Exploring Spontaneous Emission Phenomenausing Ultracold Atomic Matter Waves
A Dissertation presentedbyLudwig Krinner
to
The Graduate Schoolin Partial Fulfillment of theRequirements
for the Degree ofDoctor of Philosophy
inPhysics

Stony Brook University

August 2018

Stony Brook University
The Graduate School

Ludwig Krinner
We, the dissertation committe for the above candidate for theDoctor of Philosophy degree, hereby recommend
acceptance of this dissertation.

Dominik Schneble - Dissertation Advisor
Associate Professor, Department of Physics and Astronomy

Philip Allen - Chairperson of Defense
Professor Emeritus, Department of Physics and Astronomy

Eden Figueroa
Assistant Professor, Department of Physics and Astronomy

Andrew Daley
Professor, Department of Physics and SUPA
University of Strathclyde

This dissertation is accepted by the Graduate School.

Dean of the Graduate Schoolii

Abstract of the Dissertation

Exploring Spontaneous Emission
Phenomena using Ultracold
Atomic Matter Waves
byLudwig KrinnerDoctor of Philosophy
inPhysics
Stony Brook University
2018

The controllability of ultracold quantum gases in optical lattices offers new
perspectives on problems in various subfields of physics. In this dissertation,
we study spontaneous emission, a fundamental concept in quantum electrodynamics, using engineered emitters, realized with 87 Rb atoms in a hyperfine
ground state selective optical lattice (a lattice potential that confines one internal hyperfine ground state without confining the other), that send single
atoms into one-dimensional matter-waveguides. On a fundamental level, eachemitter is described by the Weisskopf-Wigner model. The system allows foreasy tunability of the coupling strength and excited state energy, allowing
the direct observation of how Markovian dynamics, giving rise to the usuallyobserved exponential decay, transform into non-Markovian dynamics as theexcited state energy approaches the coupling strength. This has allowed thefirst direct observation of a bound state, which was long predicted to exist
for spontaneous emission in the context of photonic bandgap materials. Weiii

study its spatial characteristics in momentum-space and are able to infer its
real-space extent.
The ability to carry out these experiments in our apparatus depends on two
technical developments. First is the development of a technique which allows
a precise characterization of magnetic fields at Gauss-level fields used in ourapparatus. This is achieved by utilizing transitions to unused hyperfine states,
which creates a ‘tag’ that characterizes the field at the position of the atoms
at the time of the experiment to an accuracy of better than 100 µG for every
single experimental repetition. This allows post-selection stabilization on alevel which is comparable to state-of-the-art active stabilization techniques.
The second development is an image analysis technique which allows for theremoval of spurious unwanted fringes in our absorption images, which allows
a reliable detection of small (on the order of several hundred) atom number.
The experimental work presented in this dissertation realizes a platform
for future studies of dissipative many-body physics in ultraold atomic gases
mirroring one of the current frontiers in quantum optics.

iv

Contents
1 Introduction

1

2 Theoretical and Experimental Basics
2.1 BEC in the Schneble Laboratory . . . . . . . . . . . . . . . .
2.1.1 Laser Cooling and Magneto Optical Trapping . . . . .
2.1.2 Magnetic Trapping . . . . . . . . . . . . . . . . . . . .
2.1.3 Optical Trapping . . . . . . . . . . . . . . . . . . . . .
2.1.4 Evaporative Cooling . . . . . . . . . . . . . . . . . . .
2.1.5 Achieving BEC . . . . . . . . . . . . . . . . . . . . . .
2.2 Condensates in Optical Lattices . . . . . . . . . . . . . . . . .
2.2.1 Optical BEC, Detection and Optical Schematic . . . .
2.2.2 Optical Lattices and Lattice Calibration Using Talbot
Kicking . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Hyperfine-State Rabi Oscillations . . . . . . . . . . . .
2.2.4 Hyperfine Structure in Magnetic Fields . . . . . . . . .
2.2.5 State-Selective Trapping Potential . . . . . . . . . . . .
2.2.6 Bose-Hubbard Model and Interacting Bosons . . . . . .

7
7
8
10
12
13
14
18
18
20
24
27
28
31

3 Experimental Upgrades
3.1 Laser Cooling and Trapping . . . . . . . . . . . . . . . . . . .
3.2 Rubidium Source . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Optical Pumping . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Quadrupole Magnetic Coils . . . . . . . . . . . . . . . . . . .
3.5 Evaporation Speed Increase . . . . . . . . . . . . . . . . . . .
3.6 Summary and Current Performance . . . . . . . . . . . . . . .

36
36
37
42
46
47
47

4 Image Analysis
4.1 Backgrounds and Fringes . . . . . . . . . . . . . . . . . . . . .
4.2 Reference Fringes . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Principal Component Analysis . . . . . . . . . . . . . . . . . .

49
49
50
50

v

4.4
4.5

Discussion of Errors . . . . . . . . . . . . . . . . . . . . . . . .
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52
55

5 In-Situ Magnetometry
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Method and Implementation . . . . . . . . . . . . . . . . . . .
5.2.1 Principle of Operation . . . . . . . . . . . . . . . . . .
5.2.2 Experimental Implementation . . . . . . . . . . . . . .
5.3 Characterization of Performance . . . . . . . . . . . . . . . . .
5.3.1 Slow Rabi Cycling . . . . . . . . . . . . . . . . . . . .
5.3.2 Theoretical Accuracy & Operation Range . . . . . . .
5.4 Application: Spectroscopy of State - Selective Optical Lattices
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61
61
62
62
64
66
66
68
69
71

6 Review of the Weisskopf-Wigner Model
6.1 Electric Dipole Hamiltonian . . . . . . . . . . . . . . . . . . .
6.2 Field Quantization and Weisskopf–Wigner Hamiltonian . . . .
6.3 Decay Rate and Lamb Shift . . . . . . . . . . . . . . . . . . .

72
72
74
75

7 Spontaneous Emission of Matter Waves from a Tunable Open
Quantum System77
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
7.2 Introduction to the System . . . . . . . . . . . . . . . . . . . .
79
7.3 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . .
81
7.4 Markovian Dynamics . . . . . . . . . . . . . . . . . . . . . . .
82
7.5 Non-Markovian Dynamics . . . . . . . . . . . . . . . . . . . .
85
7.6 Experimental Details . . . . . . . . . . . . . . . . . . . . . . .
88
7.6.1 Experimental Procedures . . . . . . . . . . . . . . . . .
88
7.6.2 Characterization of Magnetic Fields and Optical Potentials 89
7.6.3 Background Subtraction . . . . . . . . . . . . . . . . .
93
7.6.4 Energy Shift Data . . . . . . . . . . . . . . . . . . . .
94
7.7 Numerical Simulation of Array Effects in Weak Harmonic Trapping Potential . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
7.8 (*) Comparison to Non-Adiabatic Diffraction . . . . . . . . . .
98
8 Conclusion and Outlook

99

Bibliography

101

Appendix A Simulation of Mean-Field Shift

123

vi

Appendix B Other Measurements
125
B.1 Controlling the Scattering Length: Interspecies Feshbach Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
B.2 Measuring Mott-Shell Populations . . . . . . . . . . . . . . . . 128
Appendix C Electronics
129
C.1 AC-line Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . 129
C.2 Schmitt-Trigger Circuit . . . . . . . . . . . . . . . . . . . . . . 129
C.3 Stable Amplified Photo-diodes. . . . . . . . . . . . . . . . . . 130
Appendix D Development of Smart Laboratory Devices
134
D.1 Introductory Project: FPGA Based Down Counter for Ultrafast
Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
D.1.1 Basic Idea: Code snippet . . . . . . . . . . . . . . . . . 137
D.1.2 Hardware and Implementation . . . . . . . . . . . . . . 138
D.1.3 Entire Source Code . . . . . . . . . . . . . . . . . . . . 138
D.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

vii

List of Figures
1.1
1.2

Spontaneous emission of matter waves. For details see chapter 7.
Characterization of magnetic fields. For details, see chapter 5.

45

2.1 Schematic of apparatus for BEC production . . . . . . . . . .
2.2 Level diagram of Rubidium . . . . . . . . . . . . . . . . . . .
2.3 Optical schematic for main experiments . . . . . . . . . . . . .
2.4 Raw camera illuminations with and without atoms . . . . . .
2.5 Typical absorption images . . . . . . . . . . . . . . . . . . . .
2.6 Result of Kapitza-Dirac diffraction . . . . . . . . . . . . . . .
2.7 Simulation of Talbot-resonant kicking . . . . . . . . . . . . . .
2.8 Rabi Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9 Rabi resonance . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Breit-Rabi diagram for 87 Rb . . . . . . . . . . . . . . . . . . .
2.11 Potential depth and scattering rate for state-selective opticalpotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.12 Bloch-bands . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.13 Wannier and Bloch Functions . . . . . . . . . . . . . . . . . .

14
16
17
19
20
22
24
26
27
28

3.1
3.2
3.3
3.4
3.5
3.6
3.7

Temperature control for Rb source . . . . . . . . . . . . . . .
Diagram for calculating pressure and flow in vacuum. . . . . .
Lifetime of Rb-atoms in vapor cell . . . . . . . . . . . . . . . .
Optical schematic for improved optical pumping . . . . . . . .
Level diagram for optical pumping . . . . . . . . . . . . . . .
Performance characterization of improved optical pumping . .
Number stability after improvements . . . . . . . . . . . . . .

38
39
41
43
44
45
48

4.1
4.2
4.3
4.4

Sample images for fringe-removal . . . . . . . . . . . . . . . .
56
Comparison of masking and not masking the atomic signal . .
57
Sample principal components and their Fourier transforms . .
58
Mean overlap of principal components with a sample set of images 58

5.1

In situ magnetometry scheme . . . . . . . . . . . . . . . . . .
viii

29
32
32

63

5.2
5.3
5.4
5.5

Measurement of AC-field fluctuations . . . . . . . . . . . . . .
Slow Rabi cycling using field post selection . . . . . . . . . . .
Simulated field reconstruction . . . . . . . . . . . . . . . . . .
Microwave spectroscopy of a free-to-bound transition into astate-selective optical lattice . . . . . . . . . . . . . . . . . . .

70

6.1

Spontaneous emission. . . . . . . . . . . . . . . . . . . . . . .

72

7.1
7.2
7.3

Realization of matter-wave emitters . . . . . . . . . . . . . . .
Study of matter-wave emitters in the Markovian regime . . . .
Study of non-Markovian dynamics and bound state of matterwave emitters . . . . . . . . . . . . . . . . . . . . . . . . . . .
Average free-to-bound microwave spectrum . . . . . . . . . . .
Background subtraction . . . . . . . . . . . . . . . . . . . . .
Raw data for energy shift . . . . . . . . . . . . . . . . . . . .
Simulated decay dynamics . . . . . . . . . . . . . . . . . . . .
Simulated time evolution in the Markovian regime showing finite size effects. . . . . . . . . . . . . . . . . . . . . . . . . . .

80
83

7.4
7.5
7.6
7.7
7.8

66
67
69

86
91
93
94
95
97

A.1 Mean Field Shifted Spectrum . . . . . . . . . . . . . . . . . .

124

B.1 Interspecies Feshbach resonance . . . . . . . . . . . . . . . . .
B.2 Two-photon spectroscopy of a Mott-insulator . . . . . . . . . .

126
127

C.1
C.2
C.3
C.4

Schematic of the setup for the line trigger . . . . . . . . . . .
Electronic schematic for Schmitt-trigger circuit . . . . . . . . .
Electronic schematic for logarithmic photo-diodes . . . . . . .
Schematic for PI-control with logarithmic photo-diodes . . . .

130
131
132
133

D.1 Photo-Diode to Trigger Circuit . . . . . . . . . . . . . . . . .
D.2 Picture of FPGA Board . . . . . . . . . . . . . . . . . . . . .
D.3 Downcounter FPGA Pinout . . . . . . . . . . . . . . . . . . .

135
136
141

ix

Acknowledgements

This is dedicated to the people who have supported me in turning my dreamof pursuing this advanced degree into reality. It is impossible to overstate theimportance of the support and help I have received from the people aroundme, allowing me to evolve both personally and professionally.
I want to first thank my advisor Dominik Schneble, who is an endlesssource of great ideas, physical intuition, helpful, wanted, and needed criticism,and most importantly ceaseless enthusiasm for the physics in our laboratory.
Witnessing his unwavering support for and belief in his students is a constantsource of encouragement, pushing us on to new experiments every day. He has
first introduced me to the field of atomic physics and ultracold atomic physics,
which I have been in love with ever since. Motivation and advice were always
welcome, and came in plentiful amounts, which made discussions with him
always intriguing, fruitful and interesting. I want to thank him for imbuing
me with the perseverance and skills needed for pursuing a career in physics
and beyond.
In the laboratory I had the incredible pleasure to work and collaborate witha plethora of outstanding people and physicist. During my first year at Stony
Brook, I enjoyed training from the renowned Bryce Gadway, who has not only
taught me to understand physics and ultracold physics, but also to pass thatknowledge along. Jeremy Reeves has taught me many of my laboratory skills
along with doing a significant part of the work on which this dissertation isbased. Without him I would not be where I am today, and I would certainly
not be as good a physicist without his help and explanations throughout myfirst four years at Stony Brook. Michael Stewart, who joined the group shortly
after me, is an incredibly talented physicist and I had the pleasure of discusswith him some of the most intriguing and difficult questions I have had during
my Ph. D. His deep knowledge and understanding of mathematics (and language) provide a novel viewpoint and understanding of most topics, which wasinvaluable for fleshing out physical ideas for publications. Arturo Pazmiño has
been a great colleague and an invaluable help during my time in the laboratory. I have always been able to count on him for helpful advice in and aroundthe lab. Joonhyuk Kwon, the most recent addition to our group, can always
be relied on to ask intriguing questions about the publications which we are
writing or about daily lab-work. He is an excellent graduate student and I
look forward to his continued great contributions and work in the upcomingxyears. Departing from such an excellent group of people leaves me with aheavy heart, but I am more than looking forward to their future work. I will
miss lunch breaks and driving to conferences with you!
The committee for my dissertation defense has accompanied and mentored
me during my Ph. D. years. For their time, mentorship, effort, help, and support, I want to thank Andrew Daley, Philip Allen, and Eden Figueroa. Thankyou for your guidance along the way and thank you for attending my defense.
My committee has accompanied my development from a young graduate student to a graduating student with many helpful questions during the yearly
updates. While Andrew Daley was not formally on my committee until thedefense, I have had the pleasure of interacting with him and learning from him
during his visits to Stony Brook and during DAMOP meetings, which makes
him an excellent choice for the external committee member. Thank you also
to professors Thomas Weinacht, Hal Metcalf and Tom Bergeman, for always
answering the tricky questions I had and for offering advice and guidance along
the way. Martin Cohen has also been a constant presence throughout my Ph.
D. and I am thankful for his guidance in technical questions, as well as his
unwavering endurance in reading and discussing the publications of our groupwith me. He is a steady source of good advice and an amazing listener.
I am grateful for the general support of the department chairs Laszlo Mihaly and Axel Drees, the graduate program director Jacobus Verbaarschot, as
well as the detailed support of the main office staff Sara Lutterbie, Donald
Sheehan, Diane Diaferia, Socoro Delquaglio, Linda Dixon, Jin Bentley andNathan Leoce-Schappin. I have received great help from the staff in the machine shop with building various prototypes, namely Mark Jablonski, Jeffrey
Slechta and Jeffrey Thomas. I am also grateful for the help I reveived from
Eugene Shafto and Charles Pancake working with electronic circuits. Thanks
also to Rich Berscak and Frank Chin for keeping the building and labs running, as well as to Andrzej Lipski and Richard Lefferts for technical help andencouraging conversations.
Outside of my group I have had the pleasure of being a part of the wonderfulcommunity that makes Stony Brook atomic physics an outstanding place topursue studies. I want to thank Spencer Horton for showing me that a world
outside ultracold physics is also interesting and for helping me with many of
my projects, either directly or indirectly. Mehdi Namazi, who is an excellent
physicist and friend, and who made long evenings at the university enjoyable. I
want to thank Bertus Jordaan and Brian Arnold for great times in the physics
basement, help with tricky projects and great road trips. Thanks to Chris
C., Peter S., John E., Oumarou N., Arthur Z., Yusong L., Brian K., Chris I.,
Taichi I., Chuan C., Mael F., Guodong C., Samet D., Steve S., Sonali G., ,xi

Thomas M., Sam R., Alessia S., and everybody that I have failed to mention
for making my time here unforgettable and great.
Outside of the AMO-basement, I want to thank my house-mates over theyears, Lukas, Johannes, Hans, Patrick, Tristan, Cameron and Niveditha. It
was always a pleasure to come home and to be greeted by great people, be able
to share a meal and discuss physics and more importantly discuss the world ingeneral. Thank you for making my stay on Long Island a pleasant one. Special
thanks to Hans, Yiqian, Patrick, Adrian and Kent for accompanying me on
the entirety of my journey and for hanging out on many evenings, allowing me
to forget the daily challenges for a bit.
I want to thank colleagues, friends and faculty from Universität Würzburg,
especially Johannes, Martin, Marvin, Verena, David, Jan and Matthäus forremaining awesome friends over the years and Prof. Assaad for leading andenabling the exchange program that got me to Stony Brook in the first place,along with all the other professors, tutors and friends at Würzburg, who havehelped make me the physicist that I am today.
I want to thank the excellent technical support staff from Toptica, TDK
Lambda, Mini-Circuits, Princeton Instruments and Coherent for answering
my odd requests and weird questions, and really helping me understand theirproducts or helping me fix a piece of equipment on site. A special thanks
to the notorious Jason Reeves from Menlo USA, for being an amazing Stony
Brook graduate and always being the gathering place at conferences (‘is he in
the audience?’).
I want to thank the great community in and around weightlifting at Stony
Brook, which I had the great pleasure of learning from and lifting with. Aftera stressful day at work, I always found great comfort in putting work aside
and being able to enjoy the pleasures of simply picking up weights and putting
them back down. Further, I would like to thank Station Pizza at Stony Brookfor being the last place still open to feed hungry graduate students at 10:30
pm when running late nights.
My family has supported me throughout my Ph. D. and I am eternally
thankful to them. I thank my parents Christine and Ludwig for being the best
parents a child could wish for, for always supporting me and for teaching me
the value of hard work, compassion, and kindness. I am eternally indebted
to both of my sisters, Cordula and Judith, for they also did raise me in away. They were stern with me when I needed it and always explained theworld around me when I asked difficult or annoying questions. They raised abrother and son that is very proud of having received their training, kindness,and support.
Last but far from least, I want to thank you Naomi, for sticking with me forxiiall these years. When times were rough you kept with me through my antics,
you helped me through my toughest times and enjoyed with me my best times,
whether you were physically there or not. With you I always had someone totalk to, and this dissertation would certainly not have come together in theway it has today. You have taught me, maybe more than anyone, about thevalue of communication in science and in general. I am lucky to have foundyou and am looking forward to our future together, in the US, Germany orwherever life may take us.

We are grateful for the generous support by the National Science Foundation of the United States of America (NSF).
xiii

Chapter 1
Introduction
The theory of an ideal monatomic gas is one of the first topics that a beginningphysics student encounters in the great (and vast) subject of statistical physics
and thermodynamics [1]. Classically one can reduce the temperature of a gas
arbitrarily without drastic changes to its behavior as long as interactions canbe neglected. With e. g. attractive interactions present, the system of atoms
should be able to change its phase from gaseous to liquid or solid. In quantum
mechanics [2] and quantum statistical mechanics however, if we reduce thetemperature of an ‘ideal’ noninteracting gas, its behavior changes due to thenotion of ‘point-particles’ breaking down. Atoms of low enough kinetic energy
gain a quantum-mechanical version of spatial extent, the (thermal) deBroglie
wavelength [3]; if it exceeds the inter-particle spacing in the gas, the quantum
statistics of the atoms (bosonic for integer spin and fermionic for half integerspin) dominate the behavior. The dilute-gas approximation breaks down inthe sense that the occupation function for each energy is no longer classical but
has to be replaced by the proper quantum analogue, either the Bose-Einstein
distribution for bosonic atoms (such as alkali atoms with an uneven number ofnucleons) or the Fermi-Dirac distribution for fermionic atoms (such as alkali
atoms with an even number of nucleons).
In AMO physics (atomic, molecular and optical physics), the realizations ofquantum gases are ultracold and dilute: Bose-Einstein condensates (BEC, forbosonic constituent particles) [4–6] or degenerate Fermi-Dirac-gases (DFG, forfermionic constituent particles) [7–9] respectively. Outside of AMO physics,
aspects of BEC physics can be found in superfluid helium [10, 11] and conventional superconductors [12] and DFGs can be found in white dwarf stars and(at least to some degree) neutron stars [13], to name just a few examples.
The first realizations of quantum degenerate atomic gases, in 1995, were
BECs of rubidium [14], lithium [15, 16] and sodium [17] atoms, a few years
1later followed by a DFG of potassium atoms [18]. Ever since these initial realizations, the field has seen explosive growth and sustained interest. Early
work in BECs studied the coherence of these gases [19] (leading to the firstatom laser [20–22]), collective excitations and properties of sound [23, 24], theformation of vortices when imparting angular momentum [25], superfluidity
and critical velocity [26, 27], superradiance [28, 29] and matter wave amplification (four-wave mixing) [30, 31], the emergence of solitons in BECs [32, 33],
the utilization of Feshbach resonances to tune their interactions [34,35], Bragg
diffraction and spectroscopy [36], and their controlled collapse and explosion(Bosenova) [37, 38], among many other studies.
Over the past two decades, many other species have been brought to quantum degeneracy, including hydrogen [39], potassium [18, 40], metastable helium [41], cesium [42], ytterbium [43], chromium [44], calcium [45], strontium [46, 47], erbium [48], and dysprosium [49], which has enabled a plethoraof other studies. The realization of bosonic molecules from two constituent
fermions [50, 51] and the study of the BEC-BCS crossover for cold fermion [9]more recently culminated in the realization of ultracold polar molecules [52–54]
and independently ultracold highly polar atoms [44, 48, 49, 55, 56] which bothallow the implementation of long-range interactions in these ultracold gases.
With the realization of BECs in optical lattices [57, 58] it soon became
obvious that ultracold atoms promise an outstanding degree of control overmicroscopic Hamiltonians and quantum states, allowing the study of engineered many-body systems [59, 60]. Among the more recent experimental
advances of the field are the realization of single site resolved detection ofbosons [61–63] and fermions [64–68] in two-dimensional lattice geometries,
the study of disordered quantum gases [69], the emulation of model systems,
such as artificial graphene [70] and the Haldane- [71] and Hofstadter models [72, 73], spin-orbit coupled systems [74–76], and very recently the realization of the Fermi-Hubbard model (FHM) and its emergent anti-ferromagnetic
phase [77–81], and studies of conduction of fermions through 2D [82] and 1D
channels [83]. In a broader sense, cold atoms in optical lattices can be used toexperimentally access hard-to-compute quantum many-body models via direct
quantum simulation [60, 84–86] and to realize models that have no analogue
or realization anywhere else [71, 87].
In our laboratory we have a long history [88–91] of work with state-selective
optical lattices, including the study of four-wave mixing of matter-waves [92],of superfluidity of bosonic mixtures [93], of (disordered) Bose-glass behavior[94], and of probing ultracold crystals using matter waves [95], as well as
studies of non-adiabatic external-internal-state dynamics [96].
In this dissertation, we now use these state-selective optical lattices to2realize and study an open quantum system [97–99], whose Hamiltonian is thatof the celebrated Weisskopf-Wigner model [100] describing how a two-level
atom in its excited state can decay to its ground state by emitting a photon.
However, we will see that in our system, what is emitted are not single photons
but single atoms.
In the conventional Weisskopf-Wigner model, the necessary transition matrix element for the decay of an excited atom is provided by the coupling of
the atomic dipole moment to the fluctuating quantum electrodynamic vacuum field. The detailed structure of the vacuum plays a major role in how
this decay proceeds. It was originally realized by Purcell [101] that structuring of the environment (e. g. structuring of the refractive index, embedding
emitters into ordered/disordered materials) modifies the vacuum. Some of themost prominent examples of modifying the QED-vacuum [102] are photonic
crystals [103–105] and cavities [106], which can have a pronounced effect on
the spontaneous emission of an atom. Taken to its extreme, modifying thevacuum can fully prohibit spontaneous emission altogether or even makes it areversible process in the form of Rabi oscillations [106].
Since the first prediction of photonic bandgap (PBG) materials (also called
photonic crystals) [103–105] and the first experimental realization of a three
dimensional PBG in the microwave [107] and optical domains [108], thesephotonic crystals have been used extensively to manipulate and control spontaneous emission. This was first done in a one dimensional PBG material [109](distributed Bragg reflector), and later using dies in colloidal opals [110] andions in inverse opals [111], and has culminated in the incorporation of a quantum well [112] and quantum dots [113] into three dimensional photonic crystals.
Researchers were since able to isolate and image single emitters (quantumdots) incorporated into photonic crystals [114], map the local density of statesusing engineered emitters [115], find spectral evidence for non-Markovian decay [116] and modify the Lamb shift [117], to name only a few of the many
advancements in this field. In more recent years, there has been a surge ofinterest in corrugated one-dimensional waveguides (i. e. photonic crystals in
one dimension rather than one-dimensional photonic crystals) as promising
candidates for future photonic devices in quantum information, starting withthe first creation of corrugated waveguides [118] and the trapping (and thereby
coupling) of atoms close to its surface [119]. This has led to the detection ofsuperradiance in these devices [120] as well as coherent atom-atom interaction
mediated by these corrugated waveguides [121]. Corrugated microwave guides
using transmon-qubits as artificial microwave atoms were recently used to indirectly detect a photon bound state [122], and interactions of two artificial
atoms in contact with a corrugated waveguide [123].
3

The focus of this dissertation is the realization of a close relative to suchphotonic systems (which in itself is of major recent theoretical interest [124–127]), in the form of a regularly spaced array of matter-wave emitters, coupled
to a one-dimensional matter-wave guide. The dispersion of our massive particles (as opposed to massless photons) mimics the approximately quadratic
dispersion relation of photons near a photonic bandgap.
As a result of the high tunability and flexibility of our system,
we are able to observe directly,
for the first time in this typeof system, non-Markovian dynamics in real time (rather than using spectroscopy [116]) and areable to directly observe an exotic
bound state, comprising evanescent waves, which, for photonic
systems, has been predicted more
Figure 1.1: Spontaneous emission of matthan 40 years ago [103, 128].
Our experiments are techni- ter waves. For details see chapter 7.
cally (but not scientifically) basedon previous work in our laboratory on non-adiabatic diffraction of matterwaves [96] where we resonantly coupled a free BEC to a lattice-trapped stateand observe diffraction beyond the usual approximation of an adiabatic potential. Now, the initial state is not a free BEC but a sparse array of localized
atoms that get released into free space. The much lower coupling strengths
used in thef present experiment require a much finer control over the resonancecondition than before and the preparation of the samples require detection ofsmall atom numbers:(1) Since a key prerequisite for the study of spontaneous emission in our
matter-wave system is the precise knowledge and control of internal levelspacings, namely the different hyperfine Zeeman level-spacings of the alkali
atom under consideration, we need to better account for the main uncertainty,
the magnitude of the magnetic field applied to the atoms. We develop amethod [129] that enables the precise knowledge of the magnetic field at thetime a measurement is taken, by employing a procedure which divides the setof all available hyperfine states into a subset that is used for the experimentand a subset which is used for determining the magnitude of the applied magnetic field. Through the use of a sequence of multiple RF-pulses, the magnetic
field at the time of measurement can be reconstructed.

42(2) The small atom numbers
F=2
required for sparse samples of0-2 -1
matter-wave emitters rely on absorption images which are freeof any artifacts, such as imaging
6.8
GHz fringes, in order to guarantee highfidelity detection. We describe amethod to remove artifacts frompictures using empty images ob-1 0
tained during data-taking, based
F=1
1
mF
on principal component analysis inorder to filter out only significant
Figure 1.2: Characterization of magnetic contributions from the empty imfields. For details, see chapter 5.
ages.
Latex is formatting dumbly so
I have to make dumbly things.
Outline of the Dissertation
In chapter 2 we will give a brief overview of the fundamental experimental
techniques, such as laser cooling, trapping of cold atoms using magnetic fields
or light, and state-selective trapping potentials. In chapter 3 we will describe
various experimental improvements and upgrades that we were able to contribute to the laboratory, which helped to achieve a more stable operationand increase the rate of data taking. Chapter 4 describes image analysis done
to remove fringe artifacts from our images. The procedure used for (postselection-) stabilization of magnetic fields is introduced in chapter 5. Chapter
6 gives a more technical introduction into the physics of spontaneous emission and spontaneous emission in matter-wave systems. Our experiments on
matter-wave emission into matter-waveguides are described in chapter 7. We
finally give an outlook into future experiments and directions in chapter 8.

1

5

Publications of this PhD work described either in part or in full:• Chapter 5
In-situ magnetometry for experiments with atomic quantum gases
L. Krinner, M. Stewart, A. Pazmiño, and D. Schneble
Review of Scientific Instruments 89, 013108 (2018)• Chapter 7
Spontaneous emission of matter waves from a tunable open quantum
system
L. Krinner, M. Stewart, A. Pazmiño, J. Kwon, and D. Schneble
Nature 559, 589 (2018)
Publications co-authored during the PhD work (covered in othertheses):• Analysis of non-Markovian coupling of a lattice-trapped atom to freespace
M. Stewart, L. Krinner, A. Pazmiño, and D. Schneble
Physical Review A 95, 013626 (2017)• Nonadiabatic diffraction of matter waves
J. