ABSTRACT
Title of dissertation: ATOM-TRAPPING AND PHOTON-COUNTING
EXPERIMENTS WITH OPTICAL NANOFIBERS
Jeffrey Aaron Grover, Doctor of Philosophy, 2015
Dissertation directed by: Professor Luis A. Orozco
Department of Physics
New effects can arise in quantum physics when there is strong coupling, ei-
ther between atoms and light or between different quantum systems. This thesis
examines an optical nanofiber atom trap as a mediator of atom-light interactions
and a potential element of a hybrid quantum system. The evanescent field around
the sub-wavelength waist of an optical nanofiber possesses a small mode area that
increases the cooperativity between atoms and the mode, in a manner analogous to
traditional cavity QED.
We demonstrate trapping of 87Rb atoms with an optical nanofiber, confining
hundreds of atoms with typical trap lifetimes of tens of milliseconds. We then
employ single photon counting techniques to study untrapped ensembles of cold
atoms around the nanofiber. A first experiment uses intensity autocorrelations of
resonance fluorescence emitted into the nanofiber mode to observe a transition from
classical to nonclassical photon statistics. Measuring the correlations on longer
timescales reveals the motion of atoms through the optical mode, and we develop a
correspondence between the transit time and atomic cloud temperature. A second
experiment measures Purcell enhancement of spontaneous emission of atoms near
the nanofiber by correlating their fluorescence with a known trigger event. The
spontaneous decay rate of an atom near a dielectric is modified by the induced
dipole and by a change in the modes of the vacuum electromagnetic field. Our
observed enhancement of 6.5±0.9% over the free-space rate matches well with what
one finds from mode simulations in our system.
ATOM-TRAPPING AND PHOTON-COUNTING EXPERIMENTS
WITH OPTICAL NANOFIBERS
by
Jeffrey Aaron Grover
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
2015
Advisory Committee:
Professor Luis A. Orozco, Chair/Advisor
Professor Steven L. Rolston, Co-Advisor
Professor Mario Dagenais, Dean’s Representative
Professor Frederick C. Wellstood
Professor Christoper J. Lobb
Professor Mohammad Hafezi
c© Copyright by
Jeffrey Aaron Grover
2015
Dedication
For Moe. Your memory preserves the light that was your life.
ii
Acknowledgments
Scientists like to overuse a phrase oft-attributed to Newton: “If I have seen
further it is by standing on the shoulders of Giants.” I have no grand delusions
about how far I have seen. But if I have seen further than I might have otherwise,
I owe it all to the optics of that old British fellow and to the giants in my life who
have carried me, limping, to this juncture.
It would be impossible to overstate the influence of my advisor, Luis Orozco.
He taught me my first lessons in atomic physics, and the teaching continues to this
day with his characteristic, indefatigable passion for his students. That instruction
has not been limited to atomic physics nor physics at all, as our weekly meetings
spanned all realms of culture and arts, often functioning as a literature exchange.
More than this, though, his confidence in me persists despite my doubts and without
that push it is certain that I would not have made it here. I am indebted to him in
more ways than I can convey here, but I hope he accepts my leaving his tutelage as
a proper first attempt at repayment.
It is rare that having two bosses is a good thing, but Steve Rolston was a great
second advisor. He has the most scarily-keen physics intuition and an encyclopedic
knowledge to match it. I’m thankful that his door was always open, where a 5-second
question often turned into a 30-minute discussion about a problem.
Fred Wellstood and Chris Lobb (and that damned dil fridge!) were often
the foils to our plans, but their presence kept us honest and made me a better
scientist. Fred demands precision and thoroughness from us befitting of his own care,
thoughtfulness, and diligence as a scientist. And Chris reminds me of a condensed-
matter version of Steve with his quick-thinking and intuitive grasp of physics. It
was also welcome to have someone to inject much-needed levity into any physics
situation.
It was often with much envy that I watched my labmates collaborate with
Fredrik Fatemi, who appeared by all firsthand accounts to be about 10 super-
postdocs wrapped into one PI. It is an uncommon privilege to have someone with
such a vast knowledge of both physics and experimental techniques be present in
the lab with you. I’m thankful to have had that privilege, no matter how short,
iii
towards the end of my thesis and regret that I don’t have more time to learn from
him.
My science education started long before I stepped foot onto Maryland’s cam-
pus. For their tireless efforts and commitment to my success, I thank Matthew
Corcoran, Norton Starr, Rob Benedetto, Jagu, and Jonathan Friedman.
Now for the younger folks. . . I’m fairly certain that Jon and I had no idea
what we were getting into when we independently signed up for this project, but
I am glad that we stumbled into it together. I could not envision spending those
far-too-many hours with a better colleague or friend. He meets problems with an
earnestness and creativity that I am supremely jealous of, never satisfied until he’s
figured it out. I hope we find that his skills serve us well when we start our cheese
farm in the mountains soon.
No one has been more fortunate than me that Pablo decided to join our group
after his internship. I’ve had the distinct honor of working closely with him for the
last two years, and his unparalleled knowledge of all areas of physics, his experi-
mental creativity, and his Zen-like calm have enriched my time beyond measure. I’ll
miss most our philosophical chats about anything while photon clicks accumulated
and the frequent occurrences of Pablo at the BoardTM, arguably the times when I’ve
learned the most.
Sylvain was a force in the lab, unrelentingly focused and exquisite in his tech-
nique, and he also shared Pablo’s gift of being able to explain any concept off-the-cuff
with striking clarity. We were doubly lucky that he, too, “chose” us from plenty of
other options. His influence extended beyond the walls of the lab, from teaching us
about Dirty French and Speculoos to simply being a reliable and hilarious friend.
Our postdocs Jongmin and Jared joined the group with an almost-frightening
intensity and work ethic that transformed it from an effective spare parts room to
a full-fledged lab. I am grateful for their mentorship and guidance as we put this
thing together. I also must thank the superconducting grad students Kristen and
Rangga for their companionship and shared head-scratching that made the lab a
fun place to be. A cohort of other Orozco lab members made my time at Maryland
more fulfilling through shared knowledge, equipment, and commiseration: David,
Andres, Burkley, and, especially, Jiehang. It would also be remiss of me to not thank
iv
Mike for the coffee and patiently entertaining a deluge of unsophisticated physics
questions.
Adam occupies a number of roles in my life: a best friend halfway across the
country, a constant GChat companion, and the best physicist I know. I owe much
of my intellectual development and emotional stability to his frequent advice, our
discussions, my visits to Boulder, and his unwavering support.
There is nothing like a coterie of close friends to help you survive a move to
a new city and to turn a low point in life into one of its highest. For that I will be
forever grateful to my skangaga crew in DC: Max, Emily, Lindsay, Mark(-Frank),
Sami, Karti, and Billy. But most of all Billy has carried me as my partner in our
oddest of pairings, with his infectious laughter and the deepest depth of caring I
have witnessed in another human. We’ve seen each other at our respective worsts,
but that has made us each our bests.
There is no good word in the English language to define what Joe is to me, so
I turn to the Turks and call him my kanka – part brother, part best friend. Through
time and across borders our closeness has endured, thanks in no small part to email
and YouTube. I love you, son.
My parents and my sister are my tireless cheerleaders and advocates, the ones
who have made any success in my life possible. They do it all simply out of love,
with the only demand being that I actually call home occasionally. Thank you for
nurturing my curiosity and supporting me, often from afar.
Stafford, you entered my life unexpectedly, but now your presence seems so
obvious it’s hard to imagine what preceded it. Building a home with you has been
one of the most fulfilling experiences of my life, and it has kept me grounded during
this simultaneous and arduous path I’m now finishing. I owe you so much for keeping
me calm, pleasantly distracted, and always excited for what the unknown future has
in store for us. Thank you for making my life more beautiful.
I also acknowledge support from the NSF through the PFC at the JQI. And it
is with great humility that I thank the JQI for its hospitality, sense of community,
unequaled opportunity, and generous funding.
v
Table of Contents
List of Tables ix
List of Figures x
List of Abbreviations xiii
1 Introduction 1
1.1 Cooperativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Optical nanofibers . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Hybrid quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Neutral atoms coupled to superconducting circuits . . . . . . . 8
1.3 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Apparatus 13
2.1 New laboratory construction . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Nanofiber pulling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 The flame-brush method . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Setup and procedure . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Laser systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.1 Cooling and probing laser: saturation spectroscopy . . . . . . 25
2.3.2 Repump laser: Doppler-free DAVLL . . . . . . . . . . . . . . . 29
2.3.3 Nanofiber “blue” laser: Ti:Sapph . . . . . . . . . . . . . . . . 31
2.3.4 Nanofiber “red” laser: Nd:YAG . . . . . . . . . . . . . . . . . 31
2.4 UHV system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.1 Science chamber . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.2 Washer optics mounts . . . . . . . . . . . . . . . . . . . . . . 34
2.4.3 Ion pump placement . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.4 Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4.5 Nanofiber mount . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5 Magnetic field control . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5.1 MOT coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
vi
2.5.2 MOT coil PID . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5.3 Shim coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5.4 Shim coil driver . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.6 Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.6.1 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.6.2 MOT beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.7 Photon counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.8 Electronics and control . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.8.1 Timing and DAQ cards . . . . . . . . . . . . . . . . . . . . . . 45
2.9 Conclusions: improvements and laboratory relocation . . . . . . . . . 46
3 Optical nanofiber trap 48
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Fitting asymmetric absorption curves . . . . . . . . . . . . . . . . . . 58
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4 Photon correlation measurements 65
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Intensity autocorrelations . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 The system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3.1 Nanofiber mode structure . . . . . . . . . . . . . . . . . . . . 68
4.3.2 Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.3 Light shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3.4 Coupling strength . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4 Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.4.1 Transit-time effects . . . . . . . . . . . . . . . . . . . . . . . . 71
4.4.2 Relating correlations to temperature . . . . . . . . . . . . . . 72
4.4.3 Simulating atomic trajectories . . . . . . . . . . . . . . . . . . 72
4.5 Experiment and results . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.5.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.5.2 Data and fitting . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5.3 Temperature extraction . . . . . . . . . . . . . . . . . . . . . 80
4.5.4 Atomic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.5.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5 Measurement of the lifetime of the 5P3/2 state of 87Rb near a nanofiber 88
5.1 Theoretical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.1.1 Dipoles near surfaces: calculating γrad . . . . . . . . . . . . . . 90
5.1.2 A dipole near a waveguide: calculating γ1D . . . . . . . . . . . 91
5.1.3 Atom density near surfaces . . . . . . . . . . . . . . . . . . . . 93
5.1.4 Expected decay rate . . . . . . . . . . . . . . . . . . . . . . . 96
5.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
vii
5.2.1 Optical pulse generation and improvement . . . . . . . . . . . 101
5.3 Data and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.3.1 Exponential fitting . . . . . . . . . . . . . . . . . . . . . . . . 103
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6 Quantum hybrid system 108
6.1 Compact atom trap: GMOT . . . . . . . . . . . . . . . . . . . . . . . 109
6.1.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 110
6.1.2 Temperature measurement . . . . . . . . . . . . . . . . . . . . 112
6.1.3 Theory: sub-Doppler cooling . . . . . . . . . . . . . . . . . . . 112
6.1.4 Atom number and density . . . . . . . . . . . . . . . . . . . . 121
6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7 Conclusions and outlook 123
7.1 Faraday spectroscopy with nanofibers . . . . . . . . . . . . . . . . . . 123
7.2 Surface physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.3 Hybrid system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.3.1 Atomic source: 2D MOT . . . . . . . . . . . . . . . . . . . . . 128
7.3.2 Interfacing fibers with superconducting circuits . . . . . . . . 128
7.4 Self-organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
A Nanofiber modes 132
A.1 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
A.2 Propagation constant . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
A.3 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
B Nanofiber vibrations 138
B.1 Transverse vibrational modes of a nanofiber . . . . . . . . . . . . . . 140
B.2 Effect of heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
B.3 Heterodyne technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
C Calculating van der Waals coefficients for 87Rb 148
C.1 Index of refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
C.2 Atomic polarizabilities . . . . . . . . . . . . . . . . . . . . . . . . . . 150
C.3 C3 and C4 coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Bibliography 153
viii
List of Tables
2.1 Nanofiber dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 Shim coil specifications . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.1 Single-stage, 50 ms far-detuned MOT parameters when scanning the
detuning of a single MOT beam . . . . . . . . . . . . . . . . . . . . . 114
6.2 Multi-stage, 60 ms far-detuned MOT and 1 ms optical molasses pa-
rameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
C.1 van der Waals and Casimir-Polder coefficients for 87Rb near fused silica152
ix
List of Figures
2.1 New laboratory space, circa September 2009 and May 2010 . . . . . . 14
2.2 Fiber pulling apparatus schematic and photograph . . . . . . . . . . 17
2.3 Flame nozzle for nanofiber pulling . . . . . . . . . . . . . . . . . . . . 18
2.4 Three-angle pull results and comparison with simulation . . . . . . . 21
2.5 SEM image of nanofiber . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 Transmission of optical nanofiber . . . . . . . . . . . . . . . . . . . . 23
2.7 Illustration of nanofiber geometry used in this thesis . . . . . . . . . . 24
2.8 D2 level structure for 87Rb with relevant transitions . . . . . . . . . . 26
2.9 Optics for saturation-free spectroscopy . . . . . . . . . . . . . . . . . 28
2.10 Electronics for Pound-Drever-Hall locking of the cooling laser . . . . . 28
2.11 Optics for DAVLL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.12 UHV science chamber without coils . . . . . . . . . . . . . . . . . . . 33
2.13 Washers for mounting optics directly to CF flanges . . . . . . . . . . 34
2.14 Washer for mounting UHV chamber to optical table . . . . . . . . . . 35
2.15 UHV manipulator and load-lock . . . . . . . . . . . . . . . . . . . . . 37
2.16 Titanium nanofiber holder for UHV . . . . . . . . . . . . . . . . . . . 38
2.17 Science chamber with magnetic coils . . . . . . . . . . . . . . . . . . 39
2.18 Vertical imaging system . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1 Illustration of trapping potential around optical nanofiber. . . . . . . 51
3.2 Characterization of Rayleigh scattering on the nanofiber waist . . . . 52
3.3 Experimental schematic for nanofiber trapping experiments . . . . . . 53
3.4 Transmission spectrum for atoms trapped along a nanofiber . . . . . 56
3.5 Nanofiber trap lifetime measurement . . . . . . . . . . . . . . . . . . 57
3.6 Calculated light shifts (including scalar, vector, and tensor shifts) for
different polarization configurations of trapping light . . . . . . . . . 59
3.7 Simulations of transmission spectra taking into account all light-shifts
and finite temperature distribution of atoms in the nanofiber trap . . 62
3.8 A more symmetric absorption curve due to better polarization align-
ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.1 Illustration of atoms moving near an optical nanofiber showing the
position-dependent coupling . . . . . . . . . . . . . . . . . . . . . . . 69
x
4.2 Experimental schematic for correlation measurements . . . . . . . . . 75
4.3 Short timescale autocorrelation function of resonance fluorescence
emitted into the nanofiber guided mode, illustrating a transition from
bunching to antibunching . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4 Autocorrelation function that displays transit-time effects of atoms
passing through the nanofiber guided mode, with fit and residuals . . 78
4.5 Extracted transit time as a function of temperature (verified by TOF) 81
4.6 Transit time as a function of 1064-nm power for different atomic cloud
temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.7 An example simulated correlation function for an atom temperature
of 787 µK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.8 Simulated correlation function width vs. sampling angle range for an
atomic temperature of 90 µK . . . . . . . . . . . . . . . . . . . . . . 85
4.9 Comparison of experimental and simulated temperature extraction . . 86
5.1 γrad for an atom near a silica surface . . . . . . . . . . . . . . . . . . 92
5.2 γ1D for a 500-nm diameter fiber . . . . . . . . . . . . . . . . . . . . . 94
5.3 Position density of thermal rubidium atoms near a silica surface . . . 97
5.4 Schematic for atomic lifetime measurement . . . . . . . . . . . . . . . 99
5.5 Timing sequence for lifetime measurements . . . . . . . . . . . . . . . 100
5.6 Optical pulse from fiber-EOM . . . . . . . . . . . . . . . . . . . . . . 102
5.7 Free-space decay histogram . . . . . . . . . . . . . . . . . . . . . . . 105
5.8 Fiber-modified decay histogram . . . . . . . . . . . . . . . . . . . . . 106
5.9 Contour plot of γ1D as a function of fiber size and atom-fiber distance 107
6.1 Conceptual sketch of hybrid system pieced together . . . . . . . . . . 109
6.2 Schematic for room-temperature GMOT experiments . . . . . . . . . 110
6.3 Temperature measurements of atoms in a GMOT . . . . . . . . . . . 113
6.4 GMOT beam geometry and resultant lattices along different crystal
axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.5 Simulated GMOT force as a function of atom velocity for different
axes and polarization configurations . . . . . . . . . . . . . . . . . . . 119
6.6 Atom number and density measurements of the GMOT . . . . . . . . 120
6.7 Atom number measurements of the GMOT as a function of magnetic
field gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.1 Free-space Faraday spectroscopy signal . . . . . . . . . . . . . . . . . 124
7.2 Transmission through nanofiber as a function of time after applying
1064-nm light to desorb atoms from surface . . . . . . . . . . . . . . 126
7.3 Heating time before onset of desorption occurs, as a function of 1064-
nm power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.4 Schematic illustration of self-organization of atoms trapped around
an optical nanofiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
A.1 Effective index of refraction as a function of V-number . . . . . . . . 135
A.2 Fundamental (HE11) mode structure of 360-nm diameter nanofiber . . 137
xi
B.1 Experimental schematic of vibration measurements based on photon
correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
B.2 Long-timescale correlation function displaying pronounced oscilla-
tions due to nanofiber vibrations . . . . . . . . . . . . . . . . . . . . 139
B.3 Geometry of the problem for a conical cantilever . . . . . . . . . . . . 142
B.4 Power spectrum of autocorrelation function, illustrating pronounced
peaks. Inset shows effect of 1064-nm light on some of the observed
vibrational modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
B.5 Fractional change in frequency as a function of 1064-nm power sent
through the nanofiber . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
C.1 Index of refraction of fused silica for λ between 0.2 and 7 µm . . . . . 149
C.2 Scalar polarizabilities for the 5S1/2 and 5P3/2 states of 87Rb as a
function of imaginary angular frequency . . . . . . . . . . . . . . . . 151
xii
List of Abbreviations
γ1D coupling rate into a 1-dimensional optical waveguide
γrad radiative decay rate of an atom
λ/4 quarter-wave plate
λ/2 half-wave plate
AOM acousto-optical modulator
APD avalanche photodiode
AR anti-reflection
ASE amplified spontaneous emission
AWG American wire gauge
BS beam-splitter
CCD charge-coupled device
CF Conflat
CW continuous-wave
DAVLL dichroic atomic vapor laser lock
DAQ data acquisition
DDS direct digital synthesis
EOM electro-optical modulator
FEM finite element model
FPGA field-programmable gate array
FWHM full width at half maximum
GMOT grating-mirror magneto-optical trap
GRIN gradient index of refraction
HWP half-wave plate
JQI Joint Quantum Institute
LHS left-hand side
LP low pass (filter)
MCS multi-channel scaler
MOT magneto-optical trap
NA numerical aperture
Nd:YAG neodymium-doped yttrium aluminium garnet
NISP Nanoscale Imaging Spectroscopy and Properties
NIST National Institute of Standards and Technology
NRL Naval Research Laboratory
NSF National Science Foundation
OD optical depth
ONF optical nanofiber
PBS polarizing beam-splitter
PD photodetector
xiii
PDH Pound-Drever-Hall
PFC Physics Frontier Center
PFI programmable function interface
PID proportional-integral-derivative
PM polarization-maintaining
Q quality factor
QED quantum electrodynamics
QWP quarter-wave plate
RHS right-hand side
SEM scanning electron microscope
SPCM single-photon counting module
SC superconducting
SS stainless steel
TCSPC time-correlated single-photon counting
Ti:Sapph titanium:sapphire
TOF time of flight
TTL transistor-transistor logic
UHV ultrahigh vacuum
xiv
Chapter 1: Introduction
A main thrust in the development of quantum systems is the generation of
strong coupling. Strong coupling can enable, for instance, high quantum efficiency,
fast exchange of information before relaxation or decoherence occurs, and strong
nonlinearities approaching the point where a single photon can saturate a single
atom. Depending on the application, the interaction might occur between con-
stituents within the same system or between different quantum systems, often me-
diated by photons.
In the case of photon-mediated interactions, the advent of cavity QED marked
a transformative milestone [1]. The cavities formed by mirrors or other structures
modify the vacuum modes of the electromagnetic field while providing a preferential
mode for the coupling. Cavity QED ushered in the ability to sufficiently isolate
a quantum system from its environment and control nearly all of its degrees of
freedom [2]. This led to the observation of increased [3] and inhibited [4] spontaneous
emission rates, and the generation of highly nonclassical photon states [5]. Cavity
QED behavior has now been realized from the microwave range [5,6] to the optical
range [7]. It is within this context that we want to frame our study of atom-light
interactions, using the notion of cooperativity to motivate our use of evanescent
1
waveguides.
1.1 Cooperativity
Consider a two-level atom with electric dipole moment ~d interacting with an
electric field ~E from a single photon. The parameter g encodes the strength of the
coupling,
g =
~d · ~E
~
. (1.1)
For an atom with decay rate γ and a field with decay rate κ, we define the single-
atom cooperativity to be [2]
C1 =
g2
κγ
. (1.2)
A cooperativity of C1 > 1 signifies a coupling rate between atom and field that
is larger than the geometric mean of the decays of both the atom and the optical
field. This places the system in the so-called strong coupling regime, which was
a longstanding goal within the quantum optics community. This regime has been
achieved in a plethora of systems, including ions [8], Rydberg atom in microwave
cavities [5], neutral atoms in optical cavities [7], and superconducting circuits in
planar waveguides [9].
To better understand how one can coerce a system into the strong coupling
regime, we can think of the cooperativity in a different way. Consider a high-finesse
Fabry-Pe`rot cavity with mirror transmission T and optical length L so that its
FWHM linewidth is κ = cT/2L. The electric field amplitude for a field with an
2
average energy of a single photon within this mode is given by
E =
√
~ω
2ε0V
, (1.3)
defining the mode volume to be V = Amode × L. The free-space linewidth of the
atom from Fermi’s Golden Rule is [10]
γ =
4ω3
3c2
d2
4piε0~c
, (1.4)
where d is the magnitude of the dipole moment of the atom and ω = 2pic/λ is the
resonant angular frequency of the decay transition. From Eqs. 4.7– 1.4 we get a
single-atom cooperativity of [11]
C1 =
Aatom
Amode
×
1
T
. (1.5)
Here we have defined the “area” of the atom Aatom to be the resonant scattering
cross section σ0 = 3λ2/2pi. Eq. 1.5 gives us a geometric framework with which to
think about cooperativity.
Efforts to increase C have followed a few different tacks. One method recog-
nizes that the total cooperativity for linear atom-light interactions is directly pro-
portional to the number of atoms interacting with the mode, C = N × C1, so that
one can simply increase the atom number. Vapor cells with high atomic densities
have used this principle to observe coherent processes such as electromagnetically-
induced transparency (EIT) [12]. Nonlinear interactions near saturation do not
3
benefit from this scaling and require other means to increase C. The Nobel Prize-
worthy efforts of Serge Haroche focused on decreasing T with microwave cavities
possessing finesses greater than 109 while making sure that the cross section of the
cavity significantly overlapped with the proprerly aligned rydberg atoms, allowing
his group to create highly nonclassical states and perform quantum non-demolition
measurements of photon jumps [5]. Recent advances in superconducting technol-
ogy have allowed physicists to create nonlinear quantum circuits that behave like
“artificial atoms” [13]. By coupling these so-called qubits to a high-quality-factor
resonator, they have engineered an analog of cavity QED dubbed circuit QED that
achieves couplings far beyond what have been realized in optical systems [6,9]. This
superconducting architecture does rely on the larger effective finesse of a microwave
cavity to increase C, but the area of their “atoms” (antennae, qubits) can easily
exceed that of the mode by a large factor.
More recently, some groups have moved away from the use of a traditional
cavity altogether, trying to increase the cooperativity of an atom in free space. They
use a parabolic mirror that focuses a laser such that the focussed beam has the same
structure as the dipole radiation pattern of a single atom, thereby increasing the
ratio of the respective areas [14–16]. Others use high-NA optical systems to focus
light to a small spot and achieve high coupling in free space [17–19].
This thesis will focus on an attempt to increase the atom-light cooperativity
that draws on a couple of these ideas and the physics of evanescent fields of nanopho-
tonic waveguides [20,21]. Nanophotonic waveguides are not like the traditional opti-
cal cavities with high finesses discussed above, but they do modify the vacuum mode
4
structure in a nontrivial way. As a result of this and the appreciable overlap between
the atomic and optical areas, one can realize strong atom-photon interactions with
these systems. We will consider waveguides formed by thinning single-mode optical
fibers to sub-wavelength diameters, so-called optical nanofibers [22–25]. There is
now an active area of research studying waveguides constructed via nanofabrica-
tion techniques that couple either to neutral atoms [26–28] or to spins that possess
atom-like structure [29].
1.1.1 Optical nanofibers
Reducing the thickness of an optical fiber to sub-wavelength diameters changes
the boundary conditions for light such that a significant fraction of the light propa-
gates in an evanescent field outside of the fiber [20]. In this regime, the optical depth
per atom can be a few percent so that a modest number of atoms can achieve large
optical thicknesses. To confine atoms along the nanofiber for appreciable lengths of
time, one can create an optical dipole potential by sending beams down the fiber
tuned to the red and blue of the relevant atomic transition [20]. The differing radial
decay lengths of the two colors, combined with the attractive van der Waals surface
interaction, produce a potential a few hundred µK deep and a few hundred nm from
the fiber [20]. Moreover, counter-propagating one of the trapping beams can form
one-dimensional lattices along the nanofiber.
This scheme has been used to trap cesium [22–24], opening the door to reach
the strong-coupling regime with these ensembles. Trapping lifetimes of tens of ms
5
and coherence times of ∼ 600µs [30] confirm that this is a viable platform for
doing atomic physics. Other research has explored forming cavities on nanofibers to
increase the cooperativity [31–33], coupling the spin and orbital degrees of freedom
of the light [34, 35], producing highly nonlinear phase shifts when atoms are in the
mode [36], coupling quantum dots to the nanofiber [37], and storing pulses in the
atoms via EIT [38,39].
Our work with nanofibers in this thesis centers on extending optical trapping
to rubidium [25] and directly measuring the cooperativity with photon counting
techniques. We also seek to interface this ensemble with another quantum system
consisting of a superconducting circuit, which we discuss in the next section.
1.2 Hybrid quantum systems
Methods to connect spatially-separated quantum systems would enable the
construction of quantum networks [40, 41]. In this scenario, photons present a
promising candidate for a “flying qubit” that can transfer information between sta-
tionary qubits, given their ability to propagate over long distances with little loss.
On the other hand, what makes a sedentary qubit good is that is not flying around
at the speed of light.
Superconducting (SC) qubits have flourished as a potentially scalable proces-
sor of quantum information, with demonstrations that include nontrivial quantum
algorithms [42,43], entangling three qubits [44], and performing a quantum simula-
tion of the Hubbard model [45]. Despite these remarkable advances, however, the
6
coherence times of SC qubits remain quite short (up to ∼ 200µs) when compared to
those achieved in atomic systems where, for example, neutral atoms can maintain
two seconds of ground-state coherence [46]. This has spurred a concerted interest
in interfacing SC circuits with other quantum systems that can better store the
information generated by a superconducting qubit [47].
SC qubits can be designed to couple to or emit microwave photons in the
GHz range. Their propagation in dielectric waveguides is lossy over distances longer
than a few centimeters, although much longer distances are achieved with super-
conducting waveguides [48]. For this reason, some hybrid system realizations aim
to upconvert these photons into the optical band for better transfer fidelity and
reversibly retrieve them at another node [49, 50]. For example, the coupling of a
SC resonator with an optomechanical oscillator recently demonstrated a conversion
efficiency of 10% and offers a potential link between the microwave and optical
regimes [51].
Other methods to produce a hybrid system involve direct coupling that takes
advantage to the fact that the hyperfine structure of alkali atoms and other spin
systems posses energy scales comparable to those of superconducting qubits. Here
the exchange of microwave photons through a magnetic dipole interaction mediates
the coupling [47]. Because the single-atom, single-photon coupling strength is typi-
cally very small (g/2pi ≈ 100 Hz), most realizations of this hybrid system will take
advantage of the ensemble enhancement of the Rabi rate for N atoms, which scales
as
√
N [1].
Solid-state spin systems such as NV centers are another potential candidate
7
for a memory for a SC circuit, as they can be placed in close proximity to the super-
conductor without the need for complicated trapping mechanisms [52–59]. These
experiments have reached the strong-coupling regime between the spins and a res-
onator [54], coupled a qubit to the spins [56,58], and demonstrated coherent storage
of microwave photons in the ensemble [57, 59]. Although this is an all-solid-state
approach, these results have pushed hybrid quantum systems towards reality.
The hybrid system that interests us is that of a SC circuit coupled to an ensem-
ble of neutral atoms, which has garnered theoretical attention [60–64] and recent
experimental progress [65–70]. This particular brand of hybrid system adds the
complication of needing either optical or magnetic trapping (or both) to bring the
atoms close enough to the superconductor such that the coupling becomes apprecia-
ble. Despite these difficulties, neutral atoms remain a promising candidate due to
the long coherences in the hyperfine clock states and the lack of large inhomogeneous
broadening mechanisms that exist in many bulk spin systems.
1.2.1 Neutral atoms coupled to superconducting circuits
The difficulty of trapping atoms near a superconductor stems from both the
cryogenic environment of the superconductor and the conditions required for trap-
ping atoms. In order to ensure that a superconducting quantum circuit with tran-
sitions in the GHz range has low thermal excitation probability, it must be housed
inside a dilution refrigerator that can reach temperatures of tens of mK. The cooling
power of our system at the base temperature stage is 200 mW (e.g. our Oxford Tri-
8
ton 200 Cryofree system will heat up to 100 mK when 200 µW of heat is applied to
the mixing chamber plate), necessitating that great care be taken when designing an
atomic trap for this environment. Moreover, superconducting qubits are sensitive
to optical photons [71, 72] and magnetic flux [73, 74], placing even more stringent
restrictions on the types of traps one can consider.
Magnetic chip traps for this style of hybrid system are an active area of re-
search [62,66–70]. The group of Jo´zsef Forta´gh at the University of Tu¨bingen has re-
cently demonstrated coherence times of seconds in small (104 atoms) BECs trapped
near superconducting microwave circuits [68] and observed the sensitivity of these
atom clouds to quantized flux in a SQUID [70], both performed in 4.2 K cryostats.
An ongoing effort in their group is performing these experiments inside a dilution
refrigerator, where they have achieved a MOT within a higher-temperature region
and eventually a scheme of magnetic transport coils will bring the atoms to the
science region at millikelvin temperatures [69].
Our proposal instead uses a low-loss optical nanofiber trap to hold an ensemble
of 87Rb atoms [65]. The superconducting circuit at present consists of a lumped-
element, LC resonator with resonant frequencies near 6 GHz and quality factors
greater than 105. The atoms will be trapped within a few µm of the inductor line of
the resonator in order to maximize the magnetic dipole coupling. Ultrahigh optical
nanofiber transmissions of 99.95% [75] place these devices in a regime where the
scattered light from the thin waist is low enough that the quality factor of the
resonator can remain high [76].
We now estimate the cooperativity of this system based on current state-of-
9
the-art numbers. Nanofiber traps typically contain 103 atoms, but implementing
deterministic loading [77] can increase that number to 104. The linewidths of the
SC resonators are about 104 kHz, whereas the best-reported decay rate of nanofiber-
trapped atoms is roughly 1 kHz [30]. Given a coupling strength of g ≈ 100 Hz, these
numbers translate to a total cooperativity of C = 10, which places this system into
the strong coupling regime for more than about 100 atoms.
1.3 Outline of thesis
This thesis is concerned with a series of experiments performed with optical
nanofibers that were ultimately intended for integration with a superconducting
device. In Chap. 2 we present the design and construction of the atomic physics
side of the experiment, where an empty room was converted into a laboratory. It
highlights the aspects of the apparatus, such as the nanofiber puller, that differ
from some of the standard features that are ubiquitous in laser cooling and trapping
experiments. Chap. 3 demonstrates the trapping of 87Rb atoms around an optical
nanofiber produced within our laboratory. This marks an important first step for
using nanofibers as part of a hybrid system. Much of this chapter is almost verbatim
from Ref. [25].
In Chap. 4 we move to the study of photon correlations in atom-ONF systems
and how they can inform our knowledge of atomic dynamics and the mode structure
of these waveguides. Measuring the transit time of atoms through the optical mode
allows us to extract the temperature of the atomic cloud. The text of this chapter
10
is almost verbatim from a paper we have recently submitted [78]. We present an ex-
tension of these correlation measurements in Chap. 5, where we use time-correlated
single-photon counting to measure the enhancement of spontaneous decay of atoms
near the nanofiber surface. This is a useful confirmation – in the time domain – of
a Purcell-like effect in this system.
Coming back to the overall hybrid system in Chap. 6, we summarize how the
various pieces will fit together. We then delve into a particular implementation of a
compact laser cooling and trapping scheme and demonstrate that it can bring atoms
to sub-Doppler temperatures. Much of this section is presented almost verbatim
from Ref. [79]. Finally, in Chap. 7, we highlight four studies with nanofibers that
are either planned or ongoing within our group and summarize our key results.
Three Appendices add discussions regarding mode structure of optical nanofibers,
correlation measurements of nanofiber vibrations, and a calculation of van der Waals
coefficients between 87Rb and fused silica.
As an editorial note, the use of “we” throughout this thesis signifies a royal
we pertaining to the members of our group within the JQI. Specifically the initial
construction of the laboratory outlined in Chap. 2 was carried out by myself and
Jonathan Hoffman. After I helped design some aspects of the fiber puller, Jonathan
and Sylvain Ravets optimized the fiber pulling process and took the profile and
transmission data presented in this thesis. I focused on the building of the laser
cooling and trapping apparatus, together with Jongmin Lee and later Pablo Solano.
Jongmin and I took the nanofiber trapping data presented in Chap. 3, and he per-
formed the simulations for the asymmetric fitting. Pablo and I set up the experiment
11
and recorded the correlation data in Chap. 4, and I carried out the trajectory sim-
ulations. Pablo and I both took the data and developed the model presented in
Chap. 5, and I did the majority of the data analysis and fitting. In Chap. 6, I built
the beam expander for the grating MOT and took temperature data with Jongmin,
while he mounted the gratings in vacuum, took atom number and density data,
and performed the simulations to verify sub-Doppler cooling. Appendices B and C
present calculations that I performed.
12
Chapter 2: Apparatus
Starting an atomic physics laboratory from an empty room requires a combina-
tion of planning, building, designing, buying, soldering, and aligning. This chapter
outlines the process of assembling an experiment from scratch that can cool, trap,
and study neutral atoms around optical nanofibers – from conception to data-taking.
It functions almost as a “parts list” for the experiment, chronicling aspects that are
particular to this setup and providing a record of how things fit together that could
be useful for future students who work on the project.
This chapter is organized in the following manner. We first summarize the
initial conversion of office space to laboratory space in Sec. 2.1. Next we discuss the
nanofiber puller and highlight the relevant results in Sec. 2.2. Sec. 2.3 outlines the
lasers and control systems used to form a MOT and a nanofiber dipole trap. We
then proceed to cover the distinctive parts of our ultrahigh vacuum (UHV) system
in Sec. 2.4. Secs. 2.5 and 2.6 catalog the magnetic and optical systems built around
this UHV environment to create and image cold atom clouds. In Sec. 2.7, we present
the electronics and detectors that facilitate the photon-counting experiments that
are central to this thesis. Sec. 2.8 lists the devices and DAQ cards that oversee
the synchronization of the devices discussed earlier in the chapter. Finally, Sec. 2.9
13
concludes and offers an outlook on the new laboratory space in the basement of the
PSC, as well as some experimental improvements in the works.
2.1 New laboratory construction
The fall of 2009 marked the beginning of the conversion of an almost empty
former office space to a cold atom and SQUID laboratory (see Fig. 2.1). At this
point, new flooring, lighting, and some electrical work were taken care of. We
began by populating the room with fire- and chemical-resistant desks from VWR.
A 4′ × 12′ × 12′′ optical table from TMC was installed in May 2010, with our first
lasers following about a month later. We built a system of 80/20 shelves around the
table to house electronics
(a) (b)
Figure 2.1: (a) Empty laboratory space in Physics 1305B before construction,
September 2009. (b) Initial optical table installation and 80/20 con-
struction.
14
2.2 Nanofiber pulling
The success of this experiment requires the fabrication of optical nanofibers
(ONF) with high transmission. High transmission ensures that sufficient optical
powers (∼ mW) can be sent through the ONF under vacuum without breaking it,
as well as minimizes the scattering of photons into the dilution refrigerator. This
section provides an overview of the general fiber pulling process as well as the specific
apparatus in our laboratory. Additional details can be found in the PhD thesis of
Jonathan Hoffman [80] and Ref. [75].
2.2.1 The flame-brush method
Many methods exist to reduce the diameter of a normal, single-mode fiber to a
subwavelength diameter through physical or chemical mechanisms [81,82]. The non-
chemical techniques rely on heating a small section of fiber and pulling on either end,
where the choice of heat source presents the biggest variation between the different
methods. In our laboratory, we use a small, hydrogen-oxygen flame to heat the
fused silica to just below its melting point (1585 ◦C). At this temperature, the glass
flows but does not melt, and conservation of volume of the glass dictates how the
fiber radius evolves during the pull [83]. After pulling at a steady rate for a time t,
the initial radius r0 of a region in the flame becomes
r = r0 exp
(
−
t vf
2L0
)
, (2.1)
15
where L0 is the flame width and vf is the pulling velocity. A stationary flame results
in an exponential taper, with the final radius dependent on the duration and velocity
of the pull. Varying the relative position of the so-called hot zone throughout the
pulling process allows for the production of controlled taper geometries [75, 83–87],
in which relatively short exponential sections are stitched together to approximate
the desired taper shape.
The shape of the tapered sections becomes important when considering ex-
perimental length constraints, which can jeopardize the conditions of total internal
reflection that ultimately govern the adiabaticity of the shape. We want to mini-
mize the excitation of higher-order modes in as short of a fiber as possible. This
issue is especially prescient for our hybrid experiment with an ONF in a dilution
refrigerator, where the lengths set by Eq. 2.1 would be too long for the area below
the mixing chamber plate. Sec. 2.2.3 will present our results for pulling fibers with
shorter, linear tapers while maintaining ultrahigh transmission.
2.2.2 Setup and procedure
Fig. 2.2 displays a schematic and photograph of our fiber puller. The entire
apparatus is under a softwall clean room (initial rating Class 100) with HEPA filters
in the ceiling to maintain a dust-free environment. Preventing dust from falling onto
the fiber, before or after fabrication, is crucial for achieving high transmission, as
we will discuss in more detail later in this section.
We use high-precision linear bearing stages (Newport XML 210) mounted to
16
8
 
5
 
7
 
6 
3 
9 10 11 12 
13 
14 15a 
15b 
1 2 4 
16 
15c
17
13 2
45
67
8
15
14
16
17
a)
b)
Figure 2.2: (a) Schematic of the fiber puller (top view). (b) Photograph of appa-
ratus. 1) Fiber motors. 2) Granite slab. 3) Optical breadboard. 4)
Adapter plates. 5) L-bracket mounts. 6) XYZ fiber alignment flexure
stages. 7) Fiber holders. 8) Adjustment screws. 9) Gas flow meters. 10)
Gas line filters. 11) Valves. 12) Flexible pipes. 13) Fine gas line filter.
14) Nozzle. 15) (a) Kohler illumination system. (b) Optical microscope.
(c) CCD camera. 16) Flame positioning motor. 17) USB camera aligned
vertically above fiber. (figure from Ref. [75])
17
2 mm
1 mm
Figure 2.3: Image of stainless steel nozzle for hydrogen-oxygen flame in the nanofiber
pulling rig. The 1× 2 mm array consists of 29 holes with a diameter of
228 µm.
a granite slab to perform the controlled pulling of the fiber. The flatness of the
granite slab allows the stages to perform within their specifications, and the mass
of the granite minimizes jerk when the motors change directions. A flame formed
by a stoichiometric ratio of hydrogen and oxygen provides the heat source so that
the only combustion product is water, reducing the potential for contamination of
nanofiber surface. The flame head is a custom-made flange with 29 holes with a
diameter of 228 µm arranged in a 1 × 2-mm hexagonal pattern (holes made by
Twin City EDM) on a stainless steel plate (see Fig. 2.3). The small hole size helps
to ensure that hydrogen gas does not back-flow into the line, but the dimensions
ensure that the Reynolds number R (R = 1.9 for the H2 and R = 6.3 for the O2) is
low enough for laminar flow (R < 2100).
18
Before each pull, the fiber-flame position is set to maintain a repeatable flame
size. The fiber ends are then aligned relative to one another using an in-situ imaging
system (NA = 0.28, 2 µm resolution), which fixes the pulling axis to be parallel
to the flame nozzle face (see Fig. 2.3). Before pulling, the plastic fiber buffer is
mechanically stripped off of the region of fiber to be stretched, and this region
is repeatedly cleaned with methanol and cleanroom-compatible wipes (Ted Pella)
until no particulate remains. The fiber is then placed into the fiber clamps, and
the cleaned fiber is imaged section by section with the microscope to verify that the
fiber is particulate- and dust-free. If anything is visible in the images, the cleaning
procedure is repeated until this test is passed.
The pull is then initiated, and the pulling motors are computer-controlled by a
Newport XPS controller, which uses pre-generated trajectories to direct the motors.
These trajectories are calculated by an algorithm (see Refs. [80,84]) that generalizes
Eq. 2.1 to create the desired taper. Note that this process involves varying the
position of the hot zone, but air currents would disturb a moving flame and degrade
the pull. Instead the flame is kept fixed, and the algorithm moves the fiber during
the pull via the pulling motors so that one leads and the other lags during each pull
step.
2.2.3 Results
As a test of the pulling apparatus and algorithm, we stretch a single-mode,
SMF-28 fiber to a final waist radius of 10 µm with three linear sections of different
19
taper angle (5, 2, and 3 mrad) [80]. The 10-µm final radius allows us to use the
imaging system within the apparatus to characterize the entirety of the fiber profile.
Fig. 2.4 displays the measured fiber geometry compared with the intended geometry.
Over the 60-mm pull, the maximum radial deviation between the desired profile to
the measure profile from the puller is ∼ 2%, confirming that this technique can
produce non-exponential tapers with high accuracy.
Verifying nanometer-scale radii is beyond the resolution of our in-situ imaging
system, and we instead employed non-optical methods. Some fibers are (destruc-
tively) coated with graphite and imaged in a scanning electron microscope (SEM).
Fig. 2.5 shows an example of one such fiber with a final waist diameter of 536± 12
nm, compared to a design diameter of 500 nm. We find that our nanofibers are
systematically ∼ 10% larger than the design size, possibly due to small air cur-
rents pushing the thin fiber into a different region of the flame or to a systematic
under-estimate of the size of the hot zone [80].
In addition to a particular waist diameter, nanofibers must possess high trans-
mission for use in optical trapping experiments. Losses arise from Rayleigh scatter-
ing in the silica, fluorescence from impurities in the glass, nonadiabatic excitation
of higher-order-modes in the taper region, and surface scattering from contaminants
on the waist. The first two loss mechanisms are inherent to the glass, while the
latter two can be eliminated through careful fabrication. We tried to reduce surface
contaminants by using the cleaning procedure outlined in Sec. 2.2.2 and keeping the
nanofiber in a dust-free environment post-pull. Gradual tapers tend to minimize the
excitation of higher-order modes, and while an optimal adiabatic taper exists [83,88],
20
20
30
40
50
60
70
Ra
diu
s (
μm
)
0 1 2 3 4 5 6−3
−2
-1
0
1
2
Fiber axis (cm)
Re
lat
ive
 D
iffe
re
nc
e (
x 1
0-2
) 
Figure 2.4: Profile of a triple-angled, linear-taper fiber. (a) Plot of fiber radius r vs.
position z along fiber from optical microscope images. The red curve is
the expected profile from simulation. The three angles of 5, 2, 3 mrad
end at respective radii of 50, 35, and 25 µm, with a final radius of 15
µm. (b) Relative error ∆r/r between measured and expected radius
(∆r = rexpect− rm), with an RMS error of 0.0187(figure from Ref. [75]).
21
Figure 2.5: SEM image of 536-nm diameter nanofiber (design diameter: 500 nm)
taken at the NISP Lab at UMD (figure from Ref. [75]).
22
0 50 100 150 200 250 300 350 400 450 5000.9800
0.9850
0.9900
0.9950
1.0000
1.0050
Tr
an
sm
iss
ion
 N
or
ma
liz
ed
Time (sec)
Figure 2.6: Nanofiber transmission T vs. time t measured during the pulling process
for a taper angle of 2 mrad from a starting radius of 62.4 µm down to 6
µm, and an exponential taper from there to the final radius of 250 nm.
The waist is 5 mm long. The vertical red line demarcates the end of the
pull. The final transmission is 99.95± 0.02%.
we find that a single-angle taper of 2 mrad works well. For example, monitoring the
transmission during the pull reveals an overall transmission of 99.95 ± 0.02% (see
Fig. 2.6) for a nanofiber with an angle of 2 mrad and 500-nm diameter waist. These
dimensions are similar to the nanofiber that we use for the subsequent experiments
in this thesis, and we summarize the geometry in Fig. 2.7 and Table 2.1.
The region between 325 s and 425 s in Fig. 2.6 exhibits pronounced oscillations
in the transmission. These oscillations are the result of beating between higher-order
modes that are excited when the fiber thins to a radius of typically about 23 µm for
SMF-28 fiber [75,88–90].
23
1 2 3 2 1
r
z
Figure 2.7: Illustration of geometry of the nanofiber used throughout this thesis.
An exponential section (region 2) connects the linear taper (region 1)
to the uniform nanofiber waist (region 3). The region in white is the
unmodified fiber. The parameters are given in Table 2.1. Drawing is
not to scale.
Table 2.1: Dimensions of the nanofiber used in this thesis. The regions refer to
Fig. 2.7.
Region Initial radius Final radius Length Taper angle
(µm) (µm) (mm) (mrad)
1 62.5 6 28.25 2
2 6 0.25 10.75 N/A
3 0.25 0.25 7 0
2.3 Laser systems
This section details the four lasers used in the experiment, two for creating
the MOT and two for forming the nanofiber dipole potential. The two MOT lasers
require frequency stabilization in order to consistently address the hyperfine levels
within the D2 transition (λ = 780.241209686(13) nm [91]) of 87Rb (see Fig. 2.8),
and so we discuss the different locking schemes for each laser. Because the nanofiber
trap lasers are far off-resonant beams (|ωtrap − ω0|  γ0, where ω0/2pi and γ0/2pi
are the atomic resonant frequency and excited-state linewidth, respectively), small
24
drifts in the frequencies are unimportant.
2.3.1 Cooling and probing laser: saturation spectroscopy
The laser beam that provides (sub-)Doppler cooling for our magneto-optical
trap originates from a CW diode with a tapered amplifier (Toptica TA Pro) that
generated roughly 1.3 W of power at 780.24 nm upon initial purchase in 2010. To
lock this laser we employ a variation of the Pound-Drever-Hall (PDH) technique [92],
which typically uses a stable optical cavity as a reference in order to feed back to the
laser. Frequency locking requires knowledge of whether the laser is above or below
the resonant frequency, and because the lineshape of a cavity or atomic resonance
is a symmetric Lorentzian, analog techniques do not enable us to use the intensity
of a transmitted or reflected signal as our error signal1. The phase of the lineshape,
however, is asymmetric about the resonance frequency, and the PDH lock measures
the phase through a clever scheme of interfering sidebands with the carrier.
Rather than lock to an optical cavity, we send the beam through a rubidium
vapor cell and perform saturated absorption spectroscopy to find our lock point
(see Fig. 2.92). Saturated absorption spectroscopy eliminates Doppler broadening
in the vapor cell, which can smear out the natural atomic linewidth of a few MHz
to hundreds of MHz [94]. Two beams are required for this purpose, one for probing
and detecting the absorption spectrum, and another, stronger beam to pump and
1The authors in Ref. [93] demonstrated that with sufficient accuracy in their analog-to-digital
conversion, they can use microcontrollers to lock directly to the resonance peak.
2This and many other figures in this thesis were made with the help of ComponentLibrary
by Alexander Franzen, under the Creative Commons Attribution-NonCommercial 3.0 Unported
License.
25
repumping
cooling
optical
pumping
2-3 crossover 133 MHz 
6.834683 GHz 
65 MHz lock point
AOM up-shift
Figure 2.8: D2 (λ = 780.241) level structure for 87Rb with transitions for laser
cooling, repumping, and optical pumping drawn between appropriate
states. The 2 − 3 crossover represents a crossover transition due to
saturation-free spectroscopy, as discussed in Sec. 2.3.1.
26
saturate the atoms. This pump beam burns a so-called hole in the spectrum by
exciting atoms with velocity v = (ωpump − ω0) /k, so that near atomic resonance it
interacts with atoms moving with velocity v ≈ 0 relative to the beam direction [94].
When the probe beam is also tuned to resonance, it sees fewer atoms in the ground
state, decreasing absorption at this frequency. The spectral width of this hole is
given by the power-broadened natural linewidth of the atom in the absence of in-
homogenous broadening from magnetic fields and light shifts, as Doppler effects are
eliminated due to this selection of only the atoms at a particular velocity. Because
the linewidth of our Toptica laser is much narrower (. 100 kHz) than the atomic
transition linewidth, we can resolve the hyperfine resonances by these holes in the
absorption.
In addition to the hyperfine states, saturation spectroscopy produces cross-over
transitions midway between the “real” transitions [94] (the dotted line in Fig. 2.8
indicates one such transition, the 2 − 3 crossover). We use the 2 − 3 cross-over
transition for our lock point, which sits 133.3 MHz to the red of the F = 2−F ′ = 3
cycling transition. The contributions from both the F ′ = 2, 3 levels make it the
strongest peak in the spectrum, and it also gives us a nice point to work from
using AOMs. In our setup, we upshift only the probe beam (see Fig. 2.9) with a
double-passed AOM by 2 × ΩAOM/2pi = 2 × 65 MHz. This scheme will burn the
saturated holes discussed above at effective cross-overs between pump and probe,
i.e. ωlaser +ΩAOM. Thus in order for these cross-overs to be resonant with the atomic
transition, the laser will sit at a frequency ωlaser = ω2−3 − ΩAOM, where ω2−3/2pi is
the frequency of the 2− 3 cross-over transition, as indicated in Fig. 2.8. Therefore
27
9.82 MHz
65 MHz
to PID
isolator
TA Pro
PBS
AOM
EOM Rb cell PD
λ/2 λ/4
pump
probe BS
beam dump
mirror
lens
Figure 2.9: Schematic of the optics to perform saturated spectroscopy and Pound-
Drever-Hall locking of the cooling laser. Note that the probe beam (solid
red) is up-shifted by 130 MHz relative to the pump beam (dashed red).
to EOM
to laser
9.82 MHz
sat spec signal
PID boxmixer
RF amp
bias-tee LP filterPD
Figure 2.10: Schematic of the electronics to perform Pound-Drever-Hall locking of
the cooling laser. The saturated spectroscopy signal detected in the
photodiode (PD) is amplified and then mixed down and added to the
9.82-MHz signal that dithers the EOM.
the laser is tuned 198 MHz to the red of the cycling transition, and a double-passed,
tunable, 100-MHz AOM can bring the MOT and probing beams to the appropriate
detuning.
An EOM phase-modulates the probe at 9.82 MHz, and we use this same RF
frequency to demodulate the saturated spectroscopy signal (see Fig. 2.10). The
mixed-down signal is then sent to a lock box (Newport LB1005), and fed back to
the laser piezo to stabilize its frequency.
28
2.3.2 Repump laser: Doppler-free DAVLL
While the cooling transition (see Fig. 2.8) is to a reasonable extent a closed
cycling transition, there is a nonzero probability for the atom to be excited to the
F ′ = 2 state, allowing it to decay to the F = 1 state in the ground-state manifold.
Here the atom is dark to the MOT cooling laser, and no further cooling occurs. In
order to remove the atoms that have fallen into this dark state, we add a second
laser (Toptica DL Pro) tuned between F = 1 and F ′ = 2 to repump the atoms back
into the cycling transition (see Fig. 2.8).
To lock the repump laser we employ a variation of the dichroic atomic vapor
laser lock (DAVLL) [95, 96] that uses saturated spectroscopy to narrow the locking
transition [97]. The DAVLL utilizes the Faraday effect [98] with a magnetic field
along the direction of light propagation in an atomic vapor to generate an error
signal, without the need for additional RF electronics. Linearly-polarized light can
be decomposed into left- and right-circularly polarized components, σ− and σ+,
respectively. When a resonant, linearly-polarized beam is sent through an atomic
medium, these two circular components are degenerate, and a magnetic field applied
along the direction of propagation lifts this degeneracy through the Zeeman effect.
Applying this field along the axis of propagation (as in Fig. 2.11) shifts the absorp-
tion curve of each component, creating elliptically polarized light at the output with
phase difference given by [98]
ϕ '
2gµB/~Γ
1 + (2gµB/~Γ)2
l
l0
, (2.2)
29
to feedback
isolator
DL Pro
PBS Rb cell
PD
λ/2
pump
λ/4
Wollaston
Figure 2.11: Schematic of optics for the DAVLL for the repump laser. The magnetic
field points in the direction of propagation of the probe beam.
where B is the magnitude of the applied field, Γ is the width of resonance peak, g
is the Lande´ factor, µ is the Bohr magneton, l is the length of the sample, and l0
is the absorption length. We separate these components with a quarter-wave plate
and Wollaston prism, and subtracting the photodetector signals yields a dispersive
curve for locking. Optimal fields for creating a linear error signal near resonance
occur when the equality 2gµB/~ = Γ holds, and the Doppler-free DAVLL scheme
requires smaller fields to shift the narrower resonances. We find that a chain of
permanent magnets appropriately oriented near the rubidium cell is sufficient to
produce a stable error signal.
Other groups have found that a DAVLL setup can have too much temperature
dependence due to the polarization beam splitters. This is one reason why we use a
Wollaston prism at the output, after the atoms have rotated the light polarization
(the input is less sensitive). The optical setup is also covered with a plastic box
to shield air currents and to help maintain stable temperatures, as temperature
changes can indeed affect the field from the permanent magnets.
30
2.3.3 Nanofiber “blue” laser: Ti:Sapph
We use a Ti:Sapph laser tuned to 750 nm to produce the repulsive, blue-
detuned beam for our nanofiber trap. This system consists of a tunable, ring cavity
Ti:Sapph (Coherent 899-01) free-running but with thick and thin etalons, pumped
by a 10-W, 532-nm beam (Coherent Verdi-V10). The large cavity length of the
Ti:Sapph reduces the amount of ASE in the beam relative to a diode laser, which
helps when filtering background in front of our detection SPCMs. Additionally, high
output powers (hundreds of mW) ensure that we have a suitable range of power at
the nanofiber waist even after heavy filtering. The wide frequency tunability (from
∼ 720 − 810 nm) allowed us to rule out Raman gain in the silica as the cause of
the background light we detect in the optical fiber, and we instead think it is likely
fluorescence from impurities.
2.3.4 Nanofiber “red” laser: Nd:YAG
An Nd:YAG laser (JDSU NPRO-126N-1064-100) generates the 1064-nm beam
to create the attractive potential for the nanofiber trap. The NPRO laser is based on
a monolithic design that ensures narrow linewidth, excellent stability, and Gaussian
optical modes (M2 ≈ 1), which allows for highly efficient fiber-coupling and AOM
diffraction, leaving us with the requisite few mW per beam to generate the standing-
wave potential along the nanofiber. Moreover, the low intensity noise (< 0.05%
rms for frequencies between 10 Hz to 2 MHz, and < −165 dB/Hz above 10 MHz)
minimizes heating in the trap. The two beams for each leg of the standing wave are
31
up-shifted by 80 MHz using two AOMs, and the RF signals to each AOM are phase-
locked. We can also control the relative frequency of these two beams to create an
optical “conveyor belt” [99, 100] along the nanofiber to transport atoms.
2.4 UHV system
Cold-atom experiments require an environment devoid of other gases, as colli-
sions with a gas would limit the amount of time that atoms would remain trapped.
To this end, the experiments are performed in UHV systems with pressures less than
10−9 mbar. Because these systems are commonplace in many laboratories, including
AMO laboratories, in this section we outline only the characteristic features of our
UHV apparatus.
2.4.1 Science chamber
A stainless steel (SS) chamber (Kimball Physics MCF800-SphSq-G2E4C4A16)
with a range of CF flange sizes forms our main science chamber (see Fig. 2.12).
Viewports on either side are AR coated (Rocky Mountain Instruments) for 780-nm
light, and their large size affords us generous optical access to the chamber for MOT
beams, imaging, optical pumping, and other probes. A reentrant viewport (MPF
Products, Inc.) on the top of the chamber provides an orthogonal imaging direction,
decreasing the working distance to the center of the chamber from about 111 mm
to 60 mm.
32
optical 
table
AR-coated 
window
mounting
flange
Figure 2.12: Photograph of UHV science chamber before coil installation. Visible
are the large, AR-coated windows and the washer mount described in
Sec. 2.4.2 to attach the chamber to the table. For scale, the spacing of
the holes on the optical table is 1”.
33
2.3
12
 bo
lt c
irc
le
1/4
-20
 ta
pp
ed
 th
ru
2.7
3
 6 
plc
s o
n 
 TH
RU
 (1
/4-
28
 cl
ea
ran
ce
)
.27
5
1.5
0
2.3
12
 bo
lt c
irc
le
1.6
70
 bo
lt c
irc
le
4-4
0 t
ap
pe
d t
hru
,
6 p
lcs
 on
 
