ABSTRACT
Title of Dissertation: Two Experiments with Cold Atoms: I. Application of
Bessel Beams for Atom Optics, and II. Spectroscopic
Measurements of Rydberg Blockade Effect
Ilya Arakelyan, Doctor of Philosophy, 2010
Dissertation directed by: Prof. Wendell Hill, III
Department of Physics
In this dissertation we report the results of two experimental projects with laser-
cooled rubidium atoms: I. Application of Bessel beams for atom optics, and II.
Spectroscopic measurements of Rydberg blockade effect.
The first part of the thesis is devoted to the development of new elements of
atom optics based on blue-detuned high-order Bessel beams. Properties of a 4th-
order Bessel beam as an atomic guide were investigated for various parameters of
the hollow beam, such as the detuning from an atomic resonance, size and the
order of the Bessel beam. We extended its application to create more complicated
interferometer-type structures by demonstrating a tunnel lock, a novel device that
can split an atomic cloud, transport it, delay, and switch its propagation direction
between two guides. We reported a first-time demonstration of an atomic beam
switch based onthe combination of two crossed Bessel beams. We achieved the 30%
efficiency of the switch limited by the geometrical overlap between the cloud and
the intersection volume of the two tunnels, and investigate the heating processes
induced by the switch. We also showed other applications of crossed Bessel beams,
such as a 3-D optical trap for atoms confined in the intersection volume of two
hollow beams and a splitter of the atomic density.
The second part of this dissertation is devoted to the spectroscopic measure-
ments of the Rydberg blockade effect, a conditional suppression of Rydberg excita-
tions depending on the state of a control atom. We assembled a narrow-linewidth,
tunable, frequency stabilized lasersystem at480nm to excite laser-cooled rubidium
atoms to Rydberg states with a high principal quantum number n?50 through a
two-photon transition. We applied the laser system to observe the Autler-Townes
splitting of the intermediate 5p3/2 state and used the broadening of the resonance
features to investigate the enhancement of Rydberg-Rydberg interactions in the
presence of an external electric field.
Two Experiments with Cold Atoms: I. Application of Bessel
Beams for Atom Optics, and II. Spectroscopic
Measurements of Rydberg Blockade Effect
by
Ilya Arakelyan
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
2010
Advisory Committee:
Prof. Wendell Hill, III, Chairman/Advisor
Prof. Steven Rolston
Prof. Michael Coplan
Prof. Ki-Yong Kim
Prof. Luis Orozco
EP Copyright by
Ilya Arakelyan
2010
ACKNOWLEDGMENTS
I am truly grateful to my advisors, Prof. Wendell Hill, III and Prof. Steven
Rolston, for the opportunity to conduct my Ph.D. research in their laboratories.
I would like to thank them for their patience, guidance and support during years
of my graduate studies at the University of Maryland. This work could not be
possible without invaluable contribution of my colleagues, Narupon Chattrapiban,
Jennifer Johnson, Emily Edwards, Matthew Beeler, Zhao-Yuan Ma and Tao Hong.
A special thanks goes to my friend and colleague Arseni Goussev for his help,
advice and optimism.
I would also like to thank the members of my dissertation committee for their
useful comments and suggestions.
ii
TABLE OF CONTENTS
List of Figures vi
Abbreviations and Acronyms ix
1 Introduction 1
1.1 Atom Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Rydberg Blockade Effect . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Overview of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Review of Guiding and Splitting Experiments 6
2.1 Magnetic Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.1 Guiding Along Wire with Current . . . . . . . . . . . . . . . 8
2.1.2 Magnetic Beam Splitter . . . . . . . . . . . . . . . . . . . . 10
2.2 All-optical Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Hollow Core Optical Fiber Guides . . . . . . . . . . . . . . . 14
2.2.2 Free-space Red-detuned Atomic Guides . . . . . . . . . . . . 16
2.2.2.1 Red-detuned Beam Splitters . . . . . . . . . . . . . 17
2.2.3 Free-space Blue-detuned Atomic Traps . . . . . . . . . . . . 18
2.2.4 Free-space Blue-detuned Atomic Guides . . . . . . . . . . . 21
3 Generation of 4th Order Bessel Beam for Atom Guiding 24
3.1 Requirements on Blue-detuned Guides . . . . . . . . . . . . . . . . 24
3.2 Generation of Bessel Beams . . . . . . . . . . . . . . . . . . . . . . 26
iii
4 Tunnel Lock 34
4.1 Single Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Atomic Beam Switch . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 Splitter and Delay Line . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.4 Conclusion and Future Directions . . . . . . . . . . . . . . . . . . . 51
5 Quantum Information Science 53
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Quantum Computing with Rydberg States . . . . . . . . . . . . . . 55
5.2.1 Properties of Rydberg-Rydberg Interactions . . . . . . . . . 58
5.2.2 Radiative Lifetime . . . . . . . . . . . . . . . . . . . . . . . 58
5.2.3 Rydberg-Rydberg Interaction Strength . . . . . . . . . . . . 59
5.3 Experimental Progress in Realization of Rydberg Quantum Gate . . 61
5.3.1 Traps for Neutral Atom Qubits . . . . . . . . . . . . . . . . 61
5.3.2 Excitation of Rydberg States . . . . . . . . . . . . . . . . . 62
5.3.3 Experimental Realization of Two-qubit Rydberg Gates . . . 64
5.3.4 Blockade Effect in Many-atom Systems . . . . . . . . . . . . 67
6 Experimental Setup 70
6.1 Cold Atom Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.1.1 Generation of Static Electric Field . . . . . . . . . . . . . . 72
6.1.2 Detection of Rydberg Atoms . . . . . . . . . . . . . . . . . . 75
6.2 Generation of Frequency-stabilized Laser Light at 480 nm . . . . . . 76
6.2.1 Stabilization of Fabry-Perot Resonator . . . . . . . . . . . . 82
6.2.2 Stabilization of Laser Light at 960 nm . . . . . . . . . . . . 86
6.2.3 Laser Linewidth Measurement . . . . . . . . . . . . . . . . . 90
6.2.4 Frequency-doubling of 960-nm Light . . . . . . . . . . . . . 93
7 First Experiments with Rydberg Atoms 96
7.1 Autler-Townes Splitting . . . . . . . . . . . . . . . . . . . . . . . . 96
iv
7.1.1 Rydberg Production Rate Measurement . . . . . . . . . . . 110
7.2 Rydberg Atoms in External Electric Field . . . . . . . . . . . . . . 111
7.3 Discussion and On-going Work . . . . . . . . . . . . . . . . . . . . . 122
8 Summary of Dissertation 125
A Appendix A. Aspects of Light-matter Interactions 127
A.1 Spontaneous Emission Force . . . . . . . . . . . . . . . . . . . . . . 127
A.1.1 Absorption from Laser Beam by Atomic Cloud . . . . . . . . 133
A.2 Velocity Dependent Force . . . . . . . . . . . . . . . . . . . . . . . 134
A.3 Optical Molasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A.4 Magneto-optic Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
A.4.1 Density and Velocity Distribution in MOT . . . . . . . . . . 139
A.5 Dipole Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
B Appendix B. Elements of the Scalar Diffraction Theory 142
B.1 Nondiffracting Solutions of Wave Equation . . . . . . . . . . . . . . 142
C Appendix C. Apparatus for Bessel Beam Experiment 146
C.1 Magneto-optical Trap Setup . . . . . . . . . . . . . . . . . . . . . . 146
D Appendix D. Rydberg Experiment 157
D.1 Autler-Townes Splitting in 3-level Atom . . . . . . . . . . . . . . . 157
D.2 Population Dynamics of Two-level System with Ionization Loss . . 159
D.3 Stark Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
v
LIST OF FIGURES
2.1 Kepler guide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Beam splitter on a chip. . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Red-detuned beam splitter. . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Red-detuned microtraps . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 Doughnut-beam trap . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6 Axicon-generated dark hollow beam . . . . . . . . . . . . . . . . . . 22
2.7 LVIS beam splitter . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1 Double-rectangular-profile and Bessel hollow beams . . . . . . . . . 25
3.2 Computer-generated holograms . . . . . . . . . . . . . . . . . . . . 29
3.3 Layout of Bessel beam generation set-up . . . . . . . . . . . . . . . 32
3.4 Transverse intensity profile of 4th-order Bessel beam . . . . . . . . . 33
4.1 Layout of imaging probe beams . . . . . . . . . . . . . . . . . . . . 36
4.2 Propagation of cold cloud in single Bessel beam . . . . . . . . . . . 37
4.3 Decay rate in single tunnel . . . . . . . . . . . . . . . . . . . . . . . 39
4.4 Schematic of tunnel lock . . . . . . . . . . . . . . . . . . . . . . . . 42
4.5 Transverse confinement in two crossed Bessel tunnels . . . . . . . . 43
4.6 Re-direction of atomic beam by tunnel lock . . . . . . . . . . . . . . 45
4.7 Cloud temperature before and after the switch . . . . . . . . . . . . 46
4.8 Schematics of Switch . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.9 Splitter and Delay line . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.1 Controlled Two-qubit Phase Gate . . . . . . . . . . . . . . . . . . . 57
vi
5.2 Rydberg Two-photon Excitation . . . . . . . . . . . . . . . . . . . . 63
5.3 Rabi Oscillations Between Ground and Rydberg Levels . . . . . . . 64
5.4 Two-atom Rydberg Blockade . . . . . . . . . . . . . . . . . . . . . 65
5.5 Collective Excitation in Blockade Regime . . . . . . . . . . . . . . . 66
5.6 Rydberg Blockade CNOT Gate . . . . . . . . . . . . . . . . . . . . 67
5.7 Dependence of Excitation Fraction on Principle Quantum Number . 69
6.1 UHV chamber for BEC experiment . . . . . . . . . . . . . . . . . . 72
6.2 Configuration of electrodes for electric field generation . . . . . . . 74
6.3 Optical Layout for Blue Light Generation Setup . . . . . . . . . . . 77
6.4 Resonances of Fabry-Perot Cavity and Corresponding Error Signal . 81
6.5 Optical Layout for Stabilization of Fabry-Perot Resonator . . . . . 83
6.6 Schematics of Fabry-Perot Cavity Locking Circuit . . . . . . . . . . 84
6.7 Schematics of Bias and Offset Circuit . . . . . . . . . . . . . . . . . 85
6.8 Schematics of 960-laser Voltage Servo Loop Locking Circuit . . . . . 87
6.9 Schematics of 960-laser Current Servo Loop Locking Circuit . . . . 88
6.10 Schematics of Self-homodyne Method of Laser Linewidth Measure-
ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.11 Self-homodyne Method Power Spectra of 780- and 960-nm Lasers . 92
6.12 Schematics of Frequency-doubler Cavity . . . . . . . . . . . . . . . 94
6.13 WaveTrain Cavity Resonances and Error Signal . . . . . . . . . . . 95
7.1 Autler-Townes Splitting of 5p3/2 . . . . . . . . . . . . . . . . . . . . 98
7.2 Autler-Townes Splitting for MOT Detuning of 16 MHz . . . . . . . 101
7.3 Autler-Townes Splitting for MOT Detuning of 20 MHz . . . . . . . 102
7.4 Autler-Townes Splitting for MOT Detuning of 24 MHz . . . . . . . 103
7.5 Autler-Townes Splitting for MOT Detuning of 26 MHz . . . . . . . 104
7.6 Autler-Townes Splitting for MOT Detuning of 32 MHz . . . . . . . 105
7.7 Autler-Townes Splitting for MOT Detuning of 36 MHz . . . . . . . 106
7.8 Autler-Townes Splitting for MOT Detuning of 40 MHz . . . . . . . 107
vii
7.9 Autler-Townes Splitting for MOT Detuning of 48 MHz . . . . . . . 108
7.10 Autler-Townes Splitting for MOT Detuning of 56 MHz . . . . . . . 109
7.11 Widthof Two-photonResonance ofAutler-Townes Splitting forVar-
ious MOT Detunings. . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.12 MOT Trap Loss for Various Excitation Times . . . . . . . . . . . . 112
7.13 Rydberg Spectroscopy in External Electric Field . . . . . . . . . . . 114
7.14 Rydberg Spectroscopy in External Electric Field at 18.5 Volts . . . 115
7.15 Rydberg Spectroscopy in External Electric Field at 29 Volts . . . . 116
7.16 Rydberg Spectroscopy in External Electric Field at 35.3 Volts . . . 117
7.17 Rydberg Spectroscopy in External Electric Field at 46.8 Volts . . . 118
7.18 Rydberg Spectroscopy in External Electric Field for Various Voltages119
7.19 Quadratic Stark Shift . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.20 Calibration of Electric Field . . . . . . . . . . . . . . . . . . . . . . 121
7.21 Stark Shift with Inhomogeneous Electric Field . . . . . . . . . . . . 124
A.1 Layout of Magneto-optical Setup . . . . . . . . . . . . . . . . . . . 139
C.1 Experimental vacuum MOT chamber . . . . . . . . . . . . . . . . . 147
C.2 Hyperfine level structure of ground and first excited states of rubidium148
C.3 Layout of optical system for laser cooling . . . . . . . . . . . . . . . 149
C.4 Saturated absorption spectroscopy of 5s1/2|F = 2?? 5p3/2|F??
and 5s1/2|F = 1??5p3/2|F??transitions . . . . . . . . . . . . . . 152
C.5 Diagram of diode laser frequency stabilization circuit. . . . . . . . . 154
D.1 Population Dynamics of Two-level System with Ionization Loss . . 160
D.2 Radial Wavefunction of 50s1/2 State . . . . . . . . . . . . . . . . . . 162
D.3 Stark Energy Shift of 56s1/2 State . . . . . . . . . . . . . . . . . . . 163
viii
Abbreviations and Acronyms
NMR Nuclear Magnetic Resonance
AMO Atomic Molecular and Optical
BEC Bose-Einstein Condensate
TEM Transverse Electromagnetic Mode
Nd:YAG Neodymium-doped Yttrium Aluminium Garnet
DHB Dark Hollow Beam
MOT Magneto-optical Trap
LGB Laguerre-Gaussian Beam
SLM Spatial Light Modulator
LVIS Low-velocity Intensity Source
CCD Charge-coupled Device
VGA Video Graphis Array
Ti:Sapphire Titanium-sapphire
SAS Saturated Absorption Setup
CNOT Controlled NOT
SI International System of Units
FORT Far-off Resonance Optical Trap
UHV Ultra-high Vacuum
MCP Microchannel Plate
AOM Acousto-optical Modulator
EOM Electro-optical Modulator
FWHM Full Width at Half Maximum
OBE Optical Bloch Equations
ix
Chapter 1
Introduction
1.1 Atom Optics
Atom optics is a study of the manipulation of atoms. In analogy to ordinary optics,
electron or neutron optics, the goal is to develop a set of tools to create beams
of atoms, reflect, focus, diffract and make the atoms interfere. Atom optics has
several advantages over the traditional photon or particle optics based on electrons
and neutrons. Unlike electrons, atoms are neutral and therefore less susceptible
to stray electromagnetic fields. Being heavier than neutrons, atoms could make
better gravitational sensors. Large mass means short de Broglie wavelength on the
order of a nanometer and below at low particle energies, opening the possibility of
using atomic beams for a high resolution lithography or as precise surface probes.
For example, X-radiation has a wavelength in the range of 0.01 to 10 nm, de
Broglie wavelength of trapped ultra-cold neutrons is below 50 nm, and cold atoms
used in our experiments are characterized by de Broglie wavelength of about 30
nm. Various atoms can be either bosons or fermions, have a large total angular
momentum or a magnetic moment compared to electrons or neutrons, and their
complex internal structure can interact with external near-resonant electromag-
netic fields. Experiments with atomic beams have almost a century-long history
[1], but the recent development of laser cooling techniques opened the possibility of
increasing beam luminosity at large atomic phase space densities significantly and
forming well-collimated, dense atomic beams with well-defined velocity. Atomic
1
beam sources created using laser cooling are relatively straightforward and inex-
pensive to implement, especially compared to the nuclear reactor or the spallation
neutron source required to create a neutron beam.
The first part of this dissertation is devoted to the development of new el-
ements of atom optics based on blue-detuned high-order Bessel beams. We ex-
tended an application of a single Bessel atomic guide to create more complicated
interferometer-type structures by demonstrating a tunnel lock, a novel device that
can split an atomic cloud, transport it, delay, and switch its propagation direction.
1.2 Rydberg Blockade Effect
Quantum information science, a merging of quantum mechanics and classical in-
formation theory, opens new exciting possibilities in data computation, communi-
cation and storage. Over the last decades, the exponential shrinking of electronic
components that process the information may be approaching its limit. The fur-
ther minituarization of transistors on a near-atomic scales will leave no room for
packing more components on a single chip. Quantum mechanics allows us to step
beyond this limit, and in combination with information science, can provide a
physical system to process information on the microscopic level. Such quantum
systems can outperform some of their classical computational counterparts tremen-
dously. Original ideas to exploit the computational power of quantum mechanics
[2] and the development of quantum algorithms [3, 4, 5] motivated significant ef-
forts toward the practical realization of a variety of physical systems applicable for
quantum information.
Efforts to realize quantum computer hardware, a physical system that provides
qubits and gates, are occurring in very different fields, such as NMR [6], linear
optics [7], quantum dots [8], cold trapped ions [9] and neutral atoms [10]. One
of the promising implementations of quantum computer is the system of neutral
atoms, weakly interacting with the environment. The internal level structure can
2
be used as states of qubits while coupling between qubits can be provided by the
dipole-dipole interactions between Rydbergs atoms.
The second part of this dissertation is devoted to the spectroscopic measure-
ments of the Rydberg blockade effect, a conditional suppression of Rydberg exci-
tations depending on the state of a control atom.
1.3 Overview of Dissertation
The Bessel beam project discussed in the first part of the dissertation concerns the
engineering of novel tools for atom optics. In particular, we applied a high-order
Bessel mode laser beam generated by a spatial light modulator to guide a slow
atomic beam of rubidium, and used a combination of two Bessel beams to switch
the direction of the atomic beam propagation from one guide to another, thus
demonstrating a necessary element for an atom interferometer. The main results
of the Bessel beam project consist of the following:
DS construction of an experimental apparatus for laser cooling and guiding
atoms
DS application of a blue-detuned hollow laser beam as an atomic guide
DS demonstration of an atomic beam switch by combining two crossed Bessel
beams
Chapter 2 of the thesis is devoted to the review of various techniques of guiding
and splitting of atomic densities. The experiments reported in this manuscript were
performed in 2003-2005, we describe advances in the field up to those years, our
contribution to the subject, and also mention most recent progress made by other
research groups.
Repulsive optical potentials based on the blue-detuned laser light are discussed
in Chapter 3. We discuss various ways to produce the potentials, their applications
3
for manipulation of atomic densities, and our approach to generate Bessel beams
with a spatial light modulator.
The subject of the dissertation, guiding and splitting of a slow atomic beam
of rubidium is reported in Chapter 4. We first discuss the application of a single
blue-detuned Bessel beam to guide an atomic beam along the direction of gravity.
The properties of this type of guide were investigated for various parameters of
the hollow beam, such as the detuning from an atomic resonance, size and the
order of the Bessel beam. We then report a first-time demonstration of an atomic
beam switch based on the combination of two crossed Bessel beams. We measure
the efficiency of the switch and investigate the heating processes induced by the
switch. We also report other applications of crossed Bessel beams, such as a 3-D
optical trap for atoms confined in the intersection volume of two hollow beams and
a splitter of the atomic density.
The Rydberg blockade project is covered by Chapters 5, 6 and 7. Chapter 5 in-
troduces the application of neutral Rydberg atoms for quantum computing. There,
we describe various approaches to enhance the atom-atom interactions required for
fast quantum gates along with the recent experimental progress.
Chapter 6 is devoted to description of the experimental apparatus we built to
excite Rydberg transitions and investigate Rydberg-Rydberg interactions.
Finally, in Chapter 7 we report the first experimental results obtained with the
our experimental apparatus. The main results of the Rydberg blockade project
consist of the following:
DS construction of a narrow-band laser system for Rydberg excitations
DS spectroscopic measurements of Rydberg states to observe signatures of the
Rydberg-Rydberg interactions
As the laser cooling technique became a standard tool of AMO Physics nowa-
days, we assume that most readers are quite familiar with the subject. Thus, the
4
details of the more common knowledge, such as main aspects of atom-light in-
teractions, selected properties of rubidium, building and control of the cold atom
apparatus are all collected in the Appendices.
5
Chapter 2
Review of Guiding and Splitting Experiments
Chapter 2 is devoted to a review of various experimental approaches to create a
set of atom optics tools that would allow manipulation of slow velocity atomic
beams. This review does not cover control over hot1 beams of atoms, although
that branch of atom optics is quite fruitful [11]. Most of the experiments described
below were aimed at designing guides for cold atoms and atom optics analogs for
optical mirrors, lenses and beam splitters. One incentive was the creation of an
atomic interferometer (using atoms of mass ma and velocity v) that can be more
sensitive to inertial or gravitational effects [12] than the traditional optical coun-
terpart (using photons of energy ?? or wavelength ?ph) of the same geometry,
by a ratio of the order of mac2/?? = ?phc/?dbv, where ?db is the de Broglie wave-
length of atoms. Large-area interferometers require not only well-controlled atomic
beam-splitting but also reliable transfer of atomic densities over large distances.
Another exciting area of application of atomic guides is quantum information sci-
ence: ensembles of atoms as collections of quantum bits can be transported in
space using atomic guides as buses for quantum computers and wires for atom
squids. Another promising application of atom optics includes lithography, where
1We call atomic beams or sources hot if their equilibrium or characteristic temperature is
close to or above the room temperature. We refer to cold atoms as to those slowed down using
laser cooling techniques. Those atoms usually have temperatures below 1 mK but their velocity
distribution is still Maxwellian (thermal) opposed to degenerate gases (such as BEC) that are
not considered in this dissertation.
6
the deposition resolution, ?/2 is not limited by optical wavelengths. We should
point out, that with the development of the field, many experiments investigated
more fundamental issues, such as properties of matter-waves among many other
results of quantum physics.
Motion of neutral atoms can be controlled through the interaction of their
internal structure with an external electromagnetic field. Absorption and consec-
utive emission of photons may change the atom velocity both in direction and a
magnitude. Light forces are considered in some detail in Appendix A. Spontaneous
emission gives rise to a radiation pressure force that may both accelerate or de-
celerate atoms, but, due to its random nature, results in the phase decoherence of
an atomic wavefunction, which can be fatal for quantum information application.
Stimulated emission produces the conservative dipole force that allows the direc-
tion of atomic velocity to be changed coherently. The radiation pressure force is
essential for laser cooling, but we will only rely on the dipole force to manipulate
atoms while trying to reduce the effects of radiation pressure as much as possible.
Another approach to control the atoms relies on the interaction of the internal
magnetic moment of an atom with an external magnetic field. In this review we
separate the all-optical control via light-matter interactions from that via mag-
netic interactions, as the two approaches have different experimental realizations
and their own intrinsic advantages and drawbacks. It is, however, sometimes ex-
perimentally advantageous to combine the two methods together by appropriate
tailoring of the magnetic and optical potentials [13].
We first briefly describe various experiments to control the motion of cold
atoms by magnetic interactions with a special attention to a splitter based on a
current-carrying wire. A direct comparison can be done between this device and
an all-optical atomic beam switch, the subject of the dissertation. The motion
control through the light-matter interactions is the major interest of the review.
Manipulation based on both the attractive red-detuned and repulsive blue-detuned
optical potentials is discussed in detail.
7
2.1 Magnetic Interactions
Ions can be successfully manipulated with external electromagnetic fields. To
control neutral atoms, one can rely on the much weaker interaction between the
atom magnetic moment ? and an external static magnetic field B that gives rise
to the interaction potential Vmag =???B. For example, this potential is below 1
K for rubidium at moderate magnetic fields of less than 2 T, and therefore such an
approach can be applied to slow atoms only, and has to be realized in combination
with laser cooling techniques. An early review of various ion/neutral atom trap
configurations is given in Ref. [14]. For Vmag < 0, the atoms can be trapped in
a local 2-dimensional field maximum while moving unconstrained along the third
dimension. Asimple example of such a neutral atomtrapconfigurationisa current-
carrying wire that was further developed into more complicated and much smaller
fabricated magnetic microtraps, or atom chips. This approach is very tempting for
atom interferometry purposes as the individual interferometer elements not only
can be precisely and reproducibly made, but also combined together on a single
chip to extend to multicomponent experiments.
2.1.1 Guiding Along Wire with Current
For a straight wire, the atom moves along the wire freely with its thermal veloc-
ity. The motion in the plane transverse to the wire is constrained by magnetic
interactions and would follow a Kepler orbit if the orbiting frequency were smaller
than the Larmor precession frequency (rotation of magnetic moment around the
direction of the magnetic field). The resulting trajectory spirals around the wire.
Trapping neutral atoms by the 1/r magnetic field of a current-carrying wire was
proposed and implemented for atom guiding by Schmiedmayer [15]. His group has
later used this approach to guide cold atoms [16]. A 1.6 mm lithium cloud was laser
cooled down to temperature of 200?K and released from the magneto-optical trap
in the vicinity of a 50-?mthick and 10-cm long tungsten wire carrying a current of
8
Figure 2.1: Density distributions of lithium atoms trapped by the magnetic field
of a current-carrying wire for various trapping times from reference [16]. Top row
of images show the distribution across the wire integrated along it; bottom images
represent the distribution along a 2-cm portion of the wire integrated across the
wire.
1 A. Atoms with the appropriate direction of the magnetic moment were trapped
by the magnetic field of the wire and then guided along the wire. Measured density
distributions along the wire and in the plane transverse to the wire are presented
on Fig. 2.1. The Kepler guide loading efficiency was about 10% (2?106 atoms).
Better control over atomic motion can be achieved by tailoring several mag-
netic potentials together. Combination of a Kepler guide with a bias magnetic
field produces so-called ?side guide? [16] where atoms precess along the field min-
imum and move alongside the wire. More complicated structures, consisting of
two wires and a bias field, were realized to provide omnidirectional guiding along
a spiral path [17]. Further miniaturization of the guiding structures created by
lithographic techniques led to microtraps/guides where atoms can be held in steep
magnetic potentials very close to material surfaces. For example, cold rubidium
9
atoms were guided 50 ?m above a curved 10-cm long track made of two 100 ?m
cross-section copper wires deposited on a glass substrate using electroplating tech-
niques [18]. Furthermore, recent advances in nano-technology suggest the possibil-
ity of trapping neutral atoms sub-micron distances away from a dielectric substrate
containing carbon nanotubes [19].
