??]+?(?)ik [m;m?]
??
?(F)kj
?2U2
X
k
Z ti
0
dtk
?
2ziz?k?(?)ik [m?;?]+zizk?(?)ik [m?;m?]+2z?i zk?(?)ik [??;m?]
?
m(F)kj
?2U2
X
k
Z ti
0
dtk
?
?(?)ik [m?;currency1]+z?i z?k
?
2?(?)ik [?;??]+2?(?)ik [m;m?]
??
m(F)kj
+2U2
X
k
Z tj
0
dtk
?
ziz?k?(F)ik [?;?]+2zizk?(F)ik [?;m?]+2z?i z?k?(F)ik [m;?]
?
?(?)kj
+2U2
X
k
Z tj
0
dtk
?
?(F)ik [?;?]+2z?i zk
?
?(F)ik [?;??]+?(F)ik [m;m?]
??
?(?)kj
+2U2
X
k
Z tj
0
dtk
?
2ziz?k?(F)ik [m?;?]+zizk?(F)ik [m?;m?]+2z?i zk?(F)ik [??;m?]
?
m(?)kj
+2U2
X
k
Z tj
0
dtk
?
?(F)ik [m?;currency1]+2z?i z?k
?
?(F)ik [?;??]+?(F)ik [m;m?]
??
m(F)kj ;
?i~@ti?(?)ij = (6.77)
?J(?(?)i+1j(ti;tj)+?(?)i?1j(ti;tj))+2U(jzij2 +?(F)ii )?(?)ij +U(m?(F)ii +z?2i )m(?)ij
?2U2
X
k
Z ti
tj
dtk
?
ziz?k?(?)ik [?;?]+2zizk?(?)ik [?;m?]+2z?i z?k?(?)ik [m;?]
?
?(?)kj
?2U2
X
k
Z ti
tj
dtk
?
?(?)ik [?;?]+2z?i zk
?
?(?)ik [?;??]+?(?)ik [m;m?]
??
?(?)kj
?2U2
X
k
Z ti
tj
dtk
?
2ziz?k?(?)ik [m?;?]+zizk?(?)ik [m?;m?]+2z?i zk?(?)ik [??;m?]
?
m(?)kj
?2U2
X
k
Z ti
tj
dtk
?
?(?)ik [m?;currency1]+2z?i z?k
?
?(?)ik [?;??]+?(?)ik [m;m?]
??
m(?)kj ;
132
i~@tim(F)ij = (6.78)
?J(m(F)i+1j(ti;tj)+m(F)i?1j(ti;tj))+2U(jzij2 +?(F)ii )m(F)ij +U(m(F)ii +z2i )?(F)ij
?2U2
X
k
Z ti
0
dtk
?
2z?i zk?(?)ik [m;??]+z?i z?k?(?)ik [m;m]+2ziz?k?(?)ik [?;m]
?
?(F)kj
?2U2
X
k
Z ti
0
dtk
?
?(?)ik [m;currency1]+2zizk
?
?(?)ik [?;??]+?(?)ik [m;m?]
??
?(F)kj
?2U2
X
k
Z ti
0
dtk
?
z?i zk?(?)ik [??;??]+2z?i z?k?(?)ik [??;m]+2zizk?(?)ik [m?;??]
?
m(F)kj
?2U2
X
k
Z ti
0
dtk
?
?(?)ik [??;?]+2ziz?k
?
?(?)ik [?;??]+?(?)ik [m;m?]
??
m(F)kj
+2U2
X
k
Z tj
0
dtk
?
2z?i zk?(F)ik [m;??]+z?i z?k?(F)ik [m;m]+2ziz?k?(F)ik [?;m]
?
?(?)kj
+2U2
X
k
Z tj
0
dtk
?
?(F)ik [m;currency1]+2zizk
?
?(F)ik [?;??]+?(F)ik [m;m?]
??
?(?)kj
+2U2
X
k
Z tj
0
dtk
?
z?i zk?(F)ik [??;??]+2z?i z?k?(F)ik [??;m]+2zizk?(F)ik [m?;??]
?
m(?)kj
+2U2
X
k
Z tj
0
dtk
?
?(F)ik [??;?]+2ziz?k
?
?(F)ik [?;??]+?(F)ik [m;m?]
??
m(?)kj ;
i~@tim(?)ij = (6.79)
?J(m(?)i+1j(ti;tj)+m(?)i?1j(ti;tj))+2U(jzij2 +?(F)ii )m(?)ij +U(m(F)ii +z2i )?(?)ij
+2U2
X
k
Z ti
tj
dtk
?
2z?i zk?(?)ik [m;??]+z?i z?k?(?)ik [m;m]+2ziz?k?(?)ik [?;m]
?
?(?)kj
+2U2
X
k
Z ti
tj
dtk
?
?(?)ik [m;currency1]+zizk
?
2?(?)ik [?;??]+2?(?)ik [m;m?]
??
?(?)kj
+2U2
X
k
Z ti
tj
dtk
?
z?i zk?(?)ik [??;??]+2z?i z?k?(?)ik [??;m]+2zizk?(?)ik [m?;??]
?
m(?)kj
+2U2
X
k
Z ti
tj
dtk
?
?(?)ik [??;?]+2ziz?k
?
?(?)ik [?;??]+?(?)ik [m;m?]
??
m(?)kj ;
with
?(F;?)ij = ?(F;?)ij [?;??]+2?(F;?)ij [m;m?]; (6.80)
currency1(F;?)ij = 2?(F;?)ij [?;??]+?(F;?)ij [m;m?]: (6.81)
133
In the above equations we adopted the notation zk meaning zk(tk) for the condensate and mkj and
?kj meaning mkj(tk;tj) and ?kj(tk;tj) respectively for the two point functions. These notation is also
going to be used in the equations for the 1/N expansion.
2PI-1/N expansion
The 2PI effective action is a singlet under O(N) rotations. It can be shown that all graphs contained in
an O(N) expansion can be built from the irreducible invariants[127]: z2; Tr(Gn) and Tr(zzGn) , with
n < N. The factors of N in a single graph contributing to the same 1/N expansion then have two origins:
a factor of N from each irreducible invariant and a factor of 1/N from each vertex. The leading order large
N approximation scales proportional to N, the next to leading order (NLO) contributions are of order 1
and so on. At leading order only the first term of Eq. (6.42) contributes. At the next to leading order level,
if we truncate up to second-order in the coupling strength, the double-bubble is totally included but only
certain parts of the setting-sun and basketball diagrams are: the first term in both of the integrals of Eq.
(6.64),
?(2)1=N2 [z;G] = i
 U
N
?2Z
dtidtjzib(t)zjb0(t0)
?
Gbb0ij (t;t0)Gijdd0(t;t0)Gdd0ij (t;t0)
?
+
i
 U
2N
?2Z
dtidt0j
?
Gijbb0(t;t0)Gbb0ij (t;t0)Gijdd0(t;t0)Gdd0ij (t;t0)
?
: (6.82)
The equations of motion under this approximation are the ones obtained for the full second-order expansion
but with ? = ? = 0, and ? = ?.
In terms of the spectral and statistical functions the equation of motion can be written as:
i~@tizi = ?J(zi+1(ti)+zi?1(ti))+U(jzij2 +2?(F)ii )zi +Um(F)ii z?i (6.83)
?U2
X
k
Z ti
0
dtk?(?)ik
?
zk?(F)ki +z?km(F)ki
?
+U2
X
k
Z ti
0
dtk?(F)ik
?
zk?(?)ki +z?km(?)ki
?
;
134
?i~@ti?(F)ij = (6.84)
?J(?(F)i+1j(ti;tj)+?(F)i?1j(ti;tj))+2U(jzij2 +?(F)ii )?(F)ij +U(m?(F)ii +z?2i )m(F)ij
?U2
X
k
Z ti
0
dtk
?
ziz?k?(?)ik [?;?]+zizk?(?)ik [?;m?]+z?i z?k?(?)ik [m;?]
?
?(F)kj
?U2
X
k
Z ti
0
dtk
?
?(?)ik [?;?]+z?i zk
?
2?(?)ik [?;??]+?(?)ik [m;m?]
??
?(F)kj
?U2
X
k
Z ti
0
dtk
?
ziz?k?(?)ik [m?;?]+zizk?(?)ik [m?;m?]+z?i zk?(?)ik [??;m?]
?
m(F)kj
?U2
X
k
Z ti
0
dtk
?
?(?)ik [m?;?]+z?i z?k
?
?(?)ik [?;??]+2?(?)ik [m;m?]
??
m(F)kj
+U2
X
k
Z tj
0
dtk
?
ziz?k?(F)ik [?;?]+zizk?(F)ik [?;m?]+z?i z?k?(F)ik [m;?]
?
?(?)kj
+U2
X
k
Z tj
0
dtk
?
?(F)ik [?;?]+z?i zk
?
2?(F)ik [?;??]+?(F)ik [m;m?]
??
?(?)kj
+U2
X
k
Z tj
0
dtk
?
ziz?k?(F)ik [m?;?]+zizk?(F)ik [m?;m?]+z?i zk?(F)ik [??;m?]
?
m(?)kj
+U2
X
k
Z tj
0
dtk
?
?(F)ik [m?;?]+z?i z?k
?
?(F)ik [?;??]+2?(F)ik [m;m?]
??
m(F)kj ;
?i~@ti?(?)ij = (6.85)
?J(?(?)i+1j(ti;tj)+?(?)i?1j(ti;tj))+2U(jzij2 +?(F)ii )?(?)ij +U(m?(F)ii +z?2i )m(?)ij
?U2
X
k
Z ti
tj
dtk
?
ziz?k?(?)ik [?;?]+zizk?(?)ik [?;m?]+z?i z?k?(?)ik [m;?]
?
?(?)kj
?U2
X
k
Z ti
tj
dtk
?
?(?)ik [?;?]+z?i zk
?
2?(?)ik [?;??]+?(?)ik [m;m?]
??
?(?)kj
?U2
X
k
Z ti
tj
dtk
?
ziz?k?(?)ik [m?;?]+zizk?(?)ik [m?;m?]+z?i zk?(?)ik [??;m?]
?
m(?)kj
?U2
X
k
Z ti
tj
dtk
?
?(?)ik [m?;?]+z?i z?k
?
?(?)ik [?;??]+2?(?)ik [m;m?]
??
m(?)kj ;
135
i~@tim(F)ij = (6.86)
?J(m(F)i+1j(ti;tj)+m(F)i?1j(ti;tj))+2U(jzij2 +?(F)ii )m(F)ij +U(m(F)ii +z2i )?(F)ij
?U2
X
k
Z ti
0
dtk
?
z?i zk?(?)ik [m;??]+z?i z?k?(?)ik [m;m]+ziz?k?(?)ik [?;m]
?
?(F)kj
?U2
X
k
Z ti
0
dtk
?
?(?)ik [m;?]+zizk
?
?(?)ik [?;??]+2?(?)ik [m;m?]
??
?(F)kj
?U2
X
k
Z ti
0
dtk
?
z?i zk?(?)ik [??;??]+z?i z?k?(?)ik [??;m]+zizk?(?)ik [m?;??]
?
m(F)kj
?U2
X
k
Z ti
0
dtk
?
?(?)ik [??;?]+ziz?k
?
2?(?)ik [?;??]+?(?)ik [m;m?]
??
m(F)kj
+U2
X
k
Z tj
0
dtk
?
z?i zk?(F)ik [m;??]+z?i z?k?(F)ik [m;m]+ziz?k?(F)ik [?;m]
?
?(?)kj
+U2
X
k
Z tj
0
dtk
?
?(F)ik [m;?]+zizk
?
2?(F)ik [?;??]+?(F)ik [m;m?]
??
?(?)kj
+U2
X
k
Z tj
0
dtk
?
z?i zk?(F)ik [??;??]+z?i z?k?(F)ik [??;m]+zizk?(F)ik [m?;??]
?
m(?)kj
+U2
X
k
Z tj
0
dtk
?
?(F)ik [??;?]+ziz?k
?
2?(F)ik [?;??]+?(F)ik [m;m?]
??
m(?)kj ;
i~@tim(?)ij = (6.87)
?J(m(?)i+1j(ti;tj)+m(?)i?1j(ti;tj))+2U(jzij2 +?(F)ii )m(?)ij +U(m(F)ii +z2i )?(?)ij
+U2
X
k
Z ti
tj
dtk
?
z?i zk?(?)ik [m;??]+z?i z?k?(?)ik [m;m]+ziz?k?(?)ik [?;m]
?
?(?)kj
+U2
X
k
Z ti
tj
dtk
?
?(?)ik [m;?]+zizk
?
?(?)ik [?;??]+2?(?)ik [m;m?]
??
?(?)kj
+U2
X
k
Z ti
tj
dtk
?
z?i zk?(?)ik [??;??]+z?i z?k?(?)ik [??;m]+zizk?(?)ik [m?;??]
?
m(?)kj
+U2
X
k
Z ti
tj
dtk
?
?(?)ik [??;?]+ziz?k
?
2?(?)ik [?;??]+?(?)ik [m;m?]
??
m(?)kj ;
with
?(F;?)ij = ?(F;?)ij [?;??]+?(F;?)ij [m;m?]: (6.88)
We end this section by emphasizing that the only approximation introduced in the derivation of the
136
equations of motion presented here is the truncation up to second-order in the interaction strength. These
equations depict the nonlinear and non-Markovian quantum dynamics, which we consider as the primary
distinguishing features of this work. It supersedes what the second-order kinetic theories currently presented
can do, their going beyond the HFB approximation notwithstanding. For example Ref. [62] presents a
kinetic theory approach that includes binary interactions to second-order in the interaction potential but
uses the Markovian approximation. In Ref. [61] the authors gave a non-Markovian generalization to the
quantum kinetic theory derived by Walser et. al.[58] by including memory effects. However in that work
symmetry breaking fields, z and anomalous fluctuations, m, are neglected.
6.5.2 Conservation laws
For a closed (isolated) system the mean total number of particles N and energy are conserved quantities as
they are the constants of motion for the dynamical equations.
Particle number conservation is a consequence of the invariance of the Hamiltonian under a global
phase change. The mean total number of particles is given by
D ^
N
E
=
X
i
D
^ayi ^ai
E
=
X
i
 
jzij2 +?(F)ii ? 12
?
(6.89)
= N:
The kinetic equation for N is then
d
dt
D ^
N(t)
E
=
X
i
2Re
 
zi(t) @@tz?i (t)
?
(6.90)
+ lim
t!t0
 @
@t?
(F)
ii (t;t
0)+ @
@t0?
?(F)
ii (t
0;t)
?
= 0:
All three approximations we have considered, namely, HFB, 1/N expansion and full second-order
expansion, conserve the particle number. This can be shown by plugging in the kinetic equation of
D ^
N(t)
E
(Eq. 6.90) the equation of motion for the mean field and the normal statistical propagator Eqs. (45),(46),
Eqs.(6.76),(6.77) and Eqs (6.85),(6.86)), and cancelling terms. It is important to note that even though
137
total population is always conserved there is always a transfer of population between condensate and non
condensate atoms.
While number conservation can be proved explicitly, to prove total energy conservation is not obvi-
ous as the Hamiltonian cannot be represented as a linear combination of the relevant operators. It is clear
that the exact solution of a closed system is unitary in time and hence disallows any dissipation. However,
the introduction of approximation schemes that truncate the infinite hierarchy of correlation functions at
some finite order with causal boundary conditions may introduce dissipation [118].
To discuss energy conservation we can use the phi-derivable criteria [138] which states that nonequi-
librium approximations in which the self energy ? is of the form ?'=?G, with ' a functional of G, conserve
particle number, energy and momentum. All the approximations we consider in this paper are phi deriv-
able and thus they obey energy, particle number and momentum conservation laws. For HFB, ' = ?(1)2 ,
for the full second-order expansion, ' = ?(1)2 + ?(2)2 and for the second-order next to leading order 1=N
expansion, ' = ?(1)2 +?(2)1=N2 . See Eqs. (6.23), (6.42), (6.64) and (6.82). For a detailed discussion of the
complete next to leading order 1=N expansion see Refs. [126, 127] and references therein.
6.6 Conclusions
In summary, we have presented a new approach for the description of the nonequilibrium dynamics of a
Bose-Einstein condensate and fluctuations in a closed quantum field system. The formalism allows one
to go beyond the well known HFB approximation and to incorporate the nonlinear and non-Markovian
aspects of the quantum dynamics as manifest in the dissipation and fluctuations phenomena. The 2PI
effective action formalism provides a useful framework, where the mean field and the correlation functions
are treated on the same footing self-consistently and which respects conservations of particle number and
energy. The CTP formalism ensures that the dynamical equations of motion are also causal. In their current
form the scattering terms are nonlocal in time and thus hard to estimate analytically, and their calculation
is numerically demanding. However, this systematic approach can be used as a quantitative means to
obtain solutions in different regimes and make comparisons with kinetic theory results where a Markovian
approximation is assumed. We postpone these discussions to chapter 8.
138
Chapter 7
Nonequilibrium dynamics of a patterned loaded optical lattice: Beyond the mean field approximation
In this chapter the two-particle irreducible (2PI) closed-time-path (CTP) effective action formalism de-
rived in chapter 6 is used to describe the nonequilibrium dynamics of a Bose Einstein condensate (BEC)
selectively loaded into every third site of a one-dimensional optical lattice.
In chapter 4 we used a mean field approach to describe the dynamics of this system. However, we
show here that even in the case when the kinetic energy is comparable to the interaction energy, interatomic
collisions play a crucial role in determining the quantum dynamics of the system, and therefore a mean field
approach is only accurate for short times. This result is demonstrated by comparison between the mean field
solutions and the exact numerical time evolution of the initial state using the Bose Hubbard Hamiltonian
for systems with small numbers of atoms (N ? 10) and lattice sites (M = 2 or 3). The exact numerical
solution is also used to test the validity of the various methods derived under the 2PI-CTP approximation.
We show that because the second-order 2PI approximations include multi-particle scattering in a
systematic way, they are able to capture damping effects exhibited in the exact solution, which the mean
field and the HFB approximations fail to reproduce. However, our numerical results also show that all of
the approximations fail at late times, when interaction effects become significant.
7.1 Mean field dynamics
In this section we review the basic features of the mean field results obtained in chapter 4, in order to
provide the context for the subsequent discussion of the dynamics in the 2PI-CTP formalism.
7.1.1 Dynamical evolution
By making the mean field ansatz in the Bose-Hubbard Hamiltonian, we replace the field operator ^ai by a
c-number zi(t). The amplitudes zi(t) satisfy the DNLSE (Eq. (4.6)).
In section 4.3.2, we treated a model case in which the initial occupancies of each third site are the
same, and in which the condensate initially has a uniform phase. Thus at ? = 0, (? ? tJ~ ), the amplitudes
zi(?) are given by z3i(0) = p3=M;z3i+1(0) = z3i+2(0) = 0, where M is the total number of lattice sites.
139
For an infinite lattice, or one with periodic boundary conditions, the amplitudes for all initially occupied
sites z3i(?) = z0(?) evolve identically in time, and the amplitudes for the initially unoccupied sites satisfy
z3i+1(?) = z3i+2(?) = for all ?. This allows us to reduce the full set of equations to a set of two coupled
equations for z0(?) and z1(?).
The solutions jz0(?)j and jz1(?)j are oscillatory functions whose amplitudes and common period,
T( ), are determined by the parameter  ? 3NUMJ = 3?. The mean field dynamical behavior can be
qualitatively divided into two regimes:
The tunneling dominated regime ( < 1): In this regime the oscillation period is essentially constant, the
role of interactions is relatively small, and the equations of motion are equivalent to those of a two-state
Rabi problem. This system will undergo Rabi oscillations whereby atoms periodically tunnel from the
initially occupied site into the two neighboring sites. For  = 0 the period of oscillation is 2?3 .
Interaction dominated regime: The effect of interactions on the mean field dynamics is to cause the en-
ergies of the initially occupied sites to shift relative to those of the unoccupied sites. As  increases the
tunneling between sites occurs at a higher frequency, but with reduced amplitude. The population of the
initially occupied sites becomes effectively self-trapped by the purely repulsive pair interaction.
7.1.2 Comparisons with the exact solution
To check the validity of the mean field approximation, we made comparisons with the exact many body
solution for N = 6 atoms and M = 3 wells. We use a modest number of atoms and lattice sites for the
comparisons, due to the fact that the Hilbert space needed for the calculations increases rapidly with the
number of atoms and wells.
Exact solution
For the exact solution we used an initial state given by
j?(0)i = (e?N=2e
pN^ay
o j0i)?j0i?j0i: (7.1)
140
The initial state represents a coherent state with an average of N atoms in the initially populated well and
zero atoms in the others. We chose this state because in the experiment the loading of the atoms was done
slow enough with respect to band excitations but fast with respect to many body excitations that at time
t = 0 most of the atoms in the initially populated wells were condensed.
The fully quantal solution was found by evolving the initial state in time with the Bose-Hubbard
Hamiltonian, so that j?(t)i = e?i~ ^Htj?(0)i. To do the numerical calculations we partitioned the Hilbert
space in subspaces with a fixed number of atoms and propagated independently the projections of the ini-
tial state on the respective subspaces. A subspace with Nn number of atoms and M wells is spanned by
(Nn+M?1)!
Nn!(M?1)! states. This procedure could be done because the Hamiltonian commutes with the number op-
erator Pi ^aiy ^ai, and therefore during the dynamics the different subspaces never get mixed. The number of
subspaces used for the numerical evolution were such that no change in plots of the dynamical observables
was detected by adding another subspace. Generally for N atoms in the initial state, this condition was
achieved by including the subspaces between N ?4pN and N +4pN atoms.
Numerical comparisons
In Fig. 7.1 we plot the average population per well
D
^ayi (t)^ai (t)
E
and the condensate population per well
jh^ai (t)ij2 and compare them with the mean field predictions, i.e. jzi(t)j2, for three different values of  .
The salient features observed in these comparisons are:
1. Weakly interacting regime ( = 0:2):
In this regime the DNLSE gives a good description of the early time dynamics. We observe in Fig. 7.1
that the total population per well predicted by the mean field solution agrees with the exact solution
and also that the condensate population remains large for the time under consideration. We expect
the semiclassical approach to be valid for time scales less than the inverse energy level spacing. In
Ref. [129] the authors show the validity of the semiclassical approach when ? < ?cl ? NM in the
case of two lattice sites. This time scale is in good agreement with the numerical results shown in in
Fig.7.1. After ?cl quantum effects become important.
2. Intermediate regime ( = 2):
141
Quantum fluctuations lead to a non-trivial modulation of the classical oscillations. In this regime the
ratio between interaction and kinetic energy is small enough to allow the atoms to tunnel but not too
small to make interaction effects negligible. Mean field results are accurate only for a short time.
In this regime, the exact solution exhibits damped oscillations of the atomic population. Quantum
scattering effects are crucial, even for rather early times.
To understand the dynamics in the weak and intermediate regimes, we have to focus on the coherent
properties of the system. Even though interactions can be strong, the ground state is indeed super-
fluid. If we look at the initial coherence of the system, determined by
D
^ayi(0)^aj(0)
E
i6=j
, it can be seen
to be zero due to the patterned loading. However, this is no longer the case for t > 0 , and non-zero
correlations are developed in the dynamics. The dynamical restoration of the phase coherence which
tends to distribute atoms uniformly among the lattice sites and to damp the oscillations characterizes
the dynamics in the superfluid regime. In Ref. [129], the authors show, not for a patterned loaded ini-
tial state but for an initial Mott state also with zero initial phase coherence, how the phase coherence
is restored dynamically.
3. Strongly correlated regime ( = 12):
The system exhibits macroscopic quantum self-trapping of the population. Qualitatively, both the
mean field and the exact solutions agree, in the sense that both predict self-trapping of atoms in the
initially populated wells, due to interactions. However, the fast decrease of the condensate population
and its subsequent revivals (as found in the exact solutions) give us an idea of the importance of
correlation effects beyond mean field. For a uniform loaded lattice, the collapse and revivals of
the condensate in this strong interacting regime and the importance of quantum effects have been
experimentally observed[47].
Even though there is minimal initial coherence between adjacent sites due to the patterned loading
procedure we are still preparing the system in a superfluid state in the initially populated well. At
time t = 0 we have a condensate fraction of order one. However, the ground state of the system is not
superfluid. It is expected then that, after some time, the phase is going to randomize and this will lead
to the collapse of the condensate population. After the collapse, the system will remain for a while
142
with zero condensate population. However, it can not remain zero forever because we are dealing
with a closed quantum system, with finite recurrence time. Therefore at some time trev we expect
the condensate to revive again. The collapses and revivals of the condensate population in the strong
interacting regime can be easily estimated by considering the energy spectrum. In this regime the
energy eigenstates of the system are almost number (Fock) states and the energy spectrum is almost
quadratic, En ? n(n?1)U=2. The dynamics of the system is described by the interference of the
different n-particle Fock states that span the coherent state of the initially populated well. At integer
values of trev = (U=h)?1, the phase factors add to an integer value of 2?, leading to a revival
of the initial state. This time scale agrees with the one estimated in Ref.[130] for a more general
situation. That reference, also shows how the collapse time, tcoll, depends on the variance of the
initial atomic distribution; it is given by: tcoll ? trev=(2? ). If the initial state is a coherent state,
the initial distribution is Poissonian and tcoll is given by tcoll ? ~=(pNU). For the parameters used
in the strongly correlated regime,  = 12 and N = 6, we observe that the estimated collapses and
revival times are in agreement with what is shown in Fig. 7.1.
7.2 2PI-CTP approximations
It was shown in the previous section that to describe the dynamics of the patterned loaded optical lattice,
approximations beyond the standard mean field theory are required. In this section we proceed to test
the validity of the different 2PI-CTP approximations derived in chapter 6, explicitly, the time dependent
HFB approximation, the full second-order approximation and the 1=N expansion up to second-order in the
interaction strength. Because in all these approximations we use U=J as an expansion parameter we will
focus our calculations on the intermediate regime, where the ratio U=J is small enough that truncation up
to second-order makes sense but not too small so that interaction effects still have to be taken into account.
We start by describing the initial conditions chosen for the numerical calculations, then we outline
the numerical algorithms used, and finally we discuss the results.
143
7.2.1 Initial conditions and parameters
To model the patterned loading, the initial conditions assumed for the numerical solutions were zi(0) =
N?i0, ?(F)ij (0;0) = 12?ij, ?(?)ij (0;0) = ?i?ij and m(F)ij (0;0)) = m(?)ij (0;0) = 0. Here zi(t) are the
condensate amplitudes and ?(F)ij (ti;tj), ?(?)ij (ti;tj), m(F)ij (ti;tj) and m(?)ij (ti;tj) are the statistical and
spectral normal and anomalous propagators as defined in chapter 6. The initial conditions correspond to an
initial coherent state with N atoms in the initially populated well.
To study the kinetic energy dominated regime we chose for the simulations three different sets of
parameters:. The first set is chosen to be in the very weak interacting regime, M = 3;N = 6;J = 1 and
U=J = 1=30. With this choice we wanted to show the validity of a mean field approach to describe this
regime and the corrections introduced by the higher order approximations. The second set of parameters is
M = 3;N = 8;J = 1 and U=J = 1=3. In this regime the kinetic energy is big enough to allow tunneling
but the effect of the interactions is crucial in the dynamics.
At the mean field level (using the DNLSE) for a given number of wells, the only relevant parameter
for describing the dynamics of the system is the ratio UN=J. This is not the case in the exact solution where
both UN=J and N are important. The larger the initial population N, the larger the initial population in
the coherent matter field, and therefore we expect better agreement of the truncated theories with the exact
solution with larger N. To study the dependence of the dynamics on the total number of atoms, the third
set of parameters in our solutions is chosen to be M = 2;J = 1=2 and NU=J was fixed to 4, but the
number of atoms was changed from 20 to 80. To increase the number of atoms in the exact calculations
we had to reduce the number of wells to two due to the fact that the dimension of the Hilbert space scales
exponentially with N and M.
7.2.2 Numerical algorithm for the approximate solution
The time evolution equations obtained in chapter 6 are nonlinear integro-differential equations. Though the
equations are very complicated, they can be solved on a computer. The important point to note is that all
equations are causal in time, and all quantities at some later time tf can be obtained by integration over the
explicitly known functions for times t ? tf.
144
For the numerical solution we employed a time discretization t = nat; t0 = mat, with n and m
integers and took the advantage that, due to the presence of the lattice, the spatial dimension is discrete
(indices i and j). The discretized equations for the time evolution of the matrices ?(F;?)ijnm; m(F;?)ijnm and zin
advance time stepwise in the n-direction for fixed m. Due to the symmetries of the matrices only half of
the (n;m) matrices have to be computed and the values ??ijnn = ?i; m?ijnn = 0 are fixed for all time due
to the bosonic commutation relations. As initial conditions one specifies ?(F;?)ij00 ; m(F;?)ij00 and zi0:
To ensure that the discretized equations retain the conservation properties present in the continuous
ones one has to be very careful in the evolution of the diagonal terms of ?(F)iinn and take the limit m ! n in
a proper way:
?(F;?)ijn+1n+1 ??(F;?)ijnn = (7.2)
?
?(F;?)ijn+1n ??(F;?)ijnn
?
?
?
??(F;?)jin+1n ???(F;?)jinn
?
;
m(F;?)ijn+1n+1 ?m(F;?)ijnn = (7.3)
?
m(F;?)ijn+1n ?m(F;?)ijnn
?
?
?
m(F;?)jin+1n ?m(F;?)jinn
?
;
with the positive sign for the statistical propagators, (F)0s; and negative sign for the spectral functions,
(?)0s: We used the fourth order Runge-Kutta algorithm to propagate the local part of the equations and
a regular one step Euler method to iterate the non local parts. For the integrals we used the standard
trapezoidal rule. Starting with n = 1, for the time step n + 1 one computes successively all entries with
m = 0::::::;n;n+1 from known functions evaluated at previous times.
The time step at was chosen small enough so that convergence was observed, that is, further decreas-
ing it did not change the results. The greater the parameter UN=J, the smaller is the time step required. The
main numerical limitation of the 2PI approximation is set by the time integrals, which make the numerical
calculations time and memory consuming. However, within a typical numerical precision it was usually
not necessary to keep all past values of the two point functions in the memory. A characteristic time, after
which the influence of the early time in the late time behavior is negligible, is given by the inverse damping
rate. This time is described by the exponential damping of the two-point correlator at time t with the initial
145
time[127]. In our numerics we extended the length of the employed time interval until the results did not
depend on it. In general, it was less than the inverse damping rate. We used for the calculations a single
PII 400 MHz workstation with 260 Mb of memory. For a typical run 1-2 days of computational time were
required.
7.2.3 Results and discussions
In Figs. 7.2 to 7.6 we show our numerical results. We focus our attention on the evolution of the condensate
population per well, jzi(t)j2, the total atomic population per well, jzij2 + ?(F)ii (t;t)? 12, the depletion per
well or atoms out of the condensate, ?(F)ii (t;t) ? 12, and the total condensate population,Pijzi(t)j2. The
total population is also explicitly shown in the figures to emphasize number conservation.
The quasi-momentum distribution of the atoms released from the lattice is important because it
is one of the most easily accessible quantities from an experiment. (see sec.5.5) The quasi-momentum
distribution function nk is defined as
nk(t) = 1M
X
i;j
eik(i?j)
D
ayi(t)aj(t)
E
; (7.4)
where the quasi-momentum k can assume discrete values which are integral multiples of 2?Ma; with M the
total number of lattice sites and a the lattice spacing. The basic features of the plots can be summarized as
follows:
The very weakly interacting regime: In Fig. 7.2 the dynamics of the atomic population per well resem-
bles the Rabi oscillation phenomenon. Notice that even though there are three wells, periodic boundary
conditions enforce equal evolution of the initially empty ones. In this regime damping effects remain very
small for the time depicted in the plots. The numerical simulations show a general agreement between the
different approaches and the exact solution. The effect of including higher order terms in the equations
of motion is to introduce small corrections which improve the agreement with the exact dynamics. This
shows up in the plots of the condensate population and depletion, where the small differences can be bet-
ter appreciated. The second-order 1/N expansion gives an improvement over the HFB and the complete
second-order perturbative expansion almost matches the exact solution perfectly. In the duration depicted in
146
the plots of Fig. 7.2 the total condensate constitutes an important fraction of the total population. Regarding
the quasimomentum distribution, we observe that similarly to the spatial distribution where the initial con-
figuration and periodic boundary conditions reduce the three well system to a double well one, they enforce
equal evolution of the ?2?3 quasimomentum intensities. The k = 0 and ?2?3 intensities oscillate with the
same frequency as the atomic population per well, both are also well described by the approximations in
consideration.
The intermediate regime: We can see the effect of the interactions in the dynamics. They modulate the
oscillations in the population per well and scatter the atoms out of the condensate.
1. In Figs. 7.3 we plot the numerical solution for the parameters M = 3;N = 8;J = 1 and U=J = 1=3.
In contrast to the case of the very weak interacting regime, it is only at early times that any of these
approximations is close to the exact solution. Even though none of them are satisfactory after the
first oscillation the HFB approximation is the only one that fails to capture the exponential decrease
of the condensate population. This is expected, because even though this approximation goes beyond
mean field theory and takes into account the most important scattering effects, it includes the effects
of collisions only indirectly through energy shifts, and breaks down outside the collisionless regime
where multiple-scattering effects are important. In contrast, the exponential decay of the condensate
is present in the second-order approximations. Non local parts of the self-energy included in them
encode scattering effects responsible for damping. It is important to point out that, even though we
observe the collapse of the condensate population, the total population is always conserved: as the
condensate population decreases, the noncondensate population increases.
2. Comparing the two second-order approaches we observe that the full second-order expansion gives
a better description of the dynamics than the 1/N solution only in the regime when the perturbative
solutions are close to the exact dynamics. As soon as the third order terms start to be important the
large 1/N expansion gives a better qualitative description. This behavior is better appreciated in figs.
7.4 to 7.6 as the number of atoms is increased (see discussion below).
We observe as a general issue in this regime that, regardless of the fact that the second-order solutions
147
capture the damping effects, as soon as the condensate population decreases to a small percentage of
the total population, they depart from the exact dynamics: the second-order approaches predict faster
damping rates. The overdamping is more severe in the dynamics of the population per well than in
the condensate dynamics. The failure can be understood under the following lines of reasoning. At
zero temperature condensate atoms represent the most ?classical? form of a matter wave. When they
decay, the role of quantum correlations becomes more important. At this point the higher order terms
neglected in the second-order approximations are the ones that lead the dynamical behavior. Thus,
to have a more accurate description of the dynamics after the coherent matter field has decayed one
needs a better treatment of correlations.
Damping effects are also quite noticeable in the quantum evolution of the quasi-momentum intensi-
ties. Similarly to what happens to the spatial observables, the HFB approximation fails completely
to capture the damping effects present in the evolution of the Fourier intensities whereas the second-
order approaches overestimate them.
3. In Figs. 7.4 to 7.6 we explore the effect of the total number of atoms on the dynamics. In the plots
we show the numerical solutions found for a double well system with fix ratio UN=J = 4 and three
different values of N : N = 20;40 and 80. We present the results obtained for the evolution of
the atomic population per well in Fig. 7.4, the condensate population per well and total condensate
population in Fig. 7.5 and the quasi-momentum intensities in Fig. 7.6. To make the comparisons
easier we scaled the numerical results obtained for the three different values of N by dividing them
by the total number of atoms. In this way for all the cases we start with an atomic population of
magnitude one in the initially populated well.
In the exact dynamics we see that as the number of atoms is increased the damping effects occur at
slower rates. This feature can be noticed in the quantum dynamics of all of the observables depicted
in the plots 7.4 to 7.6. The decrease of the damping rates as the number of atoms is increased is
not surprising because by changing the number of atoms we affect the quantum coherence properties
of the system. As discussed above the collapse time of the condensate population is approximately
given by tcoll ? trev2?pN . The revival time is proportional to U?1 and varies with N for fixed UN=J
148
as trev / NJ , thus tcoll / pN increases with N as observed in the numerical calculations. Besides
damping rates, the qualitative behavior of the exact quantum dynamics is not much affected as the
number of atoms is increased for a fixed UN=J.
The improvement of the 2PI approximations as N is increased, as a result of the increase in the initial
number of coherent atoms is in fact observed in the plots. Even though the problem of underdamping
in the HFB approximation and the overdamping in the second-order approaches are not cured, as the
number of atoms is increased, we do observe a better matching with the full quantal solution. The
1/N expansion shows the fastest convergence. Perhaps this issue can be more easily observed in the
quasi-momentum distribution plots, Fig. 7.6. The better agreement of the 1/N expansion relies on
the fact that even though the number of fields is only two in our calculations the 1/N expansion is an
expansion about a strong quasiclassical field configuration.
7.3 Conclusions
In this work we have used the CTP functional formalism for 2PI Green?s functions to describe the nonequi-
librium dynamics of a condensate loaded in an optical lattice on every third lattice site. We have carried out
the analysis up to second-order in the interaction strength.This approximation is introduced so as to make
the numerical solution manageable, but it is sufficient to account for dissipative effects due to multiparticle
scattering that are crucial even at early times. Our formulation is capable of capturing the salient features
of the system dynamics in the regime under consideration, such as the decay of the condensate population
and the damping of the oscillations of the quasi-momentum and population per well unaccounted for in
the HFB approximation. However, at the point where an important fraction of the condensate population
has been scattered out, the second-order approximations used here predict an overdamped dynamics. To
improve on this a better treatment of higher correlations is required. One might try to include the full next
to leading order large N expansion without the truncation to second-order as done in Ref. [127] but it is
not obvious that this will lead to the required improvement. Alternatively, one may try to adopt a stochastic
approach, but the challenge will be shifted to the derivation of a noise term (which is likely to be both
colored and multiplicative) which contains the effects of these higher correlations and the solution of the
149
stochastic equations.
Even though, as is clear that the second-order 2PI approximations fail to capture the fully corre-
lated dynamics in the system, it has been proved to work at intermediate times when correlations are not
negligible and standard mean field techniques fail poorly. Because of their success in describing moder-
ately correlated regimes, the second-order approximations could become a useful tool for describing other
experimental situations.
150
?=0.2
0 2 4 6 8 10
t
0
2
4
6
#of
atoms
?=2
0
2
4
6
#of
atoms
Conden.
DNLSE
Popul.
?=12
0
2
4
6
#of
atoms
Figure 7.1: Comparisons between the exact and the DNLSE solutions for six atoms and three wells. The
time is given in units of ~=J. Top panel, strongly correlated regime ( = 12); middle panel, intermediate
regime ( = 2); bottom panel, weakly interacting regime ( = 0:2). The solid line is the DNLSE prediction
for the population per well: jz0(t)j2 and jz1;2(t)j2, the triangles are used to represent the exact solution for
the population per well calculated using the Bose Hubbard Hamiltonian: h^ay0^a0i, h^ay1;2^a1;2i. The pentagons
show the condensate population per well calculated from the exact solution: jh^a0ij2 and jh^a1;2ij2. Due to
the symmetry of the initial periodic conditions the curves for the i = 1 and 2 wells are the same in all
depicted curves .
151
0 2 4 6 8 10
t
0
0.2
0.4
0.6
0.8
Depletion/
 well
0 2 4 6 8 10
t
1.9
1.95
2
2.05
momentum
0
2
4
6
Population/
 well
0
2
4
6
condensate/
 well
5
5.5
6
Total
Condensate HFB1 /N2nd
Exact
5
5.25
5.5
5.75
6
Total
Number
Figure 7.2: Comparisons between the exact solution(solid line), the HFB approximation (boxes), the
second-order large N approximation (pentagons) and the full 2PI second-order approximation(crosses)
for the very weak interacting regime. The parameters used were M = 3;N = 6;J = 1 and U=J = 1=30.
The time is given in units of ~=J. In the plots where the population, condensate and depletion per well
are depicted the top curves correspond to the initially populated well solutions and the lower to the initially
empty wells. Notice the different scale used in the depletion plot. In the momentum distribution plot the
upper curve corresponds to the k = ?2?=3 intensities and the lower one to the k = 0 quasi-momentum
intensity.
152
k = 0
0 2 4 6 8 10t0
1
2
3
q-
momentum
k = q/ 3
0 2 4 6 8 10t
2.5
3
3.5
2 nd Population
0
1
2
3
#of
atoms
2 nd Condensate
0 2 4 6 8 10t
2 nd Depletion
0
1
2
3
1 st Population
0
2
4
6
8
#of
atoms
1 st Condensate 1 st Depletion
0
2
4
6
8
Total Number
0
2
4
6
8
#of
atoms
Total Condensate HFB
1 N
2nd
Exact
Figure 7.3: Comparisons for the case M = 3;N = 8;J = 1 and U=J = 1=3. The time is given in units of
~=J. In the plots the abbreviation 1st is used for the initially occupied well and 2nd for the initially empty
wells. In the quasimomentum plots k = 2?=a is the reciprocal lattice vector with a the lattice spacing.
153
P2, N20
0 2 4 6 8 10
t
0
0.2
0.4
0.6
0.8
1
(#
atoms
) /N
P2, N40
0 2 4 6 8 10
t
P2, N80
0 2 4 6 8 10
t
P1, N20
0
0.2
0.4
0.6
0.8
1
(#
atoms
) /N
P1, N40 P1, N80
HFB
1 /N
2nd
Exact
Figure 7.4: Comparison between the evolution of the atomic population per well for M = 2;J = 1=2,
NU=J = 4 and N = 20;40 and 80. Time is in units of ~=J. In the plots P1 stands for the fractional
atomic population in the initially populated wells and P2 for the population in the initially empty wells.
The number of atoms N is explicitly shown in each panel.
154
CT, N20
0 2 4 6 8 10
t
0
0.2
0.4
0.6
0.8
1
(#
atoms
) /N
CT, N40
0 2 4 6 8 10
t
CT, N80
0 2 4 6 8 10
t
C2, N20
0
0.2
0.4
0.6
0.8
1
(#
atoms
) /N
C2, N40 C2, N80
C1, N20
0
0.2
0.4
0.6
0.8
1
(#
atoms
) /N
C1, N40
C1, N80
HFB
1 /N
2nd
Exact
Figure 7.5: Time evolution of the condensate population per well and the total condensate population,for
the same parameters as Fig. 7.4. Time is in units of~=J. In the plots C1 stands for the fractional condensate
population in the initially populated well, C2 for the fractional condensate population in the initially empty
one and CT for the total condensate fraction.
155
ko, N20
0 2 4 6 8 10
t
0.2
0.3
0.4
0.5
0.6
0.7
q-
momentum
ko, N40
0 2 4 6 8 10
t
ko, N80
0 2 4 6 8 10
t
k1, N20
0.3
0.4
0.5
0.6
0.7
0.8
q-
momentum
k1, N40 k1, N80
HFB
1 N
2nd
Exact
Figure 7.6: Dynamical evolution of the quasi-momentum intensities. The parameters used were M =
2;J = 1=2, NU=J = 4 and N = 20;40 and 80. Time is in units of~=J. In the plots ko denotes the k = 0
quasi-momentum component and k1 the k = ?=a one (a the lattice spacing). The plots are scaled to set the
integrated quasi-momentum density to one for all N .
156
Chapter 8
From the 2PI-CTP approximations to kinetic theories and local equilibrium solutions
In chapter 6 we used the CTP functional formalism for 2PI Green?s functions to derive dynamical equations
of motion. We have carried out the analysis up to second-order in the interaction strength. In chapter 7 we
used the formalism to investigate the nonequilibrium dynamics of a condensate loaded on every third site
of an optical lattice. We showed that the formalism allowed us to go beyond the HFB approximation and
to incorporate the nonlinear and non-Markovian aspects of the quantum dynamics as manifested in the
dissipation and fluctuation phenomena.
However, in the present form the equations of motion are complicated nonlocal nonlinear equations
from which we can hardly get any physical information regarding the system behavior except by numerical
solution. Moreover, the numerical solutions are complicated enough that application to real system with
many lattice sites are presently beyond the scope of standard computation capabilities. Nevertheless, when
the perturbation induces disturbances in the system of wave length longer than thermal wave lengths and
frequencies much smaller than characteristic particle energies then the system is in a regime where standard
kinetic theories give a good description of the dynamics. The purpose of this chapter is then to simplify
the complicated 2PI-CTP equations and show that they in fact reproduce, in the slowly varying regime,
standard kinetic theories and equilibrium solutions for weakly interacting gases well known in the literature
since the late 1950?s.
8.1 Rewriting the 2PI-CTP second-order equations
In order to make the comparisons with standard approaches we will start by getting rid of the matrix in-
dices that were useful to derive the equations of motion but which complicate the notation. We define the
quantities
157
z(ti) ? zai (t); (8.1)
G(ti;t0j) ? ?iGijab(t;t0); (8.2)
H(ti;t0j) ? ?izai (t)zjb(t0); (8.3)
G>(ti;t0j) ? ?iG>ijac(t;t0); (8.4)
G<(ti;t0j) ? ?iG<ijac(t;t0) = ?iG>jica(t0;t): (8.5)
Here we used the notation introduced in Eqs. (6.26) and (6.27). With these definitions we now explicitly
separate the single particle, the HFB and the second-order contributions in the equations of motion for the
condensate and the propagators (Eq. (6.22) and Eq. (6.24)).
X
k
Z
dt00?D?1o (ti;t00k)?SHFB(ti;t00k)?z (t00k) =
X
k
Z
dt00S(ti;t00k)z(t00k); (8.6)
X
k
Z
dt00?D?1o (ti;t00k)??HFB(ti;t00k)?G(t00k;t0j) =
X
k
Z
dt00?(ti;t00k)G(t00k;t0j)
??ij?C(t?t0); (8.7)
where D?1o (ti;t0j) is the inverse free particle propagator given by:
D?1o (ti;t0j) ? (i?ij z@t +J(?i+1j +?i?1j)??ijVi)?(t?t0); (8.8)
with  being the Pauli matrices:
 z =
0
BB
@
1 0
0 ?1
1
CC
A;  x =
0
BB
@
0 1
1 0
1
CC
A: (8.9)
In D?1o (ti;t0j) we have allowed the presence of an external potential Vi. The label HFB stands for the HFB
contribution. Using the definition (8.1)-(8.5) in Eqs. (6.43) and (6.45) they can then be rewritten as:
?HFB(ti;t0j) ? (8.10)
iUN ?Tr?H(ti;t0j)+G(ti;t0j)?I +2?H(ti;t0j)+G(ti;t0j)???(t?t0)?ij;;
SHFB(ti;t0j) ? iUN ?Tr?H(ti;t0j)I +G(ti;t0j)?+2G(ti;t0j)??(t?t0)?ij: (8.11)
158
where I is the identity matrix. To evaluate the second-order contribution, we use Eqs. (6.31) and (6.32) but
instead of setting the initial time to zero we choose it to be ?1 and in this way we recover the standard
Kadanoff and Baym integration contours [137]:
X
k
Z 1
?1
dt00?D?1o (ti;t00k)?SHFB(ti;t00k)?H(t00k;t0j)?
Z t
?1
dt00 (ti;t00k)H(t00k;t0j)
= 0; (8.12)
X
k
Z 1
?1
dt00H(ti;t00k)?D?1o (t00k;t0j)?SHFB(t00k;t0j)?+
Z t
?1
dt00H(ti;t00k) (t00k;t0j)
= 0; (8.13)
X
k
Z 1
?1
dt00?D?1o (ti;t00k)??HFB(ti;t00k)?G(?)(t00k;t0j) = (8.14)
X
k
Z t
?1
dt00?(ti;t00k)G(?)(t00k;t0j)?
X
k
Z t0
?1
dt00?(?)(ti;t00k)A(t00k;t0j);
X
k
Z 1
?1
dt00G(?)(ti;t00k)?D?1o (t00k;t0j)??HFB(t00k;t0j)? = (8.15)
X
k
Z t
?1
dt00A(ti;t00k)?(?)(t00k;t0j)?
X
k
Z t0
?1
dt00G(?)(ti;t00k)?(t00k;t0j):
In the above equations, Eq. (8.15) is the hermitian conjugate of Eq. (8.14), Eq. (8.13) is the hermitian
conjugate of Eq. (8.12) and we have introduced the spectral functions
 (ti;t0j) ? (S>(ti;t0j)?S<(ti;t0j)); (8.16)
?(ti;t00j) ? (?>(ti;t00j)??<(ti;t00j)); (8.17)
A(ti;t00j) ? (G>(ti;t00j)?G<(ti;t00j)): (8.18)
Notice that A(ti;t00j) is just the spectral function defined in Eq.(6.34) multiplied by a minus sign, however,
we adopted the notation  , ? and A to be consistent with standard Kadanoff and Baym notation [137].
If we use the full second-order expansion, Eqs.(6.71 and 6.72), it can be shown that the matrices
S(?) and ?(?) are given by
S(?)(ti;t0j) ? ?12
 2U
N
?2h
G(?)(ti;t0j)Tr
?
G(?)(ti;t0j)G(7)(t0j;ti)
?
+ 2G(?)(ti;t0j)G(7)(t0j;ti)G(?)(ti;t0j)
i
; (8.19)
159
?(?)(ti;t0j) ??12
 2U
N
?2
? (8.20)
n
H(ti;t0j)Tr
?
G(?)(ti;t0j)G(7)ik (t0j;ti)
?
+2H(ti;t0j)G(7)(t0j;ti)G(?)(ti;t0j)
+2G(?)(ti;t0j)
?
H(t0j;ti)G(?)(ti;t0j)+G(7)(t0j;ti)H(ti;t0j)+G(7)(t0j;ti)G(?)(ti;t0j)
?
G(?)(ti;t0j)Tr
?
H(t0j;ti)G(?)(ti;t0j)+G(7)(t0j;ti)H(ti;t0j)+G(7)(t0j;ti)G(?)(ti;t0j)
?o
:
It is convenient to decompose the above equations in their matrix components. To do that we intro-
duce the definitions
G>(ti;t0j) = ?i
0
BB
@
e?ij mij
m?ji ?ji
1
CC
A; (8.21)
G<(ti;t0j) = ?i
0
BB
@
?ij mji
m?ij e?ji
1
CC
A: (8.22)
Using Eqs. (8.21) and (8.22) into the self-energy equations we get
?HFB(ti;t0j) = 2UN
0
BB
@
2jzij2 +?ii + e?ii z2i +mii
z?2i +m?ii 2jzij2 +?ii + e?ii
1
CC
A?(t?t0)?ij; (8.23)
SHFB(ti;t0j) = 2UN
0
BB
@
jzij2 +?ii + e?ii mii
m?ii jzij2 +?ii + e?ii
1
CC
A?(t?t0)?ij; (8.24)
S>11(ti;t0j) = ?i8U
2
N2 e?ij(2mijm
?
ji + e?ij?ji); (8.25)
S>12(ti;t0j) = ?i8U
2
N2 mij(mijm
?
ji +2e?ij?ji); (8.26)
S<11(ti;t0j) = ?i8U
2
N2 ?ij(2mjim
?
ij + e?ji?ij); (8.27)
S<12(ti;t0j) = ?i8U
2
N2 mji(mijm
?
ji +2e?ji?ij); (8.28)
160
?>11(ti;t0j) = ?i8U
2
N2 (?jie?
2
ij +2mije?ijmji
? +2e?ijmji?zizj + (8.29)
e?2ijzjzi? +2?jie?ijzizj? +2mijmji?zj?zi +2mije?ijzi?zj?);
?>12(ti;t0j) = ?i8U
2
N2 (2?jimije?ij +2?jie?ijzizj +mij
2mji? + (8.30)
2mijzizjmji? +2mije?ijzjzi? +2?jimijzizj? +mij2zi?zj?);
?<11(ti;t0j) = ?i8U
2
N2 (?ij
2e?ji +2?ijmjimij? +2?ijzizjmij? +?2
ijzjzi
? +
2?ije?jizizj? +2mjizimij?zj? +2?ijmjizi?zj?); (8.31)
?<12(ti;t0j) = ?i8U
2
N2 (2?ijmjie?ji +2?ije?jizizj +mji
2mij? + (8.32)
2mjizizjmij? +2?ijmjizjzi? +2mjie?jizizj? +mji2zi?zj?);
and
S(?)22 (ti;t0j) = S(?)11 (ti;t0j)f?ij 7! e?ijg; (8.33)
S(?)21 (ti;t0j) = S(?)12 (ti;t0j)fmji 7! mij?g; (8.34)
?(?)22 (ti;t0j) = ?(?)11 (ti;t0j)fzi 7! zj;?ij 7! e?ijg; (8.35)
?(?)21 (ti;t0j) = ?(?)12 (ti;t0j)fzi 7! zj?;mji 7! mij?g: (8.36)
The above expressions for the self-energy exactly agree with the ones used in Refs. [59, 60, 62]. In Refs.
[59, 62] the authors used these equations as their starting point before applying the Markovian approxima-
tion.
8.2 Boltzmann equations
From previous sections it can be observed that the equations of motion at second-order are quite involved:
they are nonlinear and nonlocal integro differential equations, not readily solvable in closed form. To
progress further we need to introduce approximations based on physical considerations. One of them is to
recognize two time scales in the system, one related to quantum processes (microscopic) which determines
the degree of quantum-mechanical entanglement of the system and one related to the statistical and kinetic
behavior (macroscopic) determined by the range of the interactions among particles. In a classical point
of view the two different scales in the system can be understood as the time (or length) scale separation
161
between the duration of a collision event (or scattering length) and the inverse collision rate (or the mean
free path). Close to equilibrium (when the external potential induces disturbances much longer than the
lattice spacing and frequencies much smaller than the characteristic particle energies) and in the weakly
interacting regime, we expect that a reasonable assumption would be to consider the kinetic scale larger
than the quantum one, which in a classical picture corresponds to assume that be the time between collisions
(or mean free path) is long compared to the reaction time (or scattering length).
Using this approximation it is possible to recast the quantum dynamics into the much simpler forms
of kinetic theory. In contrast to normal systems were there is not a condensate, in condensed systems
making the scale separation requires first to do a gauge transformation which makes it easier to identify
(and coarse-grain away) the fast variations induced by the rapid change of the condensate phase. Following
Ref. [139] we introduce the gauge transformation
zi(t) = ei i(t)
p
no(ti); (8.37)
G(?)(ti;t0j) = ei i(t) z ~G(?)(ti;t0j)e?i j(t0) z; (8.38)
where pno(ti) and  i(t) are real. In equilibrium, ? i ? ( i+1=2 ? i?1=2)=a is related to the superfluid
velocity and the time derivative of the phase to the chemical potential. Extending these identifications to
the nonequilibrium system we define the chemical potential and superfluid velocity as
?i = ?~@t i ? ~
2
4Ja2v
s
i
2 ?Vi; (8.39)
~vsi = 2J? ia2: (8.40)
After the gauge transformation the variables no, ~G , vs and ? are expected to be slowly varying functions
of
R = (i+j)=2; T = (t+t0)=2; (8.41)
and peaked about the zeros of
r = (i?j); ? = (t?t0); (8.42)
necessary conditions to derive Boltzmann-type kinetic equations.
The equations of motion are invariant under the phase transformation if we replace D?1o by ~D?1o :
162
~D?1o (ti;t0j) = (~?ij(i z@t ?@t i)??ijVi+ (8.43)
J(ei z? i+1=2?i+1j +e?i z? i?1=2?i?1j)??(t?t0):
Since hereafter we will use the gauge-transformed functions exclusively, in the following the tildes
will be dropped to simplify the notation. To obtain the kinetic equations we Fourier transform the functions
G and no with respect to r and ? . The Fourier transform reads
G(?)(ti;t0j) = G(?)
 ?
T + ?2
?
R+r=2
;
?
T ? ?2
?
R?r=2
?
? ?i 12?M
X
q
Z
d!e(iqr?i!?)G(?)(Rq;T;!); (8.44)
H(ti;t0j) = H
 ?
T + ?2
?
R+r=2
;
?
T ? ?2
?
R?r=2
?
? ?i 12?M
X
q
Z
d!e(iqr?i!?)H(Rq;T;!); (8.45)
Neglecting second-order variations of no(Rq;T;!) we can approximate it as
H(Rq;T;!) = 2?M (I + x)no(R;T)?(!)?q0: (8.46)
In Eq.(8.46), the quantity no(R;T) is just related to the condensate density of atoms at the space time point
(Ra;T). In Eq.(8.44), the two-point function G(<)11 (Rq;T;!) corresponds to the well known Wigner distri-
bution function [140]. It can be interpreted as the density of noncondensed particles with quasimomentum
q and energy ~! at the lattice site R and time T. On the other hand, G(>)11 (Rq;T;!) is essentially the
density of states available to a particle that is added to the system at (Ra;T) with quasimomentum q and
energy ~!. As opposed to a normal system, the presence of the condensate gives non zero values to the
off diagonal terms of the functions G(?)12 (Rq;T;!). We refer to them as the anomalous contributions to the
respective two point functions.
The generalized Boltzmann equations can be obtained as the Fourier transform of the equations
of motion for the case in which the variations in R and T are very slow. In particular when the inverse
propagator D?1o and the self energies vary very little as Ra is changed by a characteristic excitation wave-
length or T is changed by an inverse excitation energy. Following Ref.[139] and assuming a (q;!;R;T)
dependence of the variables, which is not explicitly written for simplicity, we get the following equations:
163
 
D?1o ?<S + i2 
?
H = ?i2 ?D?1o ;H?+ i2 [<S;H]+ 14 [ ;H]; (8.47)
H
 
D?1o ?<S ? i2 
?
= ?i2 ?H;D?1o ?+ i2 [H;<S]? 14 [H; ]; (8.48)
 
D?1o ?<?+ i2?
?
G(?) ? ?(?)
 
<G+ i2A
?
= ?i2
h
D?1o ;G(?)
i
+ i2
h
<?;G(?)
i
+
i
2
h
?(?);<G
i
+ 14
h
?;G(?)
i
? 14
h
?(?);A
i
; (8.49)
G(?)
 
D?1o ?<?? i2?
?
?
 
<G? i2A
?
?(?) = ?i2
h
G(?);D?1o
i
+ i2
h
G(?);<?
i
+
i
2
h
<G;?(?)
i
? 14
h
G(?);?
i
+ 14
h
A;?(?)
i
; (8.50)
with
D?1o (Rq;T;!) ? ( z (~! ?vs2J sin(qa))+(2J cos(qa)+?R(T))I): (8.51)
Because all the quantities are slowly varying functions of R and T, in Eqs (8.47-8.50) we approximated
the discretized equation to continuous differential equations. The brackets denote the generalized Poisson
brackets defined as:
[A;B] = @A@! @B@T ? @A@T @B@! +@RA@qB ?@qA@RB: (8.52)
In the equations we have also introduced the following functions:
<S(Rq;T;!) = SHF(Rq;T;!)+<SB(Rq;T;!); (8.53)
<?(Rq;T;!) = ?HF(Rq;T;!)+<?B(Rq;T;!); (8.54)
<SB(Rq;T;!) = P
Z d!0
2?
 (Rq;T;!)
! ?!0 ; (8.55)
<?B(Rq;T;!) = P
Z d!0
2?
?(qR;!0)
! ?!0 ; (8.56)
<G = P
Z d!0
2?
A(Rq;T;!)
! ?!0 : (8.57)
with P denoting the Cauchy principal value and  (Rq;T;!), ?(Rq;T;!), SHF(Rq;T;!), ?HF(Rq;T;!)
and A(Rq;T;!) understood as Fourier transforms of the functions  (ti;t0j), ?(ti;t0j), SHF(ti;t0j), ?HF(ti;t0j)
and A(ti;t0j) respectively.
164
If we define the statistical variables as:
F = G
> +G<
2 ; (8.58)
? = ?
> +?<
2 : (8.59)
Eq. (8.49)and Eq. (8.50), can be rewritten in term of statistical and spectral functions as:
 
D?1o ?<?+ i2?
?
F ? ?
 
<G+ i2A
?
= (8.60)
?i2
??
D?1o ?<?+ i2?;F
?
?
?
?;<G+ i2A
? 
;
F
 
D?1o ?<?? i2?
?
?
 
<G? i2A
?
? = (8.61)
?i2
??
F;D?1o ?<?? i2?
?
?
?
<G? i2A;?
? 
;
?D?1
o ?<?
?A??<G = ?i
2
'?D?1
o ?<?;A
??[?;<G]?; (8.62)
A?D?1o ?<???<G? = ?i2 '?A;D?1o ?<???[<G;?]?: (8.63)
Eqs. (8.47), (8.48) and (8.60)- (8.63) are our passage to the Boltzmann equations. They describe the state
of the gas at a given time. In contrast to the HFB equations, they include collisional integrals to describe
binary interactions.
To progress further we can introduce even more simplifications based on physical assumptions. The
ordinary Boltzmann equation emerges from the approximation in which the self energies that appear on
the left side of Eqs.(8.47), (8.48) and (8.60)- (8.63) are handled differently from those which appear on
the right. These two appearances of the self-energy play a different physical role in the description of the
dynamics [137]. The self energies on the right hand side describe the dynamical effects of collisions, i.e.,
how the collisions transfer particles from one energy-momenta configuration to another. On the other hand,
the self energies on the left describe the quantum kinetic effects due to interactions, i.e. how interaction
effects change the energy momentum dispersion relations from that of free particles to a more complicated
spectrum. Because these two effects are physically so different, we can treat the left and the right hand
sides in different ways.
165
In the derivation of the ordinary Boltzmann equations, one completely neglects all the kinetic effects
in the second-order self energies (the dependence on T and R in the second-order self energy terms on the
right hand side) and retain dynamical effects (T and R dependence on the left hand side). In this way, we
get the familiar Boltzmann equations which describe the particles as free particles in between collisions.
This is a reasonable assumption in dilute weakly interacting gases in which the duration of a collision is
very short compared to the essentially interaction-free dynamics between isolated collisions. Neglecting
kinetic effects in the second-order self energies, Eqs.(8.47), (8.48) and (8.60)-(8.63) can be approximated
to
 
D?1o ?<S + i2 
?
H = ?i2 ?D?1o ?SHF;H?; (8.64)
H
 
D?1o ?<S ? i2 
?
= ?i2 ?H;D?1o ?SHF?; (8.65)
 
D?1o ?<?+ i2?
?
F ??
 
<G+ i2A
?
= ?i2 ?D?1o ??HF;F?; (8.66)
?D?1
o ?<?
?A??<G = ?i
2
?D?1
o ??
HF;A?; (8.67)
F
 
D?1o ?<?? i2?
?
?
 
<G? i2A
?
? = ?i2 ?F;D?1o ??HF?; (8.68)
A?D?1o ?<???<G? = ?i2 ?A;D?1o ??HF?: (8.69)
If we take the trace of the sum and the difference of each one of the above equations with its hermitian
conjugate, they can be simplified to :
Tr'?D?1o ?<S?H? = 0; (8.70)
Tr'?D?1o ?<??F ??<G? = 0; (8.71)
Tr'?D?1o ?<??A??<G? = 0; (8.72)
Tr?D?1o ?SHF;H? = ?Tr( H); (8.73)
Tr?D?1o ??HF;F? = ?Tr(?F ??A): (8.74)
Tr?D?1o ??HF;A? = 0 (8.75)
Moreover, if we define the operator krM = M12 + M?21 and apply it again to the sum and the difference
166
of each one of the equations (8.60) to (8.63) with its transpose we also get:
Re?kr'?D?1o ?<S?H?? = 12Im?kr?D?1o ?SHF;H?+k( H)?; (8.76)
Re?kr'?D?1o ?<??F??<G?? = 12Im?kr?D?1o ??HF;F?+kr(?F??A)?; (8.77)
Re?kr'?D?1o ?<??A??<G?? = 12Im?kr?D?1o ??HF;A??; (8.78)
Im?kr'?D?1o ?<S?H?? = ?12Re?kr?D?1o ?SHF;H?+k( H)?; (8.79)
Im?kr'?D?1o ?<??F??<G?? = ?12Re?kr?D?1o ??HF;F?+kr(?F??A)?; (8.80)
Im?kr'?D?1o ?<??A??<G?? = ?12Re?kr?D?1o ??HF;A??; (8.81)
with Re and Im denoting the real and imaginary parts. To close this set of equations we need an equation
of motion for the superfluid velocity which can be found from the definitions Eq.(8.39) and Eq.(8.40) to be:
~
2Ja2
@vs
@T = ?
@
@R
 
(?+V)+ ~4Ja2vs2
?
: (8.82)
Eqs.(8.70-8.81) together with Eq.(8.82) form a closed set of equations that describe the state of the
gas at a given time. Equations (8.70-8.72) and (8.76-8.78) are usually called gap equations. They describe
the quantum properties of the gas which is evolving according to the Boltmaznn equations (8.73-8.75) and
(8.79-8.81). Under the derived formalism the Boltzmann and gap equations form a coupled set of equations
which replaces the original dynamics. The equations have to be solved self consistently for any analysis.
8.3 Equilibrium properties for a homogeneous system
In this section we will show how the nonequilibrium Boltzmann equations lead, in a special case, to the lin-
ear equilibrium solutions discussed in chapter 5 upgraded with second-order corrections in U not included
in the quadratic approximations. There are two situations in which we expect an equilibrium solution to
come from the Boltzmann equations. Firstly, when the system has never been disturbed it remains in its
equilibrium state. Secondly, when the system has had sufficient time to relax after an applied perturbation.
At equilibrium, in the absence of any external potential, the functions G? and H are completely
independent of R and T. In this case the generalized Poisson brakets terms are zero and Eqs.(8.72, 8.75)
167
and (8.78, 8.75) imply that:
A?D?1o ?<???(<G)? = 0: (8.83)
Because <G(q;!) is determined by A(q;!) as indicated in Eq. (8.57), Eq. (8.83) is satisfied when A(q;!)
is given by
A(q;!) = i
(?
D?1o ?<S + i2?
??1
?
?
D?1o ?<S ? i2?
??1)
; (8.84)
and the function <G(q;!) given by
<G(q;!) = P
Z d!0
2?
A(q;!0)
! ?!0
= 12
(?
D?1o ?<S + i2?
??1
+
?
D?1o ?<S ? i2?
??1)
: (8.85)
From Eqs.(8.73), (8.79),(8.74) and (8.80)) we also get that at equilibrium
 = 0; (8.86)
?F ??A = 0: (8.87)
Eqs. (8.86) and (8.87) are just the mathematical statement of detailed balance. They just represent the
physical condition that at equilibrium the net rate of change of the density of particles with momentum q
and energy ! is zero. Since it is always possible to write [137]
F(q;!) =
 
nq(!)+ 12
?
A(q;!); (8.88)
then Eq.(8.87) can only be satisfied if
?(q;!) =
 
nq(!)+ 12
?
?(q;!); (8.89)
is satisfied. Detailed study of the structure of the self energy indicates that nq(!) is related to the Bose-
Einstein distribution, nq(!) = 1efl~!?1 with fl interpreted as the local inverse temperature in energy units
[137, 139].
168
Since H contains delta functions in momentum and energy at equilibrium, we get from Eq. (8.70):
? = ?2J +<S11(0;0)+<S12(0;0): (8.90)
8.3.1 Quasiparticle formalism
In the non interacting case the diagonal terms of A(q;!) are just delta functions with peaks at values of~!
that matches the possible energy difference which results from adding a single particle with quasimomen-
tum q to the system. In the many body system the energy spectrum is sufficiently complex so that the diag-
onal elements of A(q;!) are not delta functions but instead continuous functions of !. However, there are
always sharp peaks in A. These sharp peaks represent the coherent and long lived excitations, which behave
like weakly interacting particles. These excitations are called quasiparticles. From Eq. (8.84) it is possible
to see that the quasiparticle decay rate is determined by ?. The quasiparticle approximation is obtained by
considering ? very small for small values of !. This assumption implies that D?1 ? D?1o ?<?? i2? is
essentially real with only an infinitesimal imaginary part. The zeros of D?1 about which A is very sharply
peaked are identified with the quasiparticle energies~!q.
Using the assumption of a very small ?, and the identity lim?!0 1!?!0+i? = P 1!?!0 ?i??(! ?!0), it is
possible to write the matrix components of D?1 as:
D?1(q;!) =~!
0
BB
@
1 0
0 ?1
1
CC
A?
0
BB
@
Lqq(q;!) Mq?q(q;!)
M?q?q(q;!) L?qq(?q;?!)
1
CC
A; (8.91)
with
Lqq(!) = ?2J cosqa??+?HFB11 (q;!)+
Z d!0
2?
?11(q;!0)
! ?!0 +i?; (8.92)
Mq?q(!) = ?HFB12 (q;!)+
Z d!0
2?
?12(q;!0)
! ?!0 +i?: (8.93)
The quasiparticle amplitudes uq and vq are the solutions to the eigenvalue problem
0
BB
@
Lqq(q;!q) Mq?q(q;!q)
M?q?q(q;!q) L?qq(?q;?!q)
1
CC
A
0
BB
@
uq
vq
1
CC
A =~!q
0
BB
@
1 0
0 ?1
1
CC
A
0
BB
@
uq
vq
1
CC
A; (8.94)
169
and satisfy the normalization condition juqj2?jvqj2 = 1. In the absence of vortices it is always possible to
find an ensemble in which the amplitudes (uq;vq) are purely real and uq = u?q; vq = v?q. In terms of
the quasiparticle amplitudes, the matrix elements of the spectral function A, Eq.(8.84), are given by:
A11(q;!) = ?2Im
"
u2q
! ?!q +i0+ ?
v2q
! ?!q +i0?
#
= 2??u2q?(! ?!q)?v2q?(! +!q)?; (8.95)
A12(q;!) = 2Im
? u
qvq
! ?!q +i0+ ?
vquq
! ?!q +i0?
?
= ?2?uqvq [?(! ?!q)??(! +!q)]; (8.96)
A22(q;!) = ?A11(?q;?!); (8.97)
A21(q;!) = A?12(q;!): (8.98)
Finally, using the definitions of F and A, we can express the matrix components: ?q(!), e?q(!) and
mq(!) defined as the Fourier transform of ?ij, e?ij and mij respectively (see Eqs. (8.21) and (8.22)) in
terms of quasiparticle amplitudes:
?q(!) = 2??u2qnq(!)?(! ?!q)+v2q(1+nq(!))?(! +!q)?; (8.99)
e?q(!) = 2??u2q(1+nq)(!)?(! ?!q)+v2qnq(!)?(! +!q)?; (8.100)
mq(!) = 2?uqvq [nq(!)?(! ?!q)?(1+nq(!))?(! +!q)]: (8.101)
8.3.2 HFB approximation
Under the HFB approximation the matrix <? and <S are just given by ?HFB and SHFB. In terms of
the quasiparticle amplitudes and setting N = 2, they can be written as:
170
?HFB = U
0
BB
@
2(no +en) no + em
no + em 2(no +en)
1
CC
A; (8.102)
SHFB = U
0
BB
@
no +2en em
em no +2en
1
CC
A: (8.103)
with
en = = 1M
X
q
(1+nq(!q))v2q +u2qnq; (8.104)
em = 1M
X
q
uqvq (2nq(!q)+1): (8.105)
In the HFB approximation, Eq.(8.94) and Eq. ( 8.90) then yield:
0
BB
@
?2J cos(qa)??+2U(no +en) U (no + em)
U (no + em) ?2J cos(qa)??+2U(no +en)
1
CC
A
0
BB
@
uq
vq
1
CC
A =~!q
0
BB
@
uq
?vq
1
CC
A;
(8.106)
? = ?2J +Uno +2Uen+Uem: (8.107)
As a final step, to fix the total density to n, the constraint
n = no +en; (8.108)
has to be satisfied.
For a given density and temperature Eqs. (8.106)- (8.108) form a closed set of equations. At zero
temperature, they reduce to the HFB equations derived in chapter 5 using the quadratic approximation.
8.3.3 Second-order and Beliaev approximations
When second-order terms are taking into account the matrixes Lqq and Mq?q become energy dependent.
For simplicity we restrict the calculations to the zero temperature case when nq = 0. In terms of the
quasiparticle amplitudes the contributions to the self-energy at second-order are given by
171
Mq?q(q;!) = Uno +Uem+ 2U
2
M~no
X
k
 2A
kBq?k +2CkAp +2CkBq?k +3CkCq?k
! ?!k ?!q?k +i?
?2BkAq?k +2CkAp +2CkBq?k +3CkCq?k! +!
k +!q?k ?i?
?
(8.109)
+ 2U
2
M2~
X
k;p
 2A
kBpCq?k?p +CkCpCq?k?p
! ?!k ?!p ?!q?k?p +i? ?
2BkApCq?k?p +CkCpCq?k?p
! +!k +!p +!q?k?p ?i?
?
;
Lqq(q;!) = ?2J cosqa??+2Uno +2Uen+
2U2no
~M
X
k
 A
kAq?k +2AkBq?k +4CkAq?k +2CkCq?k
! ?!k ?!q?k +i?
? BkBq?k +2(BkAq?k)+4CkBq?k +2CkCq?k! +!
k +!q?k ?i?
?
(8.110)
+ 2U
2
~M2
X
k;p
 A
kApBq?k?p +2AkCpCq?k?p
! ?!k ?!p ?!q?k?p +i?
?
?
 B
kBpAq?k?p +2BkCpCq?k?p
! +!k +!p +!q?k?p ?i?
?
;
? = ?2J +Uno +2Uen+Uem (8.111)
? 2U
2
~M2
X
k;p
 2A
kBpCk+p +2BkApCk+p +2CkCpCk+p
!k +!p +!k+p
?
? 2U
2
~M2
X
k;p
 2A
kCpCk+p +AkApBk+p +2BkCpCk+p +BkBpAk+p
!k +!p +!k+p
?
;
where the quantities A, B and C are defined as
Ak = u2k; Bk = v2k; Ck = ?ukvk: (8.112)
The inclusion of second-order terms modifies the structure of the HFB equations. The matrix that
we need to diagonalize to find the quasiparticle energies depends now on the quasiparticle mode in consid-
eration. This means that a separate nonlinear problem must be solved for every quasiparticle state, whereas
the solution of the HFB equations yields the whole quasiparticle spectrum. The matrix which is to be diag-
onalized also become intrinsically non-local and to solve for a quasiparticle state with quasimomentum q
we have to sum over all different quasimomenta. Finally, the diagonal elements are no longer equal as was
always the case in all the quadratic approximations considered in chapter 5.
If we omit the second-order terms containing no condensate amplitudes, the equations that we get
are the tight-binding version of the ones originally derived by Beliaev [63]:
172
Mq?q(q;!) = Uno +Uem + 2U
2
~M no
X
k
 2A
kBq?k +2CkAp +2CkBq?k +3CkCq?k
! ?!k ?!q?k +i?
? 2BkAq?k +2CkAp +2CkBq?k +3CkCq?k! +!
k +!q?k ?i?
?
; (8.113)
Lqq(q;!) = 4J sin2(qa=2)+Uno ?Uem+ (8.114)
2U2no
~M
X
k
 A
kAq?k +2AkBq?k +4CkAq?k +2CkCq?k
! ?!k ?!q?k +i?
? BkBq?k +2(BkAq?k)+4CkBq?k +2CkCq?k! +!
k +!q?k ?i?
?
; (8.115)
? = ?2J +Uno +2Uen+Uem: (8.116)
As discussed in chapter 5, the HFB approximation has the problem that it is not gapless. It was
shown by Beliaev that when second-order Beliaev contributions are included the gap problem disappears.
The reason is that as em introduces the many-body scattering matrix in the off-diagonal terms, second-
order corrections introduce the many-body scattering matrix in the diagonal terms. This can be seen by
considering the second-order Beliaev corrections in the particle approximation, uq ! 1;vq ! 0. In this
case the second-order terms in Eq.(8.115) become
L(2)qq (q;!) = ?2U
2no
~M
X
k
1
! ?!k ?!q?k ?i?: (8.117)
Comparison with Eq. (5.121) shows that L(2)qq (q;!) is the second order contribution to the many body
scattering matrix.
When both HFB and second-order corrections are included the quasiparticle energies not only are
shifted with respect to the BdG quasiparticle energies but they also become complex. The imaginary part
that the quasiparticle energies acquire comes from the poles of the second-order terms and it is associated
with a damping rate. The physical meaning is that when the energy denominator in the second-order terms
vanishes a real decay process is energetically allowed. The damping mechanism in which a quasiparticle
decay into two of lower energy is known as Beliaev damping and was calculated by Beliaev in the case of
a uniform Bose superfluid [63].
173
8.4 Conclusions
In summary, we showed in this chapter that the complicated non local non Markovian second-order solu-
tions derived under the 2PI-CTP formalism, actually reduce to the standard kinetic theory solutions when
the scale separation assumption is made. We also showed that at equilibrium the full second-order 2PI
equations reproduce the second-order corrections to the self energy well known since Beliaev.
174
Chapter 9
Characterizing the Mott Insulator Phase
Besides the superfluid regime, the other regime where analytic solutions become relatively simple is the
Mott insulator phase. Deep in the Mott insulator regime the kinetic energy term in the Hamiltonian, which
delocalizes the atoms, can be treated as a perturbation. In this chapter we use perturbation theory to describe
the basic properties of the Mott phase. We study both a translationally invariant system when no other
external potential is present and the case when there is an external harmonic confinement, as is the case
in real experimental situations. Specifically, the trapped system is studied assuming a parameter regime
where multi-occupancy in inhibited which is the relevant regime for the lattice based quantum computation
proposals.
9.1 Commensurate translationally invariant case
9.1.1 Perturbation theory
In this section we assume a commensurately filled one dimensional lattice with integer filling factor N=M =
g, periodic boundary conditions and a parameter regime where U=J  (U=J)c with (U=J)c the critical
point, so that we are deep in the insulating phase. In this parameter regime, it is fair to consider the kinetic
term of Bose-Hubbard Hamiltonian as a perturbation:
^H = ^H(0) +? ^H; (9.1)
^H(0) = 1
2U
X
n
^ayn^ayn^an^an; (9.2)
? ^H = ?J
X
n
(^ayn ^an+1 + ^ayn+1 ^an ): (9.3)
The unperturbed Hamiltonian includes only the on-site interaction term, which is diagonal in a number
Fock state basis, and to zeroth order the ground state j'0i is the Fock state with g atoms in every lattice
site. We refer to this state as the target state jTi. The lowest lying excitations are described by the Fock
states, j?nmi, that have g + 1 particles at site n, g ?1 particles at site m, and exactly g particles in every
other site. We call these states one particle-hole (1-ph) excitations . There are M(M ?1) 1-ph excitations
175
and because of the translational symmetry of the system they are degenerate at zeroth order. All of them
have an energy U above the ground state. This energy gap is one of the most important characteristics of
the Mott insulator phase.
flfl
fl'(0)0
E
? jTi?jg;g;::::;g;gi; (9.4)
flfl
fl'(0)i
E
? j?nmin6=m i = 1;:::;M(M ?1) (9.5)
=
flfl
flfl
flflg;:::;g +1|{z}
n
;:::;g ?1|{z}
m
;::;g
+
:
The unperturbed energies are
E(0)0 = U2 Mg(g ?1); (9.6)
E(0)i = U +E(0)0 i = 1;:::;M(M ?1): (9.7)
At first order the kinetic energy term mixes the target state with 1-ph excitations with the particle
and the hole at adjacent sites. The first order ground state wave-function is
flfl
fl'(1)0
E
=
 
jTi+ JU
p
2Mg(g +1)jSi
?
; (9.8)
with jSi the normalized translationally invariant state with a particle and hole at adjacent sites:
jSi?
MX
n=1
j?nn+1i+j?nn?1ip
2M : (9.9)
Notice the factor M that appears in the first order correction. It gives a restriction on the validity of a
perturbative treatment as the number of lattice sites increases. In order for perturbation theory to be valid,
the parameter JgU pM has to be small. The first order correction to the ground state energy vanishes. The
second order correction is given by
E(2)0 = ?2J
2M
U g(g +1): (9.10)
Because of the degeneracy of the unperturbed states, in order to find first order corrections to the
176
M(M ? 1) low lying excited states we must diagonalize the kinetic energy Hamiltonian within the 1-ph
subspace. If we span the eigenstates as a linear combinations of 1-ph excitations,
flfl
fl'(1)i
E
=
MX
n;m=1;m6=n
Cinmj?nmi; (9.11)
the necessary and sufficient conditions that the coefficients Cinm have to fulfill in order to diagonalize the
kinetic energy operator are given by the following equations
?(g +1)(Cin+1m +Cin?1m)?g(Cinm+1 +Cinm?1) = eEiCinm; (9.12)
with E(1)i = U ?J eEi. Besides Eq. (9.12), the amplitudes have to fulfill two other equations:
Cinn = 0; (9.13)
Cin+Mm = Cinm+M = Cinm: (9.14)
Eq. (9.12) is analogous to the tight-binding Schr?odinger equation of a two dimensional square lattice in the
x?y plane, with the x direction associated with the position of the extra particle and the y direction with
the position of the hole. The different weights g + 1 and g can be understood in the 2D-lattice model as
different effective masses in the two directions. Eq. (9.14) imposes periodic boundary conditions, whereas
the constraint Cinn = 0 takes into account the requirement that the extra particle and the hole at the same
site annihilate each other. It can be thought of as a hard wall in the x = y axis. The solutions are not
straightforward due to the fact that the effective mass difference breaks the lattice symmetry around the
x = y axis and makes the hard wall constraint hard to fulfill.
We now describe a procedure for solving the eigenvalue equations and evaluating limiting cases
where it is possible to find analytic solutions. Without any constraint, a general solution of Eq. (9.12) has
the form
Cnm / einzeimt; (9.15)
with
eEi = ?2((g +1)cos(z)+gcos(t)): (9.16)
177
To satisfy the constraint Cnn = 0 we use the linear character of the problem and look for a linear combi-
nation of solutions which have the same eigenvalue eEi but which also fulfill the hard wall constraint. We
look then for a general solution of the form:
Cnm / sin(r(n?m))eimleins; (9.17)
and determine the free parameters, l, r, s by forcing the solution to satisfy Eq. (9.12). This procedure leads
to the following equation:
(g +1)cos(r +s)+gcos(l?r) = (g +1)cos(s?r)+gcos(l +r): (9.18)
Eq. (9.18) is satisfied if we choose s to be given by:
s = Arcsin
 g
g+1 sinl
?
: (9.19)
The tricky part is trying to satisfy also the periodic boundary conditions, Eq.(9.14), because in general
solutions of the form Eq. (9.17) are not necessarily compatible with the periodic boundary conditions in
the two variables n and m. However, there are some limiting cases in which both conditions are satisfied.
Such cases are discussed in the remainder of the section.
? Case l = 0
If we choose l = 0 and use Eq. (9.19) and Eq. (9.17), we can find M ? 1 solutions of Eq. (9.12)
which satisfy all the boundary conditions and are linearly independent. These are
Crnm =
8
>><
>>:
p2
M sin
??r
Mjn?mj
?; r odd
p2
M sin
?2?r
M (n?m)
?; r even (9.20)
E(1)r = U ?2J(2g +1)cos
??r
M
?
; r = 1;::M ?1 (9.21)
Inside the 1-ph subspace all the translationally invariant states are spanned by the states whose ampli-
tudes Crnm are the r 2 odd solutions of Eq. (9.20). Because in the absence of an external confinement
the many body Hamiltonian is translationally invariant, the ground state must also be and it is only
coupled through the Hamiltonian to the subspace spanned by these translationally invariant states.
178
The dispersion relation, Eq. (9.21), agrees to first order in J to the mean-field solution found in
[77]. We want to point out that this dispersion relation only describes the spectrum of M ?1 of the
M(M ?1) low tying excitations.
When l 6= 0 the derivation of analytic solutions is more elaborate. However, in the limiting cases of
high filling factor, g  1, it is still easy to find an analytic solution.
? Case g  1
In this limiting situation s t l, (see Eq.(9.19)) and the periodic boundary conditions are satisfied if
s, l and r are integer multiples of ?M . The eigenvalues and an orthonormal set of modes in the high
filling factor regime can be chosen to be:
E(1)rR = U ?2J(2g +1)cos
??r
M
?
cos
 ?R
M
?
; (9.22)
Cr;R6=0nm =
8>
><
>>:
2
M sin(
?r
Mjn?mj)sin(
?R0
M (n+m)+firR);
2
M sin(
?r
M (n?m))sin(
?R00
M (n+m)+flrR);
Cr;0nm =
8
>><
>>:
p2
M sin
??r
Mjn?mj
?; r odd
p2
M sin
??r
M (n?m)
?; r even (9.23)
with r = 1;:::M ? 1 and R = 0;:::M ? 1. The notation R0 restricts the values of R to the
ones where R + r is an odd number and R00 to the values where R + r is even. The constants
firR = ?(r ? R + 1)=4 and flrR = ?(r + R ? 1 + M)=4 guarantee the orthogonality of the
eigenmodes. In Fig. 9.1 we show a contour plot of the two dimensional 1-ph band.
To check the range of validity of our analytical solutions, in Fig.9.2 we plot comparisons between
the first order energy shifts calculated by the exact diagonalization of the Bose-Hubbard Hamiltonian but
restricting the Hilbert space to the 1-ph subspace and Eq. (9.22). We label the eigenvalues in ascending
order. The parameters used for the comparisons were M = 11 and filling factors g = 1 and 4. We notice
that even though the analytic expression was calculated under the high filling factor assumption, for values
of g = 4, the agreement between the two solutions is very good. For the case g = 1 the two spectra do not
agree perfectly, however, we can say that the analytic solution is fair even in this low density regime.
179
0 ?FractionBarExtFractionBarExtFractionBarExtFractionBarExtFractionBarExt4 ?FractionBarExtFractionBarExtFractionBarExtFractionBarExtFractionBarExt2 3?FractionBarExtFractionBarExtFractionBarExtFractionBarExtFractionBarExtFractionBarExtFractionBarExtFractionBarExtFractionBarExtFractionBarExt
4
?
kx
0
?FractionBarExtFractionBarExtFractionBarExtFractionBarExtFractionBarExt
4
?FractionBarExtFractionBarExtFractionBarExtFractionBarExtFractionBarExt
2
3?FractionBarExtFractionBarExtFractionBarExtFractionBarExtFractionBarExtFractionBarExtFractionBarExtFractionBarExtFractionBarExtFractionBarExt
4
?
ky
P.h. Band
Figure 9.1: Contour plot of the two dimensional band of the 1-ph excitations to fist order in perturbation
theory. In the plot the brighter the color the higher the energy. The labels are ky = 2?=MR and kx =
2?=MR.
In Fig. 9.2 we checked for the validity of the analytic solution Eq. (9.22) when the Hilbert space is
restricted to the 1-ph subspace. However, to restrict the low lying excitations to the 1-ph excitations is only
good as long as first order perturbation theory is valid. This implies that the parameter JgU pM is small.
In Fig. 9.3, we explore the range of validity of perturbation theory by plotting the spectrum for different
values of U=J and filling factors. The solutions presented in the plots are the eigenenergies found by the
exact diagonalization of the Hamiltonian (red), by diagonalization in the restricted 1-ph subspace (green)
and the analytic solution, Eq. (9.22) (blue).
We observe in the plots that for low filling factors (g = 1) the analytic solution is not as good as
it is for high filling factors in reproducing the 1-ph subspace spectrum. Nevertheless, at low densities the
validity of perturbation theory holds for a larger range of U=J values. On the other hand, when the filling
is high the analytic solution gives a pretty good description of the 1-ph subspace but the parameter U=J
must be approximately g times higher than the unit filled case to ensure that higher order corrections are
180
0 20 40 60 80 100
nth
Minus6
Minus4
Minus2
0
2
4
6
CapDeltaE
nLParen11RParen1
gEqual1
analyticphsubspace
0 20 40 60 80 100
nth
Minus15Minus10
Minus50
510
15
CapDeltaE
nLParen11RParen1
gEqual4
analyticphsubspace
Figure 9.2: Comparisons between the first order corrections to the 1-ph excitations calculated by diagonal-
izing the Bose-Hubbard Hamiltonian inside the 1-ph subspace and the analytic solution Eq. (9.22). The
number of sites used for the plot was M = 11. Energies are in units of J.
negligible. The number of wells chosen for the plots was small because of the exponentially scaling of the
Hilbert space. As the number of wells is increased, the ratio of U=J required for perturbation theory to be
valid becomes larger.
Using the perturbation theory results, it is possible to calculate expectation values of physical ob-
servables that are relevant to the experiments. For example, the quasimomentum distribution (Eq. (5.57))
to first order in perturbation theory is given by:
nq = g + 4JU g(g +1)cos(qa): (9.24)
Here q is the quasimomentum q = 2?jaM , j = 0;1;:::M ?1 and a the lattice spacing.
The interference pattern after ballistic expansion is closely related to the quasimomentum distribu-
tion [103]. In the superfluid regime the intensity of the principal interference peak is proportional to the
occupation number of the q = 0 quasimomentum component, which describes the number of particles in
the condensate. The washing out of the interference peaks in the Mott phase is linked to the redistribution
of the population from the condensate to states with higher quasimomenta. When the system is in the Mott
insulator regime, instead of a macroscopically occupied state, at zeroth order in J=U, the momentum dis-
tribution is flat and all the quasimomentum states of the lowest band are uniformly occupied. This feature
is indicated in Eq. (9.24). At first order in J=U, we take into account the nearest neighbors? remaining
181
0 5 10 15 20 25 30
nth
24
26
28
30
32
E nMinus
E 0
gEqual1 KEqual6 USlash1JEqual10
phsubanal
exact
0 5 10 15 20
nth
2025
3035
4045
50
E nMinus
E 0
gEqual4 KEqual5 USlash1JEqual27
phsubanal
exact
0 5 10 15 20 25 30
nth
40
42
44
46
48
50
E nMinus
E 0
gEqual1 KEqual6 USlash1JEqual45
phsubanal
exact
0 5 10 15 20
nth
9095
100105
110115
E nMinus
E 0
gEqual4 KEqual5 USlash1JEqual100
phsubanal
exact
Figure 9.3: Comparisons between the energy eigenvalues calculated by the exact diagonalization of the
Hamiltonian (red dots), the diagonalization of the Hamiltonian in the restricted 1-ph subspace (green trian-
gles) and the analytic solution (blue crosses), Eq. (9.22). The index n labels the eigenvalues in increasing
order of energy. The filling factor g, the number of wells M and the ratio U=J is indicated in each plot.
Energies are in units of J.
coherence, always present in the Mott ground state, which creates a small cosinusoidal modulation of the
flat quasimomentum background.
In Fig. 9.4, we plot the quasimomentum distribution as a function of U=J for a unit filled lattice
with six wells. The exact solution is depicted with solid lines and the first order perturbative results with
dots. The decrease of the zero momentum population and the tendency towards a flat distribution for large
U=J ratios is observed in the plot.
Another characteristic feature of the Mott insulator phase is the reduction of the number fluctuations.
The first non vanishing number fluctuations (see Eq. (5.56)) are quadratic in J=U and given by:
182
10 20 30 40 50 60 70 80
USlash1J
0.5
1
1.5
2
n q
gEqual1
qEqual0
qEqual?Slash13
qEqual2?Slash13
qEqual?
Figure 9.4: Quasimomentum distribution as a function of U=J for a unit filled lattice with six wells (M =
6). The exact solutions are displayed with solid lines and the first order perturbative results with dotted
lines. Due to the lattice symmetry n4?=3 = n2?=3 and n5?=3 = n?=3.
?n = 4g(g +1)J
2
U2: (9.25)
We omitted the site index in ?n because of the translational symmetry. Notice that due to the fact that the
true Mott ground state is not a Fock state, the number fluctuations decrease as the ratio U=J increases, but
they are not exactly zero. This is a problem for the lattice based quantum computer proposals and we will
try to correct for it in chapter 11.
We conclude this section by using the first order perturbation theory results, to crudely estimate the
critical transition point. Quantum phase transitions such as the superfluid to Mott insulator transition can be
understood based on the phenomena of level crossing: as a characteristic parameter is changed (in this case
U=J), at some critical point a state that was an excited state becomes the new ground state of the system.
If we use this criteria to calculate the critical point and use Eq. (9.22) we get
183
(U=J)c ? 2(2g +1) ? 4g (9.26)
This result only applies for the one dimensional case. It is a factor of two smaller than the calculated
value from mean field theory (Eq. 3.9) but it is in better agreement with numerical Monte Carlo simulations
and strong coupling expansions which predict a critical point for a unit filled lattice of (U=J)c ? 4:65
[78, 79].
9.2 Harmonic confinement plus lattice
In this section we consider a one dimensional optical lattice in the presence of a magnetic confinement with
oscillation frequency !T . We assume that the magnetic trap has its minimum at the lattice site n = 0.
The magnetic confinement introduces a characteristic energy scale ? = ma2!T=2, so that Vn = ?n2 (see
Eq.(6.1)). We first focus on the case in which the system has an odd number of atoms.
For the analysis we are also going to consider a parameter regime where
U > ?(N ?1)
2
4 ; (9.27)
?N > J: (9.28)
The first condition expresses the requirements that the on site interaction energy U must be bigger than the
trapping energy of the most externally trapped atom, so multiple atom occupation in any well is inhibited.
The second states the requirement that the kinetic energy must be smaller than the potential energy cost for
an atom at the edge of the atomic cloud to tunnel to the next unoccupied site. Thus it is energetically costly
for the atoms to tunnel.
In this parameter regime the Bose-Hubbard Hamiltonian can be split into two parts: an unperturbed
part which includes the interaction and potential energy and which is diagonal in a Fock state representation,
and a kinetic energy part which is going to be treated as a perturbation.
184
^H = ^H(0) +? ^H; (9.29)
^H(0) = ?X
n
n2^ayn ^an + 12U
X
n
^ayn^ayn^an^an; (9.30)
? ^H = ?J
X
n
(^ayn ^an+1 + ^ayn+1 ^an ): (9.31)
With the constraints Eq. (9.27) and Eq. (9.28) we are guaranteed that to zeroth order in perturbation theory,
the ground state of the system consists of a unit filled central core comprising N wells in the central region
of the trap, surrounded by external empty sites, which in solid state language can be called a sea of holes.
If the number of atoms is odd, due to the the assumed trap symmetry, at zero order in perturbation theory
the ground state is unique and given by:
flfl
fl'(0)0
E
=
flfl
flfl
flfl0;::;0;1;1;::::;1;1| {z }
N sites
;0;::;0
+
: (9.32)
The unperturbed ground state energy is
E(0)0 = ?24N(N2 ?1): (9.33)
The zeroth order low lying excitations consist of 1-ph excitations and another kind of excitations which we
refer as n-hh excitations. In the following we proceed to describe them in more detail:
? One particle hole excitation (1-ph)
The 1-ph excitations are described by the Fock states, flfl?phnmfi, with an extra particle at site m and a
hole at the site n, both inside the central core.
flfl
fl'(0)iph
E
? flfl?phmnfi (9.34)
=
flfl
flfl
flfl0;::;0;1;:::; 2|{z}
m
;:::; 0|{z}
n
;::;1;0;::;0
+
:
185
As opposed to the translationally invariant system, the presence of the trap breaks the degeneracy and
the 1-ph excitations are not degenerate at zero order. Their energy is given by the interaction energy
cost U to create a 1-ph pair plus the potential energy cost due to the trap:
Eph(0)nm = E(0)0 +U +?(m2 ?n2): (9.35)
with n and m integers, n;m = ?N?12 ;:::; N?12 :
? Hole hopping excitations (n-hh)
These excitations are described by the states with a maximum of one particle per well and holes
inside the inner central core. They have to be included in the trapped system because of the reservoir
of holes surrounding the central core which brings an extra degree of delocalization. Because the
n-hh excitations are only due to the transfer of holes to the inner core, they don?t have the interaction
energy cost U for having two particles at the same site and therefore they can be lower in energy than
the 1-ph excitations. We label the 1-hh, states with only one hole inside the central core, by flfl?hhmnfi.
Explicitly they are given by:
flfl
fl'(0)ihh
E
? flfl?hhmnfi (9.36)
=
flfl
flfl
flfl0;::; 1|{z}
n
;::;0;1;:::; 0|{z}
m
;::;1;0;::;0
+
:
The zero order energy cost to create 1-hh excitations is:
Ehh(0)nm = E(0)0 +?(m2 ?n2); (9.37)
with n and m integers such that jnj > (N ?1)=2 and jmj < (N ?1)=2
At first order the kinetic energy term mixes the unperturbed ground state with the 1-ph excitation
which have the extra particle and the hole at adjacent sites and with the 1-hh excitations were the most
external trapped atom tunnels to the first available vacant site. The ground state wave function at first order
is then given by
186
flfl
fl'(1)0
E
=
flfl
fl'(0)0
E
+2J
(N?3)=2X
n=?(N?1)=2
flfl
fl?phnn+1
E
+
flfl
fl?ph?n?(n+1)
E
p2(U ??(1+2n))
+
p2J
?N
flfl?hh
nn+1
fi+flflfl?hh
?n?(n+1)
E
p2 : (9.38)
The first order wave functions of the excitations directly coupled to the unperturbed ground state are
flfl
fl'(1)?nph
E
=
flfl
fl?phnn?1
E
?p2J
flfl
fl'(0)0
E
(U ??(1?2n)); (9.39)
flfl
fl'(1)?hh
E
=
flfl
fl?hh?L?(L+1)
E
?J
flfl
fl'(0)0
E
(?N) ; (9.40)
with L = (N ?1)=2.
In Fig. 9.5 we plot comparisons between the analytic (blue) and exact (red) excitation spectra. The
index n labels the eigenenergies in increasing energy order. The parameters used for the plots were U=J =
40 and 60, ?=J = 1:875, N = 9 and M = 11. The analytical spectrum was calculated to zeroth order in
perturbation theory. The convergence of perturbative results to the exact ones as the interaction parameter
U=J is increased can be seen in the plots. The lines indicate the energies of the states that are coupled
to the zero order ground state. The first two lines correspond to the two degenerate 1-hh excitation states
flfl
fl?hh?L?(L+1)
E
with energy ?N, the others correspond to 1-ph excitations
flfl
fl?phnn?1
E
with energies (U ?
?(1?2n)).
When the number of atoms is even, the trap symmetry is broken and the ground state becomes
degenerate. In this case, at zeroth order, the degenerate ground states are:
flfl
fl'(0)0?
E
= 1p2
?flfl
fl'(0)L
E
?
flfl
fl'(0)R
E?
; (9.41)
flfl
fl'(0)L
E
=
flfl
flfl
flfl0;:::;0;1;1;1;::::;1;1| {z }
N?1 sites
;0;::;0
+
;
flfl
fl'(0)R
E
=
flfl
flfl
flfl0;::;0;1;1;::::;1;1| {z }
N?1 sites
;1;0;::;0
+
:
When the system has an even number of atoms, there are always going to be a nonzero number fluctuation
187
0 50 100 150 200
n
0
20
40
60
En
Slash1J
USlash1JEqual60.
0 50 100 150 200
n
0
10
20
30
40
50
En
Slash1J
USlash1JEqual40.
Figure 9.5: Comparisons between the exact (red) and the perturbative (blue) spectra for a trapped system
with N = 9, M = 11 and ?=J = 1:875 deep in the Mott regime. The lines indicate the energies of the
states that are coupled directly to the ground state.
at the trap edge. Because the states
flfl
fl'(0)L
E
and
flfl
fl'(0)R
E
are not coupled by the Bose-Hubbard Hamiltonian at
first order, the previous analysis we did for the system with an odd number of atoms can be straightforwardly
extended to the even case.
The constraint ?N > J used for the analysis above can be an important experimental restriction
if the total number of trapped atoms is large. In the regime where ?N < J, but still U > ?(N=2)2, the
energy splitting between 1-ph excitations induced by the trap is small and degenerate perturbation theory
must be used at first order. An analytic treatment becomes really difficult, nevertheless, it is possible to
have a qualitative analysis of the system. In this case, the system can be divided in two parts: a central unit
filled core comprising K < N wells at the center of the trap and a superfluid layer surrounding the central
core with maximum one atom per lattice site where the mobility of the holes is high. The size K of the unit
filled central core is determined by the strength of the magnetic confinement, K ? J=?. The properties of
188
the unit filled core can be described to a good approximation by the commensurate homogeneous results.
On the other hand, the atoms surrounding the central core are almost free to tunnel and they have superfluid
like properties.
9.3 Conclusions
In summary, in this chapter we have described the basic properties of the Mott insulating phase both in
translationally invariant and harmonically trapped systems. All the results derived here are going to be used
as the starting point for the calculations in the following chapters.
189
Chapter 10
Bragg Spectroscopy
To date, the primary observable used to study ultra cold atoms in an optical lattice has been the momentum
distribution of the system, observed after ballistic expansion. In particular, this type of measurement has
been used to reveal the phase coherence between sites in the lattice and has been a really useful technique to
characterize the superfluid phase. However, the disappearance of the interference pattern is not a conclusive
diagnostic of the Mott phase. Recent analysis shows that this is more correctly related to the degree of
condensate depletion. Indeed, for this reason, the diagnostic tool used in the experiments by Greiner et al.
[46], to prove the achievement of the Mott phase, was to apply a potential gradient to the lattice and show
the presence of a gap in the excitation spectrum.
Moreover, because the usual procedure for producing the Mott insulator state is to begin with a
magnetically trapped Bose-Einstein condensate (with no discernible thermal component), and slowly load
it into an optical lattice, finite temperature effects or non-adiabatic ramping of the lattice may lead to the
production of an imperfect Mott phase. As more elaborate experiments are undertaken with optical lattices
in the strongly correlated regime it will be crucially important to diagnose the properties of the states
produced.
The results we present here suggest that Bragg spectroscopy can not only give information about
the excitation spectrum, which is a crucial diagnostic of the Mott insulator phase as characterized by the
opening of a gap, but that it can also be used in the Mott regime to estimate the temperature of the system.
Specifically, we study the linear response to Bragg spectroscopy of cold atoms loaded in a one dimensional
optical lattice based on a perturbative analysis. We obtain analytical expressions for the dynamical struc-
ture factor and use them to calculate the energy deposited into the system. We also test the accuracy of our
approximations by comparing them with numerical solutions obtained by diagonalizing the Bose-Hubbard
Hamiltonian for moderate number of atoms and wells. The calculations are done for translationally invari-
ant lattices and in the presence of an harmonic external potential. First we start by reviewing the basic
formalism that describes Bragg spectroscopy.
190
10.1 Formalism
The application of Bragg spectroscopy to study cold atomic systems was first pioneered by the NIST and
MIT groups [65, 66]. Their scheme involved using a pair of interfering laser fields to Bragg scatter atoms
into a higher momentum state. Since those experiments Bragg spectroscopy has been established as a
versatile tool for probing Bose-Einstein condensates in a wide range of situations (e.g see [141, 142, 143,
144]).
Figure 10.1: Bragg spectroscopic scheme considered in this paper. The lattice potential with atoms loaded
into the ground band is perturbed by a shallow running wave perturbation (the Bragg potential).
Two-photon Bragg spectroscopy in a lattice is performed by superimposing a periodic travelling
wave potential on the lattice potential. This could be arranged by modulating the lattice amplitude, as done
by St?oferle et al. [145]. The more general situation arises when the Bragg potential is produced by an
additional pair of independent laser fields, chosen to produce a shallow potential that is not necessarily
commensurate with lattice potential (see Fig. 10.1). Specifically, the atomic sample is illuminated with
two additional laser beams with wave vectors of magnitudes k1 and k2 in the direction of the lattice and
frequency difference !: The intersecting beams create a periodic potential parallel to the lattice with trav-
elling intensity modulation given by Vo cos(qx?!t). The difference in wave vectors of the beams defines
the Bragg momentum q = k1 ?k2 and the difference in frequency the Bragg energy ~!. Atoms exposed
to these beams can undergo stimulated light scattering by absorbing a photon from one of the beams and
emitting a photon into the other.
We consider a one dimensional optical lattice with M lattice sites, sufficiently deep that the tight-
binding approximation is valid, and assume that we can restrict the dynamics of the atoms to the lowest
vibrational band. This is guaranteed if the loading is done slow enough to avoid band excitations,i.e. if the
191
energy of the Bragg perturbation ~! is less than the energy gap to the second band and if the momentum
transfer q is contained within the first Brillioun zone. A detailed analysis of the validity of the first band
approximation is found in Ref.[148]. There the authors, using a mean field approach combined with Bogoli-
ubov analysis, extended the existing theory of Bragg spectroscopy of magnetically trapped BEC [146, 147]
to lattice systems.
As shown in chapter 3, under the single-band approximation the system is described by the Bose-
Hubbard Hamiltonian, which we will denote as ^Ho. Hereafter we assume, unless otherwise specified, a
harmonic magnetic confinement such that Vn = ?n2. In the tight-binding approximation, the Hamiltonian
describing the Bragg perturbation is given by
^HB = 1
2Vo(^?
y
qe
?i!t + ^?qei!t): (10.1)
The density fluctuation operator ^?yq is defined as
^?yq =
M?1X
m;n=0
In;mq ^aym^an; (10.2)
with
In;mq =
Z
dxeiqxw?0(x?am)w0(x?an); (10.3)
where w0(x) is the Wannier orbital centered at the origin of the lowest vibrational band. To calculate
In;mq we approximate the Wannier orbitals by Gaussians localized at the bottom of the potential wells ,
`0(x?an) ? exp[?(x?an)2=2a2ho]=(?1=4paho); where the width aho can be estimated by minimizing
the ground state energy. For a lattice potential of the form ERVlat sin2(?x=a); with ER the recoil energy of
the atoms, aho scales like aho ? (Vlat)?1=4a=?. Neglecting the overlap between Wannier orbitals located
in different sites we get In;mq ? Iq?n;m exp(ia(n + m)q=2), where ?n;m is the Kronecker delta function.
Then
Iq = exp
 
? 1pV
lat
?qa
2?
?2?
: (10.4)
Notice that Iq approaches one for small qa and deep lattices. Using the approximate expression for In;mq
we can rewrite the density operator as:
192
^?yq = Iq
M?1X
m=0
^aym^ameiqma: (10.5)
To simplify the notation and to have a better physical understanding of the excitations induced by
the Bragg perturbation, it is convenient to introduce the quasimomentum field operator ^bk which annihilates
an atom with quasimomentum k, and is defined as the Fourier transform of the field operator ^an, i.e.,
^bk = 1p
M
PM
n=1 e
ikna^an. The quasimomentum k can assume only discrete values which are integer
multiples of 2?Ma:
In the quasimomentum basis the operator ^?yq can be written as
^?yq = Iq
M?1X
k=0
^byk+q^bk; (10.6)
where we have assumed that q can be also expressed as an integer multiple of 2?Ma. By writing the Bragg
perturbation potential in the quasimomentum basis it can be seen that the two means by which momentum
is imparted to the atomic sample,is either by promoting an atom from quasimomentum k to k + q or by
demoting a particle with momentum k to k?q:
10.2 Observables
Useful experimental observables in Bragg spectroscopy are related to the density-density response function.
Here we focus our attention on the imparted momentum and energy, which can be measured by time-of-
flight techniques. In this section we elaborate on the form of these observables.
In the tight-binding approximations the total momentum is given by
^P = X
k
~k^byk^bk: (10.7)
From the Heisenberg equations of motion we get
d^P
dt =
i
~
h^
H; ^P
i
= i~
?h^
HB; ^P
i
+
h^
Ho; ^P
i?
(10.8)
= ?iqVo2 (^?yqe?i!t ? ^?qei!t)?2? ^X: (10.9)
193
where ^H is the total Hamiltonian, ^H = ^Ho + ^HB, and ^X is the center of mass position given by ^X =
P
n(na)^a
yn^an. The second term in the momentum rate equation is just related to the force felt by atoms due
to the presence of an harmonic external confinement. On the other hand, the first term takes into account
the rate of change of the momentum due to the Bragg perturbation. If the perturbation is applied for a time
short compared to the oscillator period tosc= 2?p4J?a the second term in the right-hand-side is small and can
be neglected in the calculations.
The energy change rate is
d ^Ho
dt =
i
~
h^
HB; ^Ho
i
= i~
h^
HB; ^H
i
; (10.10)
= ?12Vo
?
e?i!td^?
yq
dt +e
i!td^?q
dt
!
:
Because the energy of a closed system is conserved, as opposed to the momentum rate equation, the energy
change rate is independent of the magnetic potential and only depends on the applied Bragg perturbation.
10.3 Linear response
The first order variation ?h ^O(t)i of the thermal average of an observable ^O(t) with respect to its mean value
due to a weak perturbation ^HB applied to a system at t = to , is given by the Kubo formula [51]
?
D^
O(t)
E
= i~
Z t
to
d?
Dh^
HB(?); ^O(t)
iE
; (10.11)
where the operators ^O(t) and ^HB(?) are in the Heisenberg representation with respect to the time indepen-
dent hamiltonian ^Ho; and the expectation value is taken using the equilibrium state of the system.
For the particular case where the observables are the momentum and energy rates, we get after some
algebra :
194
?
*
d^P
dt
+
=~q
 V
o
2~
?2
? (10.12)
Z t
to
d?
?
e?i!(??t)??^?yq(?); ^?q(t)?fi?ei!(??t)??^?q(?); ^?yq(t)?fi
+e?i!(?+t)??^?yq(?); ^?yq(t)?fi?ei!(?+t)h[^?q(?); ^?q(t)]i
?
?2??
D ^
X
E
;
?
*
d ^H
dt
+
= ?i~
 V
o
2
?2
? (10.13)
Z t
to
d?
 
e?i!(??t) ddt ??^?yq(?); ^?q(t)?fi?ei!(??t) ddt ??^?q(?); ^?yq(t)?fi
+e?i!(?+t) ddt ??^?yq(?); ^?yq(t)?fi?ei!(?+t) ddt h[^?q(?); ^?q(t)]i
?
:
We now introduce the functions S(q;!) and R(q;!) given by
?^?y
q(?)^?q(t)
fi = Z e?i!(??t)S(q;!)d!; (10.14)
?^?y
q(?)^?
y
q(t)
fi = Z ei!(?+t)R(q;!)d!: (10.15)
In terms of these functions it can be seen that since ??^?yq(?); ^?yq(t)?fi = ??^?q(?); ^?q(t)?fi = 0 Eq. (10.13)
and Eq. (10.14) reduce to:
?
*
d^P
dt
+
= 2~q
 V
o
2~
?2Z
d!0(S(q;!0)?S(?q;?!0))?
 sin((! ?!0)t)
(! ?!0)
?
?2??
D ^
X
E
;
?
*
d ^H
dt
+
= 2~
 V
o
2
?2Z
d!0!0(S(q;!0)?S(?q;?!0))? (10.16)
 sin((! ?!0)t)
(! ?!0)
?
:
where we have set to = 0. The quantity S(q;!), the Fourier transform of density correlations, is the so
called dynamic structure factor and is given by
195
S(q;!) = 1Z
X
ij
e?flEi jhij ^?q jjij2 ?(! ?!ij) (10.17)
?
X
ij
Sij(q)?(! ?!ij): (10.18)
where jii and Ei are eigenstates and eigenenergies of the unperturbed Hamiltonian, e?flEi is the usual
Boltzmann factor with fl = 1=kBT where kB is Boltzmann?s constant and T the temperature, Z is the
canonical partition function,~!ij = Ej ?Ei and Sij(q) ? 1Ze?flEi jhij ^?q jjij2 :
Notice that because atoms could be scattered by absorbing a photon from either one of the laser
beams the response of the system is determined not only by S(q;!) but by the combination S(q;!0) ?
S(?q;?!0): If the perturbation is applied for a time long with respect to the frequency of the applied field
the momentum and energy rates approach the golden rule results.
Performing the frequency integration we get
?
*
d^P
dt
+
= 2~q
 V
o
2~
?2X
ij
 
Sij(q)
 sin((! ?!
ij)t)
! ?!ij
?
(10.19)
?Sij(?q)
 sin((! +!
ij)t)
! +!ij
??
?2??
D ^
X(t)
E
;
?
*
d ^H
dt
+
= 2
 V
o
2~
?2X
ij
~!ij
 
Sij(q)
 sin((! ?!
ij)t)
! ?!ij
?
(10.20)
+Sij(?q)
 sin((! +!
ij)t)
! +!ij
?
):
The total momentum and energy transfer after applying the Bragg perturbation can be obtained by integrat-
ing the above equations. We get as a final result that
?
D^
P
E
= ~q
 V
o
~
?2X
ij
?
Sij(q)
 sin((! ?!
ij)?=2)
! ?!ij
?2
(10.21)
?Sij(?q)
 sin((! +!
ij)?=2)
! +!ij
?2!
?2?=a
Z ?
0
?
D ^
X(t)
E
dt
?
D^
H
E
=
 V
o
~
?2X
ij
~!ij
?
Sij(q)
 sin((! ?!
ij)?=2)
! ?!ij
?2
(10.22)
+Sij(?q)
 sin((! +!
ij)?=2)
! +!ij
?2!
;
196
where ? is the duration of the perturbation.
10.4 Zero-temperature regime
10.4.1 Bogoliubov approach
In this section we use the zero temperature Bogoliubov approximation for atoms in an optical lattice to study
the dynamical structure factor in the superfluid regime. Because quadratic approximations were discussed
in detail in in chapter 5, we refer the reader to this chapter for details.
Under the Bogoliubov approximation the field operator at lattice site n is written as a complex
number zn plus small fluctuations ^? = PMs6=0(usnbfis ?v?sn bfiys ), with fusn;vsng the quasiparticle amplitudes.
Using this ansatz in Eq.(10.17) we get an expression for the dynamical structure factor given by
ST=0(q;!) = S0(q)?(!)+
X
i
Si(q)?(! ?!Bi ); (10.23)
S0(q) ?
flfl
flfl
fl
MX
n=1
jznj2eiqan
flfl
flfl
fl
2
; (10.24)
Si(q) ?
flfl
flfl
fl
X
n
(znu?in ?z?nv?in )eiqan
flfl
flfl
fl
2
: (10.25)
Translationally invariant system
To understand many-body effects included in the Bogoliubov approximation we start by studying the case
of a translationally invariant lattice (Vn = 0) with N atoms, M wells and periodic boundary conditions.
Using the results for the quasiparticle amplitudes and energies calculated using the improved Popov ap-
proximation, which was shown to be the best quadratic approximation we get:
S(q;!) = N2?(!)+Mno "q!B
q
?(! ?!Bq =~) (10.26)
= N2?(!)+MnoS(q)?(! ?!Bq =~); (10.27)
with no given by Eq. (5.136), "k = 4J sin2(ak=2) and !Bk = p"2k +2nU"k. Here n is the total den-
sity n = N=M. Notice that due to the translational invariance of the system, the quasimomentum is a
197
good quantum number and only quasiparticle states that have the same quasimomentum as the transferred
momentum q are excited.
For small q, the structure factor behaves like
q
J
2Unq which is just the free particle expression with
the bare particle mass m replaced by the effective mass, m? =~2=2Ja2 (a the lattice spacing). Suppression
of S(q) in the phonon regime is a direct consequence of the importance of quantum correlations in the long
wave-length limit. On the other hand, away from the phonon regime, the presence of the optical lattice
changes the behavior of S(q) in a drastic way. As opposed to the free case where at high transferred
momenta !Bq ? q2 and S(q) approaches one, the periodic potential modifies the single particle dispersion
relation, which becomes always bounded by 4J. The term proportional to 2nU"k dominates even at high
momenta and many-body effects play a crucial role.
In Fig.10.2 we show the energy ?h ^Hi imparted to the system as a function of the frequency ! of
the Bragg probe, for different values of and U=J and of the momentum q. The blue curve corresponds
to the solution using the Bogoliubov approximation and the red one to the solution obtained by the exact
diagonalization of the Bose-Hubbard Hamiltonian. The number of lattice sites and wells used in the plots
were M = N = 9. For U=J = 0:001 the Bogoliubov and exact curves perfectly overlap. The agreement
is expected as the non-interacting Hamiltonian is quadratic and therefore exactly matches the Bogoliubov
Hamiltonian. The almost non interacting regime may be experimentally achieved by varying the onsite
interaction energy U independently of J, by means of a Feshbach resonance that changes the scattering
length [20, 21]. For U=J = 1 the agreement is good only for the low momentum q = 2?=9. The agreement
for low momenta can be understood by the fact that for low q the Bragg spectroscopy is probing the long
wave-length modes. These modes have a phonon-like dispersion relation and are almost not affected by the
presence of the lattice. In general Bogoliubov quasi-particle states correspond to solutions of the approxi-
mate Hamiltonian with a plane-wave character which are only valid in the weakly interacting regime. As
quantum correlations become important, the exact eigenstates of the interacting system do not necessarily
have the simple plane wave character,(see Ref. Lieb and Liniger [149]), especially in the commensurate
filling situation where for a critical value of U=J the system exhibits the superfluid-Mott insulator transi-
tion. However the unit filled commensurate case is the worse scenario, and as discussed in chapter 5, for
198
non commensurate fillings or commensurate systems with larger filling factor we expect the validity of the
Bogoliubov approximation to extend over a larger range of U=J values.
As a final remark, it is important to point out that the small oscillations in Fig. 10.2 are due to
the square shape of the applied Bragg pulse. These oscillations would disappear if instead a pulse with
Gaussian profile is applied.
0 2 4 6 8 1012?Slash1J0
2040
6080
?ELParen1Vo
2 Slash1JRParen1
USlash1JEqual2.qEqual2?Slash19
0 2 4 6 8 1012?Slash1J0
200400
600
?ELParen1Vo
2 Slash1JRParen1
USlash1JEqual2.qEqual8?Slash19
0 2 4 6 8?Slash1J0
2040
6080
100
?ELParen1Vo
2 Slash1JRParen1
USlash1JEqual1.qEqual2?Slash19
0 2 4 6 8?Slash1J0
200400
600800
?ELParen1Vo
2 Slash1JRParen1
USlash1JEqual1.qEqual8?Slash19
0 1 2 3 4 5 6?Slash1J0
2040
6080
100
?ELParen1Vo
2 Slash1JRParen1
USlash1JEqual0.001qEqual2?Slash19
0 1 2 3 4 5 6?Slash1J0
200400
600800
?ELParen1Vo
2 Slash1JRParen1
USlash1JEqual0.001qEqual8?Slash19
Figure 10.2: Imparted energy: Comparisons between the the exact and Bogoliubov approximation for
N = M = 9. The horizontal axis is in units of~. For the plot we used J?=~= 10. See text Eq. (10.23).
10.4.2 Inhomogeneous system
Using the quasiparticle amplitudes derived under the T.F. approximation (see section 5.6.2), we get an
expression for the dynamical structure factor given by
S(q;!) = 9N2
 sin(x)?xcos(x)
x3
?2
?(!)
+~!
?
8U
X
i
(i(i+1)(2i+1))jePi(x)j2?(! ?!?
p
i(i+1)=2) (10.28)
where x ? qaRTF and ePn(x) = R1?1 Pn(u)eiuxdu.
199
An alternative way to calculate the dynamical structure factor is to use the so called local density
approximation (LDA). This approximation is valid for large condensates, where the density profile varies
in a smooth way and the system behaves locally as an uniform gas. Using a LDA the dynamical structure
factor can be written as
SLDA(q;!) =
?X
m
z(m)eiqam
!
?(!)+"s
X
m
jz(m)j2
~!Bq (m)?(~! ?!
B
q (m)): (10.29)
Here !Bq (m) is the local homogeneous Bogoliubov dispersion relation with n replaced by the local density
in the trap. The LDA ignores the Doppler effect associated with the spreading of the momentum distribution
of the condensate and therefore it is not valid for large momentum transfers.
To test the validity of the HFB-Popov approximation to describe the energy imparted to the system
by Bragg spectroscopy in the trapped case we use a system with 11 lattice sites and 9 atoms. Unfortunately
for such a small number of atoms and wells the assumptions under which the analytic Thomas Fermi
expressions were developed, i.e. large number of atoms and smooth variations of the density profile, do
not apply. Therefore for the case with M = 11;N = 9 we restrict ourselves to comparisons between
numerical solutions of the HFB-Popov equations and solutions found by the exact diagonalization of the
many-body Hamiltonian. We solve numerically Eqs.(5.113) -(5.117) to obtain the quasi-particle energy and
amplitudes. We then use them to calculate the imparted energy. In the presence of the trap, the number
of occupied wells for a fixed number of atoms depends on the trap frequency, particles are not necessarily
uniformly distributed throughout the lattice, and therefore commensurability is only meaningful locally.
This explains why in contrast to the commensurate homogeneous system, the agreement between the two
solutions is good up to U=J = 5 for the chosen parameters, as shown in Fig. 10.3. The failure of the
HFB-Popov approximation for U=J ? 10 can also be observed in the density plots, as shown in Fig. 10.4.
The multiple peaks depicted in the plots are due to the external trapping potential, which not only dis-
cretizes the spectrum but also changes the plane wave character of the quasiparticle amplitudes. Therefore,
for a given transfer momentum q different quasiparticle modes can be excited. In Fig.10.3, we explicitly
show with a line the position of the quasiparticle energies.Similar to the homogeneous case, the lower the
transfer momentum q, the better is the agreement between the HFB-Popov results and the exact solution.
200
For small systems such as the one in consideration discretization effects can not be ignored. On the
other hand, if the inverse of the transfer momentum is bigger than the size of the condensate wave function,
then to first approximation the discretization in the excitation spectrum can be safely ignored and the system
can be treated as locally uniform. Instead of distinguishable peaks the response curve becomes smooth.
0 5 10152025?Slash1J02.5
57.5
1012.5
15
?ELParen1Vo
2 Slash1JRParen1
USlash1JEqual10qEqual2?Slash111
0 5 10152025?Slash1J0
2040
6080
100120
?ELParen1Vo
2 Slash1JRParen1
USlash1JEqual10qEqual6?Slash111
0 5 10 15 20?Slash1J0
510
1520
2530
?ELParen1Vo
2 Slash1JRParen1
USlash1JEqual5qEqual2?Slash111
0 5 10 15 20?Slash1J020
406080
100120140
?ELParen1Vo
2 Slash1JRParen1
USlash1JEqual5qEqual6?Slash111
0 2 4 6 8101214?Slash1J0
1020
3040
?ELParen1Vo
2 Slash1JRParen1
USlash1JEqual0.1qEqual2?Slash111
0 2 4 6 8101214?Slash1J0
50100
150200
?ELParen1Vo
2 Slash1JRParen1
USlash1JEqual0.1qEqual6?Slash111
Figure 10.3: Comparisons of the imparted energy vs. Bragg frequency curves calculated from the exact
diagonalization of the manybody Hamiltonian (red) and the from the HFB-Popov approximation(blue) for
a trapped system with N = 9, M = 11 and ?=J = 1:875. The horizontal axis is in units of ~. The lines
depicted in the plot are located at the different quasiparticle energies excited by the Bragg perturbation.
To explore the validity of Eq.(10.28) derived under the Thomas-Fermi approximation we plot in
Fig. 10.5 S(q;!) vs. ! for different transfer momenta. The parameters used for the plot were U=J = 0:2,
?=J = 9:5 ? 10?4 and N = 100. For these parameters we believe the HFB-Popov should give a good
description of the system. In chapter 5 we showed that in the regime when the TF approximation is valid, it
gives a very good description of the low lying excitations. However it fails to describe high energy modes.
To illustrate this issue we compare in Fig. 10.6 the TF excitation spectrum with the one found by solving
the HFB-Popov Equations. The quasiparticle energies are labelled in increasing energy order. When the
Thomas-Fermi approximation is used in S(q;!), we expect then to obtain a fair description of it only if
201
Minus4 Minus2 0 2 4n0
12
34
56
Density
USlash1JEqual0.1
ExactBog.
Minus4 Minus2 0 2 4n0
0.51
1.52
2.5
Density
USlash1JEqual5.05
ExactBog.
Minus4 Minus2 0 2 4n0
0.5
1
1.5
2
Density
USlash1JEqual10.
ExactBog.
Figure 10.4: Density profile calculated from the exact diagonalization of the many-body Hamiltonian (red)
and from the HFB-Popov approximation (blue) for a trapped system with N = 9, M = 11 and ?=J =
1:875. n labels the lattice site.
low-lying modes are probed. Low-lying modes are probed if the inverse of the transfer momentum q?1
is big compared to the size of the condensate wave function. This is explicitly shown in Fig. 10.5, where
only for small values of q the two solutions agree. For higher Bragg momenta, qRTF > 1, higher modes
contribute to S(q=!) and the TF approximation gives a poor description of S(q;!).
10.5 Mott insulator regime
In this section we derive expressions for the zero temperature dynamical structure factor in the Mott regime
for both the homogeneous and trapped systems and compare them with the solutions obtained by exact
diagonalization of the Bose-Hubbard Hamiltonian.
202
0.2 0.4 0.6 0.8
?Slash1J
02.5
57.5
1012.5
15
SLParen1q
,?RParen1
qEqual 0.17?
0.2 0.4 0.6 0.8
?Slash1J
0
5
10
15
20
SLParen1q
,?RParen1
qEqual 0.24?
0.2 0.4 0.6 0.8
?Slash1J
01
23
45
SLParen1q
,?RParen1
qEqual 0.034?
0.2 0.4 0.6 0.8
?Slash1J
02
46
810
1214
SLParen1q
,?RParen1
qEqual 0.1?
Figure 10.5: Dynamical structure factor vs. ~!=J in the Thomas-Fermi (blue) and HFB-Popov (red)
approximations. The horizontal axis is in units of ~. The system parameters are U=J = 0:2, ?=J =
9:5?10?4 and N = 100.
10.5.1 Commensurate homogeneous system
In this section we assume a commensurately filled lattice with filling factor N=M = g, periodic boundary
conditions and a parameter regime where U  J. Using the eigenvalues and eigenstates calculated in
section 9.1 with first order perturbation theory in the high filling factor limit, we get an expression for the
dynamical structure factor given by
S(T=0)(q;!) = N2?(!)?q0 + J
2g(g +1)
~U2 ? (10.30)
X
r;R
?(~! ?E(1)rR)
flfl
flfl
fl
MX
m=1
eiqam?CrRmm+1 +CrRmm?1 ?CrRm+1m ?CrRm?1m?
flfl
flfl
fl
2
= N2?(!)?q0 + 32J
2g(g +1)
~U2 sin
2
?qa
2
?X
r
0sin??r
M
?2
?(~! ?E(1)req ) (10.31)
with qa = 2?eq=M, and the prime in the sum meaning that it is constrained over the r?s with eq + r even.
Is important to emphasize that only the states with R = 0 have a dispersion relation which agrees to first
order in J to the mean-field solution found in Ref. [151]. However, these states give zero contribution to
203
5 10 15 20
ntheigenvalue
0.2
0.4
0.6
0.8
1
1.2
1.4
?nSlash1
J
USlash1JEqual0.2CapOmegaSlash1JEqual9.510Minus4 NEqual100
TF
BdG
Figure 10.6: Eigenvalues of the quasiparticle spectrum plotted in increasing energy order. The vertical scale
is in units of~
the sum of Eq. (10.31).
In Fig. 10.7 we compare the imparted energy as a function of the Bragg frequency calculated using
the expression of the dynamical structure factor Eq. (10.31) to results obtained by the exact diagonalization
of the Bose-Hubbard Hamiltonian with parameters N = M = 9 and U=J = 45. We show the response
for two different Bragg momenta q = 2?=9 and 8?=9. In contrast to the superfluid regime, where Bragg
spectroscopy excites only the quasiparticle state with quasimomentum q, in the Mott regime we observe
M ?1 peaks centered around U. The different peaks are due to the two dimensional character of the 1-ph
dispersion relation. The Bragg momentum q fixes one quantum number R but the other can take M ? 1
different values. In the analytic solution due to the constraint in Eq. (10.31), eq + r even, only (M ?1)=2
of the possible M ?1 peaks are present. The constraint is a consequence of the extra symmetry introduced
in the high filling factor solution where similar ?effective masses? are assumed. Nevertheless, Fig. 10.7
shows that even in the situation with g = 1, the peaks captured by the analytic solution are the most relevant
ones. As g is increased, as long as the perturbation analysis is valid, we expect even better agreement. The
interval of frequencies over which the system responds has an approximate width of 4J(2g+1)cos(qa=2).
The cos(qa=2) dependence indicates that as q approaches ? the response width is minimized. This behavior
can be observed in Fig. 10.7, when we plot the response for q = 8?=9. It is important to point out that the
presence of distinguishable peaks in Fig. 10.7 is a result of the finite system size. As the number of lattice
204
sites is increased, more peaks are contained in the interval 4J(2g + 1)cos(qa=2). The separation between
one peak and the next decreases and instead of individual peaks a continuous envelope is approached.
39 43 47 50
?Slash1J
10
20
30
?E
LParen1Vo
2 Slash1J
RParen1
USlash1JEqual45. qEqual2?Slash19
Perturb.
Exact
39 43 47 51
?Slash1J
0
30
60
90
120
?E
LParen1Vo
2 Slash1J
RParen1
USlash1JEqual45. qEqual8?Slash19
Figure 10.7: Transfer energy vs. Bragg frequency for a zero temperature homogeneous system with M =
N = 9 and different transfer momenta q. The horizontal axis is in units of~. The perturbation time was set
to J?=~= 10. Note the change of the vertical scale between the two panels.
10.5.2 Inhomogeneous system
10 20 30 40 50 60
?Slash1J
0.5
1
1.5
2
?E
LParen1Vo
2 Slash1
JRParen1
USlash1JEqual45. qEqual2?Slash111
Perturb
Exact
10 20 30 40 50 60
?Slash1J
4
8
12
?E
LParen1Vo
2 Slash1
JRParen1
USlash1JEqual45. qEqual10?Slash111
Figure 10.8: Transfer energy vs. Bragg frequency for a zero temperature trapped system with parameters
U=J=45, ?=J = 1:857, M = 11 and N = 9 and different transfer momenta q. The horizontal axis is in
units of ~. The perturbation time was set to J?=~= 10. Note the change of the vertical scale between the
two panels.
In this section we consider a one dimensional optical lattice in the presence of a magnetic confine-
ment. We assume that the magnetic trap has its minimum at the lattice site n = 0. In the regime where
205
U > ?(N?1)24 and ?N > J assuming an odd number of atoms, we can use the eigenvalues and eigenmodes
derived to first order in perturbation theory in section 9.2 in the expression for the dynamical structure factor
to get:
S(T=0)(q;!) = N2?(!)?q0 + 8J
2 sin2(qa=2)
~(N?)2 ?(~! ?N?)
+
L?1X
m=?L
16J2 sin2(qa=2)
~(U ??(1+2m))2?(~! ?U +?(1+2m)); (10.32)
where L = (N ?1)=2. In Fig. 10.8 we plot the imparted energy as a function of the Bragg frequency ~!
and Bragg momenta q = 2?=11, 10?=11. The system parameters are ? = 1:8J, U=J = 45, M = 11
and N = 9. We observe nine peaks both in the perturbative and exact solutions. The peak at lower
frequency, N? ? 18J, is described by the first term in Eq. (10.32) and corresponds to the two degenerated
1-hh excitations created when a hole tunnels into one of the most externally occupied sites. The other
N ?1 peaks, separated by ?, are described by the second term in Eq. (10.32) and correspond to the 1-ph
excitations with the particle and hole at adjacent lattice sites. To explicitly show the sin(qa=2) dependence
of the response of the system, we plot the imparted energy for two different transfer momenta q, one close
to zero the other close to ?. Even in the presence of the trap S(q;!) tends to 0 as qapproaches an integer
value of 2?.
If the number of atoms is even, the trap symmetry is broken and the ground state becomes degener-
ated. To first order in perturbation theory the dynamical structure factor in this case is given by
S(T=0)(q;!) = 2N2?(!)?q0 +8J2 sin2(q=2)
 ?(~! ?(N +1)?)
~((N +1)?)2
+?(~! ?(N ?1)?)~((N ?1)?)2 +4
N=2?1X
m=N=2
?(~! ?(U ??(1+2m))
~(U ??(1+2m))2
1
A: (10.33)
In this case instead of a single excitation peak at ~! = N? we have two almost degenerate peaks at
~! = (N ?1)?. Apart from this, the response of the system is analogous to the odd number case.
All the previous analysis was done for the case ?N > J. This can be a significant constraint if the
total number of atoms is large. In the opposite case when ?N < J but still U > ?(N=2)2, the energy
206
splitting between one particle hole excitations is small and degenerate perturbation theory must be used at
first order. In this case the system can be divided in a central unit filled core, which to a good approxima-
tion has a response which can be described by the commensurate homogeneous results, surrounded by a
superfluid region whose excitations are mainly due hole hopping excitations. We expect that the response
to Bragg spectroscopy of the system in this parameter regime has peaks localized around U, which probes
the one particle hole excitation band analogous to the commensurate translationally invariant case. We also
expect lower frequency peaks around ?N associated with hole hopping excitations inside the superfluid
region that surrounds the unit filled central core.
Be aware that the parameter regime chosen for all the plots in the Mott phase lies in the regime
where first order perturbation theory is valid and therefore peaks at 2U due to higher order excitations are
suppressed even in the exact numerical results.
10.6 Finite temperature
In most of the experiments the way to load ultra-cold atoms in an optical lattice is by first forming a Bose-
Einstein condensate in a weak magnetic trap and then adiabatically turning on the lattice by slowly ramping
up the intensity of the laser beams. Ideally this kind of process should end up with the atoms in the many-
body ground state.However, that the Bose-Einstein condensate used as a starting point is not produced
exactly at T = 0, and the final temperature is not trivially related to the initial one. It has been shown
[150] that even for an ideal non interacting gas there are different regimes where the atomic sample can be
significantly heated or cooled by adiabatically changing the lattice depth. The role of interactions is and
how many-body effects affect the final temperature of the sample are not yet well understood.
Besides this issue, there remains the point of how long one must wait to be truly adiabatic with
respect to many-body excitations. Interactions are essential for establishing equilibrium in the system, and
understanding them in detail will be really important to determine the time scales for adiabatic loading. For
a deep lattice the tunneling rate seems to be the most restrictive time scale for maintaining adiabaticity. If
adiabaticity were determined by the tunneling time~=J, it would be very hard to be satisfied experimentally
because J decreases exponentially with the lattice depth.
207
Because of these issues, it would be very interesting from an experimental point of view to have
a mechanism for determining the temperature of the system. What we show in this section is that Bragg
spectroscopy could give information about the temperature, at least deep in the Mott regime. As we did for
the zero temperature case, we start by describing the superfluid case, then the deep Mott insulator case, and
finally showing numerical calculations in the regime where the Mott transition takes place.
? Superfluid Regime
At finite temperature both quantum and thermal fluctuations contribute to the depletion of the con-
densate. We expect the Bogoliubov approach to be a good description of the many-body system for
moderate lattice depths and small thermal depletion. At finite temperature the occupation number
of a quasiparticle state is determined by Bose statistics, ns = ?^fiys^fisfi with ns the Bose distribution
function ns = 1efl!B
s ?1
. Taking into account thermal depletion, the expression for the dynamical
structure factor is
S(q;!) = S0(q)?(!)+
X
i
(ni +1)Si(q)?(! ?!Bi )
+
X
i
niSi(?q)?(! +!Bi ) (10.34)
with S0(q) and Si(q) defined in Eqs. (10.24) and (10.25).
Even though the finite temperature dynamic structure factor is in fact different from the zero tem-
perature one, observables such as the imparted momentum and imparted energy do not depend on
S(q;!) but on S(q;!)?S(?q;?!). This difference is independent of the temperature, at least at
the order of approximation at which Eq.(10.34) was calculated [15, 147, 148].
Therefore, Bogoliubov analysis shows that, provided the temperature is sufficient low and the in-
teractions are weak enough that Eq.(10.34) is valid, Bragg spectroscopy is not a good method for
determining the temperature of the gas.
? Mott Regime
In the homogeneous Mott phase all the 1-ph excitations have an energy separation of order U from the
208
ground state but splitting between them of order J. If the temperature of the system is kBT & U=5,
1-ph excitations begin to contribute to the dynamical structure factor (Eq. (10.17)) and it starts to
develop frequency peaks at frequencies resonant with the energy difference between two 1-ph exci-
tations. These peaks survive even when the difference, S(q;!)?S(?q;?!), is taken and therefore
when Bragg spectroscopy is performed a small frequency response is observed. In the presence of
the trap the analysis is more complicated because also n-hh excitations have to be taken into account.
The relevant temperature scale in this case is kBT & ?N=5 and the small frequency peaks probe the
energy difference between two n-hh, two 1-ph or one n-hh and one 1-ph excitations.
In Figs. 10.9 and 10.10 we plot the imparted energy as a function of the Bragg frequency calculated
from the exact diagonalization of the Bose-Hubbard Hamiltonian for different temperatures and different
ratios of U=J. In Fig. 10.9 we use a homogeneous system with N = M = 9 and in Fig. 10.10 a trapped
system with N = 9, M = 11 at different temperatures. The response is consistent with the previous
analysis: a response almost independent of the temperature in the superfluid regime and the appearance
of low energy Bragg peaks in the Mott regime. We observe some dependence on the temperature in the
intermediate region but it is not as pronounced as the one observed deep in the Mott regime.
10.7 Conclusions
In recent experiments [145], Bragg spectroscopy was performed using a setup where the laser beams for
the Bragg perturbation were the same as those used to create the lattice potential, and the response was
observed. Previous linear response analysis done for translationally invariant systems [148, 151]) found no
scattering when the Bragg momentum equals the lattice momentum. In ref. [151], the authors attributed the
signal observed in the experiments to nonlinear response or to effects of inhomogeneity or finite system size.
Our linear response calculations considering small trapped systems still show no scattering at q = 2?=a
and therefore indicate nonlinear response as the most plausible explanation for the experimental results.
The validity of a linear response treatment can be checked by verifying the linear dependence of the Bragg
signal upon the intensity of the Bragg beams.
In summary, we have studied the linear response of cold atoms loaded in a one dimensional opti-
209
0 5 10 15 20
?Slash1J
0
5
10
15
20
25
30
?E
LParen1Vo
2 Slash1J
2 RParen1
USlash1JEqual15.3 qEqual8?Slash19
0.2
0.4
0.8
2.
3.4KTSlash1J
0 10 20 30 40 50
?Slash1J
0
2
4
6
8
10
?E
LParen1Vo
2 Slash1J
2 RParen1
USlash1JEqual34.7 qEqual8?Slash19
0.5
0.9
2.
4.
7.5KTSlash1J
0 2 4 6 8 10
?Slash1J
020
4060
80100
120140
?E
LParen1Vo
2 Slash1J
2 RParen1
USlash1JEqual1.02 qEqual8?Slash19
0.02
0.04
0.08
0.2
0.33KTSlash1J
0 2 4 6 8 10 12
?Slash1J
0
20
40
60
80
?E
LParen1Vo
2 Slash1J
2 RParen1
USlash1JEqual5.1 qEqual8?Slash19
0.04
0.08
0.2
0.3
0.67KTSlash1J
Figure 10.9: Transfer energy as a function of the temperature for different U=J parameters. The plots are
for a homogeneous system with N = M = 9. The horizontal axis in in units of~.
cal lattice to Bragg spectroscopy and showed that it can be used to probe the excitation spectrum of the
system, with and without harmonic confinement. In the superfluid regime we showed the validity of the
Bogoliubov approximation to describe the very weakly interacting regime and its breakdown as quantum
correlations become important. A new theory beyond the simple Bogoliubov approximation is required
to describe regimes beyond the very weakly interacting one. In the Mott insulator phase we showed how
Bragg spectroscopy can be used to get information about the excitation spectrum: Bragg peaks are centered
around the characteristic Mott excitation gap, contained in an interval whose width is proportional to the
1-ph excitation band width and have an average height which is maximized when the Bragg momentum ap-
proaches ?=a. In the trapped case, Bragg peaks at lower energy reveal information about 1-hh excitations.
Finally, we also discussed how Bragg spectroscopy can be an important experimental tool to determine the
temperature of the system in the Mott insulator phase.
210
0 5 10 15 20 25 30 35
?Slash1J
0
10
20
30
40
?E
LParen1Vo
2 Slash1
J2 RParen1
USlash1JEqual22.5 qEqual6?Slash111
0.20.4
0.71.
3.KTSlash1J
0 10 20 30 40 50 60
?Slash1J
01
23
45
6
?E
LParen1Vo
2 Slash1
J2 RParen1
USlash1JEqual45. qEqual6?Slash111
0.30.7
1.3.
5.KTSlash1J
0 1 2 3 4 5
?Slash1J
0
10
20
30
40
50
?E
LParen1Vo
2 Slash1
J2 RParen1
USlash1JEqual0.1 qEqual6?Slash111
0.08
0.2
0.3
0.6
KTSlash1J
0 2 4 6 8 10 12 14
?Slash1J
05
1015
2025
3035
?E
LParen1Vo
2 Slash1
J2 RParen1
USlash1JEqual5.05 qEqual6?Slash111
0.070.1
0.30.6
1.KTSlash1J
Figure 10.10: Transfer energy as a function of the temperature for different U=J parameters. The plots are
for an trapped system with M = 11, N = 9 and ?=J = 1:875. The horizontal axis in in units of~.
211
Chapter 11
Scalable register initialization for quantum computing in an optical lattice
The goal of building a quantum computer has spurred tremendous progress in coherent control and mea-
surement of small quantum systems. Neutral atoms in optical lattices have been proposed as a suitable
candidate for quantum computing implementation. The appeal of this system stems from the defect-free
nature of the lattice potential, and the long coherence times of the constituent atoms. In order to fully realize
the promised computational speedup of a quantum device, the underlying system should be scalable to a
large number of information carriers or qubits. Indeed, the first two criteria delineated by DiVincenzo [152]
for scalable quantum computation are:
? A scalable physical system with well characterized qubits
? The ability to initialize the state of the qubits to a simple fiducial state.
In many systems, the first criterion can be met by increasing the number of storage components for
the qubits, e.g. in solid state systems the number of dopant qubits in the bulk material could be increased,
in optical lattices the number of trapped atoms could be increases and in ion systems large scale micro-trap
arrays have been proposed [153]. There are two main approaches to satisfy the second criterion [152]. One
is to allow the system to interact with the environment and ?naturally? cool to its ground state and thereafter
use this state as the initial state. The other is to actively cool each qubit by projective measurement to a
fiducial state j0i. A problem arises with these approaches when the ground state of the many body Hamil-
tonian is not a suitable initial state. This is the case for bosonic qubits embedded in systems with periodic
confinement, for example in Josephson junction arrays [154], neutral atoms trapped in electromagnetic mi-
crotraps [155] or optical lattices [46]. For these systems, the underlying dynamics is Bose-Hubbard-like
and the ground state contains residual coherences described by non-zero number fluctuations in each mode.
As shown in Eq.(9.8), these fluctuations scale as gJ=UpM, so if not corrected, the dynamics will impart a
constraint to scalability. In this chapter we show how this can be corrected in two steps, first by introducing
an inhomogeneity to the lattice using a quadratic trapping potential and second by projecting out compo-
nents of the many-body wavefunction with multiply occupied lattice sites by selective measurements on a
molecular photo-associative transition. Provided the measurement strength is sufficiently large, the system
212
does not evolve out of the restricted basis and the measurement can maintain a unit filled register for the
duration of quantum computation. This strategy allows the Mott insulator transition to become a robust
mechanism for register initialization.
11.1 Homogeneous dynamics
It was recognized early on that the Mott insulator transition might be an efficient way to initialize a register
of atomic qubits in an optical lattice for use in quantum information processing. A key advantage of
loading from a BEC is the availability of an initially high phase space density which can be frozen to the
Mott insulator state with atoms occupying most lattice sites. For the homogeneous system (Vi = 0), only
commensurate fillings give rise to a Mott insulator transition. For the purposes of quantum computation,
one particle per well is desirable. In practice, this is difficult to achieve directly because the precise number
of atoms is unknown and the lattice strength is not perfectly uniform on the boundaries. There are proposals
to prepare unit filled lattices using dissipative techniques involving filling the lattice atom by atom [156] or
using Raman side-band cooling [157]. Additionally, it has been shown that one can repair imperfect filling
from a BEC via an adiabatic transfer mechanism between two sublevels of each atom [48]. While these
techniques can initialize the lattice, any mechanism used to prepare a register of qubits in the unit filled
state will suffer a degradation in fidelity, defined as the population in the unit filled state. The degradation
comes from the coupling between the desired unit filled state and undesired states that pertain to the exact
Mott ground state.
In this section we derive the time dependent fidelity in a homogeneous commensurately filled system
prepared at time t = 0 in the unit filled Fock statejTi(See Eq.(9.4)). While our results can be generalized to
higher dimensions, we consider only a one dimension system. We assume an idealized homogeneous lattice
with periodic boundaries and unit filling, g = N=M = 1, where N is the number of atoms and M is the
number of wells. The regime of interest is the strong coupling limit, where first order perturbation theory
is valid. This regime is experimentally achievable in an optical lattice because the tunneling decreases
exponentially with the trap depth.
To study the time dependent fidelity at first order in perturbation theory, in principle we have to
213
consider all the M(M ?1) eigenmodes that span the one particle hole subspace. However, by translational
symmetry only the translationally invariant states are coupled to jTi, and only they have to be considered
for the time evolution. As shown in chapter 9, the translationally invariant states inside the one particle hole
subspace are the bM=2c states in the subspace, given by:
flfl
fl'(1)r
E
=
MX
n;m6=n
Crnmj?nmi;
Crnm =
p2
M sin
 ?(2r?1)jn?mj
M
?
; (11.1)
E(1)r = U ?6J cos
 (2r?1)?
M
?
; r = 1;:::b(M)=2c: (11.2)
The state vector at any time can be written as:
j?(t)i = c0(t)jTi+
bM=2cX
r=1
cr(t)
flfl
fl'(1)r
E
with c0(0) = 1: (11.3)
By solving the Schr?odinger equation to first order in perturbation theory, the time dependent fidelity
to be in the unit filled state F(t) = jc0(t)j2 can be estimated to be:
F(t) = 1?
bM=2cX
r=0
jcr(t)j2; (11.4)
jcr(t)j2 ? 64
 J
E(1)r
?2
sin2
 ?(2r?1)
M
?
sin2
?
E(1)r t=~
2
!
: (11.5)
If the number of wells M  1, the sum in Eq. (11.4) can be approximated by an integral. In this case we
get an expression for the fidelity of the form:
F(t) ? 1?64J
2M
U2?
Z ?
0
dxsin2(x)sin2
 (U ?6J cos(x))t
2~
?
= 1?8
 J
U
?2
M
 
1? cos(Ut=~)J1(6tJ=~)3Jt=~
?
; (11.6)
where Jn is the nth Bessel function of the first kind.
To test the validity of the approximations, we first compare in Fig.11.1 the percentage fidelity given
by the Eq. (11.4) (red) and the numerical solution calculated by the exact diagonalization of the Bose-
Hubbard Hamiltonian (dots). Due to the exponential scalability of the Hilbert space we restrict the calcula-
214
0 0.5 1 1.5 2 2.5
t JSlash1HBar
99.2
99.4
99.6
99.8
100
%Fidelity
MEqual5, gEqual1 , Radical1Radical1ExtensRadical1ExtensRadical1ExtensRadical1ExtensRadical1ExtensRadical1ExtensMJSlash1UEqual0.02
Figure 11.1: Comparisons between the time evolution of the percentage fidelity calculated by diagonalizing
the Bose-Hubbard Hamiltonian (dots), the perturbative solution inside the one particle hole subspace (red)
and the analytic solution Eq. 11.6 (blue).
tions to only 5 wells. We also show comparisons with the analytic solution, Eq. (11.6) (blue). The on site
interaction strength used was U=J = 100.
We observe a very nice agreement between the perturbative and the exact solutions. The agreement
confirms the validity of restricting the evolution to the one particle hole subspace. On the other hand,
because of the small number of wells used for the simulations, we do not expect Eq. (11.6) to be a very
good approximation. Nevertheless, we observe that for short times the analytic solution is a fair description
of the dynamics.
In Fig.11.2 we compare the analytic solution, Eq. (11.6) (blue) with the perturbative solution given
by Eq. (11.4) (red) for a larger system with M = 31. Even though for this large number of wells an exact
solution is not available, we ensure the validity of perturbation theory by choosing the same pMJ=U than
the one in Fig.11.1. In this M = 31 case, which lies in a regime where we expect Eq. (11.6) to be valid, the
agreement between the blue and red curves is very good.
In general the behavior of F(t) consists of fast oscillations with frequency equal to U=~, modulated
215
0 0.5 1 1.5 2
t JSlash1HBar
99.2
99.4
99.6
99.8
100
%Fidelity
MEqual31,gEqual1 , Radical1Radical1ExtensRadical1ExtensRadical1ExtensRadical1ExtensRadical1ExtensRadical1ExtensMJSlash1UEqual0.02
Figure 11.2: Percentage fidelity as a function of time calculated using the perturbative solution 11.4 (red)
and the analytic solution Eq. 11.6 (blue).The number of sites used for the plot is M = 31
by longer oscillations with frequency determined by the zeros of J1. For short times, Jt=~? 1, the fidelity
is
F(t) ? 1?16
 J
U
?2
M sin2
 Ut=~
2
?
; (11.7)
which corresponds to the Rabi oscillations of an effective two level system spanned by jTi and jSi, with
jSi the translationally invariant state directly coupled to the first order ground state (see Eq. (9.9)). For later
times the coupling to other states becomes important. Notice that the time average of the fidelity over many
oscillation periods is hF(t)i = 1 ? 8(J=U)2M. Consequently, the deviation from a perfectly prepared
register, described by 1 ?hF(t)i, is twice as bad as if the system were prepared in the stationary ground
state of the Bose-Hubbard Hamiltonian (Eq. (9.8)). This indicates that a lattice filled with a commensurate
number of atoms in the Mott insulator state may create a more robust quantum computer register than one
prepared dissipatively.
216
Figure 11.3: Schematic of an inhomogeneous lattice filled with N qubits with an onsite interaction energy
U. An externally applied trapping potential of strength Vj = ?j2, e.g. due to a magnetic field, acts to fill
gaps in the central region of the trap. The center subspace R of the lattice defines the quantum computer
register containing K < N qubits.
11.2 Dynamics in presence of the external trap
As mentioned before, in practice it is difficult to prepare an optical lattice with exactly one atom per well. To
arrange the desired configuration, we propose to use an inhomogeneous lattice with open boundaries created
by a weak quadratic magnetic trap. We require fewer number of atoms than available sites, N < M. The
addition of the trap acts to collect atoms near the potential minimum and leaves empty wells (holes) at the
edges. In experiments where the optical lattice is loaded from a BEC, the external trap is already present
to confine the condensate. For simplicity we assume a one dimensional trap with oscillation frequency
given by !T . The magnetic confinement introduces a characteristic energy scale ? = m=2a2!T , so that
Vn = ?n2 (see Eq. (6.1)). To inhibit multiple atom occupation in any well in the ground state configuration,
we require the on site interaction energy U to be larger than the trapping energy of the most externally
trapped atoms, U > ?(N?1)24 .
We define our register as the subspace R comprising K < N wells in the center region of the
the trap (see Fig. 11.3). The barrier space flanking R will act to suppress percolation of holes from the
edges to the center. The estimated probability for holes in R due to tunneling through the barrier is ph ?
Q(N+1)=2
j=(K?1)=2(J=?(2j + 1))
2 = (J=2?)N?K+4(?[K=2]=?[N=2 + 1])2, which is negligible provided the
barrier region is sufficiently large and J=K? < 1.
217
Hereafter, we restrict our attention to the dynamics of the reduced state of the register obtained by
the tracing over spatial modes outside the register subspace of the entire many body wavefunction. The
degree of inhomogeneity is quantified by the ratio ?=J. For 0 < ?=J ? 1, the energy splitting between
Fock states describing particle-hole pairs is small and the model of homogeneous dynamics is valid.
For ?=J & 1, to first order in perturbation theory, the state space is spanned by the unit filled state
in the register subspace, jTiR, and the 2K nearest neighbor particle-hole pairs jS?j i:
jTiR = j1;1;1; :::; ;1;1;1i| {z }
K sites
; (11.8)
jS+j iR = j1;:::; 2|{z}
j
; 0|{z}
j+1
;:::;1i: jjj < (K ?1)=2; (11.9)
jS?j iR = j1;:::; 2|{z}
j?1
; 0|{z}
j
;:::;1i: (11.10)
Here we have introduced coordinates with the site j = 0 coincident with the trap minimum. For each j the
states jS?j iR are distinguished by the two energetically distinct orientations of a doubly occupied site and
its neighboring hole with energies, E(S?j ) = U(1currency1 ?U(2j ?1)). We define the zero of energy coincident
with the state jTiR.
With an eye to the projective measurement, it is important to understand the free dynamics when the
trap is present and the register is initialized in the pure state jTiR inside the register, RhTj?(0)iR = 1.
Here j?(0)iR is the projection of the initial many body state j?(0)i in the register subspace, i.e. j?(0)iR =
jPihPj?(0)i with jPihPj the projection operator that traces over spatial modes outside the register. We
proceed to estimate the fidelity to be in the unit filled state in the register: FR(t) = jRhTjPihjPj?(t)ij2 ?
jRhTj?(t)iRj.
The quantity FR(t) is generally difficult to compute because atoms couple into and out of the reg-
ister. Instead, we will first solve the simpler problem of the fidelity to be in the target state of a com-
mensurately filled inhomogeneous lattice with K sites. We will use this fidelity to estimate FR(t). In the
commensurately filled inhomogeneous system, to first order in J=U, the dynamics can be restricted to the
Hilbert space spanned by the state vectors fjTiR;jS?j iRg, and therefore the time dependent state can be
218
written as :
j?(t)icom = cT(t)jTiR +
X
j;?
e?iE(S?j )t=~cS?
j
(t)jS?j iR: (11.11)
The evolution of the amplitudes fcT(t);cS?
j
(t)g is dictated by the following equations of motion:
i~@@tcT(t) = ?Jp2
X
j;?
e?iE(S?j )t=~cS?
j
(t); (11.12)
i~@@tcS?
j
(t) = ?Jp2e?iE(S?j )t=~cT(t): (11.13)
Using the fact that J=U ? 1, for times Jt .~we can set the value of cT(t) in Eq. (11.13) to 1. Integrating
the equation we obtain
jcS?
j
(t)j2 = 8
 J
U
?2
sin2
?
E(S?j )t
2~
!
; (11.14)
jcT(t)j2 = 1?8
 J
U
?2X
j;?
flfl
flcS?
j
(t)
flfl
fl
2
: (11.15)
After performing the sum, the fidelity Fcom(K;t) ?jcT(t)j2 is :
Fcom(K;t) ? 1?8J
2
U2
 
K ?cos(Ut=~)
 
1+ sin(?(K ?1)t=~)sin(?t=~)
??
: (11.16)
Notice that for ?t ! 0, we recover the short time fidelity in the homogeneous case, Eq. (11.7).
Once Fcom(K;t) is calculated, we can bound the fidelity FR(t) inside the register. Provided N > K
the following inequalities on the time averaged fidelities hold:
hFcom(N;t)i?hFR(t)i?hFcom(K;t)i: (11.17)
The lower bound arises because the probability to be in the unit filled state of a large commensurately filled
lattice, given an initial state which is unit filled, is always less than or equal to the probability to be unit
filled over a smaller subspace of K < N of a non commensurately filled lattice whose register is prepared
in the unit filled state. This inequality holds provided the probability for holes to tunnel into the register
R is small over relevant time scales. The upper bound is a consequence of the fact that unit filling in the
register state is degraded because particles can tunnel in and out of the register. Therefore its fidelity is less
than that of a commensurately filled lattice of the same size K prepared in the unit filled state.
219
0 0.5 1 1.5 2
t JSlash1HBar
99.2
99.4
99.6
99.8
100
%
F com
LParen1K
Equal5,t
RParen1
gEqual1 ,USlash1JEqual100, CapOmegaSlash1JEqual2.7
Figure 11.4: Comparisons between the time evolution of the percentage fidelity Fcom(K = 5;t) calculated
by evolving the initial target state using the Bose-Hubbard Hamiltonian (dots), the restricted one particle
hole basis (red) and the analytic solution Eq. 11.16 (blue). In the plot we assumed a commensurate unit
filled lattice with five sites and infinitely high boundaries.
In Fig. 11.4 we compare this analytic solution, Eq. 11.16 (blue), with the solutions found by propagating
the initial state using the exact Bose-Hubbard Hamiltonian (dots) and by using a restricted basis set of
dimension K(K ? 1) + 1 consisting of the target state and all particle-hole pairs (red). Infinitely high
boundaries were assumed for the numerical solutions. The parameters used for the plot were U=J =
100,?=J = 2:7 and K = 5. We observe complete agreement between the exact points and the red curve,
which justifies the restriction of the dynamics to the one-particle hole subspace. On the other hand, the
analytic approximation accurately reproduces the exact dynamics only for times t . 0:5~=J. This is not
unexpected because the analytic solution only takes into account the states with no separation between the
atomic pair and the hole which are the ones coupled to first order in J to the ground state and sets cT(t)
in Eq. (11.13) to one. In Fig. 11.5 we plot the fidelity for a lattice with more sites, K = 31. Because the
Hilbert space is too large to have an exact solution, we decrease J to get the same JpK=U ratio as the one
used in Fig. 11.4. In this way we can be sure of the validity of the solution constrained to the one-particle
220
0 0.5 1 1.5 2
t JSlash1HBar
99.2
99.4
99.6
99.8
100
%F
com
LParen1KEqual
31,t
RParen1
gEqual1 USlash1JEqual250 CapOmegaSlash1JEqual0.675
Figure 11.5: Percentage fidelity as a function time calculated using the approximated solution 11.16 (blue)
and the fidelity found numerically by restricting the dynamics to the one particle hole subspace. We assumed
infinity high boundaries and K = 31.
hole subspace which is plotted in red. We observe for this parameter regime that the analytic solution (blue)
gives a very good description of the quantum dynamics. This is consistent with the fact that as the ratio
J=U is decreased, the role in the dynamics of other modes different from the jS?j iR modes becomes less
important.
11.3 Measurement
In Sec. 11.2 we showed that the system dynamics can be restricted to a set of two level couplings fjTiR !
jS?j ig. We now sketch how to perform a continuous measurement to drive the register into the unit filled
?target? state jTiR. The full details are contained in [158]. The idea is to apply an external control field
that is resonant with a coupling between the ?faulty? states fjS?j ig and a set of excited states fjM?j iRg.
These states are described by K ?2 atoms trapped in the lattice and a dipole-dipole molecular S +P state
at site j +(1currency11)=2. The bound molecular state is chosen such that the catalysis laser is far off resonance
from other bound states and repulsive potentials, see Fig. 11.6. If the population in the excited states is
221
|T?{|S?j?}
?2J
0?g(P3/2)
{|M?j?}??M S+P3/2
E(S?j)+|Vc|
?M
Figure 11.6: Schematic of the relevant couplings in the problem. The unit filled state jTi describing a
target quantum register and the states jS?j i having one doubly occupied lattice site and a neighboring hole
are coupled to first order in J. A catalysis laser resonantly couples the ground states jS?j i to the excited
states jM?j i describing a bound molecule at the doubly occupied site. The bound states quickly decay and
give the possibility of monitoring population in the ?faulty? register states jS?j i.
easily measurable, for instance by the emission of photons during decay, then the presence of population
in the particle-hole states can be monitored. It is vital that the coupling field be able to spectroscopically
resolve the measurement transition without exciting the target state. This is possible if the atoms in multiply
occupied wells see a shifted excited state, E(M?j ) = E(S?j )+E?U, where for example E is the energy
of a dipole-dipole molecular state. The ?bare? energy Hamiltonian of the system is:
H0 +HBH = Pj;?E(S?j )jS?j iRRhS?j j+E(M?j )jM?j iRRhM?j j
?p2JPj;?(jS?j iRRhTj+jTiRRhS?j j):
(11.18)
When the catalysis laser is turned on, for the bound states of interest, such as the long range
bound states of the 0?g (P3=2) potential [159], the detuning from atomic resonances is several thousands
of linewidths, meaning the atomic saturation is low. In this case, the excited atomic states can be adiabat-
ically eliminated and each atom in a singly occupied well experiences a light shift equal to Vc. Because
there are K singly occupied wells in the jTi state its total single atom light shift is equal to KVc. The jS?j i
222
Figure 11.7: Population in the unit filled register state jTiR during continuous measurement of the register
beginning in the Bose-Hubbard ground state j?gi. The plots show dynamics appropriate to tunneling in
one dimension with U=J = 500. (a) Quantum trajectories corresponding to a null measurement result for
three different register sizes K. The time scale to saturate the target state is independent of the number of
qubits: tsat ? ??1. (b) Long time dynamics for K = 501, N = 551 and finite detector efficiencies ?.
The population in jTiR for ? = 1 is indistinguishable from one. Also shown is the oscillatory dynamics at
fundamental frequency U described by Eq. (11.16) if the measurement is turned off after the target state is
reached. The arrow indicates ?T;T(0).
states have K ?2 singly occupied wells giving a corresponding light shift of (K ?2)Vc. The differential
single atom light shift between these states is then 2jVcj. Therefore, when the control Hamiltonian with
Rabi frequency ?M is turned on resonant, the total Hamiltonian in the rotating frame is:
HI = Pj;?(2jVcj+E(S?j ))jS?j iRRhS?j j+(j2Vcj+E(S?j )?U)jM?j iRRhM?j j
?p2J(jS?j iRRhTj+jTiRRhS?j j)+~?M=2(jM?j iRRhS?j j+jS?j iRRhM?j j));
(11.19)
Any population in the bound molecular states will decay at a rate  M ? 2?, where ? is the single
atom decay rate. For molecular photo-association by red detuned light, the decay products are typically
ground state molecular species or ?hot? atoms meaning the atoms escape the trapped ground states described
by Eq. (11.10). It is possible then to model the system according to a trace non-preserving master equation:
223
_? = ?i=~[HI;?]? M=2
X
j;?
(jM?j iRRhM?j j?+?jM?j iRRhM?j j): (11.20)
Assuming low saturation of the excited states, the dynamics in the ground state is
_?S?
j ;T
= ?i?S?
j ;T
(E(S?j )+jVcj)=~+i(?T;T ??S?
j ;S
?
j
)p2J=~??S?
j ;T
?
_?T;T = i(?S?
j ;T
??T;S?
j
)p2J=~
_?S?
j ;S
?
j
= ?i(?S?
j ;T
??T;S?
j
)p2J=~?2?S?
j ;S
?
j
?:
(11.21)
These equations describe the Bose-Hubbard coupled states with a decay in population of each state
with an atomic pair at a rate 2? = ?M2 M=(4(U=~)2 +( M=2)2), and decay of coherences between each
of these states and state jTi at a rate ?.
This type of evolution characterized by measurement induced phase damping was studied exten-
sively by Gagen and Milburn [160]. We now show that our system can satisfy the conditions for this effect
and in particular can be driven to the jTi state by monitoring the environment for a signature of decay from
the molecular bound state.
For the inhomogeneous system, the state jTi couples to 2K distinguishable states jS?j i. However,
we can define an effective Rabi frequency between the state jTi and the subspace spanned by jS?J i. This
frequency is close to the coupling matrix element between the statejTiand the statejSiin the homogeneous
system, namely 2pKJ. The coherences between the two subspaces decay at a rate ?, and the population
in the subspace fjS?j ig decays at a rate 2?. The ?good? measurement regime as derived in [160] is then:
?M= M ? 1 <~?=2pKJ: (11.22)
The left side inequality ensures that the excited states jM?j i are weakly populated (equivalent to the
condition for adiabatic elimination of these states). The right side inequality ensures that measurement is
sufficiently strong to damp coherences on the time scale that they develop due to tunneling.
When the environment is monitored, for instance by looking for photon scattering from the bound
molecular state, the evolution of ground states can be modelled using quantum trajectories. The success or
failure of the preparation is conclusive with failure probability pfail = 1??T;T(0). Real experiments will
be constrained to finite detector efficiencies ?. For ? = 0, corresponding to nonselective measurement, the
224
system dynamics evolve according to Eq. (11.21). In the case of finite detector efficiencies, we can express
the approximate fidelity to be in the target state is F(?;t) = ?T;T(t) = ? + (1??)??=0T;T (t). Simulations
of successful register preparation and maintenance by measurement with different efficiencies are shown in
Fig. 11.7.
It is necessary to keep the measurement on during a computation to maintain high fidelity in the unit
filled state. If instead the catalysis field is turned off after the target state is reached, the system will freely
evolve as shown in Sec. 11.2. To illustrate this we show in Fig. 11.7 the evolution of the fidelity described
by Eq. (11.16) if the measurement is turned off after the target state is reached.
11.4 Conclusions
In summary, we have shown that efforts to prepare a register of atomic qubits in an optical lattice suffer
from errors inherent in the underlying many body dynamics. By considering the free dynamics in both
homogeneous and trapped systems we have shown that there is a loss of fidelity of initialization of the
unit filled register which scales with the number of qubits because it is not a stationary state. Because our
analysis is based on first order perturbation theory we established the parameter regime where our analytic
approach is valid by comparing it with exact numerical results for moderate numbers of atoms and wells. To
make the Mott insulator transition a robust mechanism for initialization, we have suggested one approach
to correct for the mentioned problems by performing a continuous measurement on the system. While our
discussion has focused on one dimensional dynamics, the method is also applicable to higher dimensions,
which is the relevant regime for scalability.
225
Chapter 12
Conlusions
This thesis gives a global understanding of the basic physics that describes bosonic cold atoms in optical
lattices, starting from the superfluid regime and going into the strongly correlated Mott insulator regime. In
the following I am going to summarize what I consider are the most relevant contributions of this work.
? The equilibrium properties of lattice systems in the superfluid regime were studied by using standard
mean field techniques and quadratic approximations.
? An explicit expression for the superfluid density based on the rigidity of the system under phase
variations was derived. This expression enabled us to explore the connection between the quantum
depletion of the condensate and the quasimomentum distribution on the one hand, and the superfluid
fraction on the other. Also, the superfluid fraction was shown to be a natural order parameter to
describe the superfluid to Mott insulator transition.
? A functional effective action approach, the 2PI-CTP formalism, capable of dealing with nonequilib-
rium situations that require a treatment beyond mean field theory, was studied. Using the patterned
loading system, this formalism was shown to be a powerful tool to go beyond the HFB approximation
and to incorporate the nonlocal and non-Markovian aspects characteristic of the quantum dynamics.
? It was shown that the complicated nonlocal and non-Markovian solutions derived using the 2PI-CTP
formalism reduce to the standard kinetic theory equations when the system dynamics admits two-time
separation.
? Bragg spectroscopy was shown to be a suitable experimental tool to characterizing the Mott insulator
phase and to estimate the temperature of the system deep in the Mott insulator regime.
? It was shown that the use of a lattice with a spatial inhomogeneity created by a quadratic magnetic
trapping potential, together with a selective measurement of atomic pairs allow for the Mott insulator
transition to become a robust mechanism for quantum register initialization.
In the recent years there has been spectacular progress in experimental and theoretical studies of atoms
loaded into an optical lattice. However, there are still many ideas and possible experiments that could help
226
us to address fundamental questions in solid state physics, atomic physics, quantum optics and quantum
information. I want to finish this work by mentioning some of them.
One can use for example multi-component ultracold atoms in optical lattices together with Feshbach
resonances to realize Hamiltonians other than the pure Bose-Hubbard one. Also by loading fermionic atoms
into an optical lattice the Hubbard Hamiltonian could be realized. This could help to answer many theoreti-
cal questions still open in fermionic systems, such as the BEC-BCS crossover and the basic physics behind
high temperature superconductivity. From the perspective of atomic and molecular physics an interesting
experiment could be the use of the Mott insulator state with two atoms per site as a mean to create a molec-
ular condensate. The creation of vortices in individual lattice sites could also allow the study of the integer
and fractional quantum Hall effects in ultracold gases. By introducing noise in a controlled way in opti-
cal lattice, one can also study disordered periodic systems, which are very important in condensed matter
physics. And finally and perhaps one of the most challenging tasks is to develop the necessary techniques
and protocols that allow neutral atoms in optical lattices as a mean to implement a quantum computer.
227
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