12
 pl
cs 
on
 
1.1
8130
.10
Figure 2.13: Drawing of custom flange used to attach 30-mm cage-mounts from
Thorlabs to a 2 3/4” CF flange, adapted from the original design of
Creston Herold.
2.4.2 Washer optics mounts
Creston Herold from the JQI Ultracold Mixtures group of Trey Porto and Steve
Rolston designed large, SS flanges that allow cage-mount optics from Thorlabs to
be mounted directly to CF flanges. We use some flanges based on his designs (see
Fig. 2.13) to mount four of MOT beams to our science chamber, and we built off the
idea to make a washer to mount the entire chamber to the optical table (see Fig. 2.14,
and visible in Fig. 2.12). These optics mounts greatly simplify the alignment of these
four crossed MOT beams, as well as free up space on the optical table.
2.4.3 Ion pump placement
Efficient devices that achieve and maintain UHV conditions with no moving
parts, ion pumps rely on a high voltage and an electron current to ionize residual
34
0.1
87
5
Ra
diu
s
2.6
9
 ar
c
5.7
5 c
irc
le
1.2
7
3.5
0
0.2
7(1
/4-
20
 cl
ea
ran
ce
) 
Ra
diu
s
67
.511
2.5
TH
RU
 AL
L
 TH
RU
 (5
/16
-24
 cl
ea
ran
ce
) 
 ar
c
 x 
0.3
3
2.2
35
4
On
On
3.6
28
 bo
lt c
irc
le
TH
RU
 AL
L
8 x
 
1.0
6
2.2
0
Figure 2.14: Drawing of custom washer, used to attach the 4 1/2” CF flange of our
science chamber to the optical table on 1” and 1.5” optical posts.
gas particles that diffuse near their entrance. The high voltage and magnetic field
are not sufficient to direct the residual electrons and/or ions to a titanium getter,
but some escape to produce a current on the order of a few pico amps [101]. We
find that if the nanofiber is placed within the line-of-sight of an ion pump, these
leakage currents cause charge buildup on the nanofiber, which eventually breaks
under strain from external electric fields. We circumvent this problem by inserting
90◦ vacuum elbows between the ion pumps and the science chamber so that the
leakage currents impinge on a grounded metal surface rather than the nanofiber. As
a further precautionary measure, we place protective vacuum screens (Pfeiffer PM
016 333) within the pump CF flange, offering another grounded surface to discharge
the leakage current without adversely affecting the conductance of the pump.
35
2.4.4 Manipulator
We added to the science chamber a load-lock system that facilitates quick
transfer of samples into UHV. A 6” cube (MDC Vacuum 408004) serves at the an-
techamber, separated from the science region by a 2 1/2” all-metal gate valve (MDC
Vacuum 302002), chosen so that the bore can accommodate mounted nanofibers (see
Fig. 2.15). Attached to the antechamber is an UHV-compatible manipulator (VG
Scienta Transax), which has 450 mm of translation parallel to the optical table and
25 mm of (radial) motion in the transverse plane. A stepper motor provides quick
motion in the parallel direction, and manual micrometers position the manipulator
transversally. Due to the large torque exerted on the mounting flange, the manip-
ulator is counter-balanced by a pulley system attached to a bucket of steel balls
on the side of the table. The manipulator has proven useful when interchanging
various GMOT prototypes and when nanofibers continually broke during our early
attempts at transferring them into UHV. One shortcoming of the system, however,
is the long lever arm of the support rod to which samples are mounted, as it couples
external vibrations to the nanofiber.
2.4.5 Nanofiber mount
Post-pull, a nanofiber must be mounted to a rigid structure before being
moved, as its thin diameter leaves it susceptible to breaking from shearing forces.
The mounts our group developed typically have a “U” shape, and the nanofiber is
glued with UV-curing epoxy to the vertical sections of the U. Fig. 2.16 is a photo-
36
gate valve
YZmicrometersantechamber
support rod
X manipulator
ion pump
Figure 2.15: Photograph of UHV manipulator attached to science chamber via a
gate valve. Note the ion pump placed at a right angle relative to the
antechamber to prevent the breaking of fibers.
graph of one of our mounts, displaying the characteristic shape and the hollowed-
out section on the bottom through which MOT light can pass. This example is
made from a titanium alloy because it is machinable, lighter than steel (less de-
flection of the manipulator support rod), non-magnetic, and has a conductivity
(σ = 2.38 × 106 Ω−1 ·m−1) 15 times smaller than that of aluminum to lessen eddy
currents when the MOT coils turn off.
2.5 Magnetic field control
Forming a MOT requires cancellation of spurious external magnetic fields, as
well as the generation of a linear magnetic gradient at the trap location. To this end
we use three pairs of coils with separation between their radius and their diameter
that produce a roughly uniform magnetic field at the center in three orthogonal
37
1 cm
Figure 2.16: Titanium nanofiber holder for UHV.
directions for cancellation, and two coils where each field points towards the center
to make the gradient. This section presents the specifications for these coils and an
overview of the circuitry that drives their currents.
2.5.1 MOT coils
Each coil for the gradient magnetic field consists of 22 turns of rectangular
wire with cross-sectional dimensions of 0.191′′ × 0.481′′. The large thermal mass of
the coils allow us to run high currents (up to 90 A) for long periods of time without
the need for water cooling. The coils have an inner diameter of approximately
8” and are separated by 7”, and typical currents (30 − 90 A) generate gradients
between 5 − 15 G · cm−1. We supply them with current via a high-current power
supply (Electronic Measurements Inc. TCR Power Supply, DC-200A), connected
by 4-AWG, super-flexible welding cable.
38
shim coils
gradient
coils
MOT optics
x
y
z
Figure 2.17: Photograph of science chamber with three pairs of shim coils and gradi-
ent coils (dark maroon with rainbow cable ties) installed. Also visible
are the MOT beam optics mounted to the special CF washers from
Sec. 2.4.2.
39
2.5.2 MOT coil PID
The 4 AWG welding cables that run from the power supply to the coils pass
through two Hall sensors (F. W. Bell, CLSM-1000), which sense the current running
through the cables. One Hall sensor acts as a monitor for diagnostic purposes, while
the other is sent to a PID servo that stabilizes the current. Four high-current
MOSFETs (E250NS10) for switching fields on and off are arranged in parallel to
distribute the current in order to prolong the life of the MOSFETs. These are heat
sunk to a thermal exchange plate with copper pipe running through it for water
cooling. The thermal exchange plate is then mounted to a large (2” in. fins) heat
sink, and a fan constantly blows air over the circuit. With all of these precautions,
the devices stay below 50◦ C when running up to 100 A.
2.5.3 Shim coils
The shim coils consists of 100 turns each of 22 AWG wire around circular
aluminum forms. Table 2.2 lists the dimensions, resistances, and inductances of the
three pairs of coils in orthogonal directions. X refers to the axis perpendicular to
the MOT coils, Y is the axis along the MOT coils, and Z is the vertical axis relative
to the optical table (see Fig. 2.17). One experimental shortcoming is the use of
continuous aluminum forms to hold the coils, as eddy currents occur in them when
changing fields quickly. This can be ameliorated by cutting a gap into the aluminum
so there is no continuous path for current to flow, but we have not found this to be
necessary given the typical timescales and currents in our experiment.
40
Table 2.2: Dimensions, resistances, and inductances of three shim coils used to can-
cel spurious fields in our system.
Direction Inner radius Separation Resistance Inductance
(in) (in) each coil (Ω) each coil (mH)
X 8 8 11/16 6.9, 7.0 5.10, 4.08
Y 4 5 13/16 3.4, 3.4 2.65, 2.61
Z 9 11 1/2 7.6, 7.7 5.68, 5.85
2.5.4 Shim coil driver
A high-voltage, high-current op-amp (Texas Instruments OPA549T) drives
each of the shim coils. Because they are all drawing current from the same 6-A
power supply, each op-amp is limited to 2 A with a clamping resistor. We set the
current with analog outputs from our DAQ cards (see Sec. 2.8.1), with the output
current varying linearly with the voltage set point.
2.6 Optics
This section briefly discusses the two imaging systems that we use to diagnose
and align our atom cloud, as well as the optics used to generate our trapping beams.
2.6.1 Imaging
Two cameras (Point Grey, Flea3 FW-03S1M-C, 5.6 µm pixels, 648x488 array)
positioned in roughly orthogonal directions allow us to measure atom cloud tem-
perature, as well as the relative position between nanofiber and cloud. These Flea3
model cameras are compact and triggerable, allowing them to be synchronized to
41
our experimental timing sequences. The camera mounted in the horizontal direction
has a magnification of 0.4, verified by tracking the position of a cloud falling under
the influence of gravity and by imaging well-known lines on the 1951 USAF test pat-
tern. This demagnification affords us more expansion time in TOF measurements
before the cloud size exceeds that of the CCD.
The vertical imaging system uses a standard f − 2f − f relay configuration,
with pairs of matched 100-mm achromat lenses (Thorlabs MAP10100100-B) for each
leg of the relay, as shown in Fig. 2.18. An adjustable iris halfway between the two
sets of matched pairs can filter stray light from the chamber. There is a lens-tube-
compatible tray housing a 780-nm bandpass filter that can be easily removed if
other wavelengths need to be imaged. Because the filter is not placed between the
matched pairs, its presence or absence shifts the imaging system focal length, which
we can adjust using a micrometer stage to which the whole setup is attached. We
also added a 70:30 (R:T) beamsplitter to send light in two orthogonal directions,
one for imaging and the other for polarization-dependent measurements. The linear
polarizer and photodetector in transmission allow us to determine polarization of
light on the nanofiber waist by detecting Rayleigh scattering (discussed further in
Chap. 3). This detection system is also easily removable, and we have interchanged
it with a multimode fiber coupler to send photons to a single photon counting module
(SPCM) for free-space atomic lifetime measurements (see Chap. 5).
42
matched 
achromat pair
iris
Flea3 CCD
camera
lens tube
removable filter
PD
rotatable
polarizer
MOT
cloud
f 2f 70(R):30(T) BS
Figure 2.18: Schematic of MOT vertical imaging system (rotated 90 degrees for
clarity). The collected light is split in a 70:30 ratio in reflection (R) and
transmission (T) by a beamsplitter (Thorlabs BS023) to divide light
between an imaging CCD and a polarization-sensitive photodetector.
The filter (dashed lines) is easily removable.
2.6.2 MOT beams
The MOT and repump beams are coupled into a 2-to-3 PM fiber splitter
(Evanescent Optics), with even power splitting of the MOT beam and a ratio of
4:1:1 for the repump beam. In this way, the cooling and repump beams are spatially
overlapped when sent into the chamber. A 75-mm focal length achromat lens (Thor-
labs AC254-075-B-ML) directly collimates the output from PM fiber, achieving a
waist diameter of approximately 13.5 mm. At the initial peak of the MOT cooling
laser’s performance, we could send up to 40 mW per beam, but after a few years of
use we typically have 19 mW (I = 13.3 mW · cm−2) per beam, which is sufficient for
cooling atoms.
2.7 Photon counting
Due to the small mode area of the evanescent field outside of the nanofiber
waist, the saturation power of atoms trapped on the nanofiber is very small, on the
43
order of tens of picowatts or less. We use SPCMs (Laser Components COUNT-250C)
operating in Geiger mode to detect these low light levels. We have three modules
on the optical table, each with dark counts lower than the specified 250 counts·s−1.
The photon detection efficiency at 780 nm is & 60%, and the chips have pre-aligned
gradient-index-of-refraction (GRIN) lenses to efficiently couple multimode fibers.
Black-clad fibers prevent stray room light from coupling into the fibers and reaching
the detectors.
The SPCMs output a transistor-transistor logic (TTL) pulse for each inci-
dent photon (afterpulsing probability is less than 0.2%), making photon counting
easy. We use a field-programmable gate array (FPGA) to operate as a multichannel
scaler (MCS) and another FPGA to time-tag and correlate clicks. The MCS FPGA
(National Instruments 779363-01) interfaces with LabView software originally de-
veloped by Michael Tandecki of the FrPNC collaboration and later expanded and
modified in our lab. We can set bin widths as narrow as 12.5 ns and as wide as
3.2 ms, offering wide functionality. This module is primarily used for nanofiber ab-
sorption measurements (see Chap. 3), as we can see shot-to-shot experimental runs
to diagnose the quality of our nanofiber trap before accumulating histograms for
analysis.
The second FPGA (KNJN Xylo-EM) uses firmware, C++ code, and LabView
VIs written by Joffrey Peters and Sergey Polyakov at NIST Gaithersburg3. An
internal 48-MHz clock sets a 21.83 ns window for performing photon coincidence
3See http://www.nist.gov/pml/div684/grp03/multicoincidence.cfm for documentation and all
necessary downloads.
44
or time-tagging measurements. We used this setup to record the data presented in
Chap. 4.
Finally, we sometimes use a digital oscilloscope (Tektronix DPO7054) for pho-
ton counting measurements requiring fine time resolution. Working in its fast ac-
quisition mode, we can histogram clicks with bin sizes as low as 100 ps. Typically
we used this oscilloscope with a bin size of 1 ns, as with the histogram in Ch. 5.
2.8 Electronics and control
Automated experimental measurement sequences are controlled via a program
called CycleX that was originally written in LabView by JQI Fellows Trey Porto
and Ian Spielman, and continually updated by students. It is a highly configurable
interface that can set digital and analog line levels, as well as set ramps based on
common functions (e.g. linear, exponential, Blackman, etc.).
2.8.1 Timing and DAQ cards
A PulseBlaster (SpinCore PB24-100-32k) DAQ card with 24 digital lines de-
termines the clock of the experiment. The internal 100 MHz clock sets a minimum
timing resolution of 10 ns. Each additional DAQ card in the control system has
a PulseBlaster channel fed into a PFI input to sync their clocks with that of the
PulseBlaster. Additional lines from the PulseBlaster are used as triggers within an
experimental sequence, but only for devices that do not need to hold TTL HIGH
when a cycle finishes, as a firmware upgrade to the PulseBlaster sets all channels to
45
TTL LOW at the end of a cycle.
Three DAQ cards (National Instruments PCIe-6353, PCI-6733 (x2)) provide
the other digital triggering lines and analog levels. They offer a total of twenty 16-
bit analog output channels for setting, e.g., coil currents and AOM RF powers. An
external box of analog buffers (Texas Instruments BUF634P) allows these channels
to drive 50 Ω loads. A small fraction of the available digital lines are broken out
from the DAQ cards so that some devices can receive logical HIGH even when the
cycle stops.
RF signals up to 171 MHz originate from three, four-channel direct digital
synthesis (DDS) boxes (Novatech 409B-AC) to drive our AOMs. Signals within
each Novatech are phase-synchronized, which is important for the AOMs that set
the frequency for the 1064-nm beams of the nanofiber trap. Frequencies on two of
the channels in each box can be changed in steps as small as 100 µs; faster switching
will require a voltage-controlled oscillator.
2.9 Conclusions: improvements and laboratory relocation
Eventually the laboratory will relocate to the basement of the new Physical
Sciences Complex. This new space will offer improvements in humidity and temper-
ature control that should eliminate the frequent realignments that we must carry
out in the current lab; even over the course of a few hours we can detect marked
polarization drift in our nanofiber trap. Dismantling and reassembling things will
also provide an opportunity to upgrade some parts of the apparatus. The highest
46
priority would be a new fiber with an optimized waist diameter for 87Rb (see the
end of Chap. 5) and a new UHV mount that will eliminate vibrations that affect
trap loading (see Appendix B).
47
Chapter 3: Optical nanofiber trap
3.1 Introduction
The small mode volume of evanescent field atom traps engenders strong atom-
light interactions without the need for a cavity [102]. The trapping of 133Cs with
an optical nanofiber (ONF) [22, 24] - and a state-insensitive variant [23] - mark
an important experimental realization of these systems. Their high optical depth
(OD) allows for efficient dispersive readout [103] and strong nonlinear interactions.
This regime of strong coupling opens the door to the study of long-range inter-
actions and the formation of so-called atomic mirrors [104] or the observation of
self-crystallization [105, 106]. Furthermore, ONFs are a crucial element of our
proposed hybrid quantum system to couple atoms to superconducting circuit el-
ements [50,62,65].
Optical dipole trapping of atoms is a well-developed technology applied to
numerous atomic species. The extension of optical trapping to evanescent fields of
an ONF shares similarities with dipole trapping with free space beams, but has one
fundamental difference – the evanescent field may have a substantial longitudinal
component of the electric field. This can lead to surprisingly large differences in the
absorption of probe light for two different species, even when they are both alkali
48
atoms (e.g. Rb and Cs), due to the effects of the vector light shift.
Our system traps atoms with two lasers, achieving trap depths of a few hun-
dreds of microKelvin. We find that we cannot simply determine atom number by
the absorption of a probe beam by an optically thick medium with a Lorentzian line
shape. Distinct asymmetries are observed that we trace to the effects of the vector
light shifts associated with the optical trapping fields, and their inherent elliptical
polarization with an appreciable component along the direction of propagation. Al-
though Rb and Cs are nominally atoms with very similar atomic structure, the light
shifts can in fact be quite different, with differential light shifts much larger in Rb
than Cs, leading to a modified absorption profile.
This chapter is organized as follows. Sec. 3.2 outlines the experimental setup.
We present experimental measurements of our trap in Sec. 3.3. We introduce a
theoretical model based on light shifts, finite atom temperature, and population
redistribution in Sec. 3.4, and use this model to understand the inhomogeneous
absorption profile. Sec. 3.5 summarizes our findings and provides an experimental
outlook.
3.2 Setup
The data presented in this chapter are taken using an ONF with a waist
diameter of 500±50 nm and length of 7 mm, with tapering regions of 28 mm in
length. A MOT loaded from a background vapor of 87Rb produces a cloud of ∼ 108
atoms. We overlap the cloud with the ONF waist using magnetic field shim coils
49
and the UHV manipulator and used our two orthogonal imaging systems to ensure
alignment. Atoms fall into the ONF trap (on throughout the experiment) after 90
ms of increased MOT detuning and a 1-ms-duration optical molasses stage. The
sub-Doppler cooling during this loading stage yields MOT temperatures of ∼ 15µK,
as determined by time-of-flight (TOF) measurements.
An ONF trap requires light that is tuned red of the resonance frequency (with
respect to the 87Rb D2 line) to provide an attractive potential and light tuned blue of
resonance to prevent atoms from striking the ONF surface (see Fig. 3.1). A 750-nm
wavelength laser provides the (blue) repulsive force, and a 1064-nm wavelength beam
in a standing wave configuration (to produce longitudinal confinement) provides the
(red) attractive potential. A potential minimum of a few hundred µK in depth is
formed ∼200 nm from the fiber surface, as calculated with a simple two level atom
and only scalar shifts (see Fig. 3.1 (e)). The 750-nm beam is intensity-stabilized.
We also use a near-resonant probe beam with a wavelength of 780.24 nm to detect
trapped atoms. We need to intensity lock the probe beam using a sample-and-hold
system to maintain stable powers when pulsed on for short (10µs to 10 ms) times.
To verify the polarization of each beam on the nanofiber waist, we take
polarization-sensitive measurements of Rayleigh scattering from the waist using the
system shown in Fig. 2.18 [23, 107]. Figure 3.2 displays one instance of this mea-
surement for a 1064-nm beam, utilizing the CCD camera instead of the PD. We
rotate a HWP before the input of the nanofiber to vary the input polarization of
the light, and we then image the Rayleigh scattering as this polarization changes.
The contrast in the oscillations of the Rayleigh scattering gives the degree to which
50
a)
b)
c)
e)
d)
0 100 200 300 400 500 600 700- 1.0
- 0.5
0.0
0.5
1.0
atom- fiber distance (nm)
po
ten
tia
l(m
K)
Figure 3.1: (a) Schematic of ONF with counter-propagating 1064-nm beams and an
orthogonally-polarized 750-nm beam. (b) Illustration of potential at the
ONF waist with lattice formed by 1064-nm beams. (c) Intensity plot of
1 mW of linearly-polarized, 1064-nm light in an ONF with diameter 500
nm. The color scale indicates increasing intensity from blue to red. (d)
Intensity profile of vertically-polarized 750-nm light through the same
ONF. (e) Total trapping potential (black) for a 500-nm diameter ONF
with contributions from 3 mW in each 1064-nm beam (red dashed), 6.5
mW of 750-nm power (blue dashed), and van der Waals (green dashed).
The potentials are calculated from only the scalar shifts. (figure adapted
from Ref. [80])
51
0 40 80 120 160
0
1
2
3
4
5
6
7
in
te
g
r
a
te
d
 C
C
D
 s
ig
n
a
l 
(
a
r
b
. 
u
n
it
s
)
 
 
angle (º)
1
2
3
4
5
6
7
8
9
10
11
12
13
Figure 3.2: An example of the characterization of Rayleigh scattering to determine
polarization of 1064-nm light on the nanofiber waist. The integrated
CCD signal is plotted as a function of input HWP angle for 13 regions
spanning about 3.6 mm of the nanofiber waist. The two pictures show
the CCD images for the points outlined by the dashed boxes. The
integration is performed along 13 vertical cuts of the images, each cut
being 50 pixels (280 µm) wide.
the light is linearly polarized (which we optimize with other bulk optics). Using the
CCD camera and integrating over 280-µm regions allowed us to confirm that the
polarization of light on the waist of the nanofiber does not rotate over a length of
3.6 mm. We then can confidently use just a PD for later polarization adjustments,
as this will integrate the signal over that entire 3.6 mm length.
We measure atomic absorption with a weak, near-resonance beam (780 nm)
coupled through the ONF, counting transmitted photons with SPCMs. Because
light levels near 10 pW saturate the APDs, we have to be careful to filter stray light
and maintain low probe power. Three narrow-line volume Bragg gratings (VBG,
52
VBG
Verdi
BP filter
long-pass filter
SPCM
Nd:YAG laser
transmit
ted 
signal
AOM
reflected 
signal
780 nm 
probe
Ti:Sapph laser
750 nm 
beam
1064 nm 
beam
dichroic
mirror
PBS
mirror
beam dump
in UHV
Figure 3.3: Experimental schematic. Beams from the Nd:YAG laser (1064 nm)
and the Ti:Sapph laser (750 nm) are coupled onto the nanofiber us-
ing dichroic mirrors. Thru-fiber absorption signals are measured with
SPCMs. Volume Bragg gratings (VBGs) offer narrow-line filtering of
spurious background the Ti:Sapph laser as well as fluorescence from im-
purities in the fiber glass. See text for details.
OptiGrate BP-785, 0.01 pm bandwidth at 785 nm) filter amplified spontaneous emis-
sion from the Ti:Sapphire laser near 780 nm (see Fig. 3.3). A fourth VBG at the
output of the nanofiber serves as a mirror to direct signal to the APDs and as an-
other filter to block background induced by the blue trapping beam. This light is
due to fluorescence from impurities in the glass, and it is the main source of back-
ground in the experiment. Two more bandpass filters further reduce background
counts, and finally long-pass color filters (Thorlabs, FGL645) directly in front of the
APD fiber couplers reduce short-wavelength background from stray light. A series
of differing optical depth neutral density filters before and after the nanofiber allow
us to vary the probe intensity while keeping light levels within the dynamic range
of the APDs. TTL pulses from the APDs are counted with a FPGA and processed
to extract absorption signals and full photon counting statistics.
53
3.3 Experiment
The absorption profiles are measured via an in-fiber analog of standard ab-
sorption spectroscopy. We use two probe pulses; the first pulse measures the atomic
absorption signal (Pat), and the second pulse is a reference signal with no trapped
atoms (P0). In between the two 780-nm probe pulses, the 1064-nm trapping beam is
turned off and a slightly blue-detuned laser from the MOT beam paths kicks away
the trapped atoms. Based on the ratio of these probe signals, we determine the
measured transmission,
T =
Pat − Pbg
P0 − Pbg
, (3.1)
where Pbg is the background APD signal with no probe light, with contributions
from detector dark counts and fiber-induced fluorescence. For a single Lorentzian
lineshape with width Γ, one can easily estimate the optical depth OD by fitting the
data to
T (ω) = exp
[
−OD
1
1 + 4 (ω − ω0)
2 /Γ2
]
, (3.2)
where ω0 is the angular frequency of a resonant photon. The total number N of
trapped atoms is then given by N = OD/OD1, where OD1 is the single-atom optical
depth. We calculate OD1 to be ∼ 2.8% by comparing the atomic cross section to
the optical nanofiber mode area.
Figure 3.4 shows a transmission spectrum, averaged over 50 experimental runs,
where the probe detuning is given relative to the bare atomic resonance. Our mea-
sured transmission profile displays a markedly asymmetric lineshape. Because of
54
this asymmetry it is not trivial to estimate the number of trapped atoms. Note
that the maximum absorption sets a lower limit on the OD and number of trapped
atoms, i.e. ODlow = −Ln[T ]. Based on our lowest transmission of 96.8 % (OD =
3.44) at a probe-detuning of 10 MHz, the absolute lower bound of trapped atom
number is N = 123 trapped atoms. Any broadening mechanisms would reduce the
maximum absorption for a given number of atoms. We will discuss this more in Sec.
3.4, where we develop a method to estimate the number of trapped atoms based for
an asymmetric absorption profile.
We perform the photon-counting equivalent of TOF imaging in this 1-dimensional
geometry and observe trapping lifetimes (without any additional cooling) of approx-
imately 23 ms (see Fig. 3.5). The lifetimes are shorter than expected based on
background gas collisions, atom temperature, trap depth, and understood scatter-
ing rates. Ref. [108] observes fiber torsional modes of several hundreds of kilohertz,
close to the trap frequencies of an ONF trap. They posit that this might be a source
of parametric heating and reduced trapping times. While we have not studied in
detail these modes in our system, we do observe that our ONF can exhibit large-
amplitude transverse vibrations near 550 Hz and the manipulator mount oscillates
at a frequency of 28 Hz (see App. B). These modes are too low in frequency to
cause parametric heating as the motion is adiabatic in terms of the optical trapping
potential. The acceleration of the fiber, on the other hand, may be high enough at
times to affect loading, as the macroscopic motion of the trap is no longer adiabatic
relative to the mean atomic motion. This lifetime appears to be typical of ONF
traps [22,23], which are generally shorter-lived than standard optical dipole traps.
55
−10 0 10 20 30 40
0
0.2
0.4
0.6
0.8
1
probe detuning (MHz)
tr
a
n
s
m
is
s
io
n
 
Figure 3.4: Atomic transmission spectra T as a function of probe detuning δ. The
detuning is defined relative to the bare atomic resonance, so the overall
shift by 10 MHz of the transmission dip is due to Stark shifts from
the trapping beams. Each dot is average from 50 experimental runs,
and the error bars represent 1σ statistical errors in the photon counting
statistics.
56
0 20 40 60 80 1000
0.5
1
1.5
2
2.5
3
time (ms)
op
tic
al
 d
ep
th
Figure 3.5: Optical depth versus time showing measurement of the trap lifetime
based on resonant absorption probing. The extracted lifetime is 23 ms
from a fit to exponential decay (red curve).
57
We also study two different polarization configurations of the red- and blue-
detuned trapping beams, one with parallel polarizations and one with cross po-
larizations. The parallel case requires lower intensities of blue-detuned light, but it
generates larger vector light shifts and produces shallower traps. The cross-polarized
configuration results in a deeper trap than the parallel case, and the potential land-
scapes differ. The parallel trap creates a ring-shaped potential around the nanofiber,
but the orthongal case has deep, localized trap sites on either side of the nanofiber.
This chapter presents results with a trapping geometry with an angle (23.5◦) between
the polarizations of red- and blue-detuned beams because we measure a maximum
higher optical depth with this configuration. We do not have a reason that this
particular polarization angle should produce the best trapping.
3.4 Fitting asymmetric absorption curves
To calculate the inhomogeneous broadening of the absorption line, we need
to include the differential light shifts for atoms trapped in the optical fiber poten-
tial. This requires appropriate weighting over the polarization of the modes, the
magnetic-sublevel populations distribution of the atoms, and the position distri-
bution of the atoms within the trap due to thermal motion. A larger fraction of
circular polarization shifts the profile to the blue due and broadens the blue-side of
the profile due to the contributions of the vector light shifts.
A complete description of the atomic absorption has contributions from homo-
geneous (natural linewidth) broadening L0(ω − ω′) and inhomogeneous broadening
58
0 200 400 600
position (nm)
(a) (b)
0 200 400 600
−0.4
−0.2
0
0.2
tr
a
p
 d
e
p
th
 (
m
K
)
0
0.5
1
position (nm)
tr
a
p
 d
e
p
th
 (
m
K
)
5S
1/2
5P
3/2
5S
1/2
5P
3/2
1.5mK
0.0
-0.5
-0.4
-0.3
-0.2
-0.1
0.0mK
0
0.5
1
−0.4
−0.2
0
0.2
Light
Shift
Light
Shift
F=3 F=3
F=2 F=2
Figure 3.6: Calculated light shifts for Zeeman sublevels with 1064-nm (3.35 mW×2,
standing wave) and 750-nm (7.4 mW) trapping beams. (a) Potentials
with linearly polarized trapping beams (top) 5P3/2, F=3 (radial-axis),
where the lines with the color of red, red-dashed, purple, black, cyan,
blue-dashed, and blue correspond to mf= +3, +2, +1, 0, -1, -2, -3, re-
spectively. (bottom) 5S1/2, F=2 states (radial-axis); Zeeman sub-levels
are degenerate because there are no vector or tensor light shifts. (b) Po-
tential shifts with circularly polarized trapping beams (top) 5P3/2, F=3
states (radial-axis) where the lines with the color of red, red-dashed,
purple, black, cyan, blue-dashed, and blue correspond to mf=+3, +2,
+1, 0, -1, -2, -3, respectively. (bottom) 5S1/2, F=2 states (radial-axis),
where the lines with the color of red, purple, black, cyan, and blue cor-
respond to mf=+2, +1, 0, -1, -2, respectively.
59
n(ω′), generally resulting in the symmetric Voigt profile [109] as follows:
I(ω) = I0
∫
n(ω′)L0(ω − ω
′)dω′, (3.3)
where I(ω) is the convolution of Lorentzian and Gaussian profiles. For the optical transi-
tion of 87Rb atoms, the inhomogeneous broadening results from the atomic temperature,
Zeeman-sublevel-dependent population distribution, and the light shifts from 5S1/2 and
5P3/2 to other upper transitions. This requires considering scalar, vector, and tensor light
shifts, which can be large due to a non-negligible axial-direction electric field component
in the fundamental mode HE11.
For a ground state |n, F,mf 〉 and an excited state |n′, F ′,m′f 〉 represented by i and
j, the inhomogeneous term nij(ω) can be defined for trapped atoms having a temperature
T as follows:
nij(ω) =
∫
Veff
1
Z
exp
(
−
Uij(~r)
kBT
)
δ(ω − ωij(~r))dV , (3.4)
where Z =
∫
Veff
exp (−Uij(~r)/(kBT )) dV ; Uij(~r) is the trapping potential of hyperfine
ground states (5S1/2); and ωij(~r) is the light-shifted optical transition frequency of 5S1/2
to 5P3/2 (n to n
′). Uij(~r) and ωij(~r), dependent on powers and polarizations of the two
trapping beams, have spatial dependence and need to be integrated over the effective
volume of a trap site. The atoms with a temperature T higher than a local trap potential
|Uij(~r)|/kB are truncated in the calculation.
Here, we define a homogeneous profile including light shift broadening as follows:
L(ω − ω′) =
1
1 + (ω − ω′)2/(Γ/2 + ∆Γ(ω′)/2)2
, (3.5)
60
where ∆Γ(ω′)/2 = ∆ω′ is the broadened width of an optical transition ω′; the standard
deviations of state-dependent light-shifted optical transitions at each location ~r are calcu-
lated for a frequency ω′ and averaged over all Veff .
The transmission T (ω) (Sec. 3.3) then can be written as
T (ω) = exp[−OD
∑
i,j
|d˜ij,q|
2fi
∫
nij(ω
′)L(ω − ω′)dω′], (3.6)
≈ exp[−N ·OD1
∫
nij(ω
′)L(ω − ω′)dω′], (3.7)
where nij(ω′) and L(ω − ω′) are defined in Eqs. 3.4 and 3.5, fi is determined by opti-
cal Bloch equations during optical pumping from the probe beam, |d˜ij,q|2 is the relative
strength of the atomic dipole moment related to the polarization state of the probe and
the population of Zeeman sub-levels, and the optical depth per atom is OD1 = σ0/Aeff ,
where Aeff is the effective mode area, and σ0 is the scattering cross-section. This can be
regarded as a constant for a given i, j, q, and assumes no light shifts from the low intensity
probe.
Given the uncertainties in the exact polarization profile of the optical modes where
the atoms are trapped, the m-state distribution of the atoms, and the degree to which
a truncated Boltzmann distribution is a correct assumption, we only use the asymmetric
profiles to qualitatively estimate the number of trapped atoms (see Fig. 3.7). We calculate
an atom number of N ' 300 for T = 55µK using the left and right tails of the absorption
profile; this corresponds to OD = 8.4 with our calculated OD1 = 0.028. Needless to say
the observed lineshape is not well-fit by the theory, so the qualitative estimate is rough.
More recently we have achieved better orthogonality of the polarization of the trap-
ping beams by adding quarter-wave plates to the existing half-wave plates. This allows us
61
  
−10 0 10 20 30 40
probe detuning (MHz)
0
0.2
0.4
0.6
0.8
1
tr
a
n
s
m
is
s
io
n
data
(cir, lin) = (1.0, 0.0)
(cir, lin) = (0.0, 1.0)
(cir, lin) = (0.4, 0.6)
(cir, lin) = (0.2, 0.8)
(cir, lin) = (0.6, 0.4)
Figure 3.7: Simulated transmission T vs. probe detuning δ at different polarizations
(red-, magenta-, black-, cyan-, blue-line) and measured asymmetric ab-
sorption data (black dots). We use 3.35 mW of power in each leg of the
red-detuned standing wave and a single 7.4-mW blue-detuned beam.
62
to correct for ellipticity in the transverse polarization of the beams at the nanofiber waist.
Fig. 3.8 displays transmission data for that configuration. The dashed line is a fit to the
Lorentzian in Eq. 3.2 but with Γ replaced by a sigmoidally-varying linewidth to model the
observed asymmetry [110]:
Γ(ω) =
2Γ0
1 + exp[a(ω − ω0)]
, (3.8)
where Γ0 is the FWHM of the symmetric lineshape, and a is the asymmetry parameter.
This model allows for the extraction of an effective optical depth from spectra with uneven
tails but is not motivated by a physical model as in the above simulations [110]. For
the fit we find
(
χ2
)
red = 1.35, suggesting that this model captures the asymmetry well.
Previously we were unable to use such simple models to obtain good fits of our asymmetric
data. The optical depth of 3.09 corresponds to about 110 atoms for this dataset.
3.5 Conclusions
This chapter presented results that confirm optical trapping of 87Rb atoms around
an optical nanofiber. We can confine ∼ 300 atoms in the evanescent mode of the ONF with
typical trap lifetimes of around 23 ms. In the future, we want to add additional cooling
protocols to prolong the trap lifetime. We also want to sensitively probe fictitious and real,
external magnetic fields using Faraday spectroscopy with trapped atoms, and we discuss
this idea in Sec. 7.1. Another effort will be to remove the axial confinement from the red-
detuned standing wave and drive the atoms with an external laser beam. The long-range
atom-atom interactions mediated by the guided mode will lead to interesting collective
effects such as self-organization [105,106]. We elaborate on this further in Sec. 7.4.
63
● ●
●
●
●
● ● ● ●
●
●
● ●
● ●
-10 0 10 20 30 400.0
0.2
0.4
0.6
0.8
1.0
detuning (MHz)
tran
smi
ssio
n
Figure 3.8: Transmission T vs. detuning δ for atoms trapped in orthogonally-
polarized configuration. Better polarization alignment yields a more
symmetric lineshape. The dashed line is a fit to Eqs. 3.2 and 3.8, with
asymmetry parameter a = −0.031 and (χ2)red = 1.35. The extracted
OD of 3.09 gives an approximate atom number of 110.
64
Chapter 4: Photon correlation measurements
4.1 Introduction
The intensity autocorrelation function, g(2)(τ), measures correlations in the fluctu-
ations of light intensity, e.g. the photon statistics [10], and can reveal both classical and
quantum aspects of the light and its sources. Here we demonstrate a method for mea-
suring the temperature of atoms by using the correlations of atomic fluorescence emitted
into the guided mode of a nanofiber. When the emitters are not stationary, the intensity
autocorrelation function is sensitive to their dynamics as well as the geometry of the mode
into which they emit [111,112]. Systems such as atomic beams [113,114], single atoms in
a MOT [115,116], and a single trapped ion [117] were used to measure these transit-time
effects. While bunched and antibunched photon statistics have been observed in the light
emitted into the ONF guided mode [118–121], the correlations related to classical atomic
dynamics near the ONF have not been previously reported. Here we measure the transit-
time envelope of the correlations for different atomic temperatures and various potentials.
The dependence of this timescale on temperature allows for a simple model to extract the
MOT temperature directly from the correlations.
This chapter is organized as follows. In Sec. 4.2 we provide a general overview
of intensity autocorrelations. Section 4.3 outlines the nanofiber mode structure, optical
dipole potential, light shifts, and coupling strength of the system. In Sec. 4.4 we briefly
65
discuss the theoretical considerations for calculating and simulating correlations. Finally
in Sec. 4.5 we present the experimental results and compare them to simulations.
4.2 Intensity autocorrelations
The intensity autocorrelation function
g(2)(τ) =
〈I(t) I(t+ τ)〉
〈I(t)〉2
, (4.1)
measures the conditional probability of measuring a photon at a time delay of τ from
recording the first photon at time t. Here 〈·〉 denotes time average, and I(t) is the intensity
of light at time t. At its core, g(2)(τ) characterizes the fluctuations in the intensity I(t). To
see this, we write the classical intensity I(t) in terms of its average value and fluctuations
about the mean, I(t) = I¯+δI(t). Consider g(2)(0), the autocorrelation at zero time delay:
g(2)(0) =
〈
I2(t)
〉
〈I(t)〉2
(4.2)
=
〈
I¯2 + 2I¯ δI(t) + δI2(t)
〉
I¯2
(4.3)
= 1 +
〈
δI2(t)
〉
I¯2
, (4.4)
where we have used the definition
〈
I¯
〉
= I¯ and the fact that the expectation value of
the fluctuations is zero (〈δI(t)〉 = 0). Equation 4.4 shows that g(2)(0) − 1 measures the
variance of the fluctuations.
The function contains contributions from different sources of fluctuations including
single-atom field-field correlations, single-atom intensity-intensity correlations, different-
atom field-field correlations, etc. Neglecting correlations between the fields of different
66
atoms, we can write g(2) as [111]
g(2)(τ) = 1 +
∣
∣
∣g
(1)
A (τ)
∣
∣
∣
2
+
1
N¯
g(2)A (τ) , (4.5)
where N¯ is the average atom number at a given time, and g(2)A (τ) and g
(1)
A (τ) are the
single-atom intensity-intensity and field-field correlations, respectively. For small atom
number N¯ , we can observe the “antibunching term” g(2)A (τ). Also note for large N¯ that
the correlation function reduces to the power spectral density of the light, |g(1)A (τ)|
2.
It is worth briefly discussing the term “anitbunching.” Sub-Poissonian statistics
gives g(2)(0) < 1 and antibunching gives g(2)(0) < g(2)(τ). These two types of fluctu-
ations readily go hand in hand, so there is often a conflation between the terms in the
literature [122, 123]. In the absence of atom-number fluctuations, antibunched photon
statistics will in general also be sub-Poissonian [124]. With atom-number fluctuations,
however, g(2)(0) can be greater than one, but one can still observe increasing coincidence
rates with increasing delay time, and this constitutes a violation of a Cauchy-Schwarz
inequality for classical fields [124]. This is precisely the situation encountered in this
chapter, where a thermal source of atoms around the fiber generates antibunching on top
of super-Poisonnian (g(2)(0) > 1) statistics.
4.3 The system
The experiment relies on two main parts: a source of cold atoms and an ONF. A
MOT provides a constant source of slowly moving atoms whose fluorescent light can couple
into the guided mode of the optical nanofiber. The nanofiber serves two purposes, as it
collects the light from the atoms and also modifies the local potential landscape through
67
which the atoms move, with typical velocities of 10 cm · s−1. Sending far-off-resonant, red-
detuned light with a wavelength of 1064 nm through the nanofiber adds a further confining
potential that we can controllably vary to systematically study atomic dynamics near the
fiber surface.
4.3.1 Nanofiber mode structure
To understand the behavior of this system, consider a single-mode nanofiber, i.e. a
fiber that is pulled to a small enough diameter such that all higher-order modes are cut
off. The mode (HE11) of such an optical nanofiber has an intensity profile given by [20]
|E(r)|2= E(r)2
[
K20 (qr) + uK
2
1 (qr) + wK
2
2 (qr)
]
, (4.6)
where E is the complex field amplitude; Ki is the modified Bessel function of the sec-
ond kind of order i; u and w are constants obtained from Maxwell’s equations; and
q =
√
β2 − k2 describes the radial field decay, β is the field propagation constant in
the nanofiber, k = 2pi/λ is the free-space wavevector, and r is the distance from the center
of the fiber.
4.3.2 Potentials
For a two-level atom, the optical dipole potential produced by a far-off-resonant
evanescent field is given by [125]
Udip(r) =
3pic2Γ
2ω30
|E(r)|2×
∑
i
c2i
∆i
, (4.7)
68
Figure 4.1: Schematic diagram of moving atoms (red streaks) near a nanofiber waist
and green shape illustrating the position-dependent coupling, G(r). This
coupling is proportional to the scattering rate into the guided mode,
γ1D(r), which is represented by the color plot.
where ω0/2pi is the atomic transition frequency (384.23 THz for the D2 line of 87Rb),
Γ/2pi is the natural linewidth of the 5P3/2 state (6.06 MHz), ∆i = ωi−ω0 is the trap laser
detuning from the excited state |ei〉 transition, ci is the line strength between the ground
state and excited state |ei〉, and |E(r)|2 is given by Eq. 4.6. We treat the ground state
as uniformly populated in the F = 2 manifold and consider coupling to both the 5P1/2
and 5P3/2 states when calculating Eq. 4.7. We approximate the nanofiber as an infinite
dielectric plane and use the Lennard-Jones form when calculating the van der Waals
potential [126–128], so that UvdW(d) = C3 × d−3 with the C3 coefficient equal to 4.94 ×
10−49 J ·m−3 for the 5S1/2 level of
87Rb (see App. C). The infinite-plane approximation
is accurate to within 20% for atom-fiber distances less than 200 nm [20].
69
4.3.3 Light shifts
The combined dipole and van der Waals potentials shift the atomic levels, and they
each depend on position. The shifts produce a spatially-varying absorption probability:
pabs (r, P ) =
s
1 + s+ 4
(
dω(r,P )+δ
Γ
)2 , (4.8)
where r is the position of the atom, s = I/Isat is the saturation parameter (Isat = 3.58 mW·
cm−2 for a uniform sublevel population distribution [91]), δ = ωL − ω0 is the detuning
of the driving beam from atomic resonance, and dω (r, P ) is the scalar light shift from a
beam with power P (with a wavelength of 1064 nm in our experiment) and van der Waals
assuming a two-level atom. Note that for the quantitative atomic cloud temperature
measurement, no 1064-nm power is used so that the shift in Eq. 4.8 arises entirely from the
van der Waals interaction. Our simulations of this temperature measurement reflect this
fact. Later we demonstrate experimentally that this timescale decreases as 1064-nm light
is sent through the nanofiber. Although we did make a quantitative comparison between
simulation and experiment in this case, I did apply a simplified level shift treatment to
check that the behavior was qualitatively similar.
4.3.4 Coupling strength
The coupling strength of an atom to the ONF is the fraction of spontaneous emission
that couples into the fiber versus into free space [129,130],
G (r) = γ1D (r) /γ0 . (4.9)
70
Fermi’s golden rule determines the form of γ1D, which follows the spatial variation of
Eq. 4.6 since the scattering rate is intensity-dependent.
Photon detection in the experiment is a joint process of absorbing a photon from the
MOT beams and emitting into the nanofiber mode, and we multiply the coupling strength
by the photon absorption probability in Eq. 4.8.
4.4 Correlations
The resonance fluorescence emitted into the fundamental mode exhibits correlations
due to transit-time effects related to the geometry of that mode. Essentially the atoms
act as beacons signaling their position while passing near the fiber waist. We write down
the full guided-mode structure and then make a series of approximations so that I can
make direct comparisons of the theory to our data. We then discuss an efficient classical
method for simulating correlation functions by modeling random trajectories, taking into
account the potential landscape Eq. 4.7, the position-dependent coupling strength Eq. 4.9,
and the position-dependent light shifts Eq. 4.8 due to dipole and surface potentials.
4.4.1 Transit-time effects
Laser-cooled atoms are not stationary emitters. Accounting for the motion of atoms
amounts to adding a temporal envelope f(τ) to Eq. 4.5 [113],
g(2)(τ) = 1 + |f(τ)g(1)A (τ)|
2 +
1
N¯
f(τ)g(2)A (τ) . (4.10)
The function f(τ) generally depends on the environment and how the emitted light couples
to the detection apparatus - it is the shape of this temporal envelope that will allow us to
71
extract information about the dynamics of atoms moving near an ONF.
4.4.2 Relating correlations to temperature
We can relate the width of the correlation function to the temperature of the atomic
cloud by noting that the temperature determines the velocity distribution of the atoms and
the velocity of the atoms determines the timescale of the interaction with the nanofiber.
The ONF mode described by Eq. 4.6 possesses a characteristic length scale of 1/q. Dividing
this length by the most probable velocity of a Maxwell-Boltzmann distribution of atoms
at a temperature T , vp =
√
2kBT/m, yields a simple relationship between transit time
and temperature:
τ0 =
a
q
√
m
2kBT
, (4.11)
where a is an overall scale factor based on the geometry of the problem. We are not able
to find an analytical form for a from simple physical considerations, but used simulations
to understand the effects produced by different choices for a (see Sec. 4.5.5).
4.4.3 Simulating atomic trajectories
We can approximate the ONF as an infinite plane when calculating the surface po-
tentials, which is a good approximation since the curvature of the ONF becomes important
only when the atom is far away from the surface where the coupling strength is small. We
also ignore momentum diffusion as the recoil velocity (∼ mm · s−1) when an atom scatters
a photon is small compared to typical atomic velocities (∼ cm · s−1), and scattering events
are not frequent enough on microsecond timescales to appreciably affect the speed of the
atom. We neglect velocity-dependent forces, as the Doppler shifts (∼ 100 kHz) are small
72
compared to the light shifts produced by the various potentials (∼ tens to hundreds of
MHz).
Trajectories are inserted into the position-dependent coupling strength in Eq. 4.9
and the position-dependent absorption probability in Eq. 4.8, which are then multiplied
together. This yields a time-dependent detection probability for each trajectory. Time-
correlating a detection probability trajectory with itself produces a signal proportional to
the intensity autocorrelation for a single atom.
4.5 Experiment and results
4.5.1 Apparatus
We load the MOT from the low-velocity tail of a background vapor of 87Rb atoms
produced by a dispenser (see details in [79]). We change the intensity and detuning of the
cooling beams in order to controllably vary the temperature of the atomic cloud between
∼ 200 − 800µK, as measured by time-of-flight expansion through fluorescence imaging.
The atomic cloud temperatures are limited by the particular time-of-flight (TOF) imaging
system in our setup, where atom numbers for colder MOTs were too low to accurately
fit the cloud width. We do not see this as a fundamental limitation to the correlation
measurement technique, but we only present data for temperatures for which we could
provide calibration to a known technique.
The optical nanofiber (ONF) is produced via the flame brushing technique, as ex-
plained in Chap. 2. [75, 83]. We do not have an accurate measure of the transmission of
the fiber used for this experiment, but it supported 40 mW of 760-nm light in UHV with-
out breaking. We glue (EPO-TEK OG116- 31) the fiber to a titanium u-shaped mount
73
for stability, and attach the mount to a UHV-compatible manipulator system (VG Scienta
Transax). The manipulator consists of a motorized stepper motor along one axis and 2D
manual translations stages along the other axes. This manipulator works in conjunction
with three pairs of magnetic shim coils to optimally overlap the nanofiber waist with the
region of highest atomic density in the cloud.
The ONF in our setup has a diameter of 500± 50 nm, uniform to within 1% over a
length of 7 mm, where the uncertainty in diameter likely arises from air currents changing
the location of the fiber in the flame near the end of the pull [75]. This fiber geometry with
the stated uncertainty accepts only one guided mode, described by Eq. 4.6 above, at both
experimentally relevant wavelengths of 780 nm and 1064 nm. Light that couples into this
mode is filtered at the output of the fiber by a volume Bragg grating (VBG, OptiGrate
BP-785), a narrow-line interference filter (Semrock LL01-780-12.5), and a long-pass color
filter (Thorlabs FGL645) before being sent to the two fiber-coupled SPCMs (see Fig. 4.2).
A FPGA stores and time-tags photon output TTL pulses from the SPCMs, which are
then post-processed and correlated. An internal clock of 48 MHz sets the minimal time
resolution to 20.83 ns. The use of two SPCMs circumvents problems near zero time delay
related to detector dead time, typically 50 ns.
Aside from varying the atom temperature, we could also change the local poten-
tial near the nanofiber surface. In particular we could create an additional attractive
potential fusing the red-detuned, 1064-nm beam to accelerate atoms near the fiber sur-
face, modifying their interaction time with the nanofiber guided mode and decreasing the
local atom density near the fused silica surface [131]. We couple a 1064-nm beam from
the Nd:YAG laser in one direction along the nanofiber, providing confinement only in
the radial and azimuthal directions. The polarization of the beam is circular in order to
74
VBG
dichroic mirror
beam
dump
BP filter
long-pass
filter
50/50 BS
MOT beams
SPCM
to correlator
MOT
cloud
1064 nm beam
780 nm 
signal
Figure 4.2: Experimental schematic. A MOT is spatially-overlapped with a
nanofiber, and the MOT beams drive resonance fluorescence that cou-
ples into the guided mode. This signal is filtered by a volume Bragg
grating (VBG), bandpass (BP) filter, and long-pass filter before being
split by a 50/50 beamsplitter (BS) and sent to two SPCMs. TTL pulses
from the SPCMs are time-tagged by an FPGA and correlated in soft-
ware.
create an azimuthally-symmetric potential, and the polarization on the waist is verified
by performing a polarization-sensitive measurement of the Rayleigh scattering of 1064-
nm light on the waist [23, 107]. We vary the power of this beam between 0 − 8 mW, as
measured at the output of the nanofiber. The resulting dipole potential has a shape that
closely resembles a decaying exponential but strictly speaking is a sum of modified Bessel
functions of order 0, 1, and 2 (see Eq. 4.6) [20]. At powers larger than 8 mW, we find
that the atomic density near the fiber decreases due to acceleration such that the thermal
bunching peak at zero time delay is unresolvable.
4.5.2 Data and fitting
For this experiment, the MOT beams are on continuously during data acquisition
and drive spontaneous emission in the atoms. We collect ∼ 2.5 ·107 photon counts for each
experimental run, corresponding to about 45 min of averaging per data point. Time-of-
flight imaging measures the temperature of the atomic cloud before and after a full scan
75
of 1064-nm laser powers. In principle the correlation measurements could be extended to
even colder clouds by averaging for longer times, but we had trouble confirming colder
temperatures via TOF due to low atom number in our MOT.
The data we get is a list of times corresponding to photon detection events. We
use this to find g(2)(τ). We do not do any further binning of the data, so that the timing
resolution of 20.83 ns is set by the internal clock in the FPGA. While this time resolu-
tion obscures details on atomic spontaneous emission timescales (tens of nanoseconds), it
provides good resolution on the timescale of a few microseconds where the atomic trajec-
tories produce signatures in the correlation function. Measurements using an oscilloscope
(Tektronix DPO 7054) with finer time resolution allowed us to observe antibunching for
low atom number.
The Rb dispenser current allows us to vary the number of atoms in the MOT, so
that we can change the average number of atoms interacting with the nanofiber mode.
Fig. 4.3 shows the transition from antibunched (increasing slope after τ = 0, estimated
atom number is ∼ 1.4) to bunched (decreasing slope after τ = 0, estimated atom number
is ∼ 6) correlations as we increase the number of atoms fluorescing into the mode of
the ONF. We can control the MOT to have, on average, only a small number (∼ 1) of
atoms interacting with the nanofiber mode on these timescales, confirming a similar result
observed in Ref. [120].
Figure 4.4 displays an example of g(2) (τ) extracted from data for an atom temper-
ature of 463 µK and 1064-nm power of 0 mW (note the very different timescale than in
Fig. 4.3). The broad bunching feature centered around zero time delay suggests a thermal
signal on top of a coherent background from scattered MOT light that couples into the
nanofiber guided mode. This signal has a characteristic width based on transit-time effects,
76
-50 0 501.15
1.20
1.25
1.30
1.35
1.40
1.45
g(2) (
τ)
τ (ns)
Figure 4.3: Second-order correlation function g(2)(τ) for light scattered into the fiber
as a function of delay time τ . The curves show data for low (blue) and
high (red) Rb dispenser currents.
which is the result of a position-dependent atom-fiber coupling strength combined with
moving atoms. An atom at a particular location will sample the mode with probability
proportional to its intensity at that position, and averaging over many atomic trajectories
will sample the entire mode. In this way, the autocorrelation function contains informa-
tion about the mode in question (the shape of g(2) (τ)) and about the dynamics of the
atoms (the decay time of g(2) (τ)).
The data is clear, but we need to make a series of approximations to the model of the
mode structure before we could compare the observed transit-time broadening to theory.
The factors u and w in Eq. 4.6 are small for a fiber radius of 250 nm and wavelength
of 780.24 nm (0.166 and 0.00875, respectively), so we neglect them and keep only the
first term proportional to K20 . As a further simplifying approximation we also take the
77
g(
2)
(τ
)
re
si
du
al
s 
τ (µs)
2
0
-2
-2 0 2-4 4
Figure 4.4: Second-order correlation function g(2) (τ) as a function of delay time τ
for an atom temperature of 463 µK and for 0 mW of 1064-nm light. The
data (blue dots) are fit (solid red line) to Eq. 4.13, with the residuals
displayed in the lower plot.
78
asymptotic form of Kα [132],
Kα(z) ∼
√
pi
2z
e−z
(
1 +
4α2
8z
+ . . .
)
, (4.12)
where α is the order of the Bessel function, and z is a complex argument satisfying |arg z| <
3pi/2. This yields a field intensity around the nanofiber proportional to exp[−2qr]/2qr.
Defining an effective index of refraction, neff = β/k, we can rewrite the propagation
constant so that the radial decay parameter becomes q = k
√
n2eff − 1, which evaluates to
0.62k for our nanofiber. We then recast the spatial dependence of the intensity into a
temporal function in order to fit the measured correlations as g(2) (τ) = 1 + f(τ)g(2)A (τ)
in Eq. 4.5 [113, 114]. Neglecting the field-field correlations that vanish on the timescales
measured in our experiment, we can write
f(τ) = A
e−2(|τ |/τ0+0.62 k r0)
(|τ |/τ0 + 0.62 k r0)
, (4.13)
where r0 = 250 nm is the fiber radius, and the absolute value reflects the time-symmetric
nature of the autocorrelation function for stationary processes. Here the fitting parameter
A is an overall amplitude determined by the signal-to-background ratio and the average
number of atoms interacting with the nanofiber mode. The parameter τ0 represents a
characteristic correlation time.
The red curve in Fig. 4.4 shows the best fit to g(2)(τ) for an atomic temperature
of approximately 463 µK, with a reduced χ2 of 1.02 for this set of data. We note that
Eq. 4.13 gets statistically better fits than an exponential decay or Gaussian, as measured
by the reduced χ2.
79
4.5.3 Temperature extraction
We extract best fit values for τ0 at different MOT atomic temperatures, with each
temperature also measured by standard TOF imaging. Fig. 4.5 shows a plot of the result-
ing best fit values τ0, where the vertical error bars are the standard errors from the fit,
and the horizontal error bars originate from a systematic uncertainty in the magnification
of the imaging system. We confirm that the horizontal uncertainties are similar (few per-
cent) to the errors obtained directly from the TOF fits. The purple line in Fig. 4.5 is a
fit to Eq. 4.11, and the shaded area represents the 5–95% confidence band. We observe
good agreement between the model and the data, as the fit has a reduced χ2 of 1.65, and
the overall scale parameter is a = 1.44± 0.04. The deviation of this scale factor from 1 is
discussed further in Sec. 4.5.5.
4.5.4 Atomic dynamics
In Sec. 4.5.3 we described measurements that used the ONF as a passive probe, with
the fiber modifying the local environment only over short distances where the surface
potential is strong. By varying the power of 1064 nm light propagating through the
ONF, we can change the potential near the nanofiber and consequently change the atomic
dynamics. Given the longer-range interaction of this attractive optical dipole potential,
the atomic trajectories are affected over longer timescales that we can then observe using
g(2)(τ) correlation measurements.
Figure 4.6 shows the correlation time τ0 for three different MOT temperatures as
a function of 1064 nm laser power. We determine τ0 from fits such as that displayed in
Fig. 4.4. There are some interesting features in this plot. The initial decrease in timescale
80
Figure 4.5: Extracted timescale τ0 vs. temperature T , measured via TOF. The ver-
tical error bars indicate standard error in the fit of Eq. 4.13, and the
horizontal error bars arise from systematic uncertainty in the magnifica-
tion of the imaging system. The purple line is a fit to Eq. 4.11, and the
shaded region is the 5–95% confidence band. The reduced χ2 is 1.65.
81
201 µK
297 µK
463 µK
Figure 4.6: Extracted correlation function width τ0 as a function of P1064 for atom
temperatures of 201 µK (purple dots), 297 µK (blue squares), and 463
µK (red diamonds). The error bars indicate one standard error. Dashed
lines are linear fits to the displayed points to illustrate the increasing
effect of the 1064-nm beam power as atom temperature decreases.
is due to acceleration of the atoms as they experience the increasingly strong confining
potential of the 1064 nm beam. Atomic temperature determines the slope of this decrease,
as slower atoms are more easily “captured” by the potential, and we see that the magnitude
of the slope decreases as the atom temperature increases. Finally we note that τ0 appears
to saturate at about 0.8 µs at high laser power, but for clarity this data is excluded from
Fig. 4.6. We discuss this saturation behavior in the next section.
4.5.5 Simulations
To better understand this data, we simulate this behavior by generating classical
atomic trajectories subject to Newton’s equations of motion [133]. These simulations
include the potentials and light shifts discussed in Sections 4.3.2 and 4.3.3. The classical
82
nature of the simulations is justified because the smallest angular momenta present in the
system are still ∼ 100 times larger than the Planck constant.
The atoms start at a radial distance r = 1500 nm away from the fiber surface. At
this distance, the coupling is weak due to the rapid decay of the mode with length scale
1/q. Symmetry of the problem allows us to restrict trajectories to the x-y plane with
initial velocities pointing in one quadrant. We need to make sure to sample the speeds
from a 3D Maxwell-Boltzmann distribution before projecting onto this plane. Trajectories
evolve for either 50 µs or when the atom strikes the fiber surface, whichever happens first.
The dipole potential used in the simulations is a numerical fit to the full, scalar
solution in order to ease integration. Similarly, the coupling strength in Eq. 4.9 is a fit to
the complete solution for a two-level atom [130]. We also assume that the orientation of
the atomic dipoles relative to the fiber surface is random, so that the coupling strength is
an effective ensemble average. Independent measurements confirm that minimal optical
pumping occurs in our MOT, and this assumption of random orientations is valid.
Once a trajectory is calculated and fed into the detection probability, we discretize
these time-dependent probabilities onto a mesh of 50 ns resolution so that calculating the
correlation function becomes a simple array operation. Experimentally measured values
for atom temperature and 1064 nm laser power are fed into the simulation, which is
averaged over 104 randomly sampled speeds and directions. The resulting correlation
function is fit to Eq. 4.13 in order to extract the decay time τ0, as shown in Fig. 4.7.
We first utilize the simulations to address the deviation from unity of the scale
factor in Eq. 4.11. Fig. 4.8 displays the dependence of the transit time on the angular
spread of the atomic trajectories for a distribution with temperature 90 µK. For an atomic
beam aimed directly at the fiber, we extract a transit time of 1.48 µs, which matches well
83
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
co
rre
lat
ion
 (a
.u.
)
τ (µs)
Figure 4.7: Simulated correlation vs. delay time τ for an atomic temperature of
787 µK. The red line is a fit to the simulated data (blue circles) using
Eq. 4.13.
the calculated time of 1.53 µs using Eq. 4.11 with a = 1. The transit time increases
slowly as we increase the angular distribution of trajectories, until it hits a critical value
of arctan(250/1500) = arctan(1/6), corresponding to the point after which not all paths
intersect with the nanofiber. Beyond this angle, atoms then interact with the fiber over
distances longer than 1/q, and the transit time consequently increases further. There is
a saturation at angles approaching pi/2, as the rapid decay in the atom-fiber coupling
strength ensures that atoms remaining many decay lengths away from the fiber surface
will contribute little to the correlation signal. Fig. 4.8 illustrates that the simulation fully
samples the interaction region with an angular spread of at least pi/6 to get reasonable
results. Moreover, we note that the ratio of the transit time for the fully-sampled sim-
ulation to the effective one-dimensional simulation with no angular spread is 1.7. These
results suggest that our observed scale parameter of a = 1.44 ± 0.04 is due to angular
84
0.0 0.1 0.2 0.3 0.4
1.6
1.8
2.0
2.2
2.4
angle rad
0
s
Figure 4.8: Simulated correlation time τ0 vs. sampling angle range ∆θ for an atomic
temperature of 90 µK. The dashed gray line indicates the critical angle
arctan(1/6) in the simulation at which not all atoms hit the fiber.
spread in the trajectories.
We did simulations for the same temperatures measured in the experiment (see
Fig. 4.5). The red curve in Fig. 4.9 displays a fit of the simulated data to Eq. 4.11,
yielding a scale parameter of a = 1.88. We observe qualitative agreement between the
simple model and full simulation of the dynamics. We also plot for comparative purposes
the fit to experimental data from Fig. 4.5 as a purple line. The discrepancy between data
and simulation may be partially due to a residual magnetic field gradient that persists
during time of flight, slightly compressing the MOT and modifying the TOF. This leads to
a systematically lower measured atom temperature, so the apparent gap between theory
and experiment should be smaller. Uncertainty in the fiber diameter also affects the
characteristic transit time of an atom through the mode, as indicated by the open circles
in Fig. 4.9. We performed the same simulation for an atomic cloud temperature of 240 µK
85
● ● ● ● ●
○
○
200 300 400 500 600 700 800
1.0
1.5
2.0
temperature (μK)
τ 0 (μs)
Figure 4.9: Simulated correlation time τ0 vs. atom temperature T . The red curve
is a fit to the simulations (blue dots) using Eq. 4.11, with the shaded
areas representing 95% confidence bands. The open circles represent a
sensitivity analysis of the fiber diameter, plotting the transit time for
a temperature of 240 µK and diameter of 450 nm (brown) and 550 nm
(green). The open square is a sensitivity analysis of the angular distri-
bution, showing the transit time for the 510 µK case for zero angular
spread. The dashed purple line is the corresponding fit to experimental
data shown in Fig. 4.5.
and fiber diameters of 450 nm (green circle) and 550 nm (brown circle), which represent the
diameters at the stated lower and upper uncertainty bounds. We have observed that fibers
can have systematically larger diameters than intended, possibly due to the flame pushing
the thin fiber at the end of a pull [75]. This suggests a potential source of disagreement
between experiment and simulation. Even so, the 30% difference between the experimental
data and the lower confidence band of the fit to simulated data is comparable to other
temperature measurement methods using optical nanofibers [134].
Simulations also confirm that the decay time decreases for the first few milliwatts
of 1064 nm laser power, after which it saturates. The saturation originates from two
86
effects: the interplay of faster speeds and longer trajectories, and atomic level shifts due
to surface and dipole potentials. While the atoms attain faster speeds due to the increasing
strength of the attractive potential for higher laser power, they experience a larger number
of orbiting trajectories that increase the interaction distance. These classical dynamics
partially offset each other, but do not fully account for the saturation we see in the data.
The light shifts alter the probability to absorb a photon from the MOT beam in such a
way that the interaction region broadens relative to the nanofiber mode structure.
4.6 Conclusions
We have presented a technique to measure the temperature of a laser-cooled atomic
cloud that is applicable to experiments with restrictive environments, such as hybrid quan-
tum systems using superconducting circuits. The method uses intensity autocorrelation
functions to extract dynamics of atoms as they pass through the ONF mode and is eas-
ily extendable to other photonic devices with different optical mode geometries. This
technique allows mapping of mode structures, which could be useful when using the next
family of higher-order modes to trap atoms near an optical nanofiber [90,135–138].
87
Chapter 5: Measurement of the lifetime of the 5P3/2 state of 87Rb
near a nanofiber
Correlation measurements need not be restricted to intensity autocorrelations. We
can instead correlate photon arrivals with a given trigger event to measure the spontaneous
decay of an excited state of atom, a method already employed to great accuracy for
measurements of rubidium [139], cesium [140], and francium [141]. In this chapter, we
apply this technique to measure the lifetime of the 5P3/2 state of
87Rb near an optical
nanofiber.
A dielectric surface modifies the dipole moment of a proximal atom, in turn changing
its rate of spontaneous emission [142, 143]. Moreover, the mode structure of a waveguide
enhances the decay rate of the atom in a manner analogous to the Purcell effect, in which
the rate γ becomes γ(1 + 2C) (where C is the cooperatively discussed in Chap. 1).
The spectral properties of atomic emission near surfaces and waveguides have been
studied extensively theoretically [102, 129, 143–147] and measured in the frequency do-
main [133, 148–152]. By using time-correlated single-photons to determine the emission
rate [28], we avoid broadening issues that can complicate linewidth measurements. Our
setup also allows us to measure both the free-space and fiber-modified rates so that we can
make a direct comparison with the same method, mitigating the influence of systematics.
88
5.1 Theoretical overview
In this section, we review the theory of spontaneous emission for an atom in free
space. This is useful background before we discuss how the presence of a nanofiber can
change this rate.
Fermi’s Golden Rule states that for an atom spontaneously decaying into a contin-
uum of states, the transition rate Wfi between some initial atomic state |i〉 and a final
atomic state |f〉 is
Wfi =
2pi
~
|〈f |Hint |i〉|
2 ρf (~ω0) , (5.1)
where Hint is the interaction Hamiltonian between the electromagnetic field and the atom,
ρf is the density of final states of the photon, and ~ω0 is the transition energy between
states |i〉 and |f〉. Note that we have already integrated over the possible photon energies.
Evaluating Eq. 5.1 in free space for a transition between an initial state with angular
momentum J and a final state with angular momentum J ′ yields
γ0 =
ω30
3piε0~c3
|〈J ||D||J ′〉|2
2J ′ + 1
, (5.2)
where ε0 is the permittivity of free space and 〈J | ||D|| |J ′〉 is the reduced electric-dipole
matrix element, and we have taken into account the two polarizations. For 87Rb, this
yields τ0 = 1/γ0 = 26.24 ns [91].
Fermi’s Golden Rule contains two factors that can be modified by the presence of
the fiber: the interaction Hamiltonian matrix element and the density of states. The
interaction Hamiltonian within the dipole approximation takes the form Hint = −d · E.
The proximity of the dielectric surface modifies the dipole moment d, which we discuss in
89
the next section. A waveguide can alter the decay rate in three ways. First it decreases
the mode volume V of the quantized electric field, |E| =
√
~ω
2ε0V
, leading to a stronger
single-photon electric field strength. Secondly the new dimensionality D of the waveguide
changes the density of states, which scales as ωD−1. And finally the dispersion relation
changes due to the non-unity propagation constant of a mode in the waveguide. These
issues will be addressed in Sec. 5.1.2.
5.1.1 Dipoles near surfaces: calculating γrad
When a radiating dipole is brought near a surface, the reflection and absorption
of the radiated field will modify the decay of the dipole [142, 143, 153]. Consider a clas-
sical dipole in a medium with dielectric constant ε1 placed a distance d away from an
infinite dielectric surface with dielectric constant ε2. Calculating the decay rate of this
atom amounts to finding the reflected electric field at its location. Refs. [143–145] solve
Maxwell’s equations and the appropriate boundary conditions to arrive at the following
rates for a dipole oriented either parallel or perpendicular to the dielectric:
γ‖rad
γ0
= 1 +
3
4
Im
[∫ ∞
0
dκ
κ
µ1
(
R⊥ + µ
2
1R‖
)
e−i 2µ1dˆ
]
(5.3)
γ⊥rad
γ0
= 1−
3
2
Im
[∫ ∞
0
dκ
κ3
µ1
R‖ e
−i 2µ1dˆ
]
, (5.4)
where µ1,2 =
√
ε1,2/ε0 − κ2, the reflection coefficients R‖ and R⊥ are given by
R‖ =
ε1µ2 − ε2µ1
ε1µ2 + ε2µ1
(5.5)
R⊥ =
µ1 − µ2
µ1 + µ2
, (5.6)
90
and dˆ is a normalized distance scaled by the wavelength of the radiation and the index of
refraction of medium 1, n1 =
√
ε1/ε0,
dˆ =
2pi
λ
√
ε1
ε0
d . (5.7)
Figure 5.1 displays the evaluation of Eqs. 5.3 (blue curve) and 5.4 (red curve) for a
87Rb atom near a fused silica surface [127] (see Appendix C for a discussion of the index of
refraction of fused silica). The oscillations in the scattering rates arise from interferences
between the original radiated field and the reflected field, with length scale given by λ/2pi.
This calculation ignores absorption (i.e. non-radiative corrections to γ), which would
cause a the atomic lifetime to go to zero at the fiber surface [143]. This is an important
effect, but because the van der Waals potential accelerates atoms as they approach the
fiber surface, their interaction time in this regime would be very short [127].
5.1.2 A dipole near a waveguide: calculating γ1D
In this section, we briefly outline the derivation of γ1D for a two-level atom near a
waveguide. We follow Ref. [130] and consider only the fundamental HE11 mode. Refs. [102,
129] provide a more complete treatment for multilevel atoms, but the analysis is beyond
the scope of the discussion here.
We proceed by decomposing the electric field E into quantized modes of the nanofiber
as
E =
∑
k
Ekak + h.c. , (5.8)
where k represents the mode index, ak is the single-photon annihilation operator in mode
k, and Ek is the single-photon electric field of mode k. The electric field in mode k has
91
0 200 400 600 800 1000
1.0
1.2
1.4
1.6
1.8
2.0
atom-surface distance (nm)
Figure 5.1: Predicted normalized radiative decay rate γ/γ0 vs. atom-surface dis-
tance d of an atomic dipole with resonant wavelength λ = 780 nm near
a fused silica surface. The two curves represent perpendicular (red) and
parallel (blue) orientations of dipole relative to the silica surface.
92
the form [10,130]
Ek = i
√
~ωk
2ε0L
E˜ ei(βk z+mϕ) , (5.9)
where the quantization box is one-dimensional and has length L; E˜ represents the normal-
ized electric field of the nanofiber mode, carrying both spatial and polarization informa-
tion; βk is the propagation constant of mode k (see Appendix A); and m represents the
angular momentum of the mode (classically, the handedness of the mode polarization).
Carrying out the standard Wigner-Weisskopf treatment of spontaneous emission, we
arrive at the following equation for the spontaneous emission rate into the fundamental
mode
γ1D = γ0 ×
3λ2β′
8pi
|E˜|2 , (5.10)
where β′ = dβ/dk is the derivative of the propagation constant with respect to wavenum-
ber k. Note that the factor 3λ2/2pi represents the classical resonant cross section of a
two-level atom. That the dependence on the propagation constant is given only by β′ is a
consequence of the system being one-dimensional.
We plot Eq. 5.10 in Fig. 5.2, normalized to the free-space value γ0, for a 87Rb atom
near a nanofiber with radius r0 = 250 nm. It reaches a maximal value of 0.18 γ0 at the
fiber surface and decays to zero with radial dependence determined by the mode intensity
|E˜|2 (see Appendix A). For atoms trapped 200 nm away from the fiber, the corresponding
increase in the decay rate due to guided-mode coupling is around 1− 2%.
5.1.3 Atom density near surfaces
The modifications to spontaneous emission discussed in Secs. 5.1.1 and 5.1.2 are
position-dependent relative to the nanofiber surface, and so we must have knowledge of
93
0 100 200 300 400 5000.00
0.05
0.10
0.15
0.20
atom-surface distance (nm)
Figure 5.2: Predicted normalized radiative decay rate γ/γ0 versus atom-surface dis-
tance d of a 87Rb atom into a 500-nm diameter optical nanofiber.
the position distribution of atoms over which to average these effects. Of course, the
nanofiber trap localizes the atoms at a particular separation from the nanofiber surface,
with some dispersion in the position due to quantum mechanical and thermal effects. One
can also consider untapped atoms that are moving in the optical potential using a thermal
distribution. Understanding this limit is useful before trying measurements with trapped
atoms.
Ref. [131] provides a thorough theoretical treatment of the problem, based on
quantum-mechanical scattering of atoms off of the surface potential. We will instead
focus on a classical approach and a thermodynamic derivation of the population distribu-
tion. We discuss why this is valid given our atom temperatures and compare the results
to the quantum results in Ref. [131].
The potential an atom sees arises from a van der Waals interaction between the silica
94
surface of the fiber and the atom, mediated by virtual photon pair exchange between two
dipoles. At larger distances from the fiber, the van der Waals interaction transitions to
a Casimir-Polder interaction, which takes into account retardation effects in the virtual
photon exchange. An approximate way to smoothly connect these two regions is to use a
phenomenological potential given by [131,147]
U(r) = −
C4
r3 (r + C4/C3)
, (5.11)
where r is the radial distance of the atom from the fiber surface, and C3 = 746 Hz · µm3
and C4 = 67 Hz · µm4 are the van der Waals and Casimir-Polder coefficients for 87Rb and
fused silica, respectively (see Appendix C). Note that Eq. 5.11 ignores the short-range,
repulsive wall included in Ref. [131], as we are not considering quantum reflections off of
the silica surface.
The attractive potential given by Eq. 5.11 accelerates atoms as they approach the
fiber. As a result of the increased speed of atoms near the surface, the atomic density of
untrapped atoms decreases near the fiber. To quickly illustrate this fact, consider a flux
of atoms, Φ moving with an average velocity v in the x-direction. Then the density of
atoms is given by Φ/v, so that the density is inversely proportional to their speed.
In order to quantify the effect of the potential on the density, consider the ideal gas
law:
PV = NkBT , (5.12)
where P is the pressure, V is the volume, N is the number of atoms, kB is Boltzmann’s
constant, and T is the temperature. Assuming constant pressure, any change in thermal
kinetic energy kBT must be balanced by an opposite change in the local atomic density
95
ρ = N/V . Denoting the total energy of an atom by E, we can write it as a sum of the
potential and kinetic energy in the system, E = 3kBT/2 + U(r), with U(r) being the
surface potential in Eq. 5.11. Then Eq. 5.12 can be recast as
P =
3
2
ρ (E − U(r)) . (5.13)
We now solve for the density, and normalize it such that it equals one at large distance
– we can safely assume that the MOT cloud density is constant far away from the fiber
surface, as its size (∼mm) is much larger than length scale over which the atom-fiber
coupling is appreciable (< µm). This yields
ρ˜(r) =
1
1− U(r)/E
, (5.14)
where ρ˜(r) is the normalized position density.
Figure 5.3 displays a plot of Eq. 5.14 for an atom temperature of 200 µK. We
see there is a sharp decrease in density near the surface, as expected. Comparing this
treatment to the exact quantum-mechanical results of Ref. [131], we see that it follows
the average of their oscillatory solutions quite well for typical MOT temperatures (tens
to hundreds of microKelvin). The oscillations are so rapid in space because of the short
de Broglie wavelength, λ = 2pi~/
√
2m [E − U(r)] ≈ 17 nm at a distance of 50 nm and
temperature of 200 µK.
5.1.4 Expected decay rate
The measured decay rate will be an average of the position-dependent rates calcu-
lated above, weighted by the population density of the atoms. Writing this average yields
96
0 100 200 300 400 500
0.0
0.2
0.4
0.6
0.8
1.0
atom-surface distance (nm)
nor
ma
lize
dp
osi
tion
den
sity
Figure 5.3: Predicted normalized position density ρ˜ of thermal rubidium atoms (T =
200µK) as a function of atom-surface distance d from a fused silica
surface.
97
γ′
γ0
=
∫
dr (γ1D + γrad) η(r)ρ˜(r)pabs(r)∫
dr γ0 η(r)ρ˜(r)pabs(r)
, (5.15)
where η(r) = γ1D(r)/γtot(r) defines the coupling efficiency into the guided mode, and
pabs(r) is the probability for an atom to absorb a photon from the probe pulse. This
absorption probability depends on the atom-fiber separation, as the van der Waals inter-
action shifts the atomic levels out of resonance with the driving beam (see Sec. 4.3.3).
5.2 Experimental setup
The experiment employs the same 500-nm diameter fiber used for the measurements
presented in Chaps. 3 and 4. A MOT is prepared around the nanofiber waist, as shown
in Fig. 5.4. The MOT and repump beams are extinguished by turning off the RF power
to their respective AOMs (not shown in Fig. 5.4), with the repump kept on for a few
microseconds longer to replenish any atoms that fell into the dark F = 1 state. After
we trigger the repump to turn off, we wait 500 µs for the background light from these
beams to go below 1% of their “on” levels (see Fig. 5.5). The pulsing sequence then
begins. A digital delay generator (Stanford Research Systems DG645) outputs a train
of 50-ns-wide square pulses with a repetition rate of 4 MHz. These pulses are sent to a
fiber-EOM (EOSPACE AZ-2K1-10-PFA-PFA-800-UL) in a Mach-Zehnder configuration
that is locked to the null point by a microcontroller (YY Labs), which feeds back to the
EOM bias port. The electronic pulse amplitude, roughly 2.3 V, is chosen to drive a 180◦
phase shift in the EOM and maximize the optical pulse size. Light for the optical pulses
originates from our cooling laser, and the AOM used for switching also sets the frequency
so that the light is on resonance.
98
SPCM
MOT
cloud
250 ns
nanofiber
x-y translation
fiber EOM
y
x
pulse generator
mirror
500 nm
780-nm pulses
imaging optics
trigger
TTL signal
histogram
Figure 5.4: Schematic for the atomic lifetime measurement. Resonant pulses 50 ns
in length are generated via a fiber EOM and interact with atoms in
the MOT. We detect free-space decay (γ0) by moving the nanofiber out
of the cloud and collecting photons on a SPCM from the side. The
fiber-modified decay (γ′) is measured through the nanofiber.
The probe beam bath is orthogonal to the nanofiber waist in order to minimize
coupling of probe photons into the guided mode and to the detector. We also have a
mirror to retroreflect the beam, which helps to keep the atoms in the region of interest
near the nanofiber for longer periods of time. The train of pulses repeats for 1 − 2 ms
before the cooling and repump beams turn back on for 15 − 25 ms to reload the MOT
before the next measurement sequence (see Fig. 5.5). The SPCMs are gated to take data
only during the pulse sequence, and we maintain low count rates (. 104 counts · s−1) to
avoid photon pileup [154,155].
There are different approaches to histogramming the arrival times of the photon
pulses, here the sync output from the delay pulse generator triggers our oscilloscope (Tek-
tronix DPO7054, see Chap. 2) to start the histogram, which correlates the decay measure-
99
MOT
Repump
Probe
5 µs 500 µs
50 ns 200 ns
1-2 ms 15-25 ms
Figure 5.5: Timing sequence for MOT (blue), repump (green), and probe (red) light
to conduct lifetime measurements. The MOT and repump beams are
turned off within 10 µs of one another, and there is a 500-µs delay before
the probe pulsing begins. 50-ns-wide pulses with a repetition rate of 4
MHz to excite the atoms for 1 − 2 ms before the MOT is loaded again
(15− 25 ms). The relevant timescales are denoted for clarity.
100
ment with a known start time. We typically use a bin size of 5 ns, which balances well our
measurement resolution with the necessary averaging time to accumulate good statistics.
Measuring the free-space decay follows the same general procedure, except the pho-
tons are collected with an imaging system that couples to the multimode fiber for the
SPCM (see Fig. 5.4 as well as Fig. 2.18, where the polarizer and PD are replaced with
fiber-coupling optics). We also use the UHV manipulator to move the nanofiber out of
the MOT and the imaging plane of the optical system so that the fiber will not influence
the atomic decay or scatter probe light into the detection path.
For all measurements, the size and density of the MOT is reduced by lowering the
magnetic field gradient and the current of the atomic dispensers. Lowering the optical
thickness of the sample allows us to minimize radiation trapping effects [156], which can
artificially lengthen the apparent lifetime of the atom due to multiple, successive absorp-
tion and emission events before photon detection. The parameters for normal, nanofiber-
trapping conditions (Chap. 3) yields measured lifetimes of ∼ 32 ns, so we decrease the
optical thickness until we achieve measurements close to the accepted value of 26.23 ns.
5.2.1 Optical pulse generation and improvement
We use fiber EOMs to generate short (∼ns) pulses with low-voltage drivers, which
are a mature technology from the telecommunications industry. The curve in Fig. 5.6 is
a fast photodiode signal of a 50-ns pulse at the output of the EOM. A “foot” at the level
of 10% between 50 and 80 ns is clearly visible. While the structure on the rising edge
is unimportant to us, accurate lifetime measurements of this nature require a clean and
sharp falling edge. Mathematically, the detected output signal f(t) is a convolution of the
101
0 50 100 150 200 2500
1
2
3
4
time (ns)
puls
eam
plitu
de(arb.u
nits)
Figure 5.6: Optical pulse at the output of the fiber-EOM, driven by the delay pulse
generator and measured on a fast photodiode.
excitation and the expected decay:
f(t) = (t)⊗
(
e−γt Θ(t)
)
, (5.16)
where (t) is the excitation applied to the atom, ⊗ denotes convolution, γ is the expected
decay rate, and Θ(t) is the Heaviside function.
The limited input power (< 10 mW) and low efficiency (∼ 10%) of this model of
EOM do not allow for concatenating devices in series to suppress low-amplitude artifacts
and still have usable power at the output. We successfully use a wideband analog pulse
multiplier (Analog Devices ADL-5391) to suppress undesired structure in the electronic
pulses from the delay pule generator. The quality of the optical pulses, however, is limited
by the EOM itself, so the improvement in electrical pulses did not result in a proportional
102
improvement optically. We are currently exploring optical alternatives to the fiber-EOM,
such a double-passed AOM in bowtie path to produce zero frequency shift, that can
generate clean, fast pulses from a CW source.
5.3 Data and results
Our analysis relies on exponential fitting of the decay, which measures a statistically
significant difference between the free-space and fiber-modified cases, but does not capture
all of the physics due to imperfect pulse shapes. We collect data until the difference
between signal and background is more than 104 counts per 5-ns bin.
Analyzing the background for both the fiber and free-space configurations, we find
that the background is flat, so background subtraction is done primarily for better visual-
izing the decay in the plots. We assume that the statistical uncertainty of each data point
before background subtraction is
√
Ni, where Ni is the number of counts in bin i, and we
use this to properly weight the fits.
5.3.1 Exponential fitting
Free atomic decay follows an exponential with an offset due to background counts:
f(t) = A exp
(
−
t− t0
τ
)
+B . (5.17)
In order to effectively fit only this free decay, we start our fits at least a few time bins after
the end of the 50-ns excitation pulse. We choose the start and end points of the fit such
that they result in the reduced chi-squared closest to 1 and so that they are in a regime
where the extracted value of τ is least sensitive to truncation [157]. Figs. 5.7 and 5.8
103
display both the data (points) and fits (red lines) for the free-space and fiber-modified
case, respectively.
We extract timescales of τ0 = 26.5 ±0.8 ns and τ ′ = 24.6 ±0.2 ns for the respective
measurements. Our free-space measurement agrees well with the reported value of τ0 =
26.20 ± 0.09 ns reported in Ref. [139], which contains a much more detailed analysis of
systematic errors. We will then use this literature value to compare the result of our
fiber-modified measurement. This yields a ratio of decay rates of
γ′
γ0
= 1.065± 0.009 . (5.18)
This compares with a theoretical prediction of 1.07 from the analysis in Sec. 5.1.4. In
calculating this number we assume that the atoms are primarily parallel to the fiber
surface, as our probe beam was linearly polarized in that orientation, preferentially driving
the atoms to align this way.
5.4 Conclusions
We have directly observed Purcell enhancement of spontaneous emission into the
guided mode of an optical nanofiber. While previous experiments have inferred this effect
by measuring the spectral linewidth of atoms near a dielectric, this is a direct measurement
in the time domain. Our observation of a 6.5 ± 0.9% enhancement is consistent with
the theoretical prediction of 7%, which takes into account radiative and guided-mode
corrections, as well as the density of a thermal distribution of atoms near the fiber surface.
Future experiments could explore how the spontaneous emission rate increases as
the fiber diameter decreases, as shown in Fig. 5.9. Simulations predict that there exists
104
time (ns)
0 50 100 150 200 250
re
sid
ua
ls
0
2
-2
flu
or
es
ce
nc
e (
ar
b. 
un
its
)
Figure 5.7: Histogram of the free-space decay measurement with bin size of 5 ns.
The red line is a fit to Eq. 5.17, with (χ2)red = 1.26. The bottom plot
displays the residuals.
105
time (ns)
0 50 100 150 200 250
re
sid
ua
ls
0
2
-2
flu
or
es
ce
nc
e (
ar
b. 
un
its
)
Figure 5.8: Histogram of the fiber-modified decay measurement with bin size of 5
ns. The red line is a fit to Eq. 5.17, with (χ2)red = 1.45. The bottom
plot displays the residuals.
106
100 150 200 250 300
0
50
100
150
200
250
300
350
0.05
0.10
0.15
0.20
0.25
fiber radius (nm)
at
om
-s
ur
fa
ce
 d
is
ta
nc
e 
(n
m
)
current nanofiber
Figure 5.9: Contour plot of γ1D as a function of fiber size and atom-fiber distance.
The white star marks the parameters of the nanofiber used in this thesis
(with the atom-fiber separation being the typical trap distance).
an optimal fiber radius of a ∼ 0.23λ = 180 nm for the coupling rate at the fiber surface.
Not only will this smaller fiber make our measured effect more pronounced, but the in-
creased coupling strength opens the door for studying interesting collective physics (see
Sec. 7.4) [28].
107
Chapter 6: Quantum hybrid system
This thesis presents a part of an overall project whose aim is to coherently couple
an atomic ensemble with a superconducting circuit. Optical nanofibers aid this end by
providing an atom trap that is in principle compatible with cryogenic systems. They are,
however, just one element of the final system. We need a method to introduce rubidium
atoms to the 10-mK stage of the dilution refrigerator without putting an undue heat load
on the system. These atoms must also be trapped and cooled before they are loaded onto
the nanofiber and transported to the superconductor; this necessitates a MOT in some
form. Finally, the superconductor and nanofiber must be aligned and positioned to within
10 µm of one another in an environment with no optical access.
Fig. 6.1 represents one conception of what the final system might look like. Atoms
are introduced via a cold atomic beam generated from an imbalanced MOT at room
temperature. Within the dilution refrigerator, a compact MOT traps and cools atoms
from this beam. This MOT is spatially close to the “science” region that houses the
superconducting resonator, but thermally anchored to a higher-temperature stage with
cooling power on the order of a Watt (as opposed to the 100 µW at base temperature). The
nanofiber then bridges the few-cm distance between the MOT and the superconductor so
that atoms can be transported to the interaction region. This chapter provides a detailed
characterization of a potential compact MOT, a so-called grating-mirror MOT (GMOT).
108
Figure 6.1: Conceptual sketch of a hybrid quantum system. A cold atomic beam
from a 2D MOT in a room-temperature chamber loads a compact grat-
ing MOT, thermally anchored to a 3 K stage in the dilution refrigera-
tor. The GMOT transfers atoms to the nanofiber trap, and an optical
conveyor belt transports the atoms to within a few micrometers of the
superconducting circuit.
6.1 Compact atom trap: GMOT
Because our nanofiber trap is a purely conservative trap, it does not cool the atoms.
A MOT provides a robust way to introduce dissipation to the system, but standard six-
beam configurations are unwieldy for the confined space at the bottom of a dilution refrig-
erator. The first single-beam MOTs, made with an axicon or pyramid of mirrors [158,159],
are compact and minimize the number of required optical elements, but the trapping vol-
ume forms within the mirrors so that overlapping the cloud with a nanofiber is difficult.
The tetrahedral MOT formed by three mirrors [160], on the other hand, creates a trapping
volume above the plane of the mirrors. Replacing the mirrors with diffraction gratings
generates the same beam geometry so that a MOT forms a few mm above the grat-
ings [79, 161, 162]. We explore this GMOT geometry, and observe sub-Doppler cooling
109
despite a polarization arrangement that does not map on to the standard configurations.
6.1.1 Experimental setup
(a)
(b) (c)
z
x
y
CCD
IP
TS
VMVMRGM
MS
IP
MC
MB
AC
MSC
GV
Figure 6.2: (a) Experimental setup for the GMOT experiments. AC: Antecham-
ber; CCD: CCD camera; GM: Grating Mirrors; GV: Gate valve; IP:
Ion pump; MB: MOT beam; MC: MOT coils; MS: Microscope slide;
MSC: Main science chamber; TS: 2-D translation stage; VM: Vacuum
manipulator; VMR: Vacuum manipulator rod. (b) Single-beam GMOT
configuration. (c) GMOT atom image with CCD camera. (figure from
Ref. [79])
110
The vacuum manipulator (see Sec. 2.4.4 and Figs. 2.15 and 6.2 (a) ) holds the
gratings and can precisely adjust the position of the gratings (450 mm total z translation
with 5µm resolution and 25 mm vectorial xy translation with 5µm resolution) to find the
optimal magnetic field value and beam overlap to reach the lowest temperature. Fig. 6.2
(b) shows the arrangement of the three commercial gratings that we use (12.7 mm ×
12.7 mm×6 mm, Edmund NT43-752, 1200 grooves/mm). The incoming beam overlapping
with the first-order reflections from the three gratings generates a capture volume of
∼ 100 mm3 with a single beam that is spatially filtered with a single-mode optical fiber.
Three gratings are glued on microscope slides (MS) with UV epoxy (EPO-TEK OG116-
31), and the slides are attached to the support rod of the manipulator. The microscope
slides are arranged such that there is a gap in the middle of the gratings to prevent
reflections that cause force imbalances in the MOT. The cooling beam (I = 2.5 mW/cm2)
is locked to the F = 2 to F ′ = 3 transition, and the repumper beam (I = 0.2 mW/cm2) is
locked to the F = 1 to F ′ = 2 transition (see Chap. 2). Both beams are sent through the
same fiber. The polarization of the single cooling beam is circular, but the polarization
of the first-order diffraction changes. The first-order diffraction efficiency for the MOT
beam is 30(5) %, leading to balanced optical molasses [161]. A GMOT requires a large
and high quality beam. We expand the beam directly out of a single-mode optical fiber
to a diameter to 3.6 cm, and this beam is collimated with a shearing interferometer. We
then finely align the beam with a tiltable mount and 3D translation stage to optimize the
GMOT.
111
6.1.2 Temperature measurement
Measuring the mean square radius of the two dimensional cloud image versus ex-
pansion time allows us to extract the atomic temperature (T = mRbσ2v/kB) from fits of
σ2 = σ20 +σ
2
vt
2. Fig. 6.3 displays the results of this measurement for different experimental
conditions. Fig. 6.3 (a, left) presents atomic cloud temperatures as measured after cooling
for 50 ms in a single-stage, far-detuned GMOT (see Table 6.1). The lowest observed tem-
perature for this procedure is 9.7 (0.3)µK and occurs at a detuning of 8.2 Γ (an example
of the fit for this detuning is given in Fig. 6.3 (a, right)).
Employing a molasses cooling stage after a multi-stage, far-detuned GMOT requires
the adjustment of the GMOT position as the magnetic field gradient decreases to zero (Fig.
6.3 (b) and Table 6.2). We optimize our bias magnetic field for each far-detuned GMOT
stage such that the laser-cooled atoms remain in the capture volume for up to 10 ms
after turning off the magnetic field. Table 6.2 summarizes the steps in this process. In
the multi-stage, far-detuned GMOT procedure without molasses, we measure an atomic
temperature of 22.5(4)µK (Fig. 6.3 (b, left), F). This temperature can be explained by
the final detuning of 6.5 Γ being closer to the resonance than that of the single-stage, far-
detuned MOT. The atomic temperature after the multi-stage, far-detuned GMOT and a
1 ms molasses stage (at a detuning of 8.2 Γ) is 7.6(0.6)µK (Fig. 6.3 (b, left), G), which is
colder than the single-stage, far-detuned GMOT without molasses described above.
6.1.3 Theory: sub-Doppler cooling
Ref. [160] describes the requirements for magneto-optical trapping in a GMOT. They
consist of finding a configuration where the optical forces sum to zero. We are interested in
112
(a)
0 10 20 30 40 500
10
20
30
40
δMOT (MHz)
T (
µK
)
A
B
C
D E
0 2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0
τ (ms)
σ2  
(10
-7 m
2 )
(b)
0 10 20 30 40 500
10
20
30
40
δMOT (MHz)
T (
µK
) F
G
0 2 4 6 80.0
0.2
0.4
0.6
0.8
1.0
G
τ (ms)
σ2  
(10
-7 m
2 )
E
10
Figure 6.3: (a) Temperature versus the detuning of the cooling beam for a single-
stage, far-detuned GMOT (see Table 6.1);
√
2σ is the 1/e radius of
atomic cloud, and we fit 1 − 9 ms time-of-flight data to σ2 = σ20 + σ
2
vt
2
(right). We estimate the atomic temperature T (= mRbσ2v/kB) from the
fits (left). (b) Temperature versus the detuning of the cooling beam for
a multi-stage, far-detuned GMOT with no molasses stage (F) and with
a molasses stage (G) (see Table 6.2). (figure from Ref. [79])
understanding sub-Doppler cooling in the polarization configuration present in the GMOT,
as it is neither Sisyphus (lin⊥lin) polarization gradient cooling, nor σ+−σ− orientational
cooling. To simplify the theory, we will assume the cold atoms are close enough to the
center of the quadrupole field so that we can neglect any Zeeman contribution to the laser
detuning. We will consider only 1D laser cooling.
There is a stable polarization configuration (relative phases between beams displace
the polarization configuration but do not change its morphology) in a four-beam configu-
ration such as the GMOT. The spatial periodicity of the underlying lattice is determined
113
Table 6.1: Single-stage, 50 ms far-detuned MOT parameters when scanning the de-
tuning of a single MOT beam (time flows downwards in the table). This
table corresponds to Fig. 6.3 (a); in this paper, the cooling process with
several stages is represented by the cooling stage time (τ), the detun-
ing of the cooling beam (δMOT , red-detuned from the cooling transition),
magnetic field gradient (dB/dz), and the relative intensity of the single
incident cooling beam (I/Isat = 2Ω2/Γ2). (table from Ref. [79])
Cooling time τ (ms) δMOT/Γ dB/dz (G · cm−1) I/Isat
τMOT 1.5 10.8 1.55
50 1.5 to 8.2 10.8 1.16
Table 6.2: Multi-stage, 60 ms far-detuned MOT and 1 ms optical molasses parame-
ters, with time flowing downwards in the table. This table corresponds
to Fig. 6.3 (b) G; the same multi-stage far-detuned MOT without 1 ms
optical molasses corresponds to Fig. 6.3 (b) F. (table from Ref. [79])
Cooling time τ (ms) δMOT/Γ dB/dz (G · cm−1) I/Isat
τMOT 1.5 10.8 1.61
30 3.2 10.8 1.40
15 4.9 6.6 1.20
15 6.5 4.5 1.20
1 8.2 0 1.20
by the geometry of four beams, with a primitive unit cell (ki − kj) of the reciprocal
lattice [163], where ki is the wavevector of the 3D beams (see Fig. 6.4).
The polarization pattern of the GMOT configuration is complicated because of the
existence of both linear and circularly polarized light. For a chosen quantization axis along
the vertical axis (z), when the σ-polarized vertical beam reflects off the diffraction gratings,
the handedness of the polarization (seen from the opposite direction of propagating beams
with k-vectors) is maintained (at the ≈ 90 % level), but in terms of the quantization axis,
the reflected beams will have σ+, σ−, and pi components. The exact composition can be
calculated by a suitable transformation matrix that connects the axes through a rotation.
114
In Fig 6.4 (a-b), a crystal axis parallel to the vertical GMOT beam with its wavevector
k1 has an angle of 109.5o from three other GMOT beams with their wavevectors of k2,
k3, and k4. For the quantization axis qz=(0,0,1), the polarization states of the vertical
GMOT beam and the three GMOT beams projected along the vertical axis correspond
to 100 %, 0 %, and 0 %; 44.4 %, 11.1 %, and 44.4 % of σ+, σ−, and pi respectively. In the
horizontal (xy) plane, the line at 60o and its perpendicular at 150o from the y-axis define
crystal axes in the system (Fig 6.4 (c)). For the quantization axis qxy = (
√
3/2,1/2,0), the
polarization states of the three beams projected to the horizontal plane with k2, k3, and
k4 correspond to 82.5 %, 0.8 %, and 16.7 %; 25 %, 25 %, and 50 %; 0.8 %, 82.5 %, and 16.7 %
of σ+, σ−, and pi respectively. A simple retroflection of a circularly polarized beam
would result in a standing wave without any polarization gradients and no sub-Doppler
cooling. Additional polarization components due to the reflection angles are critical for a
sub-Doppler cooling mechanism.
We numerically calculate the force on the atoms versus atom velocity along the
z-axis of the xz plane (Fig. 6.4 (b)) and the diagonal axes of the xy plane (Fig. 6.4
(c)). In the simulation, we include the multi-level structure of a 87Rb atom, such as
the transitions from the F = 2 Zeeman sub-states to the F
′
= 3 Zeeman sub-states.
The steady state solution of the master equation, dρˆdt = −
i
~ [Hˆ, ρˆ] + Γρˆ, is solved by the
matrix continued fraction method [164]. Treating the beams as classical optical fields, the
raising (Aˆ†+, Aˆ
†
−, Aˆ
†
0) and the lowering (Aˆ+, Aˆ−, Aˆ0) operators correspond to the optical
pumping and spontaneous emission of the transitions of σ+-, σ−-, and pi-polarized lights,
respectively, and the Clebsch-Gordon coefficients for those operators define the transition
strength between each hyperfine ground and excited state. The atom-light interaction
Hamiltonian is Hˆint = −12(Ω+(r)Aˆ+ + Ω−(r)Aˆ−+ Ω0(r)Aˆ0) +h.c., where Ω+, Ω−, and Ω0
115
k1
k4k3
k2 120o
θ
(a)
x z
y
(b)
x
z
x
y
(c)
60o
Figure 6.4: (a) Tetrahedral MOT configuration; | cos θ| = 1/3 and
∑4
i=1 kiIi = 0.
(b) Optical lattices in the xz plane (solid line: 1D optical lattice along
the z-axis of the xz plane). (c) Optical lattices in the xy plane (solid
line: 1D optical lattice along the diagonal axis with an angle of 60o
relative to the y-axis, dashed line: 1D optical lattice along the diagonal
axis with an angle of 150o relative to the y-axis). (figure from Ref. [79])
116
are Rabi frequencies for σ+-, σ−-, and pi-polarized lights, respectively. The force operator
is Fˆ = −∇(Hˆint). After using the master equation to calculate the expectation value of
Fˆ as a function of atomic velocity, we observe a sub-Doppler cooling signature (a steep
slope of force vs. velocity) at low atom velocities for both crystal axes (Fig. 6.5 (a) and
(b)).
We calculate the force for different combinations of polarization in order to under-
stand its role. If we have imbalanced σ-polarizations with no pi-component, the narrow
feature is present, but the point of zero force may not be contained within the feature,
preventing sub-Doppler temperatures. This can be understood in the following way: there
is orientational cooling for the part of the σ+ component that balances the σ− component
present, and then the force versus velocity curve is displaced vertically by the remain-
ing unbalanced σ+ component. When there is pi polarization present, we recover a force
versus velocity curve that should produce good sub-Doppler cooling with a narrow ve-
locity feature centered on the zero-force point. This arises from coherent, two-photon,
velocity-selective resonances between ground-state sublevels, coherent two-photon Raman
transitions of σ+-pi and pi-σ− that become resonant when the energy difference between
two sublevels is equal to the sum of opposite Doppler shifts of the two laser beams. The
simulations show that the narrow velocity feature shifts horizontally away from zero veloc-
ity when a longitudinal magnetic field is present, similar to traditional σ+-σ− orientational
cooling [165]. The horizontal shift of the force versus velocity curve is also accompanied
by a vertical displacement as the magnetic field increases and negates the sub-Doppler
cooling at higher magnetic fields.
Figure 6.5 shows results of our model for different axes and polarization configura-
tions present in the GMOT. The left column of the figure shows the broad features, while
117
the right is a zoom on the region around zero. The atomic temperature T (= Dp/kBα)
is determined by the momentum diffusion coefficient Dp, related to heating and the mo-
mentum friction coefficient α, related to cooling. If the spacing of the 1D optical lattice
becomes more dense for a constant Dp, α increases because of the more frequent cooling
events (note that our simulation does not calculate Dp, so we are unable to calculate
actual temperatures). Assuming an isotropic diffusion constant, we expect the atomic
temperature in the vertical direction, Tz, to be colder than the temperature in the hori-
zontal direction, Txy, based on the steeper slope of the force curve near zero velocity (see
Figs. 6.5 (a-b), where I/Isat=1.2 and δ=-1.5 Γ).
In the experimental run with a multi-stage, far-detuned GMOT and a 1ms optical
molasses (Fig. 6.3 (b) G), Tz is 1.5(0.25) times lower than Txy; a recent GMOT experiment
observes similar anisotropic sub-Doppler cooling [162]. Given our 1D simplification, this
can be considered a qualitative agreement. As a reference to compare the force vs. velocity
features, we simulated the sub-Doppler cooling process of σ+-σ− orientational cooling and
lin⊥lin polarization gradient cooling (Fig. 6.5 (c-d)). The slopes of vertical direction
GMOT and σ+-σ− cooling are similar (Fig. 6.5 (a) and (c)). In addition, the amplitude
of the Doppler cooling feature to capture atoms along the horizontal direction is lower
than in the vertical direction because the intensities of the three GMOT beams along the
horizontal axis are reduced compared to those along the vertical axis. Experimentally,
we also observe an atom cloud squeezed along the vertical direction (Fig. 6.2 (c)). If we
assume comparable diffusion constants between the GMOT and traditional sub-Doppler
mechanisms, our expectations and observations are similar.
118
forc
e (ħ
kΓ
/2)
(a)
forc
e (ħ
kΓ
/2)
(c)
forc
e (ħ
kΓ
/2)
forc
e (ħ
kΓ
/2)
(b)
velocity (Γ/k) velocity (Γ/k)
(d)
−0.2
0
0.2
−0.1
0
0.1
−5 0 5
−0.2
0
0.2
−0.2 0 0.2
−0.1
0
0.1
−0.2
0
0.2
−0.1
0
0.1
−0.2
0
0.2
−0.1
0
0.1
Figure 6.5: Calculated force on atoms as a function of atom velocity for different
axes and polarization configurations of the GMOT. (a) Vertical axis of
the GMOT, the z-axis of the xz plane (Fig. 6.4 (b)). (b) Horizontal
axes of the GMOT, the diagonal axes of the xy plane. solid line: the
axis at an angle of 60o relative to the y-axis, dashed line: the axis at
an angle of 150o relative to the y-axis (see Fig. 6.4 (c)). (c) σ+-σ−
orientational cooling. (d) lin-⊥-lin Sisyphus cooling where I/Isat=1.2
and δ=-1.5 Γ. The right column shows a zoom of the region where the
slope is largest around zero velocity. (figure from Ref. [79])
119
0 2 4 6 8 10 12
0.0
0.2
0.4
0.6
0.8
N a
tom
(10
6 ) 
dB/dz [G.cm-1]
(a)
0 2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
dB/dz (G.cm-1)
n at
om
 (1
011
cm
-3 ) 
(b)
Figure 6.6: Atom number (Natom) and atomic peak density (natom) as a function
of the magnetic field gradient (dB/dz). Each data point has the same
initial MOT atom number extracted simultaneously from a series of
trials, and we vary dB/dz during the far-detuned MOT process. (figure
from Ref. [79])
120
6.1.4 Atom number and density
The success of loading cold atoms into the small (order λ) wells around the nanofiber
requires many cold atoms at high density. We next study atom number (Natom) and atomic
peak density (natom) as a function of magnetic field gradient (Fig. 6.6) after cooling the
atoms for 50 ms in a far-detuned GMOT. The experimental parameters are: (τ (ms),
δMOT /Γ, dB/dz
(
G · cm−1
)
, I/Isat) = (τMOT , 1.5, 10.8, 1.29) → (50, 3.9, 0.4 to 10.8,
0.96). Then, as dB/dz increases, Natom decreases, as seen in Refs. [166, 167]. In addi-
tion, natom also increases linearly as a function of dB/dz, but at a certain peak density,
the linear scaling does not work anymore because the light pressure from reradiated pho-
tons limits the atomic density [166–168]. Reabsorption of scattered photons within the
trapped cloud becomes important above 1011atoms · cm−3. In this regime, natom, which
is nearly independent of Natom, cannot be simply modeled due to the effective repulsive
force between atoms. The atomic density decreases above the peak density because the
multiple scattering of photons prevents further compression of the atomic cloud. Multiple
scattering results in the heating of atoms because of increased momentum diffusion and
reduced friction even with the restoring and friction forces of sub-Doppler cooling [166].
A GMOT with no sub-Doppler cooling captures more atoms from the background
atomic vapor as we increase the magnetic field gradient (see Fig. 6.7). The capture
velocity of the GMOT increases when the magnetic fields in the GMOT shift the energy
levels of atoms entering the trap from all directions. This is a mechanism similar to that
of a Zeeman slower, which has a spatially-varying magnetic field to tune the atoms back
into resonance as they decelerate and their Doppler shift changes. The parameters of the
experiments are: (τ (ms), δMOT /Γ, dB/dz
(
G · cm−1
)
, I/Isat) = (τMOT , 1.5, 6.6 to 19,
121
0 5 10 15 200.0
0.2
0.4
0.6
0.8
1.0
dB/dz (G.cm-1)
N a
tom
 (1
06 )
 
Figure 6.7: Atom number in a GMOT (without sub-Doppler cooling) as a function
of the magnetic field gradient. (figure from Ref. [79])
1.35). The total number of atoms is smaller by two orders of magnitude than in typical
MOTs.
6.2 Conclusions
The GMOT presents a viable option for laser cooling atoms in a dilution refrigerator
due to its simplified optics and compact size. We have reached sub-Doppler temperatures
with this system, a criterion that is crucial for the efficient loading of a nanofiber trap.
Future work will focus on making a robust structure for use in a cryogenic environment,
as well as loading a nanofiber trap from a GMOT. Recent results in Ref. [162] also suggest
that atom number can be improved to ∼ 107 atoms by using nanofabricated gratings
optimized for this trap geometry. We present an overview of the other components of the
hybrid system in progress in the next chapter.
122
Chapter 7: Conclusions and outlook
This thesis presents work with optical nanofibers, focusing on trapping 87Rb atoms
in their evanescent field and using photon-counting techniques to measure atom dynamics
and excited-state lifetimes near them. It also outlines an ongoing effort to create a hybrid
system consisting of an ensemble of 87Rb atoms coupled to a superconducting circuit. As
part of that effort, the optical nanofiber is a potentially cryogenically-compatible atom
trap, and we have demonstrated the functionality of a compact MOT to load atoms onto
the nanofiber within the spatially- and optically-constrained environment of a dilution
refrigerator.
Much work remains to piece together this hybrid system, but luckily there is an
array of exciting physics to explore with nanofibers on their own. In this chapter we
outline three ongoing and one future project out of the many possible ones, both known
and not yet conceived.
7.1 Faraday spectroscopy with nanofibers
The Faraday effect [98], which we use to lock our repump laser (see Chap. 2), also
has applications in spectroscopy of magnetic fields. It has been used in both high-precision
atomic magnetometry [169], as well as spatial resolution of magnetic field profiles [170,171].
Nanofibers could aid this effort given their ability to be brought close to surfaces and their
123
0 200 400 600 800 1000
- 3
- 2
- 1
0
1
2
3
time (μs)
sig
na
l(V
)
Figure 7.1: Faraday rotation signal of 87Rb atoms measured on a balanced photode-
tector. The atoms are optically pumped into the |F, mF 〉 = |2, 2〉 state.
The observed frequency of 58± 2 kHz corresponds to a field of 85 mG.
large optical depth with only a small number of atoms.
An ongoing collaboration with Fredrik Fatemi of NRL seeks to perform Faraday
spectroscopy with nanofiber-trapped atoms. Fig. 7.1 displays a free-space Faraday rotation
signal in our setup, with a frequency of 58 ± 2 kHz, corresponding to a field of about 83
mG. We have observed free-space Faraday signals as small as 20 kHz. We are currently
working to observe the same signal using atoms trapped around the nanofiber. It is likely
that this measurement will also shed light on the longitudinal light polarization present
in the nanofiber mode, which creates so-called fictitious magnetic fields [172].
124
7.2 Surface physics
The background vapor generated by our atomic dispensers coats the nanofiber with
rubidium. Measuring in real time the transmission of a weak resonant probe reveals
interesting dynamics of how atoms adsorb and desorb from the silica surface.
We find that, after a few seconds in a background rubidium pressure of ∼ 10−9
mbar, the coating on the nanofiber can extinguish a resonant probe of a few microwatts to
the level of the dark counts (102 counts · s−1) in our SPCM. This represents an extinction
of 100 dB, and we posit it is due either to direct absorption by the atomic layer or to the
changing of the boundary condition of the waveguide to a metal such that the evanescent
wave cannot propagate.
Sending higher powers of 1064-nm (or 750-nm) light through the nanofiber heats
the waist such that atoms will desorb from its surface. Fig. 7.2 shows two time traces
of count rates of the weak probe after turning on the 1064-nm beam, using powers of
0.48 mW (dark red) and 0.12 mW (blue). There is a clear dependence of the heating
time before the onset of desorption occurs, td, on the power of the light propagating
through the fiber. We approximately extract those times and plot them in Fig. 7.3.
This large change in the desorption time for weak transmitted powers might allow us to
quantitatively add to the discussion of thermalization processes of nanofibers in Ref. [173],
but in a different parameter regime where the temperature changes are much smaller. A
separate experiment involving heterodyne detection, where the local oscillator propagates
in free-space and the signal through the nanofiber. This method is able to sensitively detect
temperature changes in the fiber by measuring the Doppler shift of the light induced by the
varying index of refraction as the nanofiber heats or cools. We can observe temperature
125
Figure 7.2: Transmission of a weak resonant probe through nanofiber as a function
of time after applying 1064-nm light to desorb atoms from the surface.
The two datasets correspond to 1064-nm powers of 0.48 mW (dark red)
and 0.12 mw (blue).
differences when only microwatts of light are sent through the nanofiber, present another
parameter regime to study.
Another experiment in the planning stage seeks to take advantage of the high ex-
tinction ratio of weak probe beams in order to create a new kind of optical switch. Placing
the nanofiber in a heated vapor cell can decrease the coating times to nanoseconds [101],
and we see in Fig. 7.3 that desorption times of microseconds are achievable with modest
powers of heating laser.
126
●●●
●
●
●
●
●
0.0 0.5 1.0 1.5 2.00
1
2
3
4
5
6
7
P1064 (mW)
t d(ms)
Figure 7.3: Extracted heating time before onset of desorption occurs, as a function
of 1064-nm power.
127
7.3 Hybrid system
These next sections sketch two projects within the group aimed at completing the
overall hybrid system. The first is a source of 87Rb atoms from which to load the GMOT
described in Chap. 6. It will consist of an atomic beam originating from an unbalanced 2D
MOT [174] in a room-temperature UHV chamber. The second is a method to accurately
position the fiber relative to the superconducting chip by studying how Rayleigh scattering
from the fiber affects the resonance frequency and phase of the LC resonator [76].
7.3.1 Atomic source: 2D MOT
This project needs a 2D MOT to provide atoms for the cryogenic MOT setup. In
this configuration, four laser beams cool atoms only in the transverse directions so that an
atomic beam with small divergence emerges along the un-cooled axis [174]. High atomic
flux rates of ∼ 109 atoms·s−1 have been reported experimentally, allowing for the formation
of large 3D MOTs with up to 1010 atoms tens of centimeters downstream [174–176]. This
makes the 2D MOT a simple alternative to the Zeeman slower, as it is more compact and
permanent magnets can be used to generate the quadrupole field [176].
7.3.2 Interfacing fibers with superconducting circuits
We must place the atoms with a few micrometers of the superconducting chip in
order to achieve the strongest magnetic dipole coupling [62, 65]. This requires precise
alignment of the optical nanofiber relative to the chip without the aid of optical access.
To this end, we have developed a technique to use the effect of scattered light on the
resonator to determine the position of the fiber, with details of the setup in Ref. [76].
128
7.4 Self-organization
The strong atom-waveguide coupling in optical nanofibers opens the door to the
collective physics of self-organization already observed in traditional cavity-QED exper-
iments [177, 178]. Driving the atoms with a near-resonant laser from the side causes a
non-negligible fraction of spontaneous emission to couple back into the guided mode, as
we saw in Chap. 5. If the atoms are also trapped along the nanofiber, then this spon-
taneous emission into the guided mode mediates a long-range interaction between the
atoms [104]. Furthermore, if one removes the axial confinement so that atoms can freely
move along the fiber axis, then this interaction creates a potential [105,106],
Udd ≈
γ1D s0
2
∑
j, j′
sin
(
k0|zj − zj′ |
)
, (7.1)
where s0 is the magnitude of the atomic coherences, j and j′ are the atomic indices,
k0 = 2pi/λ0 is the free-space wavenumber of the light emitted by the atoms, and zj
is the position of atom j. A self-ordered arrangement of N atoms into a lattice with
spacing λ0 (1− 1/2N) minimizes this potential [105], as schematically depicted in Fig. 7.4.
Observing this collective behavior either in the reflection spectrum or in the correlations
of the light emitted by the atoms would be an interesting application of this system and
remains a priority of future experiments. An ongoing collaboration with JQI Fellow Alexey
Gorshkov and his student also explores interesting ways to engineer center-of-mass cooling
by introducing bandgaps and how [34, 179, 180] the chirality of the system could lead to
different self-organized arrangements.
129
pump
Figure 7.4: Schematic illustration of self-organization of atoms trapped around an
optical nanofiber. An external pump beam drives spontaneous emission
in atoms trapped around the nanofiber but without axial confinement.
The resulting long-range interaction results in a dipole-dipole potential
Udd with lattice spacing λ0(1−1/2N), where N is the number of atoms.
130
7.5 Conclusions
Fibers are a potential platform for many interesting quantum optics and quantum
many-body studies now that we are gaining control over modes, transmission [75, 90],
detection, and atom-trapping [22–25]. Recent experiments in other groups have used
nanofibers, for example, to store optical pulses in an atomic ensemble with EIT [38, 39],
to couple the spin and orbital angular momentum of light [34,35], and to create nonlinear
optical elements with single atoms [36, 181]. These developments point to a rich future
of utilizing nanofibers to explore physics within quantum information science and the
engineering of novel interactions.
131
Appendix A: Nanofiber modes
For a cylindrical, step-index waveguide with radius a, and with index n1 for r <
a and n2 for r > a, the equations for the fundamental mode structure are in many
references. [182–185]. Here I present, for the ease of the reader, the derivation from the
thesis of J. E. Hoffmann [80]. We will apply them to the case of a fused silica (n1 = 1.45367)
nanofiber (a = 250 nm) in air (n2 = 1).
A.1 Field equations
This section lists the electric and magnetic fields in cylindrical coordinates for the
geometry stated above. For r < a:
Er,± =
−iβ
h2
[
±
iµ0ωl
βr
BJl(hr) +AhJ
′
l (hr)
]
ei(ωt−βz±lϕ) (A.1)
Eφ,± =
−iβ
h2
[
±
il
r
AJl(hr)−
µ0ωh
β
BJ ′l (hr)
]
ei(ωt−βz±lϕ) (A.2)
Ez,± = AJl(hr)e
i(ωt−βz±lϕ) (A.3)
Hr,± =
−iβ
h2
[
∓
iε1ωl
βr
AJl(hr) +BhJ
′
l (hr)
]
ei(ωt−βz±lϕ) (A.4)
Hφ,± =
−iβ
h2
[
ε1ω
β
AhJ ′l (hr)±
il
r
BJl(hr)
]
ei(ωt−βz±lϕ) (A.5)
Hz,± = BJl(hr)e
i(ωt−βz±lϕ) , (A.6)
132
and r < a,
Er,± =
iβ
q2
[
±
iµ0ωl
βr
DKl(qr) + ChK
′
l(qr)
]
ei(ωt−βz±lϕ) (A.7)
Eφ,± =
iβ
q2
[
±
il
r
CKl(qr)−
µ0ωh
β
DK ′l(qr)
]
ei(ωt−βz±lϕ) (A.8)
Ez,± = CKl(qr)e
i(ωt−βz±lϕ) (A.9)
Hr,± =
iβ
q2
[
∓
iε2ωl
βr
CKl(qr) +DqK
′
l(qr)
]
ei(ωt−βz±lϕ) (A.10)
Hφ,± =
iβ
q2
[
ε2ω
β
CqJ ′l (qr)±
il
r
DKl(qr)
]
ei(ωt−βz±lϕ) (A.11)
Hz,± = DKl(qr)e
i(ωt−βz±lϕ) , (A.12)
where β is the mode propagation constant, h =
√
k2 − β2, q =
√
β2 − k2, k = 2pi/λ is
the wavenumber, and εi gives the dielectric constant in regions i = 1, 2. The parameter
l is a nonnegative integer. We also use the notation J ′l (hr) = ∂Jl(hr)/∂(hr), K
′
l(qr) =
∂Kl(qr)/∂(qr) for derivatives of Bessel functions Jl and modified Bessel functions of the
second kind Kl of order l.
Boundary conditions impose the following relations for the constants A, B, C, and
D:
B
A
= ±
[(
1
ha
)2
+
(
1
qa
)2
] [
J ′l (ha)
haJl(ha)
+
K ′l(qa)
qaKl(qa)
]−1
(A.13)
C
A
=
Jl(ha)
Kl(qa)
(A.14)
D
A
=
B
A
Jl(ha)
Kl(qa)
−
l2β2
k20
[(
1
ha
)2
+
(
1
qa
)2
]2
, (A.15)
so that the knowledge of A (normalization, Sec. A.3) and the propagation constants β will
completely define the system.
133
A.2 Propagation constant
An eigenvalue equation determines the propagation constants:
Jl−1(ha)
haJl(ha)
=
(
n21 + n
2
2
)
4n21
[
Kl−1(qa) +Kl+1(qa)
qaKl(qa)
]
+ ±R (A.16)
R =
√
√
√
√
(
n21 − n
2
2
)2
(
4n21
)2
[
Kl−1(qa) +Kl+1(qa)
qaKl(qa)
]2
+
l2β2
n21k
2
0
[(
1
ha
)2
+
(
1
qa
)2
]2
,
(A.17)
and the ±R solutions correspond to EH and HE modes, respectively. A normalized fre-
quency called the V−number is defined by the relation V = (2pi/λ)a
√
n21 − n
2
2, which
scales the optical frequency by the fiber radius and its index of refraction (
√
n21 − n
2
2). We
can numerically solve Eq. A.16 for a particular V−number and l by finding the points of
intersection of its LHS and RHS. Modes are labeled with subscripts lm, e.g. HElm, where
for a given l, the successive points of intersection signify increasing m. Fig. A.1 plots
the result of this calculation (where neff = β/k) as a function of V−number for various
families of modes. Note that the cutoff occurs at V = 2.405 and that the fundamental
HE11 mode propagates for any V > 0.
A.3 Normalization
The last parameter to determine is A, which is calculated using energy conservation
considerations. We normalize the time-averaged Poynting vector in the z-direction relative
to the input power,
P = 〈Sz〉t = A
2pi (Din +Dout) , (A.18)
134
0 2 4 6 8 10
1.0
1.1
1.2
1.3
1.4
Figure A.1: Effective index of refraction as a function of V-number. The families
of modes and their colors are HE (blue), EH (black), TE (red), TM
(green) (figure from Ref. [80]).
135
where Din and Dout will be found analytically. For the HElm and EHlm modes (this thesis
does not deal with TE and TM modes, so we omit these solutions), these parameters are
Din =
piaβ2
4µ0ω
β
h2
[(1 + sl)(N21 + sl)[J
2
l+1(ha)− Jl(ha)Jl+2(ha)]
+ (1− sl)(N21 − sl)[J
2
l−1(ha)− Jl(ha)Jl−2(ha)]] (A.19)
Dout =
−piaβ2
4µ0ω
β
q2
(
Jl(ha)
Kl(qa)
)2
[(1 + sl)(N22 + sl)[K
2
l+1(qa)−Kl(qa)Kl+2(ha)]
+ (1− sl)(N22 − sl)[K
2
l−1(qa)−Kl(qa)Kl−2(qa)]] , (A.20)
where Ni = nik/β and s = Bµ0ω/(ilβ) (with B given by Eq. A.13).
Fig. A.2 offers a summary of this Appendix by plotting the mode structure of the
HE11 mode, showing the intensity as well as the norm of each electric field component
normalized to their value at the fiber surface. These values were calculated for a 360-nm
diameter fiber with index of refraction n1 = 1.45367 and propagating wavelength of 780
nm. Note the sharp discontinuity at the fiber surface, as well as the sizable longitudinal
component (Ez).
136
-200
200
0
-200
200
0
-200
200
0
-200
200
0
-20
0
200
0
-20
0
200
0
-20
0
200
0
-20
0
200
0
x (nm)
x (nm) x (nm)
x (nm)
y (n
m)
y (n
m)
y (n
m)
y (n
m)0.0
0.4
0.8
0.0
0.4
0.8
0.0
0.4
0.8
0.0
0.5
1.0
a) b)
c) d)
  
|Ex(r,ⱷ)|2
|Ex(a,ⱷ)|2
  
|Ey(r,ⱷ)|2
|Ey(a,ⱷ)|2
  
|Ez(r,ⱷ)|2
|Ez(a,ⱷ)|2
Int (a.u.)
Figure A.2: Fundamental (HE11) mode structure of 360-nm diameter nanofiber.
(figure from Ref. [80])
137
Appendix B: Nanofiber vibrations
This appendix presents a first tentative analysis of something that we discovered
while performing the correlation measurements in Chap. 4. Though we are still figuring
out the whole picture, we think that this can provide us much quantitative information
about the nanofiber and its motion.
A nice feature of TCSPC measurements is that the same time series of photon clicks
can be used to study markedly different dynamics. We have seen in Chap. 4 that the
autocorrelation function reveals quantum effects on timescales associated with internal
atomic dynamics (tens of nanoseconds) and correlations resulting from classical atomic
dynamics near the nanofiber (microseconds). Taking the autocorrelation of the same data
but for millisecond timescales uncovers information regarding the motion of the ONF.
Light from one of the MOT beams couples into the guided mode of the ONF and into
the detection path of the SPCMs (see Fig. B.1). This coupling is time-dependent due to
vibrations of the nanofiber, and its signature is imprinted in the fluctuations of the light, as
seen in Fig. B.2. We also use correlations to measure heating of the fiber from absorption
of light propagating through it, as the frequency of vibration lessens with increasing laser
power.
138
filters 50/50 BS
780 nm 
beam SPCM
to correlator
1064 nm
beam
780 nm 
signal
45°
85 mm
Figure B.1: Experimental setup. The MOT beam at 45◦ relative to the nanofiber
couples into the waist and photon-detection path. Vibrations of the
nanofiber modulate the coupling as a function of time, which is mea-
sured via correlations.
-10 -5 0 5 10
1.00
1.05
1.10
1.15
1.20
1.25
1.30
g(2)
(τ)
τ (ms)
Figure B.2: Correlation function, g(2)(τ), as a function of delay for long time. The
triangular shape of the data is an artifact of finite window effects.
139
B.1 Transverse vibrational modes of a nanofiber
Finding the vibrational modes of the nanofiber is difficult because the cross-sectional
area changes by four orders of magnitude over its profile. The fiber is fixed in the axial
direction by the holder so that there is no axial acceleration, and the axial forces must be
constant. This leads to vastly different axial stresses in different parts of the fiber given
the area variation described above, and the magnitude of the local stress determines the
relevant dynamics. In the thicker taper regions, the so-called bending (or cantilever) modes
are important because of the non-negligible second moment of area and its resistance to
flexing [186]. Along the 500-nm-diameter waist, however, the fiber is well-approximated
by a string whose motion is dictated instead by the large axial stresses [186]. This yields
the following wave equation for the displacement, w(z, t), of the nanofiber [187]:
−
∂2
∂z2
(
E I(z)
∂2
∂z2
w(z, t)
)
+ Faxial
∂2
∂z2
w(z, t) = µ(z)
∂2
∂t2
w(z, t) , (B.1)
where the first term on the LHS comes from the Euler-Bernoulli beam equation (assuming
no transverse load), and the second term on the LHS is the standard string wave equation.
E is the Young’s modulus (71.7 GPa for fused silica), I(z) = pir4(z)/4 is the second
moment of area for a cylinder, Faxial is the axial force (or tension), and µ(z) = ρA(z) is
the linear mass density. The transversal waves described by Eq. B.1 are distinct from the
compressional and torsional waves studied in detail in Refs. [108,186]. Those other waves
tend to be much higher frequency (∼ 100 kHz compared to ∼ 100 Hz) and couple to the
polarization of the light propagating in the fiber, pointing to a potential candidate for
parametric heating in nanofiber traps [108].
140
Given the observed frequencies of hundreds of Hz (see Fig. B.4), we suspect that the
modulations in the correlation function arise primarily from bending modes in the tapers.
Solutions to the conical, tapered cantilever vibrational modes exist, but generally for
boundary conditions where one end is free [188–194]. In our case, the thin waist connects
the tapers so that the typically free end is now under tension. The complicated nature of
this problem makes it more amenable to numerical FEM solutions, though finding a proper
mesh for the disparately sized regions is challenging [186]. We are currently working on
the full FEM simulations in COMSOL, but they remain incomplete as a result of the
difficulty of properly defining the tension.
We gain some insight, however, by first neglecting the waist and solving the conical
cantilever problem, following the treatments in Refs. [188–194]. Ignoring the string term
in Eq. B.1, we consider the geometry depicted in Fig. B.3, where a cantilever with circular
cross-section tapers from a radius of r0 to r1 over a distance l. This simplified picture
approximates the exponential section to be linear so that the fiber taper connects directly
to the waist. The exponential horn plays a much stronger role in the case of torsional
modes [108].
The material has constant density so that the linear density is µ = ρA(z) = pir2(z).
Since we are finding normal modes of vibration, we separate the displacement function into
time- and space-dependent terms, w(z, t) = h(z) sin(ωt), where ω is the angular frequency
of the normal mode. Rescaling the problem in terms of a dimensionless distance u = x/l,
we arrive at the following differential equation:
∂4uh− 8
1− α
1− (1− α)u
∂3uh+ 12
(
1− α
1− (1− α)u
)2
∂2uh =
(lk)4
(1− (1− α)u)2
h , (B.2)
141
Figure B.3: Geometry of the problem of a conical cantilever. The cantilever tapers
from radius r0 to r1 over a distance l and has a circular cross-section.
The large radius end at z = 0 obeys a fixed boundary condition. The
displacement is given by w(z, t), and the dotted line displays an exag-
gerated vibrational mode.
142
where ∂i denotes derivation with respect to coordinate i, α = r1/r0, and k = 4ρω2/Er20.
To find the normal modes, we solve this eigenvalue equation numerically in Mathematica,
applying clamped (h = ∂uh = 0) boundary conditions at u = 0 and sliding boundary
conditions
(
∂uh = ∂3uh = 0
)
at u = 1. We consider a cantilever made out of fused silica
(ρ = 2.203 g ·cm3, E = 71.7 GPa) with length l = 39 mm, and with large and small radii of
r0 = 62.5µm and r1 = 250 nm, respectively. This yields frequencies of ω/2pi = 161.5, 392.3
Hz for the first two normal modes. We confirm that these frequencies match the FEM
simulations to within 2 Hz for the case of an untensioned cantilever with these dimensions.
The FEM simulations also illustrate that the transversal modes have the same fre-
quency for the full fiber with two tapers connected by the nanofiber waist. In the case of
no tension, the two cantilevers are essentially independent. Adding tension increases the
vibrational frequencies, and we can approximate the effect by following the treatment in
Refs. [193,195,196]. The normal mode frequencies ωn for small n become
ω′n = ωn
√
1 + Un , (B.3)
where
Un =
4
(2n− 1)2pi2
Faxiall2
EI
. (B.4)
We will use this result to characterize our observed vibrations.
Fig. B.4 displays the power spectrum of an exemplary long-time correlation mea-
surement. We can clearly identify three features: the leftmost peak corresponds to the
slow oscillations of the manipulator mount that holds the fiber, and the latter two are
nanofiber vibrations. We note that sidebands from the manipulator vibrations are visible
on the peak at approximately 555 Hz.
143
We do not have a good physical reason for why the first two peaks corresponding
to nanofiber vibrations look like harmonics, which should not be the case for cantilever
modes. One hypothesis is that the “fundamental” vibration occurs when the cantilevers
oscillate in phase, and the “harmonic” arises when the cantilevers are out of phase so that
the waist moves through its optimally-coupled position twice per oscillation. The other
explanation is that the cantilever modes begin to exhibit string-like behavior as more
tension is applied, and the frequency spacing between modes becomes more harmonic.
FEM simulations with tension should help to elucidate this question.
Assuming that the peak at 277 Hz is the first vibrational mode of the fiber, we use
Eq. B.4 to estimate the tension of the nanofiber. We use an average second moment of area,
I, and n = 1 to arrive at Faxial ≈ 0.14 mN. Translating this to total initial elongation
(see Chap. 2 for a discussion of post-pull tensioning) amounts to integrating the local
strain, (z) over the length of the fiber. We use the nonlinear stress-strain equation,
σ(z) = E (z)(1 + α(z)/2), with α = 6, and define the strain σ(z) = Faxial/A(z) [197].
Solving for (z) and integrating over our fiber geometry with Faxial = 0.14 mN, we obtain
a total initial elongation of 69 µm. We do not have an exact value from our nanofiber
tensioning process, but this is a reasonable estimate based on how far the motors are
moved after the pull.
B.2 Effect of heating
When laser light is sent through the nanofiber, it heats as a result of surface ab-
sorption [173,186]. The thermal expansion of the nanofiber relaxes the axial load, causing
the fiber vibration frequency to decrease. We observe this effect in our correlation mea-
surements when 1064-nm light is transmitted through the nanofiber, as illustrated in the
144
0 200 400 600 800
0
2
4
6
8
frequency (Hz)
|S
(f
)|
 (
ar
b.
 u
ni
ts
)
450 500 550 600 650
0
1
2
3
4
5
6
increasing P
1064
Figure B.4: Power spectrum of the long-time correlation function, such as in
Fig. B.2. Inset shows the power spectra on a zoomed frequency axis
for 1064 nm laser powers ranging from 0 mW to 7.5 mW (red arrow
indicates direction of increasing power).
inset of Fig. B.4. We extract the fractional change in the mode frequency as a function of
transmitted 1064-nm laser power and plot it in Fig. B.5. The largest fractional change of
12% for 7.5 mW of 1064-nm light corresponds to change of 23 µm relative to the initial,
pre-tensioned fiber, following the above analysis. The fractional frequency resolution of
about 0.5% translates to detecting fiber-length changes of about 2 µm.
B.3 Heterodyne technique
The heterodyne technique mentioned briefly in Chap. 7 has also confirmed the pres-
ence of these vibrations. A local oscillator beam is sent down a reference path, and a
roughly 1-MHz-detuned signal beam propagates down the fiber. These are overlapped in
145
Figure B.5: Fractional change in nanofiber vibrational frequency as a function of
1064 nm laser power. The error bars represent the 3 dB point of the
peaks in the power spectrum. The different colors correspond to data
taken on different days.
146
the same spatial mode, and the beatnote is detected with a lock-in amplifier. As discussed
earlier, this is a highly sensitive measurement scheme that has been able to detect, in real
time, the Doppler shift of the signal light as the thermo-optic effect changes the optical
path length. High signal-to-noise is maintained when sending only microwatts of heating
power through it.
During the writing of this thesis, we have also seen that when the UHV manip-
ulator is tapped, low-frequency signals appear in the lock-in signal. They correspond
precisely with the 277 Hz and 555 Hz signals mentioned above, among other frequencies
not detected with the correlation technique. We can also discern signals that appear to be
torsional modes, starting at 180 kHz. The heterodyne scheme will enhance our capability
to distinguish nanofiber vibrations modes and how they evolve during heating and cooling.
It is much simpler than using correlations, as it requires very little averaging and minimal
post-processing of the data; the signal is observed directly on an oscilloscope.
147
Appendix C: Calculating van der Waals coefficients for 87Rb
Knowing well the surface interactions is important for calculating the full potential
that nanofiber-trapped atoms see, as well as to understand the dynamics of untrapped
atoms moving near the nanofiber. Numbers for the van der Waals coefficients of cesium
atoms near fused silica are given, for example, in Ref. [20], but we have not found consistent
values for rubidium [198,199]. This Appendix outlines the calculation of these coefficients,
following the treatment in Refs. [20, 184].
C.1 Index of refraction
The Sellmeier equation for fused silica calculates the index of refraction for wave-
lengths in range of 0.2−7µm, over which the absorption coefficient is negligibly small [200]:
n2(λ) ≡
ε(λ)
ε0
= 1 +
0.6961663λ2
λ2 − 0.06840432
+
0.4079426λ2
λ2 − 0.11624142
+
0.8974794λ2
λ2 − 9.8961612
, (C.1)
where λ is in micrometers. Fig. C.1 illustrates the behavior of this function from 0.2 to 7
µm. Because we will later need to integrate the dielectric response function over imaginary
frequencies, we recast Eq. C.1 from wavelength to these imaginary frequencies.
148
1 2 3 4 5 6 71.0
1.2
1.4
1.6
1.8
λ (μm)
Re
frac
tive
Ind
ex,
n
Figure C.1: Index of refraction of fused silica for λ between 0.2 and 7 µm, calculated
using the Sellmeier equation (Eq. C.1).
149
C.2 Atomic polarizabilities
We also need the dynamical scalar polarizability of 87Rb, which we calculate follow-
ing the treatment in Ref. [201], but using the reduced dipole matrix element values and
energy levels for the 87Rb D2 line from Refs. [202, 203]. The scalar polarizability for an
atom in the hyperfine level |nJF 〉 is [201]
αsnJF =
1
√
3(2J + 1)
(−1)J+1
∑
n′J ′
(−1)J
′



1 0 1
J J ′ J



|〈n′J ′||d||nJ〉|2 (C.2)
×
1
~
Re
(
1
ωn′J ′nJ − ω − iγn′J ′nJ/2
+
1
ωn′J ′nJ + ω + iγn′J ′nJ/2
)
, (C.3)
where we sum over transitions to allowed levels |n′J ′〉, and 〈n′J ′||d||nJ〉 is the reduced
dipole matrix element. The symbol
{
j1 j2 j3
j4 j5 j6
}
represents the Wigner 6-j symbol, and
ωn′J ′nJ and γn′J ′nJ correspond to the frequency and linewidth, respectively, of transitions
between states |nJ〉 and |n′J ′〉.
In Fig. C.2 we plot the scalar polarizabilities for the or the 5S1/2 (blue) and 5P3/2
(orange) states of 87Rb as a function of imaginary angular frequency. For use in numerical
integration later, we generate interpolation functions for both of these lists that are valid
out to values of angular frequency of ∼ 1036 rad · s−1.
150
1015 1018 1021 10240
200
400
600
800
Re[ⅈω] (rad s-1)
pol
ariz
abi
lity
(e2 a 02 E
h-1 )
Figure C.2: Scalar polarizabilities for the 5S1/2 (blue) and 5P3/2 (orange) states of
87Rb as a function of imaginary angular frequency. The polarizability is
given in atomic units of e2a20/Eh, with Eh = mee
4/(4piε0~)2 = mec2α2
being the Hartree energy (approximately twice the ionization energy of
ground-state hydrogen). Note the logarithmic horizontal axis.
151
C.3 C3 and C4 coefficients
We insert the above results for the dielectric response and polarizabilities into the
following integral to calculate C3 [20, 184]:
C3 =
~
16pi2ε0
∫ ∞
0
dω αsnJF (iω)
ε(iω)− ε0
ε0 + ε(iω)
, (C.4)
where we use the appropriate polarizability for either the 5S1/2 or 5P3/2 state. Note
that we are also ignoring geometric effects due to the curvature of the nanofiber, which is
unimportant for small distances (see Refs. [20,127,128,184] for more details). To calculate
the Casimir-Polder C4 coefficients, we make use of an approximate formula assuming an
infinite dielectric [184,204]:
C4 =
3
8pi
~αsnJF (0)
4piε0
n2 − 1
n2 + (30/23)n+ 7/23
, (C.5)
where we use the static polarizability (Eq. C.2 for ω = 0). Table C.1 summarizes our
results. As a check, we verified our calculation for cesium against the values quoted in the
literature [20].
State C3 (×10−49 J · µm3) C4 (×10−56 J · µm4)
5S1/2 4.94 4.47
5P3/2 7.05 12.2
Table C.1: van der Waals and Casimir-Polder coefficients for 87Rb near fused silica
for both the 5S1/2 or 5P3/2 states.
152
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