2.1.2 Magnetic Beam Splitter
One example of magnetic microtraps is a magnetic beam splitter [20]. Its purpose
is similar to the subject of this dissertation, the atomic beam switch, and therefore
it is interesting to discuss the properties of this device. A side guide arrangement
was extended to a Y-shaped wire in the presence of a bias magnetic field. As a
result, the atoms could either follow a specific arm of the wire or be split between
the two arms. Fig. 2.2 presents a photograph of fluorescing atoms for different
modes of operation of the device (panel (b)). The blue images show the atoms
following either one of the arms and a 50/50 splitting between the two arms. The
splitting ratio is determined by the current ratio between two arms and is presented
on the right panel of Fig. 2.2. The separation angle between two arms was 15?
and more than than 1?106 lithium atoms were loaded and split by this device.
It is worth mentioning, that this device can be used to split atomic density; the
splitting of the wavefunction would require a single mode operation of the device
for a given de Broglie wavelength.
Atom interferometry requires bright atomic sources and Bose-Einstein conden-
sates (BEC) are ideal for this purpose. Guiding BEC clouds on a microtrap has
been demonstrated along with the splitting by a double-well potential [21, 22],
and the splitting, reflecting, and recombining of condensate atoms by a standing-
wave light field [23]. The area of the prototypes was quite small, which reduces
the sensitivity of an interferometer. For example, a phase shift, ?s, between two
arms of the Sagnac atom interferometer is proportional to its area, A, according
10
Figure 2.2: Beam splitter on a chip: (a) chip schematic (right) and (b) fluorescence
images of guided atoms (right). Atoms can be guided along either of arm of the Y-
shaped wire or split into both sides as they move along the wire with their thermal
velocity [20].
to ??s = 2m?A/planckover2pi1, where m is the atomic mass, and ? is the rotation frequency.
In additional, guiding along a microchip has a serious problem: for a small cross-
section wire, any material imperfection causes deviation of the currents from a
straight line that results in fluctuation of the magnetic field large enough to pro-
duce a break-up of the BEC cloud. Close proximity of the cloud to the material
of the chip is another issue: van der Waals interactions can cause decoherence
(e.g., spin flips) of the atomic ensemble. Microtraps imply permanent design and
thus lack some flexibility as the fabricated chip must be positioned appropriately
with respect to the source of the cold atoms in advance to optimize the loading
efficiency; after that, any modification of the experimental setup that is under
ultra-high vacuum might be challenging. Finally, only atoms with the specific
magnetic moment can be manipulated by the external magnetic field.
11
2.2 All-optical Control
A different approach to controlling the atomic motion relies on the interactions
between an external electric field (laser light) and the induced dipole moment of
the atom. This approach reduces the coupling between the investigated system
and the environment as atoms can be manipulated far away (compared to short
ranges of the van der Waals potentials) from the materials of the chamber walls;
the arrangement of the laser beams can be modified outside the vacuum system,
optical potentials of arbitrary shapes can be modified in real time, for example,
using a spatial light modulator. Optical dipole traps are especially interesting
for trapping atoms with arbitrary magnetic sublevels, or trapping several atomic
species simultaneously. Optical control has a strong limitation: laser beams prop-
agate along straight lines in free space. Various techniques to manipulate atoms
by light are described in the next section.
A microscopic particle suspended in a medium of refractive index n0 and polar-
ized in the presence of the external electric field E is subject to a force due to the
interaction between the field and the induced electric dipole moment2. Classically,
the force expressed as a gradient of the interaction potential is given by
Fg =?12n0??E2, (2.1)
where ? is the particle?s polarizability and n0 is the refractive index of the medium
in which the particle is suspended. With laser light sources, the gradient force
(Eq. 2.1) traps microscopic dielectric particles in a tightly focused Gaussian laser
beam [24]. For a large electric field gradient the particles are attracted to the beam
focus, and the gradient force can dominate gravitational forces, Brownian motion,
and forward-scattering light force, and trap the particle in 3-D [25]. This type of
trap, referred to as optical tweezers, was later extended to hollow laser beams [26].
Optical tweezer is a non-contact method to manipulate macroscopic objects and is
2Electric field is low, such that the higher order processes can be neglected
12
especially powerful for biological applications [27]. Some hollow laser beams, such
as Bessel beams, possess orbital angular momentum that can be transferred to the
particles and cause them to rotate (optical spanners) [28]. Blue-detuned optical
potentials are especially promising as optical tweezers: the null intensity of the on-
axis region reduces the light-scattering force. That minimizes the optical damage
to the trapped particles, which is particularly important for trapping biological
samples.
A microscopic particle, such as an atom, in the presence of a near-resonant laser
light field is subjected to the dipole force as a result of the interaction between the
field and the induced dipole moment of the atom (for details refer to Appendix A).
The force can be either attractive or repulsive depending on the sign of the light
detuning from an atomic resonance. The magnitude of the dipole force depends
on the spatial gradient of the light intensity. This leads to the possibility of either
trapping atoms in the regions of the maximum light intensity (attractive potential)
or confining them in the dark region of the minimum intensity surrounded by light
(repulsive potential); but in both cases the spatial variation of the light intensity
has to be large. For example, atoms placed in a Gaussian (TEM00) laser beam will
be pulled to the beam axis in the transverse plane of the beam and attracted to
the beam focus in the longitudinal direction as described in Ref. [29]; for a higher-
order Bessel (dark-core hollow beam) atoms can be confined in the dark core and
move freely along the beam as reported in Chapter 4 of this thesis.
In most cases one tries to suppress the absorption from the trap light by de-
tuning its frequency, ?, far from an atomic resonant frequency, ?a (on the scale of
the inverse lifetime, ?, of an upper level of a two-level system), because the photon
scattering rate scales as 1/?2 (Eq. A.26), where detuning ? = ???a. For the
moderate light intensities, I(r), the expression for the potential associated with
the dipole force exerted on a two-level atom (Eq. A.45) can be simplified to
13
U(r) = 12??I(r)/Is4(?/?)2, (2.2)
where the saturation intensity Is ? h?c?/3?3 [30], and ? is the wavelength of
light. The dipole potential scales as 1/?, and for a large detuning and a given
light intensity one can achieve the desired value of the potential, while suppressing
heating due to the absorption over the time of an experiment.
The dipole force as the gradient of the potential relies on the spatial variation
of the light intensity I(r) that allows for various trap configurations. For example,
atoms can be moved along a straight line in free space using a red-detuned Gaussian
laser beam. In this configuration the atoms are trapped in three dimensions at
the beam focus: a Gaussian intensity distribution characterized by the beam?s
waist size provides a tight confinement in the transverse plane while a less tightly
confining optical potential (limited by the beam?s Rayleigh range) holds the atoms
in the longitudinal direction. The position of the beam focus and thus the trapped
cloud can be moved by means of a relay optics. As discussed in Section 2.2, for
large enough detunings the scattering rate can be kept low, while the required
optical dipole potential can be achieved for higher laser light intensities according
to Eq. 2.2. For instance, a BEC cloud of 106 rubidium atoms can be trapped at
a focus of a 15 W fiber laser at 1567 nm with the waist of 40 ?m (the Rayleigh
range of 3.2 mm) and routinely moved over a distance of tens of centimeters [29].
2.2.1 Hollow Core Optical Fiber Guides
Guiding atoms along laser beams in free space is limited to straight lines. One
can avoid this constraint by employing a hollow-core optical fiber. For efficient
guiding, the atoms must be kept away from the dielectric material that contains
the light in the fiber. That implies that the optical potential has to be higher than
the sum of the kinetic energy of the atoms in the plane transverse to the fiber3
3Assuming that longitudinal and transverse velocities are comparable
14
and the short-ranged van der Waals attractive potential between the dielectric
and the atoms. Light can be mode-matched with a hollow core fiber to either
propagate along the core, in vacuum, surrounded by a cylinder of the dielectric or
can be internally reflected by the dielectric and travel inside the walls of the hollow
cylinder of a dielectric material. The first configuration in combination with the
red-detuned light creates an optical potential attracting atoms to the dark core
of the fiber, while the second pushes the atoms away from the dielectric walls for
the blue-detuned laser light. The red-detuned guiding of atoms inside a hollow
core optical fiber is similar to that by a Gaussian beam: intensity gradient in the
transverse plane attracts atoms to the maximum of the light field creating a 2-
dimensional trap. As atoms are mostly trapped along the axis of the fiber and far
from the walls, the van der Waals interaction is low. The red-detuned guiding was
experimentally demonstrated by Renn, et al. [31], who used a 3 cm long capillary
hollow core fiber to transport hot atoms from one vacuum chamber to another.
Supported light modes for this type of the optical fiber decay exponentially with
the fiber length thus severely limiting the guiding distance, while a high light
intensity required for the on-axis trapping leads to a significant heating of the
atoms. A strong electric field attenuation can be avoided by the use of a photonic
crystal fiber. The band gap structure of the fiber reduces the propagation loss thus
potentially improving the red-detuned guiding distances. Preliminary experiments
with microscopic dielectric particles [32] show some promise to apply the photonic
crystal fibers for atom guiding.
For the case of the blue-detuned hollow fiber guide, light propagates inside the
dielectric and its intensity exponentially vanishes into the dark core over a distance
of about?/2?. Such evanescent light fields characterized by a large spatial gradient
can be used to expel the atoms from the high-intensity region near the fiber wall
and guide them along the fiber [33]. In the cited experiment, the laser power of 500
mW tuned blue of rubidium D2 line provided a maximum optical potential depth
of 22 mK large enough to constrain the transverse atomic motion and exceed the
15
attractive van der Waals potential. The guiding experiment was later extended to
a laser-cooled atomic sample [34]. In contrast with the red-detuned fiber guides,
the evanescent field potential is generated by lossless modes of light making the
application of long blue-detuned fiber guides more practical. Moreover, as with any
blue-detuned optical potential, atoms spend most of their time in the low-intensity
region, and therefore decoherence or heating of atoms due to spontaneous emission
can be kept small for the evanescent field guides. In addition to the evanescent
field potential, it should be possible to propagate doughnut-type modes in photonic
hollow core fibers. We made a few unsuccessful attempts to exploit that idea, which
is promising and should be investigated more in the future.
The use of hollow core optical fibers offer several advantages over other methods
of guiding atoms. A guiding path is not limited by a straight line as the fiber can be
bent4. A hollow fiber guide can be used to connect two separate vacuum chambers.
Moreover, as the fiber dielectric is transparent at optical frequencies, the atoms
inside the fiber can be probed outside of the vacuum system by a high numerical
aperture optical system. Unfortunately, the input/output coupling of both atoms
and the light with the fiber is quite challenging as it has to be done inside the
vacuum chamber thus limiting the flexibility of the experimental setup.
2.2.2 Free-space Red-detuned Atomic Guides
Early guiding of an atomic beam along a red-detuned Gaussian beam was reported
by Bjorkholm, et al. [35]. A thermal beam of sodium atoms was focused and
steered by a co-propagating red-detuned laser beam, and de-focused by a blue-
detuned beam. The experimental arrangement was limited by small detunings
4The transverse guiding force must provide adequate centripetal force. The tightest bend
radius is given by Rmin =mr0v2z/U(0), where m is the mass of the atom, r0 is the radius of the
light mode propagating in the fiber, vz is the atomic velocity along the fiber, and U(0) the value
of the optical potential on the axis of the fiber [31]
16
(less than 103? in Eq. 2.2) at a relatively low laser power (50 mW) resulted in
significant heating. With the technological advances in laser manufacturing, the
problem of heating was addressed by using far-detuned high power laser beams to
guide atoms loaded from an atomic fountain [36] and a MOT [37]. In the second
experiment, rubidium atoms were released from optical molasses and fell in the
gravitational field of the Earth in a red-detuned Nd:YAG laser beam at 1064 nm
with a corresponding detuning of 1.6?107?. A 40 % loading efficiency was reported.
The laser power of 15 W focused into a beam waist of 100 ?m provided a dipole
potential of 100 ?K in the transverse plane with the maximum scattering rate5 of
1 s?1.
2.2.2.1 Red-detuned Beam Splitters
A combination of two red-detuned Gaussian beams can be applied to guide the
atoms along one of the beams and then split the cloud into two separate propa-
gation directions by a second beam [38]. In this demonstration a rubidium cloud
of 107 atoms cooled down to 14 ?K was released from optical molasses and fell
in a vertical guide (Fig. 2.3) composed of a fundamental mode of Nd:YAG laser.
The 14 W collimated beam of 400 ?m in diameter created an optical potential
of 30 ?K in the transverse plane and provided a loading efficiency of 10% from
the molasses. In 29 ms after the cloud release, the second, oblique guide (10 W,
600 ?m diameter collimated Nd:YAG laser beam) was turned on, when the cloud
reached the intersection region of the two beams, 4 mm below the molasses. As
an atom oscillated across the intersection region in a two-well potential, it could
end up in either of the guides. For the two beams crossed at about 7EW, this device
provided a splitting efficiency of 29%.
The two-crossed guides experiment was later extended to demonstrate more
5Number of photons absorbed and spontaneously emitted per unit time per atom. Note, that
Erec = 3.6 ?K for rubidium D2 line
17
Figure 2.3: The schematics of the red-detuned beam splitter (left), and the pictures
of the cloud, imaged for the vertical guide (a) and for two guides (b). Reproduced
from Ref. [38].
complicated structures such as an array of crossed red-detuned laser beams for
atom guiding [39]. A cylindrical lens transforms a round Gaussian beam into an
elliptical onewith a largeaspect ratio, and can be used as anatom guide in thefocal
plane of the lens. A set of cylindrical microlenses provides a parallel array of the
guides while the second set rotated with respect of the first creates a 2-dimensional
mesh of light that can be used to split, guide and recombine the atomic densities.
A fluorescence image of atoms loaded onto such an interferometer-type structure is
presented in Fig. 2.4. By proper control of the intensity at the intersection points
onecan employ this structure as eitherMach-Zehnder orMichelson interferometers.
The angle between the arms of the interferometer could be adjusted by mutual
rotation of the lens arrays, but the interferometer area of 1 mm2 was quite small.
2.2.3 Free-space Blue-detuned Atomic Traps
In case of the blue-detuned laser light, the corresponding optical potential is re-
pulsive, and atoms can be confined within the dark region surrounded by light.
Practically, this can be achieved by coupling atoms with laser beams of zero central
18
Figure 2.4: Fluorescence image of rubidium atoms loaded from a MOT into a
multiple-path Mach-Zehnder type guiding structure createdby thecombined dipole
potentials of two sets of microfabricated arrays of cylindrical lenses. Reproduced
from Ref. [39].
intensity, so-called dark hollow beams (DHBs). Among those are high-order Bessel-
Gaussian, Laguerre-Gaussian, high-order Mathieu and doughnut hollow beams.
The properties of the various DHBs and techniques of their generation were dis-
cussed in some detail by Yin, et al. [40]. Some aspects of the diffraction theory
applied to the hollow beam generation are discussed in Appendix B. As atoms
in the blue-detuned optical potentials are trapped in the dark regions, they ex-
perience a low photon-scattering and photon-assisted-collision rates, and suffer
minimal light-shift effects of atomic levels. Being all-optical, DHBs can trap and
transport atoms far from material walls thus suppressing attractive van der Waals
interactions.
The first applications of the DHBs as atom traps were experimentally demon-
strated with the development of laser cooling techniques. In a doughnut-beam
trapping scheme [41], a 3-dimensional dipole trap consisted of a blue-detuned
19
Figure 2.5: Schematic illustration of the novel optical trap: an atom cloud is
confined transversely inside a Laguerre-Gaussian beam and longitudinally by two
blue-detuned plugging beams. Reproduced from Ref. [41].
Laguerre-Gaussian beam and two blue-detuned plug beams transverse to the hol-
low beam (Fig. 2.5). Such configuration provided a maximum optical potential
of 40 ?K, high enough to trap 108 rubidium atoms laser-cooled to 10 ?K. In
yet another variation, the gravito-optical surface trap, cesium atoms were confined
inside a DHB and held by a blue-detuned evanescent light in the gravity field [42].
Cold metastable xenon atoms were captured in a dark region of space inside a
single blue-detuned hollow laser beam formed by an axicon [43], a lens which has
a conical surface. The trap region was an elongated diamond of revolution with a
diameter of roughly 1.5 mm and 150 mm of length corresponding to a volume of
about 80 mm2, where up to one million atoms were confined. Since then, various
configurations for the blue-detuned atom traps and refrigerators were proposed
and demonstrated [40] to investigate the possibility of effective evaporative cooling
towards BEC. However the achieved phase-space densities of?3?10?4 are too low
for BEC production, and the blue-detuned traps do not compete with the standard
evaporation cooling techniques in magnetic [29] or red-detuned dipole [44] traps.
20
2.2.4 Free-space Blue-detuned Atomic Guides
Cylindrically symmetric blue-detuned DHBareidealcandidates to transportatoms
along their dark core. Various types of hollow beams can be applied as atomic wave
guides, atomic funnels, atomic motors, beam collimators and splitters. In one of
the earliest demonstrations, a hologram generated LG-beam was employed to guide
an atomic beam of neon for about 30 cm and focus it to a radius of 6.5?m[45]. At
the same time, rubidium atoms from a MOT fell in the gravitational field inside a
LP01 beam generated from an output of a hollow-core optical fiber [46]. Guiding
cesium atoms over a 20 cm distance was reported for an axicon-generated hollow
beam with a 10% efficiency limited by the collisions with the background vapor[47].
A TEM00 laser beam was converted into a DHB with three axicons (A1,2,3) and a
spherical lens (L) as shown on the top inset of Fig. 2.6 along with a photograph
of the transverse beam profile, and the corresponding intensity distribution. A
guiding efficiency of 50% was achieved for a LP01 beam generated from an output
of a hollow-core optical fiber for the atoms falling in gravity over a 10 cm distance
[48].
An axicon-generated DHB, being a mixture of high order Bessel modes, pre-
serves its shape over a limited distance and does not posses phase singularity on its
axis, the property required for non-vanishing null on-axis intensity of the tightly
focused beam. On the contrary, a pure high order Bessel beam is diffraction-free,
has on-axis phase singularity, and thus is very useful as a waveguide and funnel. A
Bessel beam can be generated with a 2-dimensional spatial light modulator (SLM)
[49], and was shown to be successfully employed as an atom tunnel [50]. More-
over, using the SLM allows creation of arbitrary 2-dimensional optical potentials6,
that can be used as billiards or scatterers for atomic wavepackets to study the
connection between classical and quantum chaos [50, 51].
6Continuity of the potential is limited by the resolution of the SLM, which is a digital instru-
ment.
21
Figure 2.6: 1-mm hollow-core beam intensity distribution at the position of the
MOT (graph and bottom inset) and optical components for hollow-core beam
generation (top inset). Reproduced from Ref. [47].
With the advent of the experimental realization of Bose-Einstein condensates,
the application of the blue-detuned optical potentials for manipulation of BEC,
especially creation of a single-mode waveguides, was investigated both theoretically
and experimentally. In particular, a blue-detuned Laguerre-Gaussian (LG) beam
was used to realize a BEC waveguide to investigate dynamic evolution of matter
waves in the loading and waveguiding processes [52]. In this demonstration, a 100%
loading efficiency from a magnetically trapped BEC to a 10 ?m LG beam was
achieved. It has been shown that due to non-adiabatic transfer, a few transverse
modes of the waveguide were occupied, leading to longitudinal heating.
For interferometry applications, one has to be able not only to guide atoms, but
also to split the wavefunction of the atom, and deflect an atomic beam. One of the
first experiments towards interferometry-type structures using blue-detuned opti-
cal potentials demonstrated guiding and splitting of a low-velocity intense source
of atoms (LVIS) [53] using an axicon generated DHB [54]. The hollow beam was
overlapped with the rubidium vapor MOT at angle of 8EW with the LVIS beam and
22
Figure 2.7: Schematic of the low-velocity beam splitter (left), and (a) CCD im-
age of rubidium beam split and guided by the DHB. (c) Spatial profile of the
rubidium beam taken at the location indicated by the arrow. For comparison, (b)
and (d) show the corresponding image and the spatial profile without the DHB.
Reproduced from Ref. [54]
.
directed along the gravity field. Atoms originally inside the hollow beam propa-
gated along it while those outside the guide followed the direction of LVIS beam
due to the radiation pressure imbalance in the MOT beams. For this experiment,
a spatial separation of the two atomic beams was demonstrated (Fig. 2.7), with
the atomic flux in the hollow beam to be 50% of that of a single LVIS beam7.
7Note, that the diameter of the guided atomic beam is smaller than that of the LVIS, and
therefore the total number of atoms in the DHB was smaller that half of the atom number in a
single LVIS.
23
Chapter 3
Generation of 4th Order Bessel Beam for Atom
Guiding
Chapter 3 describes approaches to generate blue-detuned opticalpotentials suitable
to transport cold atoms over long distances. Here we discuss advantages and
limitations of the various hollow beam generation techniques, and described the
experimental setup we used to create a 4th-order Bessel beam.
3.1 Requirements on Blue-detuned Guides
A blue-detuned optical potential is repulsive: atoms are confined in a dark region
surrounded by light. The simple way to create such an intensity pattern is to
block the central part of a Gaussian beam with an opaque disk or by using a
mirror with a small hole to reflect a Gaussian beam. This approach has two main
disadvantages that make this approach impractical. Because of the power loss, the
conversion efficiency is low, and the radial intensity distribution along the beam is
not preserved, as its propagation is governed by Fresnel or Fraunhofer diffraction
of the disk (hole), while the distribution of the remaining part is still Gaussian.
When choosing an appropriate hollow beam to guide atoms one wants to avoid
those limitations and have both the high conversion efficiency and conservation of
the beam transverse profile along the propagation direction. The most common
way to generate a dark hollow beam (DHB) relies on a mode conversion of a
TEM00 laser cavity mode into a desired superposition of another complete set of
24
Figure 3.1: Ideal dark hollow beam of inner diameter 2a with double-rectangular
(b?a) wide transverse intensity profile I(r) (left), and a normalized transverse
profile of a 1 mm diameter, 4th-order Bessel beam at 780 nm (right).
optical modes. However, a hollow beam can be generated directly from a properly
designed laser cavity if the TEM?011 is predominantly excited [55]. In addition, the
magnitude of the repulsive force is larger for a steeper optical potential and thicker
guide walls reduce probability of tunneling through the walls. The ideal candidate
for an atomic guide would be a double-rectangular-profile hollow beam with the
rectangular transverse intensity distribution, I(r) invariant along the beam axis
(Fig. 3.1). Such a beam has not been generated yet. The practically generated
hollow beams are designed to have transverse profiles similar to the rectangular
beam, and are characterized by the intensity profile I(r), and the inner and outer
diameters a and b. As an example, a normalized transverse profile of a 1 mm
diameter, 4th-order Bessel beam at 780 nm is plotted in Fig. 3.1, right. The beam
radius is defined as the position of the first maximum of the Bessel function.
1The TEM?01 is a superposition of TEM01 and TEM10 modes with a relative phase shift of
pi/2
25
For some applications, such as lithography, it might be desirable to be able
to focus a hollow beam while preserving zero electric field on the beam axis. In
general, this can only be done for a hollow beam with a phase singularity along its
axis, such as a high-order Bessel beam. A high-order Bessel beam is a diffraction-
free, hollow laser beam, and its field amplitude is given by
En(r,?,z) = Aw(z)Jn(krr)eikzzein?, (3.1)
where A is a normalization constant, w(z) = w0radicalbig1 + (?z/?w20)2 is the beam
waist at z (w0 being the beam waist at z = 0), kr and kz are the transverse
and longitudinal components of the wavevector, and Jn is the nth-order Bessel
function. This beam carries infinite amount of energy. Practically, the transverse
profile should be modified by a Gaussian function, thus forming a real Bessel-
Gaussian beam (BGB)2[56]. The electric field amplitude for a linearly polarized
nth-order BGB is given by
En(r,?,z) = Aw(z)Jn(krr)eikzzein?e? r
2
w2(z). (3.2)
Atoms are usually confined within the first ring of light in guiding experiments,
and for a nearly collimated Bessel beam of a 1 mm diameter its propagation is
nearly diffraction-free as w(z) ? r0, where r0 is radius of the first ring. Under
these experimental conditions, the first ring of the Bessel-Gaussian beam matches
that of the Bessel beam.
3.2 Generation of Bessel Beams
There are three main approaches to generate a Bessel-Gaussian beam: the geomet-
ric optical method, the optical holographic method, and the computer-generated
2This is an approximation. The electric field given by Eq. 3.2 does not satisfy the wave
equation.
26
holographic method. They rely on a mode conversion of a Gaussian or a Laguerre-
Gaussian (LGB) laser beam into the Bessel beam. All three methods can be
understood in terms of the scalar diffraction theory outlined in Appendix B. Here
we briefly describe the basics of each generation technique with the emphasis on
the versatile computer-generated holographic method implemented with a spatial
light modulator. This particular approach has been used to create the Bessel
beams discussed in the dissertation.
The geometric optical method can be applied to generate a Bessel beam by
illuminating an axicon, a lens with a conical surface, with a LGB [57]. In this case,
the field distribution behind the axicon placed at the waist of the LGB is given
by the Fresnel diffraction integral, and the resulting intensity distribution in the
transverse plane behind the axicon is proportional to
I(r,z)?z2n+1exp(?2z
2
z2max)J
2
n(krr), (3.3)
where zmax is a ?propagation distance? defined by
zmax = w0k/kr, kr = k(n?1)?a, (3.4)
where n and ?a are the refractive index and a small internal angle of the axicon,
respectively. According the Equation 3.3, the generated beam approximates a
Bessel beam of order n close to the optical axis and within the ?propagation
distance? zmax.
The optical holographic method is based on the mathematical connection be-
tween Bessel beams of the adjacent order. In the integral representation, the
zeroth-order Bessel function can be written as
J0 = 12?
integraldisplay 2pi
0
eixsin?d?, (3.5)
while the nth-order Bessel function is given by
27
Jn = 12?
integraldisplay 2pi
0
ei(xsin??n?)d?. (3.6)
In particular, the first-order Bessel beam can be expressed as
J1 = 12?
integraldisplay 2pi
0
e?i?eixsin?d?. (3.7)
Experimentally, the e?i? factor in the integrand of the Eq. 3.7 can be obtained
by adding a circular wedged phase plate (e.g., a transparent plate of a variable
thickness) to the configuration of J0 beam [58]. In this case, the phase plate could
be a disk whose optical thickness increases linearly with the polar angle.
A plane-wave (large collimated Gaussian laser beam) illuminating a phase mask
given by a transmission function
T(r,?,0) =
?
??
??
ein?e?ikrr r?r0,
0 r>r0,
(3.8)
is converted into a beam-like wave field with a Bessel transverse intensity profile
|Jn(krr)|2 within the propagation-invariant distance given by
zmax = 1k
r
?D
? . (3.9)
Here D is the diameter of the hologram, ? is the wavelength of light, (r,?) are the
polar coordinates in the plane z = 0 of the hologram, and kr is a radial spatial
frequency of the Bessel beam of order n, as defined in Eq. 3.4.
The transmission function (3.8) can be realized by a dielectric disk whose opti-
cal thickness varies as (?krr+n?) in polar coordinates. The order n of the Bessel
beam is determined by the integral number of full revolutions around the mask
center to produce the phase shift of 2?. The radial spatial frequency kr sets the
position of local maxima of the Bessel function, and thus the diameter of the first
ring. A separate dielectric disk needs to be manufactured for each set of mask
parameters (n,kr).
28
Figure 3.2: Phase masks - contour plots of the transmission function (3.8) (bottom
row) and the corresponding surfaces of the constant phase (top row) of a first order
Bessel beam. The total phase (right bottom) is a sum of the angular term n? (left
bottom) and the radial term?krr (middle bottom).
A more versatile method to generate the phase mask involves application of a
spatial light modulator (SLM), which allows dynamical adjustment of both n and
kr by a computer-generated hologram. An example of a 2-dimensional represen-
tation of the dielectric disk would be a contour graph shown in Fig. 3.2 (bottom
row) for a first order Bessel beam. The gray scale of the graph varies from white
to black, which corresponds to a phase change of the wave field from 0 to 2?. Now,
instead of using a dielectric disk of a variable thickness, we employ a thin phase
mask which shifts the phase of the light field at each point of the (r,?) plane of
the mask according to (?krr+n?) for a given kr and n. The left mask in Fig. 3.2
corresponds to the angular increment of the phase n?, the middle - to the radial
?krr, and the right is the sum of the two. The resulting surfaces of constant phase
of the diffracted light fields are shown in the top row. Practically, the phase mask
is discrete and has a finite resolution; therefore, one can only adjust the phase of
the light at each pixel of the mask, and the part of the incoming light field within
that pixel area would be phase-shifted by the same value. The value of the shift
is also discrete and can be changed from 0 to 2? in discrete steps determined by
29
a dynamic range of the mask. Thus, the higher the resolution and the dynamic
range of the generated mask the better it represents the continuous transmission
function (3.8). The phase mask can be realized as a liquid crystal matrix (the
active element of the SLM), a 2-dimensional array of polar molecules transparent
to the incoming light. The orientation of the molecules of each pixel, and thus the
local index of refraction responsible for the phase shift, can be controlled by an ex-
ternal electric field through an applied voltage. The voltage is computer-generated
for each element of the liquid crystal matrix (pixel of the SLM) in accordance with
the required transmission function (3.8).
In our experiments we used a Spatial Light Modulator produced by the Hama-
matsu Corp. A grey-scale contour plot similar to that in Fig. 3.2 was computer
generated for a given order of the Bessel beam n and a specific diameter of the
first ring set by kr. The plot?s 8-bit image as an array of voltages was sent through
a video-graphic array (VGA) port of the computer to the SLM, thus controlling
the phase shift at each element of the liquid crystal matrix with a 256-level depth.
The physical size of the pixel was set by the active square cross-section area of the
modulator of 2.5cm?2.5cm and the resolution of 768?768 pixels. Technically,
the control voltage for each pixel is created in the following way. The voltage from
the computer sets the power of the SLM?s internal laser. The internal laser beam
propagates onto an array of photoelectrodes, which place voltages across a liquid
crystal matrix. As the voltage is applied across the crystal, the polar molecules
re-orient themselves. When a molecule in the liquid crystal re-orients, the index
of refraction changes in that region of the crystal, thus retarding the phase of the
corresponding portion of the external light field [59]. More details on the inter-
nal structure and the functionality of a particular model of the SLM used in our
experiments can be found in reference [60].
The optical setup to create the Bessel beams used in our experiments is pre-
sented in Fig. 3.3. A Coherent 899 Ti:Sapphire laser pumped by a Coherent Innova
400 Ar+ laser generated a Gaussian laser beam at 780 nm with the output power
30
of 500 mW. A small portion of the light was reflected from the main beam and
coupled with a saturated absorption setup (SAS) to calibrate the laser frequency.
The main beam was horizontally polarized by a half-wave plate, expanded by a
telescope consisting of two spherical lenses L1 and L2 to cover the active area of the
SLM. As our SLM was reflective, we used long optical paths of several meters be-
tween mirrors M1, M2 and the SLM to reduce the angle between beams incident to
and diffracted from the SLM. The computer-generated phase mask corresponding
to the transmission function (3.8) assumes normal incidence. The small deviation
from the normal incidence gives rise to a slight ellipticity of the Bessel rings that
can be compensated by changing the aspect ratio of the computer-generated holo-
gram. Also, note that the incident light has to be linearly polarized along the
orientation axis of the liquid crystal molecules at zero external voltage. This was
achieved by a half-wave retardation plate that set the linear polarization of the
Ti:Sapph beam to provide the maximum conversion efficiency from the Gaussian
to Bessel transverse optical modes. As an example, a photograph of a transverse
intensity profile of the 4th-order Bessel beam is shown in Fig. 3.4.
We have done an extensive investigation of the transverse profiles of Bessel
beams of various orders n and transverse spatial frequencies kr generated for dif-
ferent resolutions and dynamic ranges of the SLM to find the best combination
suitable for guiding cold atoms inside the first ring of the hollow beam. In particu-
lar, we tracked three main parameters important for the guiding: the transverse in-
tensity profile, the maximum propagation-invariant distance and the optical mode
conversion efficiency. The transverse profile determines the repulsive optical poten-
tial for the atoms and its quality is important for atom confinement. It is essential
to minimize the amount of leakage light inside the dark core of the beam to re-
duce the photon scattering rate, to ensure the homogeneity of the light intensity
distribution in the first ring as a function of the polar angle, and to create a large
radial intensity gradient that provides a higher repulsive force. The larger intensity
gradient corresponds to a smaller width of the first Bessel ring at the same optical
31
Figure 3.3: Optical layout of the Bessel beam generation setup. The Gaussian
output of a Ti:Sapphire laser is expanded by spherical lenses L1 and L2, polarized
by a half-waveplate (?/2) and diffracted from the surface of the SLM to form a
Bessel beam. A part of the Gaussian beam is split by a polarizing beam splitter
(PBS) and sent to a saturated absorption spectroscopy setup (SAS) to monitor its
detuning. The Ti:Sapphire laser is pumped by a 6 W argon laser. M, M1 and M2
are dielectric mirrors.
power, and can be increased with the order of the Bessel beam. For example,
an n = 20 beam provides a much more attractive confining potential than one of
n = 4, but has a much larger radius of the first ring for a fixed spatial frequency kr.
The ring?s radius can be reduced with kr, but at some point the beam quality be-
comes limited by the resolution of the SLM, and we could only use Bessel beams of
low order guides for small atomic clouds. The propagation-invariant distance sets
the quality of the Bessel beam as an atom guide of a constant cross-section. This
distance is set by the size of the experimental vacuum chamber or the desired dis-
tance over which the atoms have to be transported. For our experimental setup we
required an invariant-propagation distance of 25 cm, while invariant-propagation
distances of over a meter were demonstrated for the applied Bessel beams. Finally,
we also tried to optimize the SLM efficiency to convert the power of the Gaussian
32
Figure 3.4: A photograph of transverse intensity profile of 4th-order Bessel beam
(left) and a one-dimensional slice of the beam?s intensity distribution along radial
direction (right).
optical mode into the Bessel?s first ring, as the higher total power in the first ring
of the hollow beam increases the height of the repulsive optical potential seen by
the atoms. The most interesting results on the properties of the hollow Bessel
beams generated by an SLM were reported in reference [50]; more technical details
and numerical simulations can be found in Ref. [60].
After investigation of the transverse profiles of various Bessel beams and their
ability to confine atoms within the first ring of the beam, we have found that the
most suitable candidate for a parallel atomic guide would be a 4th-order Bessel
beam with the transverse spatial frequency kr = 10635m?1. The corresponding
transverse intensity profile is presented in Fig. 3.4; it is characterized by the radius
of the first ring (position of the first maximum of the Bessel function) of 0.5 mm
and the total width of the first ring of 0.7 mm. The invariant-propagation distance
of the tunnel was over 1 m. We found that the SLM conversion efficiency under
those conditions was about 20 %, when 500 mW of the Gaussian laser beam were
converted by the SLM into 100 mW of power in the first ring of the Bessel beam.
Various devices based on the generated Bessel beam were demonstrated and are
described in Chapter 4.
33
Chapter 4
Tunnel Lock
As discussed in Chapter 2, optical dipole potentials are useful for such applications
as atom interferometry, lithography, quantum information and biological physics.
Repulsive blue-detuned optical potentials are especially promising for trapping
particles in the null intensity on-axis regions, where light scattering is greatly
reduced. A high-order Bessel mode, non-diffracting solution of the wave equation
withthe transverse profilegiven by a Bessel function, isideal asanatomicguide ofa
constant cross-section to transport cold atoms over large distances. The generation
of the Bessel beams suitable foratomoptics was discussed in Chapter 3. In Chapter
4 we will show how onecan apply a Bessel beamas a building block to create several
elements from the atom optics toolbox: a single Bessel beam can be used as a guide
to transport atom; a tunnel lock, proper combination of two beams allows us to
switch the direction of an atomic beam, to split an atomic density between two
beams, and delay the propagation of the atomic beam.
4.1 Single Tunnel
We first consider an application of a single blue-detuned Bessel beam for guiding
laser-cooled rubidium atoms. As described in Chapter 3, we chose a 4th-order
Bessel beam with the transverse spatial frequency kr = 10635m?1. The beam had
100 mW of power in the first ring of 1 mm in diameter that set the ring intensity,
and therefore the dipole optical potential, at a fixed detuning. As discussed in
34
Chapter 2 (Sec. 2.2), the height of the dipole optical potential is inversely propor-
tional to detuning from an atomic resonance at high values of the detuning and
moderate laser intensities. Lower detunings provide higher optical potentials for
the atoms, but increase the photon scattering rate, which results in heating and
acceleration of the cold atomic sample. For example, the detuning by + 4 GHz1
from the 5s1/2?5p3/2 transition in rubidium provides the maximum value of the
optical potential of 300 ?K for the Bessel beam of our choice, high enough to
confine atoms laser-cooled and confined by a magneto-optical trap (MOT). The
corresponding heating rate due to the absorption is less than 1?K per millisecond.
The height of the optical potential and the photon scattering rate are calculated
in Appendix A.
We used a standard magneto-optical trap as a source of cold atoms. The
principle of the MOT operation is discussed in Appendix A, while the particular
design of our experimental apparatus for laser cooling is presented in full detail in
Appendix C. Here we only mention the MOT parameters relevant to the guiding
experiments. Our demonstrations of all Bessel beam applications were performed
with a 1 mm diameter Rb cloud collected from a vapor of 10?8 torr into a MOT
held at 250 ?K. A Bessel beam tunnel of 1 mm in diameter was blue-detuned 4
GHz from 5s1/2|F = 2??5p3/2|F? = 3?rubidium cycling transition. Due to the
absorption of light, the atoms can decay to the lower hyperfine ground state level
5s1/2|F = 1?, which is - 6.8 GHz from the cycling transition. For those atoms,
the tunnel is red-detuned, and the corresponding optical potential attract atoms
to the high-intensity walls of the tunnel, where they can be heated and pumped
back to the cycling transition.
The propagation direction of the Bessel beam was close to vertical, 6EW from
vertical. The guide was aligned with the trapped MOT cloud by maximizing the
number of atoms inside the dark core of the hollow beam. To load the tunnel,
1The ?+?-sign indicates that the laser frequency is blue-detuned from the atomic resonance.
35
Figure 4.1: Two probe beams were used to capture the cloud dynamics along and
transverse to the tunnel.
MOT laser beams were blocked by a mechanical shutter and the Bessel beam was
turned on. This event set time t = 0 for the evolution of the atomic density in
the presence of the Bessel beam. The atomic density distribution was monitored
by an absorption imaging technique with two probe laser beams, one along and
the other one across the tunnel. The schematic of the imaging system is presented
in Fig. 4.1. To improve the signal-to-noise ratio we took a background image
with no atoms in the field of view and subtracted it from the image with the
cloud. The experimental sequence, MOT loading, propagation in the tunnel for a
fixed time t, and cloud imaging were repeated many times. The resulting column
density distributions averaged for 40 individual photoshots at each evolution time
36
Figure 4.2: Dynamics of a cold rubidium cloud in the presence of a single tunnel.
The top row of images shows propagation along the tunnel; bottom row confirms
the confinement in the transverse direction.
are presented in Fig. 4.2.
The images in the top row were taken across the tunnel: the initial round cloud
at t = 0 starts falling in the gravitational field, expands according to its thermal
energy, while experiencing the dipole force from the optical potential of the Bessel
beam. The atoms inside the tunnel move freely along the tunnel while confined
by the tunnel walls in the transverse direction, when their corresponding kinetic
energy at the wall was less then the height of the optical potential. The rest of the
atoms, up to 40 % of the initial number of atoms in the MOT, too energetic to
be trapped by the tunnel walls, leave the first ring of the Bessel beam and are not
taken into account when investigating the properties of the atomic guide. At t = 14
ms there is a noticeable separation between atoms trapped inside and those outside
the tunnel, and the atomic cloud takes the shape of the tunnel. This separation is
more pronounced when the transverse density profile is monitored along the tunnel
(the probe beam for absorption imaging and the Bessel beam were aligned) as seen
37
on the bottom row of images in Fig. 4.2. At t = 8 ms the separation between the
trapped atoms and those outside of the tunnel is quite clear. Also note, the cloud?s
optical density, when imaged along the tunnel, is much higher than when imaged
across the tunnel, which provides a high signal-to-noise ratio even at longer times.
We calculated the relative number of atoms trapped in the tunnel at a given time
as a column density integrated along the direction of the probe beam. The absolute
number of atoms was deduced by calibrating the relative number of atoms in the
freely expanding MOT cloud as described in Appendix A.
We have tracked the atomic cloud confined in the transverse plane of the Bessel
beam and free-falling along the beam for up to 70 ms. The time of the experi-
ment and the propagation distance were limited by the collisions with the room-
temperature background gas due to a relatively high rubidium vapor pressure in
the experimental chamber. We estimated the total single tunnel loss rate due to
the background gas collisions together with the leak of hot atoms through the tun-
nel from a series of images similar to those in Fig. 4.5 (bottom row). The number
of atoms in the tunnel as a function of time is plotted in Fig. 4.3 (top), along with
an exponential fit of the data taken between 6 and 70 ms, and normalized residuals
from the fit (bottom). In a simplified model, the total decay rate is determined by
the background collisions and the leakage through the tunnel walls, as discussed
in Ref. [60]. Here we treat the decay as a single exponential, y = Aex/? + y0,
to use the resulted decay time in calculations of the switch efficiency. From the
fit, we found the 1/e-decay time ? = (15.9?1.41) ms and the reduced chi-square
?2red = 4.71.
As atoms propagate along the tunnel, they scatter photons and are accelerated.
Depending on the detuning, atoms with high kinetic energy are not trapped by
the tunnel, resulting in a cut-off of the high-energy tail of the Maxwellian velocity
distribution of the remaining cloud in the transverse plane. At small detunings the
dipole potential is high, but so is the scattering rate. Due to light absorption, the
atoms are heated and accelerated along the tunnel, their kinetic energy exceeding
38
Figure 4.3: The number of atoms confined in a single tunnel is shown as a function
of time along with an exponential fit at the detuning of + 4 GHz (top), and
normalized residuals resulted from the fit (bottom). The error bars represent the
standard deviation in the mean of forty runs. The exponential fit yields the 1/e-
decay time ? = (15.9?1.41) ms and the reduced chi-square ?2red = 4.71.
the height of the tunnel potential. At larger detunings, the scattering rate, ?,
and therefore heating of the atoms, drops faster (? 1/?2) than the height of the
dipole potential (?1/?), and atoms can be confined in the tunnel for longer times.
To find the optimum detuning, we compared the number of atoms left in the
tunnel at t = 16 ms for various detunings. The choice of time was determined
by the duration of the switch experiment. We found, that the number of confined
39
atoms reached its maximum value at the detuning of about + 4 GHz. At higher
detunings, atoms gain less thermal energy from the absorption, but the reduced
height of the trapping potential results in the leakage loss. In this case, the atom
number in the tunnel will drop faster than at short times as the most energetic
atoms will leave the tunnel in several milliseconds. The number of atoms left in the
tunnel will decay due to collisions with the background gas similar to the tail of
Fig. 4.3. As the number of trapped atoms dropped with the detuning, we observed
the number to be about 10% of its maximum value at a detuning of + 5 GHz.
However, further detuning gave rise to an increase of the atom number to about
30% of the maximum value at a detuning of + 6 GHz, and the number gradually
fell for higher ?. This local maximum as a function of the detuning is thought to
be due to the internal level structure of rubidium-872. As discussed before, atoms
can decay into 5s1/2|F = 1?ground state. The closer the detuning of the Bessel
beam to the hyperfine splitting of the ground state, the more the tunnel pumps the
population back to the 5s1/2|F = 2?state, acting similar to the repump beam of
a standard MOT setup. Thus, we attribute the increase of the number of trapped
atoms to the repumping effect of the Bessel beam at the detuning of + 6 GHz.
The most effective repumping occurs on resonance, when the detuning is equal to
+ 6.8 GHz, the hyperfine splitting of the ground state. However, as the detuning
is increased from 6 to 6.8 GHz, the tunnel starts acting as a red-detuned optical
potential for the atoms in|F = 1?state, and those atoms will be attracted to the
tunnel walls by the potential. For the case of a multi-level atom, Eq. 2.2, valid
for the two-level atom, has to be generalized as the dipole potential depends on
the particular sub-state of the atom [61]. In some situations, the complex internal
structure can be used to our advantage, for instance, in the splitting experiment
with 85Rb as described at the end of this Chapter.
An extensive experimental investigation of dynamics of atoms trapped in the
2Rubidium level structure is discussed in Appendix C.
40
Bessel beam, theoretical analysis and numerical simulations are discussed in details
in reference [60].
4.2 Atomic Beam Switch
After characterizing the properties of the Bessel beam as an atomic guide, we
may apply it to demonstrate more complicated devices. For interferometry-type
structures or Quantum Information applications, an atomic beam may need to
be switched from one direction to another, an atomic density split into separate
atomic beams, or the propagation of an atomic cloud delayed. All three scenarios
can be realized by a tunnel lock, consisting of two properly gated hollow Bessel
beams. Consider two crossed, near-vertical atomic guides as shown in Fig. 4.4.
Let the atoms start propagating along tunnel A. If both tunnels are on (solid red,
middle cross in Fig 4.4) the atoms can not penetrate the intersection region of the
two Bessels beams3. If the intensity of the second tunnel B is lowered (entrance
gate open; left cross in Fig 4.4) the atoms propagate into the intersection region
to be positioned in both tunnels simultaneously. When the intensity of the second
tunnel is raised (entrance gate closed; middle cross in Fig 4.4), the cloud will be
trapped in three dimensions, and its propagation delayed. If the intensity of the
second tunnel is reduced (left cross in Fig 4.4), the atoms will follow the original
propagation direction after some delay thus realizing a delay line. If instead the
intensity of the first tunnel is reduced (exit gate opened; right cross in Fig 4.4),
the cloud will propagate along tunnel B, a different direction with respect to the
original propagation, thus realizing an atomic beam switch.
The prototype tunnel lock was used to divide and switch the direction of a
moving ensemble of cold atoms. We demonstrated the switch by launching a
portion of a MOT cloud into a tunnel and then deviating the direction of the
3We assume that the optical potential due to the Bessel beam is much higher than the kinetic
energy of the atoms.
41
Figure 4.4: Schematic of the tunnel lock, formed by crossing two tunnels, and cloud
images taken along and across the tunnels. Red (clear) cylinders indicate tunnels
being on (off). From left to right: a) only tunnel A is on (atoms enter overlap
volume), b) both tunnels are on (atoms trapped in the intersection volume), c)
only tunnel B is on (atoms leave overlap volume). Appropriate switching of the
tunnels generates a beam switch, a splitter and/or a delay line.
moving cloud with another tunnel. The tunnels were created by splitting the Bessel
beam diffracted from the SLM into two beams with an optical beam splitting cube.
Both tunnels were aligned with the MOT cloud and crossed at an angle of 12EW .
Both beams were nearly in the vertical direction as shown in Fig. 4.4. In order
to distinguish cloud propagation in tunnel A from that in tunnel B, we monitored
the atom distribution in each hollow beam in the vertical direction using only
one probe beam directed along tunnel B. The corresponding shadow images are
presented in Fig. 4.5 for the cloud trapped in tunnel A (top row) and tunnel B
(bottom row) at various time. For example, as atoms move in tunnel B, the cloud
elongates along the tunnel and projects an image that is circular with a constant
42
Figure 4.5: Cloud evolution in a single tunnel as imaged by a probe beam directed
along tunnel B. Top row of images show the propagation in tunnel A, bottom row
- in tunnel B. The bottom row of images is the same as in Fig. 4.2. The shape
change from a stubby oval to a circle at longer times allows to distinguish the
propagation in tunnel A from that in tunnel B.
diameter at all times (Fig. 4.5, top row). By contrast, the atoms in tunnel B
propagate at a 12 EWangle with respect to the probe beam direction, and as the
cloud spreads, its projection changes from near circular to that of a stubby oval.
The longer the evolution time the larger the cloud aspect ratio is as seen in the
top row of images in Fig. 4.5. Therefore a switch of the atomic beam direction
from tunnel A to tunnel B results in a change of the cloud?s shape from an oval to
a circle as in our imaging system.
Now we are set to describe the demonstration of the atomic beam switch when
both tunnels were used in the same experiment. We changed the direction of the
atomic beam with the following lock operations. At t = 0, the atomic cloud was
launched in tunnel A with the entrance gate open (tunnel A on and tunnel B off).
At t = 8 ms, the entrance gate was closed (tunnel B on) and the exit gate was
opened (tunnel A off) simultaneously with two synchronized mechanical shutters.
The speed of the shutters determined the gate switch time of 1.6 ms. As with single
tunnel experiments, the cloud propagation was monitored by two probe beams, one
43
along the axis of tunnel B, and the other perpendicular to the geometrical plane
defined by the axes of the two tunnels. The shadow images of the corresponding
density distributions are presented in Fig. 4.6 for the probe directed across the
plane of the two tunnels (top row) and the probe along the tunnel B (bottom
row).
First, let us consider the top row of images. We see that 4 ms after the switch
was activated (the 12 ms panel), the cloud is no longer aligned with tunnel A (tun-
nel A off and tunnel B already fully on at this point). It takes several milliseconds
for the majority of atoms to encounter the walls of tunnel B and to start propa-
gating along the tunnel. At this stage a significant part of the cloud is confined by
tunnel B; the remaining images show the trapped cloud aligning itself with tunnel
B and propagating along it. The re-direction of the cloud can be monitored as well
from the other probe beam, aligned with the axis of tunnel B, and is presented
in the lower row in Fig. 4.6. By 8 ms, the cloud starts taking an oval shape and
would continue stretching similar to the top row of Fig. 4.5 if it were to propagate
in tunnel A alone. Instead, after the switch from tunnel A to tunnel B is activated,
we observe the distinct shape change from a stubby oval in the top row of Fig. 4.5
to a circle in the bottom row of Fig. 4.5 at t > 8 ms indicating the atomic beam
switch from tunnel A to tunnel B.
To characterize the switch we measured its efficiency and the heating effects
due to the abrupt change of the optical potential and the re-direction of the cloud.
We define the switching efficiency as the ratio of the number of atoms in tunnel A
before the switch (at t = 8 ms) to the number in tunnel B after the switch (t = 10
ms). The number of atoms before the switch can be determined directly from the
absorption image at 8 ms (Fig. 4.5, bottom row). During the switch, a significant
fraction of the atoms was outside of the intersection volume of the two tunnels and
could not be re-trapped by tunnel B while still contributing to the shadow images
right after the switch. Therefore the images corresponding to 10 and 12 ms were
not used to determine the efficiency. Instead we measured the number of atoms
44
Figure 4.6: Shadow images of rubidium atoms passing through the tunnel lock.
Top row shows the cloud initially moving downward and to the right in tunnel A
and switching direction to propagate downward and to the left along tunnel B as
observed by probe 1 (refer to Fig. 4.4). Bottom row of images shows the same
switching as detected by probe 2. The cloud, starting in tunnel A, should stretch
according to the top sequence of images in Fig. 4.5. After the switch was activated,
the cloud takes and preserves a round shape similar to the bottom row of images in
Fig. 4.5, an indication of propagation in tunnel B consistent with switching from
one tunnel to another. The black arrow indicates the moment of switching.
at t> 14 ms, long enough after switching for the non-trapped atoms to leave the
imaged volume. We used the single tunnel loss rate (Fig. 4.3) to extrapolate back
to the number of atoms at t = 10 ms. We found the switching efficiency to be
about (33?5)%. The resulting efficiency was mainly limited by a relatively large
size of the atomic cloud compared to the intersection volume of the two tunnels.
The efficiency can be increased by using smaller and colder clouds, and switching
of the whole cloud should be possible.
We investigated switch-induced heating and acceleration by measuring the tem-
45
Figure 4.7: Cloud evolution in a single tunnel and through the switch where t is
the time after the cloud is released from the trap and ? is the width of a Gaussian
fit of the cloud density along either tunnel. The error bars represent the standard
deviation of the mean of five runs.
perature and kinetic energy of the cloud before and after the switch in the longitu-
dinal direction. There was no measurable enhancement to the center of mass accel-
eration over acceleration due to gravity after the switch was activated. Since the
cloud?s motion was not constrained along the tunnel, it preserved its original near-
Gaussian density distribution expanded freely in 1-D according to ?2 = ?20 +v2t2,
where ?2 is the initial width of the cloud, and v2 is the mean square velocity of
the Maxwellian distribution. The sizes of the cloud propagating in a single tunnel
and switching from one tunnel to another are presented in Fig. 4.7.
The slope of a curve in Fig. 4.7 is v2 and proportional to the temperature. For
the case of a single tunnel (blue triangles in Fig. 4.7), the slope decreases between
0 and 5 ms and after that becomes constant corresponding to the temperature of
(150?7)?K. The confined cloud is colder than the source atoms. The temperature
reduction may be attributed to the inability of the tunnel to confine the hottest
46
Figure 4.8: Schematics of the cloud re-direction upon the switch of tunnel B. The
wings of the cloud, spread after thermal expansion along tunnel A, are outside the
intersection volume of the two tunnel, and cannot be re-trapped.
atoms since the initial cloud temperature is comparable with the optical potential
of the tunnel. In the switch experiments (red circles in Fig. 4.7), the evolution
of the cloud is identical to that of a single tunnel up to 8 ms when the switch is
activated. Shortly after the switch is activated, the length of the cloud is reduced
by about 25% as the intersection of the two tunnels is smaller than the size of
the cloud, and the ends of the expanded cloud are not re-trapped by the second
tunnel. It is clear that between 8 and 10 ms the evolution of the imaged atomic
density does not correspond the Maxwellian velocity distribution of the cloud; ? is
nearly constant while the cloud re-adjusts. After 10 ms, when the near-Gaussian
distribution is re-established, ?2 becomes linear with t2, giving a temperature of
about 118 ?K.
The measured reduction of the cloud temperature reflects the fact that hot
47
atoms occupied the ends of the cloud that were truncated by the switch as illus-
trated in Fig. 4.8. The atoms were originally confined in tunnel A, with their
transverse kinetic energy smaller than the height of the corresponding optical po-
tential. The cloud?s transverse size across the tunnel was set by the 1 mm diameter
of the tunnel, while the length along the tunnel, resulting from the free expansion
of the cloud, was ? 4 mm. When the second tunnel, B, was turned on, only a
fraction of the cloud was inside the intersection volume of the two tunnels, giving
byV = 163 r3/sin?for the perfect overlap, wherer is the radius of a tunnel, and?is
the angle between the two tunnels. The atoms outside the intersection volume were
lost for the second tunnel. Atoms at the edge of the cloud were the hottest as they
moved furthest from the center of the cloud. In addition, some of the atoms had
kinetic energy along tunnel A higher than the confining optical potential. When
tunnel B was turned on at an angle with respect to tunnel A, those atoms were not
re-directed by the barrier of tunnel B, but penetrated the walls of tunnel B, thus
contributing to cooling the rest of the cloud. In the real experiment, the tunnel
cross was not perfectly overlapped with the central portion of the cloud, and some
of the colder atoms closer to the center were also ejected upon switch activation.
We studied the temperature change numerically by truncating all but the central
spatial part of the cloud, such that only 30% of atoms were left in the tunnel
after the truncation. We assumed a Gaussian density and Maxwellian velocity
distributions of the cloud and the cloud temperature of 150 ?K before the switch.
Keeping the central portion led to a 60-?K cloud after re-thermalization without
taking into account the change of the cloud propagation direction by a small angle
of 12EW. To verify the validity of this approach, we eliminated the central portion of
the cloud containing 70% of the atoms. The resulted re-thermalization gave rise
to a 1500-?K cloud. This simple consideration is consistent with our observation
that we truncated both hot and cold atoms in our experiment, but mostly hot.
More importantly, our results suggest no measurable heating of the cloud upon
switch activation due to absorption from the tunnels. Although our simulations
48
were overly simplified, not accounting for the full 3D behavior, they did identify
an important contribution to cooling after the switch. Furthermore, they revealed
the importance of the overlap geometry, something that will have to be addressed
by any beam-splitter design based on hollow tunnels.
4.3 Splitter and Delay Line
We can extend the application of the prototype lock device to an atomic splitter
and delay line by crossing the two tunnels at the MOT cloud maximizing the initial
number of atoms in the tunnel overlap volume. In this configuration, both Bessel
beams were on simultaneously after the MOT was switched off. The evolution of
the atoms along the plane of the two tunnels is presented in Fig. 4.9. Consider
three right frames, which show the evolution of three distinct clouds with different
momenta. Atoms with transverse and longitudinal velocities below 15 cm/s (250
?K) form the delayed cloud trapped in the intersection volume of the two tunnels.
The traveling clouds have higher longitudinal velocities and propagate along each
tunnel thus forming two atomic beams split from the main cloud. Before the
tunnel was loaded, the MOT atoms were in the upper hyperfine level of the ground
state (5s1/2|F = 2?). The tunnel loss via the lower hyperfine level(5s1/2|F =
1?), and thus the image contrast (the number of trapped atoms), is sensitive to
detuning. For 87Rb, the best image contrast (4.9, two left frames) was obtained
at a detuning of + 6 GHz relative to the F = 2 ? F? = 3 transition (refer in
Appendix C). This detuning places the laser light frequency 0.2 - 0.4 GHz red of
the F = 1 ? F? = 0,1,2 transitions. Therefore the Bessel beam turns out to
be blue-detuned with respect to some near-resonant atomic transitions and red-
detuned with respect to the others: atoms in the 5s1/2|F = 2?are repelled by the
tunnel walls but those in the 5s1/2|F = 1?are drawn in by the tunnel walls and
get accelerated by a high photon scattering rate. In addition, a significant fraction
of these atoms will be lost because, when they are pumped back to the upper
49
Figure 4.9: Splitter/delay line realized with the tunnel lock. Tunnels A and B are
on simultaneously and crossed at the MOT location. The three(four) left (right)
panels correspond to 87Rb (85Rb) with a tunnel detuning of +6 GHz (+3 GHz).
Three momenta are generated corresponding to clouds moving downward to th left
and to the right, or trapped in the overlap volume of the two tunnels.
hyperfine level, they can be expelled from the walls both back into and out of the
tunnel. Smaller detunings increase the scattering rate as well as the decay rate
into the 5s1/2|F = 1?state thus reducing the image contrast. We can eliminate
the lower hyperfine leak by tuning the tunnel to the blue of the re-pump frequency
as well. However, above + 6.6 GHz detuning, we could only generate a potential of
?150?K or lower as the Bessel beam power was limited by the damage threshold
of the SLM. Such an optical potential is too low to contain most of the atoms.
Instead of increasing the detuning, we achieved better performance by using 85Rb
isotope at the detuning of + 3 GHz, which is blue of both sets of resonances (three
right frames on Fig. 4.9). We have not performed any quantitative measurement
to characterize the properties of the splitter and the delay line, as the main goal
of this demonstration was to show the versatility of the tunnel lock for various
applications.
50
4.4 Conclusion and Future Directions
This part of the dissertation was devoted to development of new tools for atom
optics, a beam switch, a splitter, and a delay line that can find applications in atom
interferometry, quantum information, lithography, and biological physics. We have
successfully employed a blue-detuned 4th-order Bessel beam as an atomic guide
to transport laser-cooled rubidium atoms. We found the application of the Bessel
beams to be quite versatile: the tunnel lock, two properly gated crossed Bessel
tunnels, was demonstrated to transport and split an atomic cloud, delay its prop-
agation, and switch the direction of an atomic beam from one tunnel to another.
The ensemble?s life time in the tunnel was limited by the collisions with the hot
background gas, and the switching efficiency by the relative sizes of the cloud and
the tunnel. The properties of the demonstrated tunnel lock are comparable with
those achieved by other methods such as guiding and splitting along a magnetic
wire, and all-optical red-detuned schemes.
Some of the applications of the demonstrated devices require decoherence-free
evolution that remains to be properly shown. Because decoherence is strongly in-
fluenced by the high scattering rate at small detunings, coherence issues will have
to be studied with larger detunings, where spontaneous emission is suppressed.
The act of altering the direction of the cloud does not lead to significant heat-
ing, it would be interesting to follow this approach to study colder samples for
interferometry purposes. A reduction in the sample temperature to less than 1?K
and large detunings would allow to virtually eliminate absorption. However, the
employed 1-mm diameter hollow beam is a multi-mode tunnel for atoms [60] that
can results in excitations of higher order modes, similar to the property of a multi-
mode optical fiber. While condensates are often envisioned for interferometry, it
was shown that high fringe visibility can be achieved with thermal clouds [62]. The
higher repetition rate is one distinct advantage of thermal clouds over condensates.
The tunnel locks demonstrated in our experiments can be applied to red-detuned
51
(filled) tunnels. Since increasing the intensity essentially deepens the potential
well for red-detuned tunnels, it is not clear how to stop a traveling cloud with a
diffractionless laser beam (i.e., J0) that does not come to a focus without intro-
ducing additional beams. This is possible with blue-detuned tunnels because blue
detuning raises the barrier. Nevertheless, intensity modulation should allow the
branching between two red tunnels to be modified as well. At the same time, inter-
ferometer structures are easier to imagine with red-detuned beams. Consequently,
for greater flexibility in free-space manipulation of neutral ensembles, elements
based on both red and blue detuning will be required.
We should also point out several promising applications of an SLM to genera-
tion of arbitrary shaped blue-detuned optical potentials on a plane in real time [50].
They might be used as billiards to study the chaotic dynamics of confined atoms
and as tweezers to manipulate macroscopic biological objects. As an example, it
was theoretically shown [51] that the escape rate of a wavepacket from a chaotic
billiard deviated from the classical value when the quantum mechanical nature
of the interaction was taken into account. The classical regime was investigated
using cold atoms in optical billiards [63, 64], while a proposed measurement of
quantum corrections would help to understand the quantum mechanical behavior
of the classically chaotic system.
52
Chapter 5
Quantum Information Science
5.1 Introduction
Quantum information science, a merging of quantum mechanics and classical in-
formation theory, opens new exciting possibilities in data computation, communi-
cation and storage. Over the last decades, the exponential shrinking of electronic
components that process the information may be approaching its limit. The fur-
ther minituarization of transistors on a near-atomic scales will leave no room for
packing more components on a single chip. Quantum mechanics allows us to step
beyond this limit, and in combination with information science, can provide a
physical system to process information on the microscopic level. Such quantum
systems can outperform some of their classical computational counterparts tremen-
dously. Original ideas to exploit the computational power of quantum mechanics
[2] and the development of quantum algorithms [3, 4, 5] promoted significant ef-
forts toward the practical realization of a variety of physical systems applicable for
quantum information.
Quantum bits, or qubits, are the states of a quantum mechanical system. If a
classical bit, the carrier of classical information, can be in either one of two states,
its quantum analog can also be in a superposition of the two states. The number of
simultaneously accessible states scales as 2N for N qubits, dramatically increasing
the potential computational power of the quantum information processor.
Two quantum algorithms are known to outperform their classical counterparts:
53
the factoring algorithm, introduced by Shor, provides an exponential growth of
computation speed compared to the classical algorithms [5]. Grover?s search algo-
rithm can find an element in an unsorted database with a quadratic increase of
speed [4].
Quantum algorithms are constructed from smaller elements, that include a
single qubit rotation between its two states and logical operations with qubits
provided by quantum gates. Quantum gates are linear unitary operators acting
on a state of the system. A set of properly combined gates that can provide any
linear unitary operation, is called universal, such as the one consisting of a single-
qubit Hadamard gate (H), a single-qubit ?/8 gate (T) and a two-qubit controlled-
NOT gate (CNOT) [65]. If |0?and|1?denote the two states of a single qubit,
|00?,|01?,|10?,|11?are the basis of the two-qubit system, then the universal set
of gates can be written as
H = 1?2
?
?1 1
1 ?1
?
?, T =
?
?1 0
0 eipi/4
?
?,
and
CNOT =
?
??
??
??
?
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
?
??
??
??
?
. (5.1)
Efforts to realize quantum computer hardware, a physical system that provides
qubits and gates, are occurring in very different fields, such as NMR [6], linear
optics [7], quantum dots [8], cold trapped ions [9] and neutral atoms [10]. Each
approach has its advantages and drawbacks, and has to meet a set of minimum
requirements, so-called DiVincenzo criteria [66]: the quantum computer must be
a scalable physical system that can be initialized to a known quantum state; qubit
coherence time must be much longer than the gate operation time; it must be
possible to implement a universal set of gates within the system, and read out the
54
results of the computation.
One of the promising implementations of quantum computer is the system of
neutral atoms. Their internal level structure, hyperfine levels in particular, can be
used as states of qubits while coupling between qubits can be provided by atom-
atom interactions. Due to their neutrality, the interaction of atoms with the envi-
ronment is weak, which results in long coherence times. In addition, well-developed
techniques for trapping cold atoms provide good control. We will consider a de-
velopment of a CNOT-phase gate based on the conditional excitation to Rydberg
states.
5.2 Quantum Computing with Rydberg States
In contrast to ions it is difficult to have strong interactions between neutral atoms
that are required for a realization of a fast quantum gate. The coupling between
two atoms can be implemented through control of their internal structure, for
example by means of collisions [67], photon exchange between atoms in a cavity
[68] and the dipole-dipole interactions between Rydbergs atoms [69].
Alkali atoms may act as quantum bits. In case of rubidium-87, the hyperfine
ground state levels can be used as states of the qubit,
|0???5s1/2|F = 1,m = 0?
|1???5s1/2|F = 2,m= 0?.
The hyperfine structure and Zeeman splitting of 87Rb are thoroughly studied and
well-understood [70]. The states can be addressed by microwave radiation or op-
tical Raman transitions, and the interaction between two ground state atoms is
negligible. The |F = 2? ground state is employed in the laser-cooling cycling
transition; atoms can be pumped to this state efficiently, and it can be naturally
considered as a prepared initial state for an ensemble of qubits. A single qubit
55
rotation with a fidelity1 of 0.99 was demonstrated recently [71].
The conditional operations required for realization of a two-qubit CNOT gate
can be achieved by altering the state of a target qubit with a control qubit. For
example, if and only if the logic state of the control qubit is|1?, the state of the
target qubit flips. Such control gate can be realized using the Rydberg blockade
effect [69]. Consider two atoms, A and B initially in their two hyperfine ground
states|0?and|1?asillustratedinFig.5.1. The atomscanbecoupled toa Rydberg
level |r? with a high principle quantum number by means of a direct optical
excitation or through an intermediate state such that states |1? and |r? form
an effective two-level system. When atom A is in the Rydberg state it shifts the
Rydberg energy of the second atom B by ?rr thus detuning the atomic resonance
of atom B with respect to the excitation laser. The system is controlled by a
sequence of ?-pulses2 according to the following protocol: (1) excite atom A to the
Rydberg state |r? with a ? pulse, (2) while atom A is the Rydberg state, apply
a 2? pulse resonant with the|0??|r?transition for atom B, and (3) de-excite
atom A to state|0?with a ? pulse. A qubit rotation between states|0?and|1?
can be achieved by a microwave radiation as necessary.
Depending on the initial state of the 2-qubit system the gate operation will
have the following result:
|00? 1??i|r0? 2??iei?|r0? 3??ei?|00?,
|01? 1??i|r1? 2??i|r0? 3??|00?,
|10? 1?|10? 2??|r0? 3??|00?,
|11? 1?|11? 2?|r0? 3?|00?,
that coincides with the CNOT-matrix in Eq. 5.1 when a single-qubit Hadamard
gate is applied to atom B. The extra phase shift term ei? is due to the off-resonant
1for two pure states ? and ?, fidelity is defined as|??,??|
2the definition of the pi-pulses is given in Appendix A
56
Figure 5.1: The controlled 2-qubit phase gate based on the Rydberg blockade
effect. Logic state of atom A controls the state of the target atom B by altering
its Rydberg state energy.
coupling between the ground and the Rydberg states of atom B when atom A is
excited. The phase shift can be evaluated from the 2-level evolution operator [72]
to find ? = ??R/2?rr, where ?R is the Rabi frequency of the Rydberg excitation
of atom B. For strong interactions, one can achieve ?rr >> ?R and the unwanted
phase shift can be suppressed. The gate operation time istop = 2?/|?A|+2?/|?B|,
where ?A,B are the Rabi frequencies for the corresponding atoms. According to the
DiVincenzo criterion, the gate operation time must be shorter than the coherence
time of the qubit, set by the radiative decay of the Rydberg state.
The idea of the two-qubit blockade gate can be extended to an ensemble of
qubits, each consisting of N atoms [73]. The extension relies on the concept of the
collective Rydberg blockade, in which excitation of a single atom to a Rydberg
57
state can block the subsequent excitation of a large number of atoms within the
interaction volume. If the blockade shift is sufficiently large, the simultaneous
excitation of two or more atoms is prevented and the enhanced Rabi frequency
is given by ?N =?N?. Two-qubit gates can then be implemented between two
ensemble qubits or between a single atom and an ensemble qubit analogous to the
single atom protocol of Fig. 5.1.
The Rydberg blockade idea spurred a large amount of theoretical and experi-
mental work and was further developed beyond the original proposals [69, 73]. For
a detailed and up-to-date review on the subject please refer to Ref. [74].
5.2.1 Properties of Rydberg-Rydberg Interactions
Rydberg atoms with high principal quantum numbernare potentially useful due to
their long radiative lifetimes and strong atom-atom interactions resulting in large
values of ?rr. In the absence of external fields, the eigen-states of a hydrogen-like
alkali atom can be represented by quantum numbers n,l,s, and j denoting the
principal, orbital angular momentum, spin angular momentum, and total angular
momentum quantum numbers. The energy levels are accurately represented by
the quantum defect theory [75]
Enlj =? Ry(n?)2, n? = n??lj(n), (5.2)
whereRy = 109737.3cm?1 is the Rydberg constant, and the quantum defect?lj(n)
is a slowly varying function of the principal quantum number [76].
5.2.2 Radiative Lifetime
The qubit coherence time is limited by the radiative decay and the black-body
ionization rates of a Rydberg state that is close to the ionization limit. The finite
temperature lifetime ?nl is [75]
1
?nl =
1
?0nl +
1
?(bb)nl
(5.3)
58
where ?0nl is radiative lifetime at 0 K, and ?(bb)nl is the finite temperature blackbody
contribution. The 0 K radiative lifetime can be parameterized by the expression
[77]
?0nl = ?0l (n?)?l, (5.4)
where the value of numerical constants ?0l and ?l are tabulated in reference [75].
For all alkali atoms, ?l ?3, for low l states. The blackbody ionization rate can be
approximated for large values of n as [75]
?(bb)nl = 4?
3kBT
3planckover2pi1n2 , (5.5)
where ? is the fine structure constant and T is the characteristic temperature
of the blackbody radiation. For n ? 50, the Rydberg state lifetime is about
100?sat room temperature, about half from blackbody and half from the radiative
contributions.
5.2.3 Rydberg-Rydberg Interaction Strength
The long-range interactions between two Rydberg atoms are extremely strong and
are the core of the quantum computation scheme discussed here. We consider the
Rydberg-Rydberg interactions in two limits: at zero electric field, where the long-
range 1/R6 van der Waals interactions are of primary importance, and in a large
external electric field, where the atoms have a dipole moment of magnitude n2ea0,
where e is the electron charge and a0 is the Bohr radius of the alkali atom.
At interatomic distancesR>>n2ea0 between two Rydberg atoms A and B, the
leading electrostatic interaction is the dipole-dipole interaction, that can written
in atomic units as
Vdd = e
2
R3
bracketleftBig
a?b?3(a? ?R)(b? ?R)
bracketrightBig
, (5.6)
where a and b are the positions of the two Rydberg electrons measured from
their respective nuclei.
59
When the two atoms are excited by light to the same fine-structure level, the
two-atom unperturbed state for R = ? can be written as |?2? = |?A?B?. At
zero field, the energy shift of the 50s states is due to the van der Waals interaction
dominated by the near resonance between the states|50s+50s?and|50p+49p?,
with the energy defect ? = E(49p)?E(50p)?2E(50s) =?3 GHz. If we neglect
the contribution from other states, the energy shift will be given by [74]
V(R) = ?2 +
radicalbigg
4U3(R)2
3 +
?2
4 , (5.7)
where U3(R) = e2?50s||r||50p??50s||r||49p?/4??0R3. For large defects ? >>U3,
the expression for the energy shift is characterized by 1/R6-dependence:
V(R) =?4U3(R)
2
3? ?
1
R6, (5.8)
The Rydberg dipole moment scales as n2, the energy defect as 1/n3, resulting in
a n11-dependent energy shift. Therefore the excitation to high Rydberg levels is
advantageous for a practical realization of the Rydberg blockade.
The 1/R6 interaction can be enhanced either by applying an electric field or
by using resonances. If the excitation levels can be chosen such that the energy
defect is small in comparison with the interaction strength (F?orster resonances),
Eq. 5.7 is reduced to the dipole-dipole 1/R3 interactions
V(R) =
radicalbigg4
3U3(R) ?
1
R3.
Note, that an external electric field can be used to tune the Rydberg states to
F?orster resonances through the Stark energy shift as discussed in the recent review
[78].
Another approach to increase the interatomic interaction and the corresponding
energy shift relies on the application of the external electrostatic field E. For small
fields, the interaction energy e?r?E between the induced dipole moment and the
field is less than the energy difference with the nearest opposite parity state, ?,
and the Stark effect is quadratic in the field. As the electric field gets higher, the
60
mixing of states with different orbital angular momenta gives rise to a Rydberg
atom with a constant electric dipole moment on the order of n2ea0, that results in
the linear Stark effect. The permanent dipole moment ? provides the strong 1/R3
dipole-dipole interaction, which from Eq. 5.6 can be written in SI-units as
Vdd = ?
2
4??0R3(1?3cos
2?), (5.9)
where ? is the angle between the interatomic axis and the electric field. Note,
that the interaction is anisotropic, which may reduce the corresponding overall
energy shift in a macroscopic sample. Control of the dipole-dipole interactions by
an external electric field is the subject of this dissertation and is considered in
more detail in Chapter 6.
5.3 Experimental Progress in Realization of Rydberg Quan-
tum Gate
5.3.1 Traps for Neutral Atom Qubits
Thw practical realization of quantum computing devices based on neutral atoms
relies on the ability to trap the atoms to constrain their motion, select and address
the states of the quantum system, and provide a conditional coupling between two
or more qubits. Individual atoms were successfully trapped in a MOT [79], and
later in far-offresonance optical traps (FORTs) [80]. Various red- and blue-detuned
optical traps were discussed in some details in the first part of this dissertation. For
large trap detunings and depths, the trap lifetime is limited by the collisions with
the background gas and can be many orders of magnitude longer than the desired
time of qubit operations. Atomic motion within the trap beam results in AC Stark
shifts of the qubit states, which can be a limiting factor for qubit coherence in
FORTs [81]. For the conditional coupling realized through the Rydberg-Rydberg
61
interaction, the high power optical trap can cause photoionization with the rate
substantially higherthanthe radiativedecay rate. Those effects canbesignificantly
reduced by confining the atom in dark optical traps where the atoms are localized
near an intensity minimum. We are not aware of any experimental demonstration
of trapping single atom qubits in blue-detuned optical potentials. Magnetic traps
are characterized by very long coherence times, and an array of such traps [82]
can be promising for scalability of trapping to many qubits. Unfortunately, it may
be difficult to combine magnetic traps with Rydberg atoms. To achieve tightly
confining magnetic traps on a chip the trap position is close to the surface of the
chip, which can lead to undesired interactions between the surface and a Rydberg
atom.
5.3.2 Excitation of Rydberg States
A trapped neutral atom can be encoded as a qubit using its hyperfine or Zeeman
ground states. Practical realization of the conditional gates based on the atom-
atom interactions requires precise excitation and deexcitation of Rydberg states.
Early work of laser excitation and spectroscopy of Rydberg atoms is reviewed in
reference [83]. One photon Rydberg excitation from the ground state of rubidium
was recently demonstrated with 297 nm light generated by a frequency-doubled dye
laser [84]. We choose to excite a Rydberg state through a two-photon transition,
specifically with the intermediate state close to 5p3/2 state. In this case, the first
photon is at 780 nm (red), and can be provided by one of the cooling lasers that
couple the ground state with the 5p3/2 state. The second photon must be at 480
nm (blue), which can be generated by a frequency-doubled diode or Ti:Sapphire
lasers. A diagram of the two-photon excitation is shown in Fig. 5.2. The first
example of Rydberg spectroscopy with narrow linewidth lasers using excitation
scheme was a study of Autler-Townes spectra in a cold 85Rb sample [85]. Our
observation of the Autler-Townes spectra in 87Rb MOT is reported in Chapter 6.
62
Figure 5.2: Schematic of the two-photon excitation of the Rydberg state|r?. The
excitation lasers at 780 (red) and 480 nm (blue) are detuned by ?p from the
intermediate level|p?to reduce its population.
Rabi oscillations between ground and Rydberg levels were recently demon-
strated for single 87Rb atoms confined to an optical trap [86]. Rydberg excitation
to 43d5/2 state was provided by ? -polarized lasers at 780 and 480 nm. The exci-
tation laser beams had waists that were a few times larger than the width of the
optical trap so the effects of spatial variation of the Rabi frequency were minimized.
The dipole trap laser was turned off before the excitation lasers were applied to
avoid the photoionization of the Rydberg atoms. After a variable length excita-
tion pulse the trap was turned on again which photoionized the Rydberg atoms
before they could radiatively decay. Loss of a Rydberg atom from the trap pro-
vided a measure of Rydberg excitation. The reported Rabi oscillations are shown
in Fig. 5.3. Collective excitation in ensembles were also observed using the stimu-
63
Figure 5.3: Rabi oscillations between ground and 43d5/2 rubidium Rydberg level.
Reproduced from reference [86].
lated Raman adiabatic passage (STIRAP) with a counterintuitive pulse sequence
where the 480 nm laser is applied before the 780 nm [87]. Excitation probabilities
of up to 70% were demonstrated.
5.3.3 Experimental Realization of Two-qubit Rydberg Gates
It has recently been shown by the research groups in Wisconsin and at the Insti-
tute of Optics, Palaiseau, France that the experimental methods of single atom
trapping, coherent excitations and control over Rydberg-Rydberg interactions can
be combined to demonstrate Rydberg blockade [88, 89], and a two-qubit Rydberg
gate [90]. The blockade experiment in Wisconsin [88] showed that excitation of a
Rb atom to the 90d5/2 state blocked the subsequent excitation of the second atom
at a distance R?10?mwith a fidelity of about 90%. Experimental data, together
with a Monte-Carlo simulation taking into account the finite blockade strength
and experimental imperfections are shown in Fig. 5.4.
64
Figure 5.4: a) The experimental data for Rydberg excitation of the target atom
with and without a control atom present. b) A Monte Carlo simulation accounting
for experimental imperfections. The amplitude of the curve fit to the blockaded
oscillations is a = 0.09 (experiment) and a = 0.11 (simulation). Reproduced from
reference [88].
65
Figure 5.5: Excitation of one atom versus collective excitation of two atoms sep-
arated by 3.6?m. The circles represent the probability to excite atom a when
atom b is absent. A fit to the data yields a frequency of this Rabi oscillation
?/2? = 7.0?0.2 MHz. The squares represent the probability to excite only one
atom when the two atoms are trapped and are exposed to the same excitation
pulse. The fit gives an oscillation frequency ?/2? = 7.0?0.2 MHz. The ratio
of the oscillation frequencies is 1.38?0.03 close to the value?2 expected for the
collective oscillation of two atoms. Reproduced from reference [89].
A closely related experiment in Palaiseau [89] used simultaneous excitation of
two Rb atoms in optical traps separated by R?3.6?m. A very strong blockade
shift was obtained by using the 58d3/2 F?orster resonance. The excitation was pro-
vided through the 5p1/2 state by 794 nm light. As shown in Fig. 5.5, the excitation
that couples the two atom state to the singly excited state was characterized by
the collectively enhanced Rabi frequency ?c =?2?.
The realization of the 2-atom Rydberg blockade was later combined with the
coherent Rabi oscillations experiment to demonstrate a CNOT gate [90]. The
probabilities of the prepared initial state and the output of the Rydberg gate are
66
Figure 5.6: Practical realization of the Rydberg CNOT gate. Measured proba-
bilities for state preparation (left) and CNOT output (right). Reproduced from
reference [90].
shown in Fig. 5.6.
Although promising, the initial results should only be considered as first steps
as they lag far behind the high fidelity results obtained with trapped ions [91].
5.3.4 Blockade Effect in Many-atom Systems
With the proposed application of the Rydberg blockade effect to quantum comput-
ing, most experimental efforts were aimed at the study of the Rydberg excitation
and interatomic interactions in cold atomic samples of many atoms. Bulk samples
are hard to envision to use for the quantum information processing but their study
provides deeper understanding of the properties of Rydberg atoms and promote
development of new techniques to realize a single 2-qubit gate and extend it to
scalable systems required for quantum computation.
Original experiments to study the behavior of cold Rydberg atoms in MOTs
showed the signatures of Rydberg-Rydberg interactions [92, 93]. In case of the
large samples, the Rydberg interactions occur on the ?m-scale, much smaller than
67
1 mm, the size of a large atomic cloud. To characterize the blockade phenomenon it
is convenient to introduce a concept of the ?blockade sphere? [94]: a single excited
atom blocks further excitations with a sphere of radius Rb. The radius is set by
the range of interatomic interactions. As the sphere contains indistinguishable Nb
atoms in the ground state, they comprise an effective ?superatom? that interacts
with the excitation light with a collective?Nb enhancement of the Rabi frequency
? [95]. The blockade radius is set by the condition that the collective Rabi fre-
quency be comparable to the dipole-dipole shift, which results in the excitation
fraction to be [96]
??
parenleftbigg ?
?2C6
parenrightbigg2/5
,
where ? is the atomic density of the sample and C6 is the amplitude of the van der
Waals interaction, which scales as n11. The dependence of the excitation fraction
ontheexcitation pulse intensity, andthusC6, is illustrated by Fig.5.7 [94]. Afactor
of 2 ratio of the excitation fraction for the 70p and 80p data was consistent with
the blockade density scalings, and the comparison of either with the excitations to
the 30p state dramatically shows the overall blockade effect.
These suppression effects could be considerably enhanced for the dipole-dipole
interactions resulted by tuning a F?orster resonance with an electric field. A di-
rect comparison between the van der Waals and F?orster-enhanced blockade was
reported [97].
Another signature of the excitation blockade in extended samples would be a
resonance broadening of the Rydberg transition. Density variations that change
interatomic distances or control over the interaction strength may give rise to a
shift of the atomic resonance with respect to the frequency of the excitation laser
for a fraction of the atomic cloud. The resulted broadening can be observed in spec-
troscopic measurements. We have assembled an experimental apparatus to excite
rubidium atoms to Rydberg states and investigate the excitation in the presence
of an electric field when the transition from the van der Waals to the dipole-dipole
68
Figure 5.7: Dependence of excitation fraction on principle quantum number for
the excitations to 30p, 70p and 80p Rydberg states. Reproduced from reference
[94].
interactions is expected to enhance the blockade effect at higher fields.
69
Chapter 6
Experimental Setup
Practical realization of quantum logic gate protocols introduced in Chapter 5 re-
lies on the efficient excitation of atoms to a specific Rydberg state. The rate of
excitation to the Rydberg state depends on the convolution of the linewidth of the
transition for a stationary atom (< 100 kHz), the Doppler broadening due to the
atom?s motion and the linewidth of the excitation laser. The Doppler broadening
can be reduced to a few hundred kHz by cooling the atoms below the temperature
of a few hundred ?K and trapping them such that their position in space is fixed
with respect to the excitation laser light. A narrow linewidth1 of the excitation
laser system is required to not limit the excitation rate and to be able to exclusively
address a desired Rydberg level. To observe the blockade effect, the resolution of
the excitation laser must be smaller than the energy shift of a few MHz due to
the Rydberg-Rydberg interactions. In addition, the laser light should be contin-
uously tunable, to make spectroscopic measurements of the Rydberg states and
investigate the Stark shift due to an external electric field.
6.1 Cold Atom Apparatus
To cool and trap neutral atoms, and excite them to Rydberg states, we used a
BEC apparatus assembled by our colleagues, Emily Edwards and Matthew Beeler,
specifically designed for the experiments with rubidium atoms trapped in optical
1Usually, a diode laser with an extended cavity has a?MHz linewidth.
70
lattices [29]. Here we briefly describe the cold atom machine, discuss its advan-
tages and the drawbacks as an apparatus to study the Rydberg states for various
parameters of the cold atomic clouds. It must be pointed out, that we did not
participate in the development of this apparatus, but benefited from its existence,
as it was used throughout the work reported here.
The apparatus, a sophisticated version of the laser cooling setup described in
Appendix A, allowed a production of a cold rubidium cloud held in a magneto-
optical trap, further cooling of the cloud in an optical molasses, its subsequent
transfer to a pure magnetic trap, radio-frequency evaporation of the cloud in the
magnetic trap to achieve quantum degeneracy of the trapped atoms, and its trans-
fer to a dipole trap created by a red-detuned focused laser beam. We investigated
different approaches to produce Rydbergs atoms in the MOT, molasses, the mag-
netic and dipole traps.
A part of the ultra-high vacuum (UHV) chamber for the BEC experiments is
presented in Fig. 6.1. The chamber pressure of?3?10?11 Torr was maintained by
a combination of an ion and a titanium sublimation pump that provided a lifetime
of many seconds for the trapped rubidium sample. A Zeeman slower was used as
a source of atoms for a mirror-MOT setup [98] and allowed fast turn on and off
times of the cloud loading. The long lifetime of the cloud allowed to excite atoms
to Rydberg states for up to 50 ms, when the laser induced excitation was the sole
mechanism responsible for the change of the ground state population. The setup
provided a collection of 107?109 atoms in a MOT, and a 50% transfer efficiency to
the magnetic trap. The magnetic confinement with a depth of 1 mK was provided
by an interaction of the magnetic moment of atoms in the|F = 2,mf = 2?ground
state with a magnetic field generated by Z-shaped conductor in vacuum and a bias
field with the components along x- and y-directions (Fig. 6.1). Further evaporative
cooling in the magnetic trap allowed us to change the density of the atomic cloud
by varying the evaporation time.
71
Figure 6.1: View from the bottom of the UHV chamber showing orientation of
a 480-nm Rydberg excitation beam (blue) and input beams for the MOT (red).
There are four MOT beams total, two from the side and two from the bottom. The
reflection off the gold mirror gives the other two required beams for 3-D cooling
in the MOT. Absorption from a probe beam (grey) is detected by a CCD camera
to measure the MOT atom number. Detailed description of the chamber can be
found in Ref. [29].
6.1.1 Generation of Static Electric Field
Both Rydberg and optical lattice experiments used the same apparatus, and modi-
fications of the existing apparatus required for the Rydberg project were not always
72
performed in the optimal way in order to keep the setup compatible with the needs
of both experiments. For example, to perform the Rydberg excitations in the pres-
ence of the static electric field we utilized already existing conductors in the vacuum
chamber, although they were not intended to be field electrodes. The field was gen-
erated between the gold mirror, which was grounded by the way it was mounted,
and two metal conductors kept at a positive voltage. We used rubidium dispensers,
already inside the chamber, as the positive electrodes. The dispensers, a legacy
atomic source that was unused after the installation of the slower, were connected
to a high-voltage source through the UHV electrical feedthroughs. The voltage was
computer generated and amplified by a high-voltage amplifier (PZD350, TREK).
The dispensers were kept at a constant potential with no current flowing through
them, such that their temperature remained constant, and no rubidium was re-
leased. In this configuration the dispensers can be treated simply as metal plates,
5 mm wide and 3 cm long. The arrangement of the electrodes and the atomic
cloud is shown in Fig. 6.2.
The asymmetric positioning of the positive electrodes with respect to the
location of the atomic cloud was set by the fixed orientation of the electrical
feedthroughs. The electrodes were placed above the level of the gold mirror not
to restrict the optical access and block the MOT laser beams. The spatial pro-
file of electric field generated by the described set of electrodes and the grounded
surface of the stainless steel vacuum chamber was quite complex and could not
be calculated precisely, as the exact locations and sizes of all nearby electrodes
and grounded surfaces were not accurately known. A simplified configuration of
the electrodes was considered, and the resulting electric field was compared to
the Stark shift data as discussed in details in Chapter 7. We note that the given
configuration of small electrodes placed far from, and not symmetric with respect
to the cloud, resulted in an inhomogeneous electric field, complicating the ex-
perimental investigation of the Rydberg excitation rate. Unfortunately, a more
significant modification of the existed apparatus to improve the field homogeneity,
73
Figure 6.2: Grounded surfaces of a 3?3 cm gold mirror, chamber walls and
the optical table, and two thin metal plates 3 cm long and 5 mm wide kept at
a constant potential V formed an electric field at the location of the cold cloud,
several millimeters below the mirror. A few field lines are shown inside the chamber
and out through the bottom glass window.
was not possible. Originally, we tried to place an electrode outside the chamber,
below the bottom glass window, to form the electric field between the electrode
and the grounded gold mirror. As the Rydberg excitation is followed by a partial
black-body ionization of the sample, it resulted in production of electric charges
accelerating toward the appropriate electrode when the field is on. In this configu-
ration, the charges moving towards the outside electrode hit the glass window and
74
accumulate on the surface of the dielectric. Indeed, we observed an accumulated
surface charge at higher fields, that was sufficient to completely shield the electric
field at the location of the atomic cloud.
6.1.2 Detection of Rydberg Atoms
Absorption imaging was used as the detection method to characterize a cold cloud
and the Rydberg excitations2. The absorption imaging allowed a measurement of
the cloud?s atom number and its spatial profile across the probe beam for the atoms
in the 5s1/2|F = 2? ground state. As the atoms are excited to a Rydberg state,
they might return back to the ground state through lower levels (e.g., 5p or 6s),
or get ionized by black-body radiation. In the case of ionization, an atom (now a
positively charged ion) is not resonant with the probe beam anymore and not held
by a cold atom trap. The resulting trap loss can be detected by the absorption
imaging technique to measure the Rydberg excitation rate. The rate of ionization is
very small in comparison with the de-excitation rate to the ground state, and long
excitation times (on the order of several milliseconds) were required to observe trap
loss above the noise level of the experiment. On this time scale some properties of
the cloud, like density, may change significantly, which complicates the analysis of
the obtained results. Excitations on much shorter time scales are usually studied
by a field-ionization of Rydberg atoms in a Microchannel Plate (MCP) detector
(plates and grids to accelerate ions to an electrode, and count the number of the
arrived charges as electric current). Unfortunately, the MCP was not an available
detection technique for our experimental setup, as it would limit optical access to
the cloud making it incompatible with the optical lattice experiment. A simplified
population dynamics of the Rydberg excitations is discussed in Appendix D.
2Absorption imaging is discussed in Appendix C
75
6.2 Generation of Frequency-stabilized Laser Light at 480
nm
The energy gap between the rubidium ground state and its excited state with
the principal quantum number n = 50 corresponds to 297 nm. There is no gain
medium available to construct a laser system at that wavelength; usually the light
in this part of the optical spectrum is generated by frequency doubling. Another
constraint is that only the lower angular momentum states can be excited from
the l = 0 ground state as allowed by the selection rules in the dipole approxima-
tion (?l = ? 1). Also, in Rb the Cooper minimum makes the cross section for
excitation quite small, requiring high laser intensities. However, an observation of
electric quadrupole transitions to Rydberg nd states was recently reported [99]. It
is advantageous to excite a Rydberg state through a two-photon transition, specif-
ically with the intermediate state close to 5p3/2 state. In this case, the first photon
is at 780 nm (red), and can be provided by one of the cooling lasers that couple
the ground state with the 5p3/2 state. The second photon must be at a more
convenient wavelength, 480 nm. Practically, we used the same laser beams at 780
nm to cool and trap rubidium atoms (DLX-110 laser, Toptica Photonics), and to
provide the first step in the two-photon ladder excitation scheme. The frequency
of the red light was controlled by a double-pass AOM, which allowed detunings of
up to 56 MHz from the 5s1/2|F = 2??5p3/2|F = 3?atomic transition. The blue
light was generated by doubling the frequency of a 960-nm laser in a non-linear
crystal in a build-up cavity.
An efficient excitation of Rydberg states requires narrow linewidth laser light,
with frequency stabilization and tuning. A diagram of the laser system we assem-
bled to generate the frequency-stabilized light at 480 nm is shown in Fig. 6.3.
The laser system was set up on a dedicated air-suspended optical table to
damp vibrations, and enclosed in a plexiglass box to protect the setup from air
76
Figure 6.3: The optical layout for the blue light generation setup. The 960-nm
laser seeds a tapered amplifier. The output of the amplifier is frequency-doubled to
generate laser light at 480 nm. The wavelength of the red light is locked to a Fabry-
Perot cavity, and monitored by a wavelength meter. Spherical lenses L1-L4 shape
the 960-nm beam, L5-L6 - the 480-nm beam. Acousto-optical modulators (AOM)
and an electro-optical modulator (EOM) are used to modulate the frequency of
the 960-beam. Half-wave plates (?/2) are used to rotate the beam polarization.
currents and reduce acoustic noise that might limit the stability of the laser. A
home-made external cavity diode laser at 960 nm pumped a tapered amplifier.
The output beam from the amplifier was shaped by a lens system to match a
77
fundamental eigenmode of a commercial frequency doubling cavity. The generated
second harmonic of the cavity at 480 nm was beam-shaped, coupled to an optical
fiber, and delivered to the optical table that hosted the cold atom apparatus. A
detailed description of the components of the laser system is provided below. We
also specify the manufactures and the part numbers of most of the components.
A laser diode (QLD-960-100S, QPhotonics) with an the external cavity in the
Littrow configuration [100] provided up to 35 mW of light at 960 nm. The diode
was controlled by a commercial DLC-200 (MOGLabs) temperature and current
controller. The laser frequency was tuned by tilting an optical grating provided
with two piezo elements, a stack piezo that allowed a frequency change of several
GHz, and a disk piezo for fast active stabilization of the laser frequency with the
tuning range of 200 MHz. The diode laser was normally operated at 18 EWC, and
a current of 196 mA to generate blue photons resonant with the 5p3/2 ? 56s1/2
Rydberg transition. The home-made 960-laser was separated from the rest of the
setup by two optical isolators with the total isolation of 59 dB to avoid possible
optical feedback that might disturb the operation of the laser. A small part of
the beam (< 1 mW) was sent to an optical wavelength meter (WA-1500, EXFO)
that was used for a crude tuning of the 960-laser frequency close to a Rydberg
transition. The accuracy of the wavemeter was 65 MHz at 960 nm, not sufficient for
a fine frequency control of the laser used to excite a sub-MHz linewidth transition.
The laser frequency slowly drifts due to thermal effects changing the length of
the external cavity, and oscillates on a shorter time scale due to acoustic noise,
changes of the air pressure at the cavity or other vibrational motion of the cavity.
The stabilization of the laser frequency using absorption spectroscopy is discussed
in Appendix C. Unfortunately, there are no simple methods to generate a narrow
absorption signal at 480 or 960 nm. Absorption spectroscopy of rubidium Rydberg
states in a glass vapor cell at 480 nm was reported [101] but required a significant
amount of blue light power due to the low Rydberg excitation rate in a vapor.
Also, a laser locked to a spectroscopic feature of an atomic resonance can not be
78
efficiently tuned far away from the resonance, while some the Rydberg blockade
experiments might need to be performed a few GHz off the intermediate 5p3/2
state to avoid its population at the price of even lower Rydberg production rate.
Instead, we used a transfer lock approach: the length of a Fabry-Perot resonator
was fixed with respect to a frequency-stabilized laser, while the frequency of the
home-made 960-laser was simultaneously locked to the resonator.
After the losses on the optical isolators and splitting of the diagnostics waveme-
ter beam, about 25 mW of power was injected into a tapered amplifier diode laser
(a converted Tiger laser, Sacher Lasertechnik) with its gain curve centered around
960 nm. The amplifier was controlled by a commercial TEC3000 temperature
and current controller (Sacher Lasertechnik). The amplifier output power of 650
mW in the seeded mode corresponded to gain factor of about 25 at the current
of 1650 mA at 19.5 EWC. The amplifier was separated from the rest of the setup in
the forward direction by an optical isolator (32 dB). The output beam was shaped
by a set of four lenses to match the fundamental mode of the double cavity. A
small fraction of the 960-beam (< 10 mW) was split from the main beam and cou-
pled to an optical fiber modulator (Lucent) to provide the frequency stabilization
and tunability of the home-made laser. The fiber modulator, an electro-optical
modulator integrated into a polarization maintaining fiber, was driven by a Fluke
6060 radio-frequency generator. The modulated output laser beam, characterized
by a carrier optical frequency plus two first-order sidebands, was sent through an
acousto-optical modulator. If P0 is the total power in the beam, then the power
in the carrier is J20(?m)P0, and the power in each sideband is J21(?m)P0, where ?m
is the modulation depth. The first order output from the AOM was coupled to a
home-made Fabry-Perot confocal resonator.
The resonator was formed by a 50-cm long invar cylinder holding two concave
mirrors anti-reflection coated at 780 and 960 nm. The cavity had a free spectral
range (FSR) of 150 MHz and a finesse of 25 at 780 nm and 783 at 960 nm. One
of the cavity?s mirrors was glued to a hollow cylindrical piezo-electric transducer
79
(PZT), which allowed us to change the length of the cavity. The light reflected from
the input mirrorwas collected ona photodiode detector. When the frequency of the
input light is an integer number times the cavity?s free spectral range, constructive
interference inside the cavity increases the transmitted power and decreases the
reflected power. When the laser frequency or the cavity length are continuously
scanned, the minima of reflected power are evenly spaced by the free spectral
range in the frequency domain. As the laser beam carried the sidebands shifted in
frequency from the unmodulated zero order, the destructive interference condition
for the sideband occurred at a cavity length different from that of the carrier. An
example of the transmitted cavity signal, where the carrier was separated from
the two sidebands by the modulation frequency, fmod, and repeated every FSR,
is shown in Fig. 6.4 (top). Assuming the cavity length is stable, the laser can be
locked to this signal: the change in laser frequency near the cavity resonance would
produce a change in the reflected signal power thus establishing the correspondence
between the laser frequency and the voltage produced by the photodiode detector.
The voltage is fed back to the laser to stabilize its frequency similar to locking a
laser to a saturated absorption setup as discussed in Appendix C. The detected
power might change not only due to the change of the laser frequency but because
of the power fluctuation of the laser itself. It is therefore, advantageous to lock the
laser on top of the resonance signal, where reflected power has its maximum value
when the laser is resonant with the cavity. Since the resonance peak is symmetric,
both positive and negative changes of the laser frequency from resonance will
result in the same decrease of the photodiode signal, which can not be used to
change the laser frequency back to resonance through a feedback loop. However,
a first derivative of the cavity peak, as shown on the bottom scan in Fig. 6.4, is
asymmetric about resonance: positive and negative frequency shifts give rise to
either increase or decrease of the derivative signal.
Practically, there are several ways to generate the derivative, or error signal,
from the cavity resonances, and feed the error signal back to the laser to achieve
80
Figure 6.4: Top frame: cavity resonances of the Fabry-Perot resonator measured by
a photodiode. The 960-laser beam coupled with the cavity was amplitude modu-
lated by a fiber-modulator. The main carrier and two sidebands at the modulation
frequency are separated from the next set of resonances by a free spectral range of
the cavity. The corresponding derivative is shown on the bottom. The error signal
was generated by a frequency modulation of each cavity resonance.
its frequency stabilization. All approaches are based on the Pound-Driver-Hall
stabilization technique [102]. If the laser frequency (or amplitude) is modulated
and above resonance, the reflected signal is amplitude modulated and in phase
81
with the frequency variation. Below resonance, the derivative is negative, and the
reflected intensity varies 180EW out of phase with respect to the input. Therefore, the
frequency modulation allows us to distinguish between the two sides of the cavity
resonance. We first consider the application of the Pound-Driver-Hall technique to
stabilize the length of our Fabry-Perot cavity to a frequency stabilized laser, and
then show how the stabilized cavity was used to lock the frequency of the 960-laser.
6.2.1 Stabilization of Fabry-Perot Resonator
The diode laser at 780 nm, used for the laser cooling in the cold atom apparatus,
was frequency-stabilized by locking it to a rubidium saturated absorption spec-
troscopy signal, a part of the existing BEC apparatus. A part of the beam (< 3
mW) was coupled to a polarization maintaining fiber and delivered to the blue light
optical table. The linearly polarized output from the fiber was mode-matched with
the Fabry-Perot resonator as shown in Fig. 6.5.
The polarization of the beam was rotated to maximize its reflection from a
polarizing beam splitter cube, and the reflected beam was sent through a stan-
dard double-pass AOM configuration. After the splitter, the beam passed through
another AOM, the first diffraction order was coupled to the resonator cavity, and
the reflected power was measured by a photodiode detector (PDA110, Thorlabs)
with a bandwidth of 1.5 MHz. The signal was similar to that presented on the
top of Fig. 6.4, but without the sideband (smaller) peaks. To realize the Pound-
Driver-Hall technique, we modulated the central frequency of the second AOM by
a local oscillator running at about 800 kHz, such that the sidebands were close
to the carrier. The modulated photodiode signal was compared with the signal
from a local oscillator with a RF-mixer. The mixer, analogous to a lock-in am-
plifier, multiplied the input signals to produce the output with the sum and the
difference of the input frequency components, e.g., the DC component and twice
the modulation frequency component. The DC (frequency difference) component
82
Figure6.5: Opticallayout forstabilization ofthe Fabry-Perotresonator. Frequency
of the 780-nm laser beam from a frequency-stabilized diode laser is shifted by a
double-pass AOM #1, fm-modulated by an AOM #2, and coupled with the Fabry-
Perot cavity. The reflected power is measured by a fast photodiode detector.
is proportional to the derivative of the unmodulated reflected intensity. A typi-
cal output from the mixer, after the high frequency components were attenuated
by a low-pass filter, is shown on the bottom part of Fig. 6.4. Thus the Pound-
Driver-Hall technique provides an asymmetric error signal, suitable for a servo-lock
system. The error signal is closed to a linear ramp around the resonant frequency,
and standard tools of control theory [103] can be applied to correct for the laser
frequency fluctuations.
The length of the resonator was controlled by an output of a high-voltage am-
83
Figure 6.6: Schematics of the Fabry-Perot cavity locking circuit. The input error
signal is offset, integrated, amplified, summed with the bias and ramp voltages,
and fed back to control the length of the cavity.
plifier (PZD350, TREK, 250 kHz bandwidth) applied to the PZT attached to one
of the cavity mirrors. The voltage was generated by an electronic circuit, whose
diagram is shown in Fig. 6.6. The error signal from the rf mixer was offset and
amplified by the input gain such that the middle point of the error signal slope
(cavity resonance) corresponded to zero volts. The resulting signal was integrated,
further amplified with a proportional gain and the voltage?s polarity was inverted,
if necessary, to provide negative feedback. The result was combined with an exter-
nally generated ramp and bias voltages, and sent to the high-voltage amplifier. The
ramp voltage from a function generator provided a continuous scan of the cavity
length around the bias voltage, similar to that Fig. 6.4. To lock the cavity length to
the frequency-stabilized 780-laser, the ramp voltage was turned off, the bias volt-
age adjusted to set the cavity length resonant with the wavelength of light, and the
84
Figure 6.7: The bias and offset voltages for the servo loops generated by a LM399H
precision reference.
feedback loop was closed to keep the resonator?s length fixed with the respect to
the wavelength of light. The offset and the bias voltages were generated according
to a circuit diagram presented in Fig. 6.7. The servo?s bandwidth of a kilohertz
was limited by mechanical response of the cavity mirror. That was high enough
to compensate for a thermal drift of the cavity length and vibrations induced by
the acoustic noise, and low enough not to transfer a possible high-frequency noise
from the 780-laser to the 960-laser that was locked to the stabilized Fabry-Perot
cavity.
As the cavity length was stabilized, it was possible to change its length con-
tinuously without unlocking the feedback loop. For that purpose, the central
frequency of the double-pass AOM #2 in Fig. 6.5 was slowly tuned, thus changing
the frequency of light injected into the resonator. The servo responded accord-
ingly controlling the cavity length to keep the laser resonant with the wavelength
of light. The tuning range of 35 MHz was limited by the bandwidth of the AOM.
85
6.2.2 Stabilization of Laser Light at 960 nm
To stabilize the frequency, achieve a narrow linewidth, and tunability of the 960-
laser, we locked the laser to the stabilized Fabry-Perot cavity. The intensity of
laser light was modulated by the fiber modulator, passed through an AOM, its
first diffraction order coupled with the resonator cavity, and the cavity resonances
measured by a photodiode detector as shown in Fig. 6.3. The central frequency
of the AOM was frequency modulated at 300 kHz by a local oscillator to generate
an error signal using the Pound-Drever-Hall technique as described in the previous
subsection. The cavity resonances and the corresponding error signal for each res-
onance dip is presented in Fig. 6.4. The error signal was fed back to the controller
of the home-made laser to keep its wavelength resonant with the length of the
resonator. The schematic of the employed servo loop is shown in Fig. 6.8.
The error signal was amplified, offset and the resulting signal was divided
between two independent servo loops. The frequency of the diode laser could
be controlled by a voltage applied to a PZT disk to change the length of the laser?s
external cavity, or by changing the current driving the laser diode. The piezo could
be used to tune the laser frequency over a range of 0.5 GHz, but the bandwidth
of such control is quite slow, below 1.5 kHz, limited by the mechanical motion of
the optical grating. The current control can be fast, with a bandwidth of several
MHz when the current is supplied directly to the diode; however, adjusting the
current allows only a small tuning of the laser frequency, about 50 MHz in our
case, before the laser hops to a different mode as the laser cavity length depends
on the properties of the diode semiconductor that are very sensitive to the driving
current3. We used the piezo voltage to control slow changes in the laser wavelength,
while current can be used to control fast changes of the laser frequency. For this
purpose, a low-pass filter with a cut-off frequency of 2??200 Hz allowed only low-
3In addition to that, significant changes of current result in large changes of the laser output
power.
86
Figure 6.8: Schematics of the 960-laser locking circuit. The input error signal is
amplified, offset, its DC component processed by a servo loop controlling the piezo
of the 960-laser.
frequency components of the error signal to pass to the voltage loop, and high-pass
filter with the cut-off frequency of 2??2 Hz blocked the DC-component of the
signal from the current servo loop.
The voltage loop acted as an integrator of the near-DC components of the error
signal; after the integration, the error signal was summed with the bias voltage.
The resulting voltage was sent to the laser?s piezo through a high voltage amplifier.
The closed voltage servo loop provided the frequency stabilization compensating
for possible long-term thermal drifts and low-frequency vibrations.
The AC-part of the error signal passed through the current loop presented in
Fig. 6.9. The output voltage, a sum of the integrated and proportionally gain
87
Figure 6.9: Schematics of the 960-laser current loop locking circuit. The AC-
component of the error signal, generated by the circuit in Fig. 6.8, is split, in-
tegrated and amplified. The resulted voltage controls the current of the diode
laser.
input signal, was sent to the diode laser current controller. The bandwidth of
the current loop was limited by the bandwidth of the current controller of 10
kHz. Closing the voltage and the current loop consecutively provided the desired
frequency stabilization of the 960-laser light.
We used one of the first-order sidebands produced by the fiber-modulator as
the cavity resonance for locking. The sideband had a frequency f = fL + fmod,
where fL is the frequency of the laser, and fmod is the modulation frequency of the
EOM. If the modulation frequency is changed, the servo loops will change the laser
frequency accordingly, such that the sum of the two, f, is resonant with the cavity.
This way the laser frequency could be tuned continuously while the laser stayed
locked over a range much larger than that provided by a double-pass AOM. The
maximum possible tuning range can be limited by the bandwidth of the EOM,
the range of frequencies generated by the local oscillator, or the mode-hop-free
range of the diode laser, whichever is smaller. We were constrained by the local
88
oscillator, Fluke 6060, with the maximum output frequency of 500 MHz at 13 dBm
of power. The free spectral range of the cavity of 150 MHz was another constraint.
Intentional misalignment of the input beam with respect to the input coupler of
the fiber modulator allowed us to suppress the transmission of the zeroth order
through the fiber while preserving the power of the first-order sidebands at a higher
modulation depth. In this case, the scan over the cavity resonances looked similar
to that in Fig. 6.4 with the carrier frequency resonances much smaller than the
side-band ones, and the two sets were separated by the free spectral range of the
cavity. If we ignore the zeroth order features, and start increasing the modulation
frequency, the first order from the left set of resonances and the negative first
order from the right set will move toward each other, and the corresponding error
signal will no longer be a linear ramp as the features get closer than their width.
This break of the error signal profile occurs at ?singularity points?, where the
modulation frequency is a multiple of 75 MHz, half of the free spectral range of
the cavity. The width of the error signal was smaller than 1 MHz, and we changed
the modulation frequency in steps of 1 MHz with the laser locked: apparently, the
change of the modulation frequency of the function generator was slow enough for
the servo loop to follow. The laser was still locked at the ?singularity points?,
but as the modulation frequency was further increased the locking point might
?jump? to another resonance from the other set. Practically, we needed to pass
the ?singularity points? several times back and forth until the laser happened to
stay locked to the same sideband before and after the ?singularity points?. This
approach allowed us to scan the 960-laser frequency over the range of 250 MHz
(and 500 MHz for the frequency-doubled blue light), with the exception of the
3 MHz regions around the ?singularity points?, where the laser frequency was
undetermined.
89
Figure 6.10: A linewidth of a laser is measured by the self-homodyne measurement
technique. Frequency fluctuations of the laser are converted to variations of inten-
sity in a Mach-Zander type interferometer. The optical fiber delay line provides
uncorrelated mixture of optical fields at the detector.
6.2.3 Laser Linewidth Measurement
The excitation rate of a very narrow Rydberg transition might be limited by the
linewidth of the excitation laser. We measured the linewidth of our diode laser
by a self-homodyne measurement technique [104]. The optical phase or frequency
fluctuations of a laser, resulting in the broadening of the Schawlow-Townes funda-
mental linewidth, can be converted into variations of light intensity in a unequal-
arm Mach-Zehnder interferometer. In the interferometer, one arm is extended by
an optical fiber, the mixture of two optical fields is measured by a fast photodi-
ode, and the power spectrum of the photodiode current fluctuations provides the
information about the laser linewidth. We used fiber splitters to divide and re-
combine optical fields as shown in Fig. 6.10. A 2-km optical fiber provided a delay
of ?0 ? 6.7 ?s in one of the arms. The mixed power was detected by a photo-
diode with the bandwidth of 20 MHz, and the photocurrent spectral distribution
detected by a power spectrum analyzer (E4440A, Agilient) over 4.6 ms sweep time.
The setup was applied to characterize two lasers that provided the Rydberg ex-
citations: the 780-nm Toptica and the tampered amplifier when seeded by the
960-nm home-made diode laser. The power spectra are presented in Fig. 6.11 on a
logarithmic scale for the frequency-stabilized Toptica laser (graph 1), free-running
90
960-laser (graph 2), and the 960-laser locked to the Fabry-Perot cavity of the sta-
bilized length (graph 3). When the delay line time, ?0 is much larger than the laser
coherence time (inverse linewidth), the interfering optical fields are uncorrelated
at the detector, and the power spectrum of the photocurrent is strictly Lorentzian
with a width twice the FWHM of the laser spectrum. That was the case for the
recorded Toptica spectrum, where a reliable Lorentzian fit gave a linewidth value
of about 1.5 MHz, with the corresponding coherence time ?c = 0.67 ?s << ?0.
When the delay time is comparable or shorter than the laser coherence time, the
phase dynamics of the interfering optical fields remains partially correlated, and the
corresponding photocurrent power spectrum deviates from the Lorentzian profile.
The power spectrum for the case of a free-running 960-laser is shown in the middle
graph in Fig. 6.11, and could not be fitted solely by a Lorentzian. The Lorentzian
fit with the FWHM of 750 kHz corresponded to 375 kHz of the laser linewidth,
or to the coherence time of 2.7 ?s, still smaller than ?0 = 6.7 ?s. When the 960-
laser was frequency-stabilized to the Fabry-Perot resonator, the appropriate power
spectrum revealed an oscillating pattern superimposed on the Lorentzian line with
the period of oscillations given by 1/?0, which was an indication that the coherence
time became comparable or larger than the delay time, and the laser linewidth was
smaller than 1/?0 = 150 kHz. In this case, our self-homodyne measurement setup
can not be used to extract the exact value of the laser linewidth as the beat signal
critically depends on the length difference between the two optical paths on the
wavelength scale, and an extension of the delay line or a modification of the setup
was required to perform an accurate measurement, for example, as described in
reference [105]. We did not pursue such measurements further, as we were satisfied
with the results: the 780-laser linewidth was measured to be 1.5 MHz, much larger
than the estimated linewidth of the 960-laser of about 150 kHz, or 300 kHz of the
blue light at 480 nm after the frequency-doubling. As both lasers were employed
for an excitation of Rydberg levels, the effective linewidth of the excitation light
was limited by that of the 780-laser, and the precise measurement of the 960-laser
91
Figure 6.11: Power spectra measured with the self-homodyne technique for the
frequency-stabilized Toptica (graph 1), the free-running home-made 960-laser
(graph 2), and the 960-laser locked to the Fabry-Perot resonator (graph 3). The
spectrum is strictly Lorentzian for Toptica when the delay line is longer than the
coherence length of the laser. For a longer coherence length the spectrum deviates
from Lorentzian (graph 2) and exhibits oscillations when the coherence length is
longer then the delay line (graph 3).
linewidth was not found necessary. A better stabilization of the Toptica laser to
reduce its linewidth was not available at the time.
92
6.2.4 Frequency-doubling of 960-nm Light
The stabilized and tunable light at 960 nm was converted into the 480-nm light
by a second harmonic generation (SHG) process in a Lithium Triborate (LBO)
nonlinear crystal in a commercial, external-cavity frequency doubler (WaveTrain,
Newport). The generation of the second harmonic of a continuous wave laser
requires significant input fundamental power, not available experimentally. The
amplification of the optical power can be achieved in an optical build-up cavity, and
the conversion efficiency from a nonlinear crystal can be increased in comparison
with a direct conversion arrangement [106]. The resonator cavity schematics uti-
lized in the WaveTrain is shown in Fig. 6.12. A ring cavity is formed by two curved
dielectric mirrors and a prism. The length of the resonator is tuned by translating
the prism along its symmetry axis via the piezoelectric transducer PZT. Both the
prism P and the nonlinear crystal X are cut for Brewster?s incidence angle to min-
imize reflection losses that limit the gain of the resonator. The 960-laser beam was
shaped by a cylindrical lens L1, and spherical lenses L2, L3 and L4 (Fig. 6.3) to
match the fundamental eigenmode of the cavity. The pump beam passed through
an EOM, that modulated the frequency of the beam at 80 MHz. A reflection
from the input cavity mirror M1 was measured by a photodiode. The resulting
cavity resonances and the corresponding Pound-Drever-Hall error signal are shown
in Fig. 6.13. The FWHM of the resonances was about 6 MHz. A servo loop con-
trolling the piezo voltage provided the locking of the cavity on resonance with the
wavelength of the pump beam to ensure the maximum build-up of power inside
the resonator. Curved cavity mirrors focused the fundamental resonator mode on
the LBO crystal. The critical phase matching, an equality of phase velocities of
the fundamental and harmonic modes, was achieved through the angular depen-
dence of the refractive indices of the crystal. The doubler cavity converted 300
mW of 960-nm light into a 30 mW of blue light at 480 nm. The elliptical profile
of the output blue beam was corrected to a round one by two cylindrical lenses
93
Figure 6.12: Two curved mirrors M1 and M2, and a prism P form a power build-up
cavity for a second harmonic generation in a LBO crystal X. The cavity length is
controlled by a transverse motion of the prism attached to a PZT. The prism and
the crystal are at Brewster?s angle to minimize the cavity loss.
L5 and L6. The beam passed through an AOM for amplitude control. The first
diffraction order was coupled with a polarization maintaining fiber and delivered
to the optical table hosting the cold atom apparatus. The central frequency of the
AOM was computer-controlled, and the modulator was used as a beam deflector
to provide a fast switching time of the blue beam. Routinely, 6.5 mW of power of
the 1.6 mm diameter laser beam at 960 nm was delivered to a cold atomic cloud
for the Rydberg excitation experiments.
94
Figure 6.13: An oscilloscope screen shot of the cavity resonances of the WaveTrain
doubler and the corresponding error signal. The FWHM of the resonance is about
6 MHz (one division corresponds to?110 MHz on a 2-ms time scale).
95
Chapter 7
First Experiments with Rydberg Atoms
As discussed in Chapter 5, the strength of the Rydberg-Rydberg interactions can
be dramatically increased in the presence of an external electric field. The pro-
duced DC Stark effect can be used to either shift the atomic energy levels to F?orster
resonances or mix the energy levels to create a field-independent permanent dipole
moment. In both cases, the application of the electric field results in the transi-
tion from 1/R6 van der Waals to 1/R3 dipole-dipole interactions, where R is the
interatomic separation. For the 50s state and R = 5 ?m the energy shift is about
1 MHz due to the van der Waals interaction and up to 150 MHz for the dipole-
dipole interaction at an external field of 5 V/cm. Required density of 4 ?m?3 is
an order of magnitude higher than the peak densities of a standard MOT setup,
but can be achieved in a tight magnetic trap after the some evaporative cooling.
A broadening of the Rydberg transition resonance from the energy shift due to
interactions can be observed with spectroscopic measurements. The first spectro-
scopic measurements obtained with our excitation laser system are reported in this
Chapter.
7.1 Autler-Townes Splitting
The linewidth of our two-photon excitation laser system, described in Chapter 6,
was measured to be about 1.5 MHz. Our first step to test the excitation system was
to investigate the Rydberg production in a MOT cloud. In this set of experiments,
96
the MOT cooling laser, red-detuned from the 5s3/2 level, was used to provide the
first excitation photons at 780 nm. I will refer to this laser as the pump. The laser
intensity was above saturation of the cooling transition resulting in the strong
pumping of the 5s3/2 state. The laser detuning from the atomic resonance, ?, was
variable fordifferent experiments. The second, 480-nm photon was delivered with a
collimated 1.6 mm diameter laser beam (probe) of 6.5 mW of power. The frequency
of the probe beam was tuned to provide the two-photon resonance between the
ground 5s1/2 and the 56s1/2 excited state. The choice of the particular excitation
level was not critical for the initial test of the system. To align the probe with
the atomic cloud we overlapped the beam with an auxiliary near-resonant 780-
beam that could destroy the trapped cloud when aligned with it. Thus, the proper
alignment of the auxiliary beam guaranteed the alignment of the blue beam.
We have conducted the spectroscopic measurements in the following way. The
MOT cloud was collected at the cooling laser detuning?/2? =?24 MHz. We used
a relatively small cloud of 107 atoms with the diameter of 0.5 mm, smaller than the
size of the blue beam. Then, the slower beam was shut off to stop the MOT loading
and the atoms were exposed to the blue light for a time tb = 50 ms, much shorter
than the trap lifetime due to the background gas collisions. As atoms are excited
to a Rydberg state, they can be ionized by the blackbody radiation and are no
longer trapped. Therefore, a net trap loss due to the excitation is proportional to
the number of excited Rydberg atoms. After the exposure, the number of atoms in
the ground state was measured by absorption imaging technique, which destroyed
the trapped cloud concluding the measurement cycle. During consecutive cycles,
loading - exposure - imaging, the frequency of the probe was incremented in steps
of 2 MHz. The relative trap loss, the number of atoms lost from the trap divided
by the initial number of atoms in the trap, around two-photon resonance along
with a two-peak Lorentzian fit is presented in Fig. 7.1 as a function of the probe
frequency.
The spectrum, an Autler-Townes doublet [107], features two resonances at fre-
97
Figure 7.1: Autler-Townes splitting of 5p3/2 as probed by transition to 56s state.
The separation between resonances is the generalized Rabi frequency of the satu-
rated pump transition.
quencies ?1/2? and ?2/2?. The Autler-Townes splitting resulting from a dipole
transition between an unperturbed state and a pair of dressed states is considered
in Appendix D. Not taking into account the width of resonances, the relative posi-
tion of the two resonances can be found from the normalized spectral distribution
function
F(?) = cos2? f(?b??)+sin2? f(?b?? + ??), (7.1)
where?b is frequency of the blue light, ? is the Rabi frequency of the 5s3/2?5p3/2
transition, the generalized Rabi frequency ?? =??2 + ?2, the dynamic Stark shift
2? = ?? + ?, tan(2?) = ??/?, ? is the detuning from the intermediate state,
and f(?) is the Dirac?s delta-function1. The spectral distribution function sets the
relative position and height of the two resonances. The spacing between resonances
1We use an unconventional notation for the delta-function to distinguish it from the detuning
?.
98
in Fig. 7.1 is ?2??1 = ??, which allows an accurate measurement of the 780-nm
Rabi frequency once the detuning ? is known. The actual spectrum in Fig. 7.1
consists of two power-broadened Lorentzian peaks with the separation between the
two of ??/2? = 27 MHz as extracted from a data fit with a sum of two Lorentzian
profiles.
At small detunings ?, the Rydberg excitation is step-wise as atoms are pumped
to the intermediate level and spend a finite amount of time, eventually to either get
excited to the Rydberg state or radiatively decay to the ground state. In this case,
the width of the Lorentzian resonances, w is dominated by the saturation of the
pump transition, such thatw???1 +s> 6 MHz, where? = 6 MHz is the natural
linewidth of the cooling transition and s is the saturation parameter. For large
detunings, ?/2?? >> 1 the Rydberg transition occurs through a virtual state, as
atoms are pumped to the Rydberg state directly through a two-photon transition.
The width of resonance is set by the convolution of the Rydberg state decay rate
and the linewidth of the excitation laser. For our experimental setup, the 1.5 MHz
width was limited by the resolution of the 780-laser. To ensure that there were no
other broadening mechanisms present, we investigated the Autler-Townes spectra
for various MOT detunings for the excitation time of 10 ms. Experiments were
conducted under the same conditions as before for a set of detunings ? between 16
and 56 MHz provided by a double-pass AOM. The maximum possible detuning was
limited by the bandwidth of the AOM: the power transmitted through the double
pass, and therefore the corresponding Rabi frequency, was just enough to provide
a detectable Rydberg excitation. The spectra for each detuning are presented in
figures 7.2-7.10. Each data point represents a single measurement of the trap loss.
We ascribed the vertical error bar to the data point based on 10-20% variation of
the MOT atom number from one loading cycle to another.
At larger detunings the dressing interaction is weaker, so that ??0, and the
spectrum reduces to a single line at the frequency of the probe light,?b. For? > 26
MHz, the smaller peak corresponding to the stepwise excitation is comparable with
99
the background level making the fitting procedure with the sum of two Lorentzian
functions dependent on the initial parameters of the fit. Therefore, we fitted the
experimental data around the taller, two-photon excitation peak with a single
Lorentzian function. For each data set, we chose a subset to fit around the peak,
limited to several data points on the lower energy side of the peak to avoid possible
contribution from the other, smaller peak. We ensured that the fit results did not
depend on a particular choice of the subset within the uncertainty of the fit results.
That was not true for higher detunings, ? > 32 MHz, where the width of the peak
of a few megahertzs was close to the scan step of 2 MHz, and only a few data points
represented the peak complicating the fitting procedure. For figures 7.6-7.10, we
show a specific subset of data that provided a converging fit resulted in a peak
height between the maximum value of the data (within the corresponding error
bar) and one, the maximum possible value of the trap loss. The corresponding fits
are shown in solid lines in each figure along with a distribution of the normalized
residuals (bottom plots). The uncertainties of the fit results were increased as
necessary to make the results independent of the data subset. The fit results,
values of the peak widths, their respective uncertainties, and the reduced ?2red
values are reported in captions of each figure.
The width ofthetaller, two-photonresonance peakasa functionof thedetuning
is presented in Fig. 7.11. We see that the width drops from about 6 MHz at
the standard MOT detunings to about 1.5 MHz at larger detunings, which is
consistent with the value of the linewidth of our excitation laser. At higher values
of the detuning, the fits are quite poor resulting in large values of uncertainties
and reduced ?2red ? 10. Nevertheless, we were satisfied with the results as no
broadening above the value of the excitation laser linewidth was observed. As an
example, the solid line in Fig. 7.11 shows the power broadening due to detuning
of a 100 kHz resonance of the two-level atom that models Rydberg excitations at
large detunings from the intermediate state.
100
Figure 7.2: Autler-Townes splitting for the MOT detuning of 16 MHz (top) and the
Lorentzian fit of the two-photon transition peak with the width, wc = (6.06?0.31)
MHz. The normalized residuals of the fit are shown on the bottom plot. The
corresponding reduced ?2red =2.42.
101
Figure 7.3: Autler-Townes splitting for the MOT detuning of 20 MHz (top) and the
Lorentzian fit of the two-photon transition peak with the width, wc = (4.13?0.77)
MHz. The normalized residuals of the fit are shown on the bottom plot. The
corresponding reduced ?2red =2.98.
102
Figure 7.4: Autler-Townes splitting for the MOT detuning of 24 MHz (top) and the
Lorentzian fit of the two-photon transition peak with the width, wc = (4.14?0.71)
MHz. The normalized residuals of the fit are shown on the bottom plot. The
corresponding reduced ?2red =2.89.
103
Figure 7.5: Autler-Townes splitting for the MOT detuning of 26 MHz (top) and the
Lorentzian fit of the two-photon transition peak with the width, wc = (3.05?0.63)
MHz. The normalized residuals of the fit are shown on the bottom plot. The
corresponding reduced ?2red =3.32.
104
Figure 7.6: Autler-Townes splitting for the MOT detuning of 32 MHz (top) and the
Lorentzian fit of the two-photon transition peak with the width, wc = (2.97?1.26)
MHz. The normalized residuals of the fit are shown on the bottom plot. The
corresponding reduced ?2red =9.38.
105
Figure 7.7: Autler-Townes splitting for the MOT detuning of 36 MHz (top) and the
Lorentzian fit of the two-photon transition peak with the width, wc = (2.18?0.82)
MHz. The normalized residuals of the fit are shown on the bottom plot. The
corresponding reduced ?2red =6.63.
106
Figure 7.8: Autler-Townes splitting for the MOT detuning of 40 MHz (top) and the
Lorentzian fit of the two-photon transition peak with the width, wc = (2.29?1.17)
MHz. The normalized residuals of the fit are shown on the bottom plot. The
corresponding reduced ?2red =7.89.
107
Figure 7.9: Autler-Townes splitting for the MOT detuning of 48 MHz (top) and the
Lorentzian fit of the two-photon transition peak with the width, wc = (2.06?1.39)
MHz. The normalized residuals of the fit are shown on the bottom plot. The
corresponding reduced ?2red =12.1.
108
Figure 7.10: Autler-Townes splitting for the MOT detuning of 56 MHz (top) and
the Lorentzian fit of the two-photon transition peak with the width, wc = (1.35?
1.30) MHz. The normalized residuals of the fit are shown on the bottom plot. The
corresponding reduced ?2red =10.9.
109
Figure 7.11: The width of the two-photon resonance peak of the Autler-Townes
splitting for various MOT detunings. The error bars represent the uncertainties of
the corresponding Lorentzian fits of the resonances. The solid line is an example
of power broadening of a 100 kHz resonance.
7.1.1 Rydberg Production Rate Measurement
Strong Rydberg-Rydberg interactions can result in a broadening of the two-photon
resonance. Another signature of the strong interactions is the reduction of the
Rydberg production rate at a fixed detuning from the intermediate level. It is
useful to be able to measure the production rate with the possibility of its further
investigation under conditions of suppressed excitations.
At large detunings, the population of the intermediate level can be adiabatically
eliminated, and three-level atom system can be considered as effective two-level
110
system. The population dynamics of a two-level system with the ionization decay is
considered in Appendix D. The 56sRydberg state radiatively decays to the ground
state with the rate 1/?0nl = 1/170?s and gets ionized by blackbody radiation with
the rate 1/?bb = 1/150?s. The trap loss is given by the number of ionized atoms,
Ni, and for the case of close radiative and ionization rates 1/?0nl ?1/?bb = ?, can
be written as
Ni = N0
parenleftbigg
? ?? +R
R
e?(?+RR)t +e??t? RR? +R
R
parenrightbigg
, (7.2)
where N0 is the initial trap number and RR is the Rydberg production rate.
We experimentally measured the number of atoms lost from the trap as a
function of the excitation time at a fixed detuning ? = 36 MHz, and the probe
frequency tuned to the two-photon resonance. The trap loss data is presented
in Fig. 7.12 (top), and was fitted with the rate Eq. 7.2 to obtain the Rydberg
production rate RR = 1/(2.62?0.16) ms with the reduced ?2red of 0.25. Each data
point represents a single measurement of the atom number left in the trap after
the excitation. We ascribed the vertical error bar to the data point based on the
10% variation of the MOT atom number from one loading cycle to another. The
normalized residuals for the fit are shown on the bottom plot of Fig. 7.12.
7.2 Rydberg Atoms in External Electric Field
As discussed in Chapter 5, the application of an external constant electric field
allows us to dramatically enhance the Rydberg-Rydberg interactions. At low fields,
the atom?s induced dipole moment interacts with the field resulting in the Stark
energy shift quadratic in field for an s state. At higher fields of about 5-6 V/cm,
the atom in n ? 50 state obtains a permanent dipole moment independent of
the field due to the opposite parity level mixing. In this work we investigated
the effect of the low fields. In particular, we measured the energy shift due to
the field and observed a resonance broadening that can be an indication of the
111
Figure 7.12: The decay of trapped atom number due to blackbody ionization (top
plot). The solid curve represents the exponential fit (Eq.7.2) that provides the
value of the Rydberg production rate RR = 1/(2.62?0.16). Normalized residuals
for the fit are shown on the bottom plot.
increased Rydberg-Rydberg interactions. The developed experimental procedure
can be directly extended to the domain of higher fields.
To investigate the effects of the external electric field we used the same exci-
112
tation scheme as before. A MOT cloud at the cooling light detuning of 40 MHz
was exposed to the blue light for 5 ms. At this detuning, the Autler-Townes split-
ting reduces to a single resonance peak. The probe frequency was scanned around
two-photon resonances in steps of 2 MHz. The electric field was generated by two
metal electrodes as described in Chapter 6. The measured spectra for several volt-
ages from 0 to 46.8 V applied to the electrodes are shown in Fig. 7.13-Fig. 7.17.
Each data point represents a single measurement of the trap loss. We ascribed
the vertical error bar to the data point based on 10-20% variation of the MOT
atom number from one loading cycle to another. To illustrate the energy shift
and broadening due to the increasing electric field we plotted all resonances on the
same graph as shown in Fig. 7.18. The energy level shift responsible for the shift
of the peak position is the expected DC Stark effect.
We used Lorentzian fits to extract the width and relative position of each
resonance. The results are reported in the caption of the corresponding figure
and are plotted in black for the peak position and in blue for resonance width in
Fig. 7.19 (top). The error bar of each data point represents its uncertainty obtained
from the Lorentzian fit of the corresponding spectroscopic feature. The data set
for the peak position was fitted with a parabola (black solid line in Fig. 7.19),
y = a(x?x0)2, where the offset of the independent variable x0 is due to possible
stray electric fields. The found the fit parametersa = (?5.18?0.12)?10?2 MHz/V2
and the electric field offset (in units of the electrode voltage) x0 = (?2.51?0.45)
V. The normalized residuals for the peak position fit are shown on the bottom of
Fig. 7.19. The data set for the resonance width was fitted with a parabola (blue
solid line in Fig. 7.19), y1 = Ax2 +y0, and the fit parameters were found to be
a = (2.06?0.15)?10?3 MHz/V2, y0 = (3.22?0.14) MHz.
The exact values of the electric field could not be calculated precisely due to
the complicated geometry of the conducting surfaces of constant potential. To
calibrate the electric field we perform numerical calculations of the Stark shift for
56s state. Forthat, we numerically evaluated the unperturbed radial wavefunctions
113
Figure 7.13: The Rydberg spectrum of the 56s state with both electrodes grounded
(top) and the Lorentzian fit of the two-photon transition peak atxc = (0.88?0.71)
MHz with the width, wc = (3.05?0.63) MHz. The normalized residuals of the fit
are shown on the bottom plot. The corresponding reduced ?2red = 5.60.
114
Figure 7.14: The Rydberg spectrum of 56s state in the presence of the external
electric field generated by two electrodes at the voltage of 18.5 V (top) and the
Lorentzian fit of the two-photon transition peak at xc = (?23.4?0.4) MHz with
the width, wc = (3.69?1.14) MHz. The normalized residuals of the fit are shown
on the bottom plot. The corresponding reduced ?2red = 7.61.
115
Figure 7.15: The Rydberg spectrum of 56s state in the presence of the external
electric field generated by two electrodes at the voltage of 29 V (top) and the
Lorentzian fit of the two-photon transition peak at xc = (?51.1?0.5) MHz with
the width, wc = (5.04?0.82) MHz. The normalized residuals of the fit are shown
on the bottom plot. The corresponding reduced ?2red = 2.35.
116
Figure 7.16: The Rydberg spectrum of 56s state in the presence of the external
electric field generated by two electrodes at the voltage of 35.3 V (top) and the
Lorentzian fit of the two-photon transition peak at xc = (?73.4?0.6) MHz with
the width, wc = (5.82?1.80) MHz. The normalized residuals of the fit are shown
on the bottom plot. The corresponding reduced ?2red = 3.71.
117
Figure 7.17: The Rydberg spectrum of 56s state in the presence of the external
electric field generated by two electrodes at the voltage of 46.8 V (top) and the
Lorentzian fit of the two-photon transition peak at xc = (?126.6?0.7) MHz with
the width, wc = (7.59?2.16) MHz. The normalized residuals of the fit are shown
on the bottom plot. The corresponding reduced ?2red = 3.11.
118
Figure 7.18: The Rydberg spectrum of 56s state for various electrode voltages
119
Figure 7.19: Dependence of the energy shift of the 56s state on the applied voltage
resulted from the quadratic Stark effect. The shift of the resonance peak position
is shown in black, peak width - in blue on the top plot. The bottom plot presents
the normalized residuals from the parabolic fit of the peak position data. The
reduced
chi2red was found to be 2.38.
120
Figure 7.20: The calculated energy shift of the 56s state due to the DC Stark
effect (solid line) the experimental data (crosses) were fit together to provide the
calibration of the electrode voltage.
of a hydrogen-type one-electron atom for the known energy eigen-values given by
the quantum defect theory [75]. The Stark shift ?th was then calculated from
the perturbation theory for a mixture of four energy levels adjacent to 56s as
discussed in Appendix D, and can be written in units of frequency as a parabolic
function ?th = ?82.5E2, where the coefficient of proportionality is in units of
MHz/V2/cm2, and E is the electric field in units of V/cm. Comparison with the
fit results for the peak position in Fig. 7.19 gives rise to the conversion factor
between the electric field and the electrode voltage, ?E?V = (2.51?0.03)?10?2
cm?1, such that E = ?E?VV, where V is the electrode voltage.
Both calculated and measured energy shifts for positive and negative values
of the electric field are presented in Fig. 7.20. Note, that the experimental data
is asymmetric with respect to the vertical axis as a result of stray fields, which
illustrates that the Stark effect can be a very sensitive tool to measure small elec-
121
tric fields. The results of the electric field calibration were qualitatively consistent
with calculations performed by the SIMION ion and electron optics simulator soft-
ware based on a simplified geometry of the electrodes. It was quite obvious that
the quadratic broadening of the resonances as shown in Fig. 7.19 was dominated
by the inhomogeneity of the electric field across the atomic cloud. To investigate
the results of the inhomogeneity we conducted the Stark shift spectroscopy mea-
surements with only one electrode kept at a constant voltage to enhance the field
inhomogeneity. The applied voltage was chosen such that the resonances occur
at the same probe frequencies as those for the case of two electrodes. Obtained
spectra are shown in Fig. 7.21 (top), along with the 2-electrode spectra (bottom)
for comparison.
It is clear that the intentionally introduced inhomogeneity dramatically broad-
ened the resonances. The SIMION simulations showed a 20% variation of the field
across a 1 mm cloud for the case of one electrode and 3% for the two-electrode
configuration qualitatively consistent with the observed broadening. Therefore it is
an open question whether the resonance broadening due to the Rydberg-Rydberg
interactions increasing with the field can be separated from the inhomogeneity-
induced width of resonance.
7.3 Discussion and On-going Work
In conclusion, we constructed a narrow linewidth laser system and successfully
used it for Rydberg excitations of rubidium. Spectroscopic measurements of Ryd-
berg resonances revealed a broadening of the resonances that could be induced by
the interactions between the Rydberg atom and the inhomogeneity of the applied
electric field. The field inhomogeneity was found to be the primary source of the
broadening, and it needs to be suppressed to separate the effects of interactions
and the inhomogeneous field. There are several ways to approach this goal. Higher
electric fields and smaller excitation volumes may increase the strength of inter-
122
actions as atoms gain permanent dipole moments, while the inhomogeneity of the
field is kept small. Smaller excitation volumes can be achieved by either focusing
the 480-beam, or conducting the excitations of a smaller cloud produced by the
evaporative cooling.
The current detection system relies on the trap loss due to ionization by black-
body radiation, which is limited by a low ionization rate. The rate can be greatly
enhanced by radiative ionization with a high power far off-resonant laser beam at
1550 nm, which is available in the laboratory.
123
Figure 7.21: Stark shifted spectra for one- and two-electrode configuration. In case
of one electrode, the generated field inhomogeneity results in a large resonance
broadening (top). The two-electrode configuration results in more homogeneous
field and smaller broadenings (bottom).
124
Chapter 8
Summary of Dissertation
In this dissertation we reported the results of two experimental projects with laser-
cooled rubidium atoms. The first part of the thesis is devoted to the development
of new elements of atom optics based on blue-detuned high-order Bessel beams. We
assembled a cold atom apparatus to produce a 1-mm cloud of about 108 rubidium
atoms in a MOT at 250?K. A 4th-order Bessel beam of 1 mm in diameter gener-
ated by an SLM was used to guide the atoms. The properties of this type of guide
were investigated for various parameters of the hollow beam, such as the detuning
from an atomic resonance, size and the order of the Bessel beam. We extended its
application to create more complicated interferometer-type structures by demon-
strating a tunnel lock, a novel device that can split an atomic cloud, transport it,
delay, and switch its propagation direction between two guides. We reported a
first-time demonstration of an atomic beam switch based on the combination of
two crossed Bessel beams. We achieved the 30% efficiency of the switch limited
by the geometrical overlap between the cloud and the intersection volume of the
two tunnels, and investigate the heating processes induced by the switch. We also
showed other applications of crossed Bessel beams, such as a 3-D optical trap for
atoms confined in the intersection volume of two hollow beams and a splitter of
the atomic density.
The second part of this dissertation is devoted to the spectroscopic measure-
ments of the Rydberg blockade effect, a conditional suppression of Rydberg excita-
tions depending on the state of a control atom. We assembled a narrow-linewidth,
125
tunable, frequency stabilized lasersystem at480nm to excite laser-cooled rubidium
atoms to Rydberg states with a high principal quantum number n?50 through a
two-photon transition. We applied the laser system to observe the Autler-Townes
splitting of the intermediate 5p3/2 state and used the broadening of the resonance
features to investigate the enhancement of Rydberg-Rydberg interactions in the
presence of an external electric field. The observed broadening was mainly due to
the inhomogeneity of the applied electric field across the atomic cloud. Experi-
ments are currently in progress to distinguish between the two sources of broad-
ening using smaller excitation volumes, higher detection sensitivity of Rydberg
excitations and the dependence of the atom-atom interactions on density.
126
Appendix A
Appendix A. Aspects of Light-matter Interactions
In this section we outline several basic concepts of light interaction with a two-
level atom. The derived light force illustrates the idea of the laser cooling and
trapping of atoms. If an atom initially in a lower energy level absorbs a photon of
momentum k from a laser beam, the former receives momentum ?k and is excited
to a higher energy level. Consecutive irradiation of a photon leads to de-excitation
of the atom and momentum transfer (recoil) from the photon with a corresponding
recoil energy of
Erec = ?
2k2
2M . (A.1)
Emission processes occur along two channels, spontaneous and stimulated. Each
one of them in combination with preceding absorption gives rise to two different
forces the light exerts on the atom, spontaneous emission and stimulated emission
forces.
A.1 Spontaneous Emission Force
In case of spontaneous emission, the angular distribution of the irradiated photons
is symmetric and scattering of a large number of photons results in zero net recoil
momentum. Indeed, momentum conservation after summation of N scattering
events
pN +N?k = p0 +
summationdisplay
i
?ki, (A.2)
127
shows that the atom gains momentum in direction of the laser beam, k, when the
last term can be neglected for a large N. In the Eq. A.2, p0, pN are corresponding
initial and final momenta of the atom, ki is the recoil momentum of theith emitted
photon. If one introduces the force as an expectation value of a force operatorF
= dp/dt, then according to the Ehrenfest theorem
F =?F?=???V(r)?, (A.3)
where V is a potential corresponding to the light-atom interaction exerting the
force F. The part of the interaction potential contributing to the expectation
value is (for a rigorous derivation, see, for example [108])
V =?eE?r, (A.4)
where E, r are the electric field and atomic dipole operators accordingly. For
optical wavelengths the spatial variation of the electric field over the atomic size is
negligible (electric dipole approximation) and the expectation value can be taken
before the gradient in Eq. A.3. We consider the electric field operator of the
monochromatic traveling wave in the form
E = E02 ?(ei(kz??t) +c.c.), (A.5)
where E0 is the amplitude, ? is the polarization vector and ? is the frequency of
the laser field.
We further ignore quantum correlations (semi-classical radiation theory) and
replace the operators in Eq. A.4 by their corresponding expectation values. In
that case,?E?is interpreted to be a classical electric field vector. It is convenient
to calculate the expectation values in (A.3) using the density matrix formalism: for
an operatorA, its expectation value?A?= Tr(?A). The density matrix operator
? evolves in time as follows:
128
i?d?dt =?[?,H], (A.6)
whereH=H0 +V is the full Hamiltonian of the system comprised of a field-free,
time-independent Hamiltonian H0 and the atom-light interaction part V . The
time-dependent Schr?odinger equation corresponding to the full Hamiltonian
i???(r,t)?t =H?(r,t) (A.7)
is equivalent to
i?dcjdt =
summationdisplay
k
ck(t)Vjk(t)ei?jkt, (A.8)
where?jk ??j??k is the difference between the eigenvalues?j of the Hamiltonian
H0, cj(t) are the expansion amplitudes of the atom wavefunctions ? in the basis of
the eigenfunctions{?j}ofH0, and Vjk are the matrix elements of the interaction
Hamiltonian in the same representation1.
For a pure state, the density matrix can be expressed in terms of the expansion
amplitudes cj,
?jk = cjc?k, (A.9)
and can be calculated after solving Eq. A.8 for cj for the case of a specific
interaction V. Knowing the matrix density operator one can obtain the expression
for the light force from Eq. A.3.
We assume that the laser light frequency ? is close to the energy difference (in
units of frequency) ?a??12 = ?21 between the two levels of the atom, j,k = 1,2,
i.e., the laser detuning from the atomic resonant frequency ?????a ??,?a.
This assumption (two-level atom model) is reasonable for a variety of experiments
in Atomic Physics and justifies the truncation of the summation of Eq. A.8 to
1Note that Vjk contains only off-diagonal elements as V is odd under parity
129
just two terms. The complete solution of the Eq. A.8 contains ?non-resonant?
1?exp(i(?0 +?a)t)
?0 +?a and ?resonant?
1?exp(i(?0??a)t)
?0??a terms, and only the
latter have the dominant contribution to the elements of the density matrix at
small detunings ?. We therefore retain only the ?resonant? terms in Eq. A.8 when
calculating the Vjkei?jkt products (rotating-wave approximation). For a rigorous
application of the rotating-wave approximation to the two-level atom refer, for
example, to Ref.[109].
Time evolution of the density matrix in terms of amplitudes cj can be obtained
by differentiating Eq. A.9
d?jk
dt = cj
dc?k
dt +c
?
k
dcj
dt. (A.10)
The semi-classical radiation theory does not include spontaneous emission, and
it must be introduced phenomenologically. We choose the lower energy level of the
atom, j = 1 to be the ground state that does not decay (?1 = 0), and describe the
spontaneous emission by an exponential decay of the upper level, j = 2
dc2(t)
dt =?
?2
2 c2(t), (A.11)
such that the upper level population decays with the rate ?2 ? ? with lifetime
? = 1/?. Spontaneous emission can now be included in Eq. A.8
i?dcjdt =
summationdisplay
k
ck(t)Vjk(t)ei?jkt?i??j2 cj(t) j,k = 1,2. (A.12)
Substitutingdcj/dtandcj from Eq. A.12 and A.9 correspondingly into Eq. A.11
give rise to the optical Bloch equations (OBE)
d??
dt =
?
??
??
??
?
0 ?i2? ?i2?? ?
?i2?? ?(?2 +i?) 0 i2??
i
2? 0 ?(
?
2 ?i?) ?
i
2?
0 i2? ?i2?? ??
?
??
??
??
?
??, (A.13)
130
where we have introduced a 4-vector ?? = (?11 ?12e?i?t ?21e?i?t ?22)T composed
of the elements of the density matrix operator, and expressed the interaction via
the Rabi frequency
???eE0? ??2|r|?1?. (A.14)
We can now re-write the expectation value in Eq. A.3 with the elements of the
density matrix and the Rabi frequency for the light force along the direction of the
laser beam ?z = k|k|
Fz = ?
parenleftbigg??
?z ?
?
21 +
???
?z ?21
parenrightbigg
. (A.15)
Off-diagonal matrix element ?21 (note, ?jk = ??kj)) can be calculated from a
steady-state solution of the optical Bloch equations (A.13). Particularly,
?21 = i?2(?/2?i?)(1 +s), (A.16)
and
?22 = s2(1+s), (A.17)
where the saturation parameter s is given by
s = 2|?|
2
4?2 +?2. (A.18)
The force can be expressed in terms of the real, qr, and imaginary, qi, parts of
the logarithmic derivative of ?, ?lg?/?z = (qr +iqi), and Eq. A.15 will be
F = ? ss+ 1(??qr + 12?qi), (A.19)
where the first term is proportional to the gradient of the electric field amplitude
and second - to the gradient of the field?s phase.
131
Forthecase ofmultiple absorptionandconsecutive spontaneous emission events,
Eq. A.2 gives rise to the expression of the spontaneous (light or radiation pressure)
force
Fsp = ?k??22 = ?k ?/21+?2/2?2 + 4?2/2?2, (A.20)
which coincides with the second term in Eq. A.19 for the case of a traveling wave.
For practical purposes, it is convenient to introduce the saturation parameter
on resonance, s0 = s|?=0, and the expression for the spontaneous force will be
Fsp = ?k?2 s01+s
0 + (2?/?)2
. (A.21)
The on-resonance saturation parameter can be expressed via the intensity of
the laser beam I,
s0?2|?|
2
?2 = I/Is, (A.22)
where we follow Citron et al. [30] definition of the saturation intensity
Is?h?c?3?3 . (A.23)
We can apply the derived expression for the scattering rate (A.17) to calculate
the heating rate of a 1 mm rubidium cloud with a Gaussian density distribution
placed inside a 1 mm hollow core of a 4th-order Bessel beam. Consider a thermal
cloud in cylindrical coordinates (r,?,z) with the Gaussian density distribution
along the z-axis, ?z, and uniform, ?r and ?? in the (r,?)-plane:
?(r,?,z) =
?
???
???
???
??
Nr/R r?R,
N?/R 0 lessorequalslant?lessorequalslant 2?,
Nze?(z?z0)2/2?2z ??<z< +?,
(A.24)
132
where Nr,N?,Nz,z0 are constants, ?z is the width of the Gaussian distribution in
z-direction, and R is the radius of the cloud in the transverse plane. If N0 is the
total number of atoms in the cloud, then normalization of the distribution (A.24)
gives rise to N0 =?2?3NrN?Nz.
The total number of photons absorbed by the whole cloud is given by the
integral
Nph =
integraldisplay
V
R?dV (A.25)
over the elementary volume V. There,R= ??22 is the photon scattering rate per
atom, which for a large detunings and high laser intensities is reduced to
R= ?2 s0(r)(2?/?)2, (A.26)
where the saturation parameter on resonance s0 = I(r)/Isat is determined by the
intensity of the Bessel beam in the transverse plane I(r) = I0[J4(krr)]2, J4(krr) is
the 4th-order Bessel function of the first kind, andkr = 10635 m?1 is the transverse
spatial frequency of the 1 mm diameter Bessel beam. For 70 mW of power in the
first ring of the Bessel beam, I0 = 6?104 mW/cm2, and the total number of the
absorbed photons (A.25) will be
Nph = 2R2N0
integraldisplay R
0
rR(r)dr.
Numerical calculation of the radial integral leads to the number of photons ab-
sorbed by each atom per second at a given detuning ?:
Nph/N0?10?3
parenleftbigg2?
?
parenrightbigg2
. (A.27)
A.1.1 Absorption from Laser Beam by Atomic Cloud
Consider a laser beam of intensity I traveling alongz-axis through an atomic cloud
of density n. The amount of scattered power per unit volume is given by ??Rn,
133
and thus dI/dz =???Rn, whereRis the scattering rate given by Eq.(A.26). For
low intensity light near the atomic resonance,
dI
dz =??egnI, (A.28)
where the two-level atom cross section ?eg for scattering light out of the beam on
resonance is given by
?eg = 3?
2
2? .
The solution of Eq.(A.28) is I(z) = I0e?
R ?
egn(z)Idz and allows to study the
density of the absorbing cloud if the laser beam intensities after the absorption,
I(z), and without absorption, I0, can be measured by a CCD array. This approach
is widely used in AMO physics, and we refer to it as the absorption imaging
technique throughout the thesis.
A.2 Velocity Dependent Force
The radiation pressure force (A.21) is conservative and does not account for the
motion of an atom. One can include the atomic velocities into the optical Bloch
equations (A.13) or consider them as small corrections to the solutions of the OBE
for the atom at rest [110]. A simpler (but less rigorous) approach is to introduce the
velocity dependence through a Doppler shift contribution to the effective detuning
of the laser beam frequency from the atomic resonance ??k?v, where v is the
velocity of the atom. We can now write the velocity-dependent expression for the
radiation pressure force
Fv = ?k?2 s01 +s
0 + 4((??k?v)/?)2
. (A.29)
For the case of an atom moving with a small velocityv in a counter-propagating
non-saturated laser beam, the last equation can be linearized in v to obtain the
expression for the velocity-dependent force
134
Fsp(v) = Fsp??v, (A.30)
where Fsp is velocity-independent force given by the Eq. A.21, and the damping
coefficient ? is given by
? =??k2 4s0(?/?)(1 +s
0 + 4(?/?)2)2
. (A.31)
For the red-detuned (? < 0) laser beam the damping coefficient is positive, thus
resulting to the deceleration of the atom. If a collection of atoms with Maxwellian
velocity distribution is exposed to a near-resonant laser beam, only a certain ve-
locity class of the ensemble will decelerate. The slow atoms that have the velocity
component in the direction opposite to the propagation direction of the laser beam
(k?v =?(kv)z) such that the effective detuning does not exceed the width of the
profile of the radiation pressure force (A.29)
(?+kv)2 <?2(1 +s0), (A.32)
and forsmall detunings only atomswith the initial velocity below a capture velocity
vc = ?k (A.33)
will undergo multiple scattering events and be efficiently decelerated to small ve-
locities. For 87Rb, vc = 4.7 m/s, while the average velocity of Maxwell distribution
at room temperature is about 170 m/s. Therefore, only a small fraction of the
ensemble can scatter photons from the laser beam. The recoil velocity is only 5.88
mm/s, and thus multiple scattering events are required to slow down the atoms
substantially.
135
A.3 Optical Molasses
Consider an atom scattering photons from two counter-propagating but otherwise
identical laser beams2. In the one-dimensional case the light pressure force from
the two beams will be, according to Eq. A.303
Fsp =?2?v. (A.34)
For the red-detuned laser beams (? > 0) the light pressure force viscously
damps the atomic motion. This cooling mechanism is called optical molasses: an
atom traveling along one of the laser beams scatters less photons from that beam
than it does from the other, the counter-propagating beam, which is closer to the
atomic resonance due to the Doppler shift. Thus the atom gains net momentum
opposing its velocity in agreement with Eq. A.34. The damping force fluctuates
around its average value due to discrete momentum transfer upon the absorp-
tion/emission events. When these events are uncorrelated, atomic momentum
undergoes a random walk, and the mean square of the momentum <p2 > (kinetic
energy) increases. Equilibration of this heating mechanism with the damping sets
the Doppler-cooling limit. In case of a single laser beam, the momentum diffusion
coefficient from the spontaneous emission
Dsp =< ?p2 >= ?2k2 ?2 s01 +s
0 + 4(?/?)2
(A.35)
is equal to the momentum diffusion coefficient from absorption Dabs in the low
intensity limit as the mean number of emitted photons is equal to that of absorbed.
Equating the heating rate with the rate of the damping cooling
< ?Eheat >= Dsp +DabsM =?< ?Ecool >= ? <v2 >, (A.36)
2Intensities are low and stimulated emission is neglected.
3This could be obtained directly from the second term of Eq. A.12 representing the electric
field in the standing wave form.
136
we obtain the cooling limit in units of temperature
kBT =???2 1 +s0 + (2?/?)
2
2?/? , (A.37)
and for the low intensity limit s<< 1 the temperature reaches its minimum value,
the Doppler-cooling limit
kBTD = ??2 (A.38)
at ? =??/2. Early time-of-flight measurements in sodium, however, revealed the
molasses temperatures of an order of magnitude lower than the expected Doppler-
cooling limit [111]: in Zeeman-degenerate atoms, sub-Doppler cooling mechanisms
permit temperatures substantially below this limit [112].
Optical molasses may slow the atoms down but do not provide a trapping
mechanism that would allow collection of a long-lived large atomic sample. The
atoms in the molasses undergo diffusion with the maximum diffusion time given
by [111]
td = 4k
2?r2?
27N2? , (A.39)
where?r2?is the mean square distance diffused and N is the molasses dimension.
For rubidium it takes about 200 ms to diffuse a 1 mm distance. Moreover, for a
static optical field that produces a radiation pressure force linearly proportional to
the light intensity (Eq.A.29), it is not possible to create a 3-D stable trap [113].
A.4 Magneto-optic Trap
Onecan builduponthe opticalmolasses setup by creating aradiationpressure force
that selectively interacts with the Zeeman sublevels of the upper state such that an
atomdisplaced fromthe originscatters photons preferablyfromthe laser beamthat
pushes it back to the origin. Such configuration was first demonstrated as a 3-D
137
trap by Raab, et al. [114]. The combination of the optical molasses and two anti-
Helmholtz coils constitutes the Magneto-optical Trap (MOT) and is schematically
presented in Fig. A.1 (left). The anti-Helmholtz coils create a magnetic field B
which is zero at the origin and linear with the distance away from the origin
in its small vicinity. The upper state Zeeman splitting for such an arrangement
in 1-D is shown in Fig. A.1 (right): the energy of m = +1 sublevel increases
linearly along the positive direction of the z-axis while the energy of the m =?1
sublevel decreases, and the frequency shift can be written as ?z = ?B/? = bz,
and b is proportional to the magnetic field gradient. Optical molasses are formed
by two counterpropagating laser beams with opposite directions of the circular
polarization ?+and ??, and the atom absorbing from the ?+-beam gets excited to
the m = +1 uppers state while the ??-beam excites the atom into the m = ?1
state. The molasses beams are red-detuned from the |g,m = 0??|e,m = 0?
transition(dashed linein Fig. 1). Therefore, anatomdisplaced byzfromtheorigin
will be preferably excited into m = ?1 state by the ??-beam (relative detuning
of ??bz) rather than into m = +1 state by the ?+-beam (relative detuning of
?+bz). The preferred scattering from the ?+-beam provides a restoring force that
pushes the atom back to origin. A similar argument applies if the atom is placed
the distance?z away from the origin.
In the low intensity limit, the total force exerted by the two beams on an atom
dispaced by a distance z is given by
F = ?k?2
bracketleftBigg
s0
1 +s0 + 4((??kv?bz)/?)2 ?
s0
1 +s0 + 4((??kv+bz)/?)2
bracketrightBigg
. (A.40)
For small velocities and a low magnetic field (kv,bz << ?), we can write the
force as a function of the velocity and the displacement
F =?2?v??r, (A.41)
138
Figure A.1: Magneto-optical trap. Proper polarization of six molasses beams in
combination with a magnetic field gradient provide a cooling mechanism to slow
atoms down and a restoring force to keep them trapped in the intersection region
of the beams (left). The right panel illustrates an energy splitting of the upper
level in the presence of a magnetic field gradient.
where the dumping coefficient ? is given by the Eq. A.31 and the spring constant
is
?= ?b?k?. (A.42)
For typical magnetic field gradients b?10 G/cm, harmonic motion of atoms
is overdumped (damping rate ?/m> 1).
A.4.1 Density and Velocity Distribution in MOT
Consider a non-interacting 1-D gas of particles of mass mcharacterized by a distri-
bution function f(r,v,t) in the phase space. The gas is subjected to a conservative
force F = F(r,t). The Boltzmann?s equation df = 0 in a steady-state ?f/?t will
be
v?f?r + Fm?f?v = 0,
139
or, expressing the force as a gradient of a potential U, the last equation will be
v?f?r ? 1m?U?r ?f?v = 0, (A.43)
with the solution
f(r,v) = f0e??U(r)e??mv22 , (A.44)
where ? is the equilibrium temperature of the Maxwellian velocity distribution,
and f0 is the normalization constant. For a cloud trapped in a harmonic oscilla-
tor potential with a spring constant k, U(r) = ?kr2/2, the spatial 1-D density
distribution in the MOT is a Gaussian,
f(r)?e?r2/2?2,
where the cloud size ? =??k.
If the MOT, initially characterized by a Gaussian density and Maxwellian ve-
locity distribution at t = 0, starts evolving freely (U(r) = 0), the corresponding
Boltzmann?s equation,
?f
?t +v
?f
?r = 0,
hasa partialsolutionf(t,r) = e?r2/?2(t), where thecloud size,?(t) = radicalbig?2(0)+v2t2,
ballistically expands in time.
A.5 Dipole Force
The first term in the Eq. A.19 is proportional to the gradient of the electric field
amplitude. This force is called the dipole (or stimulated) force and is usually
expressed as a gradient of a potential
U = ??2 ln
bracketleftbigg
1 + s01 + (2?/?)2
bracketrightbigg
, (A.45)
140
where the sign of the detuning ? determines if the dipole potential is repulsive
(? < 0) or attractive (?> 0), while the spatial variation of the light intensity, s0 =
I(r)/Is sets the shape of the potential. Note that the dipole force is conservative
and due to the absorption and the consecutive stimulated emission. The original
derivation of the Eq. A.45 can be found in Ref. [115].
141
Appendix B
Appendix B. Elements of the Scalar Diffraction
Theory
Appendix B is devoted to the scalar diffraction theory applied to the generation
of the non-diffracting beams.
B.1 Nondiffracting Solutions of Wave Equation
We look for monochromatic solutions of the scalar wave equation in free space
parenleftbigg
?2? 1c2 ?
2
?t2
parenrightbigg
E(r,t) = 0, (B.1)
which represent nondiffracting wavefields, whose corresponding intensity distribu-
tion have the property that
I(x,y,z?0) = I(x,y,0), (B.2)
along the propagation direction z.
The wave equation can be reduced to the two-dimensional Helmholtz equation
if we consider solutions of frequency ? in the form
E(r,t) = U(x,y)Z(z)e?i?t, (B.3)
such that|Z|is independent of z to satisfy relation (B.2).
Consecutively, we obtain
142
d2Z
dz2 +?
2Z = 0, (B.4)
for the longitudinal direction, and
parenleftbig?2
? +?
2parenrightbigU(x,y) = 0, (B.5)
transverse plane accordingly. There, ?2 +?2 = k2 = (?/c)2, and?2? = ?2/?x2 +
?2/?y2.
We limit the family of solutions of Eq. (B.4) along the propagation direction
by the Sommerfeld radiation condition to find
Z(z) = Cei?z, (B.6)
where C is a constant, and ? is real and positive to satisfy the condition (B.2).
The solution for the Eq. (B.5) can be written in an integral form [116] as
U(x,y) =
integraldisplay 2pi
0
F?(u)ei?(xcosu+ysinu)du, (B.7)
where F? is an arbitrary complex function of a real parameter u = [0..2?].
We can re-write the wave equation nondiffracting solution (B.3) as
E(r,t) = ei(?z??t)
integraldisplay 2pi
0
F(u)ei?(xcosu+ysinu)du, (B.8)
where F(u) is an arbitrary and well-behaved complex function. The nondiffracting
wave field of this type were originally introduced by Durnin [117]. For a special case
of F(u) = exp(inu)1, we obtain an axially symmetric solution with the transverse
profile given by a nth-order Bessel function of the first kind
E(r,t) = 2?ei(?z??t)Jn(??), (B.9)
where ?2 = x2 +y2.
1If we treat the parameter u as a polar angle of the cylindrical coordinate system.
143
It is impossible to produce an ideal Bessel beam since the wave field (B.9)
carries the infinite amount of energy. One, however, can create such a beam over a
finite area: we will show that the finite-aperture approximation of a wave field with
a Bessel transverse profile similar to Eq. B.9 can be generated from a collimated
laser beam with a help of single hologram[118]. Consider a unit-amplitude plane
wave illuminating a thin circular finite-size hologram of radius r0 characterized by
the following transmission function
T(r,?,0) =
?
??
??
ein?e?ikrr r?r0,
0 r>r0,
(B.10)
where (r,?) are the polar coordinates in the plane z = 0 of the hologram, and kr
is a radial spatial frequency.
The wave field behind the hologram is given by the Kirchhoff integral in the
Fresnel approximation
U(r,?,z) = 1i?zeik(z+r22z)
integraldisplay r0
0
r?eikr?22z
bracketleftbiggintegraldisplay 2pi
0
T(r?,??,0)e?ikrr
? cos(????)
z d??
bracketrightbigg
dr?.
(B.11)
Substituting the transmission function (B.10) into the last equation and eval-
uating the angular integral yields
U(r,?,z) = 2??zeik(z+r22z)ein?
integraldisplay r0
0
r?Jn(krr
?
z )exp
bracketleftbigg
i
parenleftbiggkr?2
2z ?krr
?
parenrightbiggbracketrightbigg
dr?. (B.12)
The radial integral can be evaluated using the principle of the stationary phase
[119]. Omitting the position-independent factors, intensity distribution behind the
hologram in the observation plane z = constant is given by [118]
I(r,?,z) ?|ein?Jn(krr)|2, (B.13)
or in Cartesian coordinates
144
I(x,y,z) ?|
integraldisplay 2pi
0
ein?eikr(xcos?+ysinu?)d?|2, (B.14)
similar to the intensity distribution of the nondiffracting solution (B.7) of the
wave equation when kr = ? is the radial spatial frequency of the Bessel wave, and
F?(?) = ein?, which is the hologram transmission function.
The finite aperture of the holographic element results in a finite propagation-
invariant distance while the ideal Bessel beam is invariant in propagation. The
maximum propagation-invariant distance derived from geometric optics is given
by
zmax = 1k
r
?D
? , (B.15)
where D is the diameter of the hologram and ? is the wavelength of light.
A hologram transmission function (B.10) can be realized as a circular wedged
dielectric disk whose optical thickness is (?krr+n?) in polar coordinates. Practi-
cally, it is convenient to adjust the phase by a 2-dimensional spatial light modula-
tor, which acts as a holographic phase mask generated by a computer. Application
of this technique is described in Chapter 3.
145
Appendix C
Appendix C. Apparatus for Bessel Beam
Experiment
Appendix C is devoted to the descriptions of the experimental apparatus assembled
for the Bessel beam experiment. The apparatus includes a laser-cooling machine to
produce cold rubidium clouds and the setup to generate the Bessel beams that were
applied to manipulate the atomic clouds. We include the names of manufacturers
and model numbers for various devices constituting the experimental setup.
C.1 Magneto-optical Trap Setup
As we show in Chapter 3, we can generate Bessel beams with corresponding optical
potentials of a few hundred microkelvins. Such potentials are high enough to
confine atoms cooled and trapped in a standard magneto-optical trap (MOT).
The basic ideas of laser cooling and trapping in a MOT are discussed in Appendix
A. Below we provide the details of our realization of the MOT.
When the MOT loading is stopped, the cloud life-time and therefore the acces-
sible time of the experiment is determined by the collisions with the background
gas, which is at room temperature. Therefore a cold atom experiment needs to be
conducted under high vacuum. The required pressure in a vacuum chamber is set
by the desired time of the experiment after the MOT loading stage (a few hundred
milliseconds), which corresponds to p ? 10?8 Torr of rubidium vapor pressure.
To reach this pressure we assembled a stainless steel chamber with several glass
146
Figure C.1: Stainless steel vacuum chamber for laser-cooling and guiding rubidium
atoms (left). Fluorescence of the cooling laser beams and the MOT in the chamber
is shown on the right.
windows to provide optical access of the cooling laser light to the atomic vapor.
The chamber consisted of a spherical octagon ultra-high vacuum chamber from
Kimball Physics that has two 6? and eight 2 3/4? conflat flanges. The 6?-flanges
were sealed by two uncoated glass viewports, one 2 3/4? flange was used to connect
the chamber to a 40 l/s ion pump (Varian Starcell), and two more were extended
by conflat tees. The rest of the open 2 3/4? flanges were sealed by glass windows
anti-reflection coated for 780 nm. A part of the chamber is presented in Fig. C.1
(left image) along with a photograph of a MOT cloud collected at the intersec-
tion of the six cooling laser beams. After the vacuum chamber was sealed, it was
evacuated by a turbo pump (Pfeiffer TPH170) to about 10?7 Torr and pumped
down below 10?8 Torr by the ion pump. The low pressure in the chamber was
maintained by the ion pump alone from that moment on, and the value of the
pressure was measured by the ion pump current.
To simplify the experimental setup we loaded the MOT from rubidium vapor
[120]. As a source of atoms, we used a glass ampule with rubidium placed in a
4-mm inner diameter copper tube. One end of the tube was pinched off and thus
vacuum sealed, while the other end was welded to a copper gasket, which sealed
147
Figure C.2: Hyperfine level structure of the ground and the first excited states of
87Rb (left) and 85Rb (right) showing the corresponding frequencies of the trapping
and the re-pumping lasers. Adopted from reference [121]
the connection between the chamber and an stainless steel end cap with an on-
axis hole. The tube - aluminum foil wrapped finger - can be seen in the left top
corner of Fig. C.1. As the base pressure in the chamber was reached, the rubidium
ampule was cracked by bending the tube. The tube was equipped with a heater
to evaporate atoms into the chamber as needed to maintain the rubidium vapor
pressure of 10?8 Torr1.
The laser cooling light was provided by two diode lasers at 780 nm. The
hyperfine structure of rubidium 5s and 5p levels is shown in Fig. C.2.
The 5s1/2|F = 2?? 5p3/2|F? = 3? transition from the ground state to the
first excited state was chosen as the cycling cooling transition and was driven by a
frequency-stabilized diode laser (Toptica, DL100). As required by the laser cooling
mechanism, the laser frequency was about 20 MHz red-detuned from the atomic
1Rubidium saturated vapor pressure at room temperature is 2?10?7 Torr
148
Figure C.3: Layout of the optical system for the laser cooling setup. Two frequency
stabilized diode lasers were used to provide six cooling and trapping beams, and a
probe (Toptica), and repumping beam (Vortex).
transition. Since the atoms may decay into 5s1/2|F = 1? state from the excited
state, they were re-pumped back to the cooling cycle by the second stabilized diode
laser (New Focus, Vortex 6000 series) driving the 5s1/2|F = 1??5p3/2|F? = 2?
transition. The energy difference between the cooling and the repumping transition
larger than 8 GHz required the usage of two separate lasers.
The cooling and the repumping light was delivered to the chamber as shown
on the optical layout in Fig. C.3. The Toptica DL100 laser produced 90 mW of
power. The laser was separated from the rest of the experiment by an external opti-
cal isolator (OI) of 32 dB isolation (Isowave, I-80-T5-H) to prevent possible optical
feedback to the laser that might disturb the laser oscillations. Several milliwatts
were sent to a saturated absorption spectroscopy setup that provided a marker to
stabilize the laser frequency. Another 2-3 mW of power were used to generate a
probe beam for the absorption imaging of the 5s1/2|F = 2?ground state popula-
149
tion2. To maximize the absorption by a low density cloud the probe beam needs to
be on resonance with an atomic transition (e.g., 5s1/2|F = 2??5p3/2|F? = 3?).
As the frequency of the Toptica beam was detuned about 20 MHz below the
transition, two acousto-optical modulators (AOM) (Isomet, 1205C-2-804B) were
employed to shift the frequency to resonance. Each AOM was characterized by
a central frequency of 80 MHz and the bandwidth of 30 MHz; the first AO was
driven at 70 MHz, its negative first diffraction output order coupled with the sec-
ond AOM driven at 90 MHz. Thus the frequency of the first order of the second
AOM output was upshifted by 20 MHz with respect to the original Toptica beam
and resonant with the cooling transition. The transverse intensity profile of the
first order was cleaned by a spatial filter, the beam diameter increased to 1 cm by
a telescope consisted of two spherical lenses, and the resulting beam was applied
as the probe beam of the shadow imaging technique.
The main Toptica beam was divided into three independent beams by a com-
bination of a half-wave plate and a polarizing beam splitting cube, each beam
expanded to a 2 cm diameter. The beams were properly circularly polarized by
quarter-wave plates, crossed at 90EW with respect to each other in the center of the
vacuum chamber. Each beam was then retro-reflected to counter-propagate itself
inside the chamber thus forming six molasses beams (Fig. C.3, right). The cooling
beams were switched on and off by a mechanical shutter (Uniblitz, LS6 T2) in
1.7 ms. The appropriate magnetic field gradient inside the chamber was provided
by two sets of coils with electric current in anti-Helmholtz configuration. Each
coil consisted of 42 loops of hollow copper refrigeration tubing wound around the
large 6? window of the vacuum chamber. The coils were arranged such that the
magnetic field in the center of the chamber was zero and increased linearly away
from the center of the chamber with the gradient of 15 G/cm at the current of 25
A. The heat produced by the current was dissipated by water running inside the
2The basics of the absorption imaging are described in Appendix A
150
copper tubing.
The re-pumping beam was provided by the 5 mW Vortex diode laser. The laser
was separated from the rest of the setup by an optical isolator (Isomet, 1205C-
2-804B). A part of the beam was sent to the saturated absorption spectroscopy
setup to stabilize the laser frequency. The main Vortex beam of about 3 mm in
diameter was aligned with the intersection region of the cooling beams.
In order to stabilize a laser one has to establish a one-to-one correspondence
between its frequency and a voltage generated by a device external to the laser.
Spectroscopy of a medium absorbing light at the wavelength of interest provides
frequency markers that are independent of the environmental conditions and re-
peatable. As our lasers had to be frequency-stabilized close to atomic transitions in
rubidium we used the appropriate spectroscopic features to establish the frequency-
voltage correspondence. A saturated rubidium vapor in a glass cell is characterized
by a Doppler-broadened absorption profile of a gigahertz as atoms from each veloc-
ity class are detuned differently from the resonant frequency of a stationary atom.
Such profile does not provide enough resolution to keep the laser frequency stable
within a linewidth of an atomic transition of several megahertz. A narrow atomic
transition can be resolved by a Doppler-free saturated absorption spectroscopy.
Consider two laser beam at the same frequency, one above (pump) and the other
below (probe) the saturation intensity, counter-propagating in a Rb glass cell such
that the power of the weaker beam is detected by a photodiode. The photodiode
current proportional to the probe beam power is converted to a voltage signal. If
the laser frequency is detuned (within the Doppler profile) with respect to reso-
nance of the stationary atom, the probe beam is resonant with a certain velocity
class and partly absorbed by the media, while the pump beam is resonant with
the opposite velocity class. As the laser detuning is zero (within the linewidth of
the laser), both beams are interacting with the same zero-velocity class, the high
intensity pump beams saturates the atomic transition, and the low intensity probe
is not absorbed by the medium any more producing a sharp dip in the Doppler
151
Figure C.4: Saturated absorption spectroscopy of 5s1/2|F = 2??5p3/2|F??(left)
and 5s1/2|F = 1?? 5p3/2|F?? (right) transitions. Cross-over peaks (denoted
by excited state quantum numbers in round brackets) do not correspond to real
atomic transitions. They are due to a reduced absorption of the probe beam when
the laser frequency is tuned to the middle between two atomic transitions. There,
both the probe and the pump interact with the same velocity classes but excite
them to different atomic transitions thus suppressing absorption of the weak probe.
The (2?3) and (1?2) peaks were used as the frequency markers to stabilize
the cooling and the re-pumping lasers accordingly.
absorption profile of the probe beam. Another weak laser beam at the same fre-
quency, notoverlapped with the probe orthe pump can be sent throughthe cell and
detected by a second photodiode. Further subtraction of the two photodiode sig-
nals leaves only the peak at the resonance frequency of the stationary atom. The
spectroscopy of the 5s1/2|F = 2?? 5p3/2|F?? and 5s1/2|F = 1?? 5p3/2|F??
transition is presented on the left and right of Fig. C.4 correspondingly.
For example, the frequency of the cooling laser has to be a few linewidths below
the frequency of the atomic transition, and on the slope of the|F = 2??|F? = 3?
peak in Fig. C.4. That frequency corresponds to a certain set-point voltage; a small
change in the laser frequency changes the absorption of the medium and results in a
different voltage generated from the photodiode in accordance with the absorption
152
profile. If the difference between the read-out voltage from the photodiode and
the set-point voltage (error signal) is properly fed-back to the laser, it forces the
latter to change its frequency back to the desired value corresponding the set-point
voltage. Thus the difference between the read-out and the set point stays zero
keeping the laser frequency stabilized even if the properties of the environment
(temperature, acoustic noise, etc.) are changing. We achieved the stabilization
by a servo loop, which circuit diagram is presented in Fig. C.5. The voltage
difference between the two photodiodes is subtracted from the set point voltage,
the result integrated, amplified, and fed back to the laser system electronics. The
internal cavity of the laser diode is extended by a diffraction grating; the first
order is diffracted of the grating back to the diode thus setting the oscillating
frequency of the laser within its gain profile. The variable tilt of the grating
allows continuous tuning of the laser?s wavelength and is practically realized by
a piezoelectric transducer (PZT). The signal from the servo loop, when added to
the bias voltage of the PZT, provides the stabilization of the laser frequency at
the given set point voltage. The tuning of the laser frequency within the slope
of a spectroscopic feature can be achieved by the appropriate adjustment the set
point voltage. As the laser frequency in controlled by a mechanical motion of the
grating, the bandwidth of the feedback control is limited by a few kilohertz, still
large enough to compensate for slow thermal effects and the acoustic noise. Faster
Fourier components in the laser frequency spectrum can be suppressed by feeding
back the error signal to the driving current of the laser diode, effectively reducing
the linewidth of the laser.
Practically, we frequency-stabilized the Vortex laser by locking the correspond-
ing photodiode signal on the side of the|F = 1??|F? = 2?(re-pump) peak in
Fig. C.4 using the circuit diagram in Fig. C.5. The cooling laser was stabilized on
the side of the|F = 2??|F? = 3?peak in Fig. C.4 using a commercial locking
circuit (Toptica, PID 100).
With the magnetic field on and the frequency of the cooling and the repumping
153
Figure C.5: The diagram of the locking circuit used to stabilize the re-pump laser
on the slope of the spectroscopic feature of the 5s1/2|F = 1?? 5p3/2|F?? tran-
sition with the values of passive and the part numbers for active elements of the
circuit.
lasers stabilized, the rubidium cloud was collected at the intersection region of
cooling lasers. The fluorescence of the MOT cloud and cooling beams (photons
spontaneously emitted by excited atoms decaying into the ground state) is shown in
Fig. C.1, photograph on the right. The steady-state number of atoms in the MOT
154
was estimated to be about 108 using the fluorescence signal from the MOT cloud.
The spontaneously emitted photons from the fluorescent cloud were collected on
an active area of a power meter by a high numerical aperture imaging system. The
measured power is proportional to the number of photons emitted into the solid
angle enclosed by the imaging system, and thus determines the number of emitters
(atoms) via the scattering rate at the known detuning. This estimated number was
used to calibrate shadow imaging signal - ourprimary detection method. The beam
after the absorption by the cold atom cloud is projected onto a charge-coupled
device (CCD) array of a 8-bit video camera (COHU, 4910) by a spherical lens that
provides one-to-one imaging of the cloud on the plane of the CCD. The CCD array
area was 4 x 6 mm with the resolution of 480 by 752. The analog signal from the
camera was sent to a computer and digitized by a frame grabber card (PIXCI,
SV4). The resulting digital image provided the information on the atom number,
cloud size in the direction transverse to the probe beam and the cloud position
with respect to the probe beam. Practically, the magnification of the imaging
system is unknown and needs to be calibrated to the known size of the CCD array.
That was done by imaging the cloud free fall in the gravity field and tracking the
postion of the cloud?s center of mass on the CCD array at various fall times. The
cloud temperature defined in Appendix A, can be estimated from the cloud?s free
expansion. The linear size of the cloud ? along a specific direction as measured
by the shadow imaging after a certain expansion time t is given by ?2(t) = ?20 +
v2t2, where ?0 is the initial size of the cloud, and v2 is the mean square velocity
of the Maxwellian distribution. As ?2 grows linearly with t2, the slope of this
dependence gives v2. Assuming anisotropic expansion, the equipartition theorem,
1/2kBT = 1/2mV2, allows the expression of the value of the kinetic energy in
units of temperature T; there kB is the Boltzmann?s constant, m is the mass of
the rubidium atom. Generally, the cloud can be characterized by three values
of temperature according to its expansion along three perpendicular directions if
imaged by two orthogonal probe beams. A typical single MOT temperature for
155
our experiments was measured to be (250?15) ?K.
156
Appendix D
Appendix D. Rydberg Experiment
D.1 Autler-Townes Splitting in 3-level Atom
According to the Wiener-Khinchine theorem, the spectrum of a steadily oscillating
system can be found from the Fourier transform of the corresponding autocorre-
lation function. For an atomic system, the relative quantity is the dipole moment
autocorrelation function,
G(?) =
angbracketleftBig?
d(?) ?d(0)
angbracketrightBig
.
In the Sch?odinger picture, the state vectors embody the time dependence, and
the autocorrelation can be written as
G(?) =
angbracketleftBig
eiH?/planckover2pi1 ?de?iH?/planckover2pi1 ?d
angbracketrightBig
.
If an atom initially, at t = 0, is in the pure quantum state ?, the dipole
autocorrelation function read
G(?) =
angbracketleftBig
?(?)|?de?iH?/planckover2pi1 ?d|?(0)
angbracketrightBig
. (D.1)
The last expression can be treated as a matrix element
G(?) =
angbracketleftBig
?(?)|?d|?(t)
angbracketrightBig
,
where states ?(t)and?(t)areeach solutionsto thesame time-dependent Sch?odinger
equation, but they evolve from different initial conditions: the state ?(t) starts
157
from the initial state ?, while ?(t) starts from the state produced from ? by the
action of the dipole moment, ?(0) =?, and ?(0) = ?d?.
We can treat the exponential Hamiltonian by introducing a complete set of
dressed eigenstates of the operator H,
(H?planckover2pi1Zk)?k = 0, ?k(t) = e?iZkt ?k,
and Eq. D.1 will be
G(?) =
summationdisplay
k
e?iZk?
angbracketleftBig
?(?)|?d|?k
angbracketrightBigangbracketleftBig
?k|?d|?(0)
angbracketrightBig
. (D.2)
Consider a 3-level system, where the Hamiltonian H strongly couples two states,
and the third state,?0 is weakly connected by a dipole transition to a dressed state.
Then, the autocorrelation function for this transition will be
G(?) =
summationdisplay
k=1,2
e?i?k0?
vextendsinglevextendsingle
vextendsingle
angbracketleftBig
?(0)|?d|?k
angbracketrightBigvextendsinglevextendsingle
vextendsingle
2, (D.3)
where ?k are the transition frequencies from the weakly-coupled state.
The dressed states and their eigenvalues can be written in terms of the unper-
turbed states ?1 and ?2
?1 = cos??1?sin??2 ?2 = sin??1 +cos??2, (D.4)
and
planckover2pi1Z1 = EA1 ?planckover2pi1? planckover2pi1Z2 = EA1 ?planckover2pi1?+ planckover2pi1??,
whereEA1 is the energy of statepsi1, the generalized Rabi frequency ?? =??2 + ?2,
? is the Rabi frequency of the dressed state, the dynamic Stark shift 2? = ??+?,
and tan(2?) = ??/?, and ? is the laser detuning from the ?0 ??1 transition.
There we assumed that the state ?0 is coupled only with the unperturbed state
?1. The normalized autocorrelation function will be
158
g(?) = G(?)G(0) = cos2? e?i(?10??)? +sin2? e?i(?b??+??)?. (D.5)
The normalized autocorrelation function is related to the normalized spectral
distribution function as
F(?) = 12?
integraldisplay +?
??
g1(?)ei??d?.
For the autocorrelation function (D.5),
F(?) = cos2? f(?b??)+sin2? f(?b?? + ??),
where f(?) is the Dirac?s delta-function1. The resulting spectrum comprises a
pair of lines (the Autler-Townes doublet) in place of the single line that would
occur in the absence of the strong interaction. When the dressing interaction is
weak (? = 0), the spectrum reduces to a single line at frequency ?10. When the
dressing interaction is resonant (? = ?/2), the doublet comprises two lines of equal
intensity, separated by the Rabi frequency ?.
D.2 Population Dynamics of Two-level System with Ioniza-
tion Loss
Consider two-level system with the ground state population, Ng, excited state
population, Ne. Population dynamics of the system is set an excitation rate, R,
from the excited state to the ground state, de-excitation rate,?1, and an ionization
rate of the excited state, ?2. For simplicity, we assume ?1 = ?2??. The dynamics
is schematically shown in Fig. D.1. The total number of atoms in the system is
N0 = Ni +Ng +Ne. We are interested in the loss of atoms due to ionization.
1Note, that the finite lifetime of an atomic level, ?0, results in the Lorentzian line shape of
the width of 1/?0. The Lorentzian spectral distribution function can be obtained by multiplying
the autocorrelation function by the factor of e??/?0.
159
Figure D.1: Population dynamics of a two-level system with ionization loss
The corresponding rate equations are
?Ni = ?Ne
?Ne = RNg?2?Ne
?Ng =?RNg +?Ne
And we look for the solutions in the form
Ne?e?t,
which give rise to the following roots of the characteristic equation:
?1,2 = ?(2? +R)2 ?
radicalbig4?2 +R2
2 ,
160
and for the case R>>?,
?1 =?(? +R), ?2 =??,
an up to numerical constants C and D,
Ni =? ?C? +Re?(?+R)t +Ce??t +D.
From the initial conditions, C = N0 and D =?RR/(? +RR).
D.3 Stark Effect
In this section we numerically evaluate the energy shift of a Rydberg state of
rubidium due to an external electric field. We first calculate the wavefunctions
of Rydberg states. Electric field E will be considered as a perturbation and the
new wavefunctions expanded in the basis of the eigenfunctions of the unperturbed
Hamiltonian. The full Hamiltonian is given by
H =??
2
2 +
1
r +Vd(r)+Ez,
where Vd(r) is the difference between the rubidium potential and the coulomb
potential, ?1/r. The eigenvalues of the unperturb Hamiltonian, HE=0, are given
by the quantum defect:
W =?nlm|HE=0|nlm?= 12(n??
l)2
.
As the Hamiltonian is given by the central potential, the separation of angular and
radial coordinates is possible, and the radial part of the Schr?odinger equation will
be:
?2R
?r2 +
2
r
?R
?r +
parenleftbigg
2W + 2r?l(l+ 1)r2
parenrightbigg
R = 0,
where R is the radial wavefunction. The angular part of the wavefunction is give
by the spherical harmonics of the hydrogenic atom Ylm [122]. The substitution
161
Figure D.2: Numerically calculated radial wavefunction of 50s1/2 state.
x =?r,Y(x) = r3/4R results in [123]
parenleftbigg
?d
2
dx2 ?8Wx
2 + (2l+ 1/2)(2l+ 3/2)
x2
parenrightbigg
Y = 8Y.
The last equation can be integrated numerically, and, as an example, the solution
for n = 50s is plotted in Fig. D.2.
In the presence of the electric field, the resulted Stark shift is given by
? =?nlm|eEcos?|n?l?m??,
where |nlm? are the calculated unperturbed wavefunctions. The radial and an-
gular parts of the expectation value can be evaluated numerically. The resulted
Stark energy shift is presented in Fig. D.3 for 56s1/2 state. Note, that the effect is
quadratic in the field at E < 2 V/cm and linear between 3 and 5 V/cm.
162
s48 s49 s50 s51 s52 s53
s45s51s57s46s51s56
s45s51s57s46s51s54
s45s51s57s46s51s52
s45s51s57s46s51s50
s45s51s57s46s51s48
s45s51s57s46s50s56
s45s51s57s46s50s54
s69
s110
s101
s114
s103
s121
s44
s32
s99s109
s45
s49
s69s108s101s99s116s114s105s99s32s102s105s101s108s100s44s32s86s47s99s109
Figure D.3: The Stark effect for 56s1/2 state. The energy shift in quadratic in field
at low fields, E < 2 V/cm and linear between 3 and 5 V/cm.
163
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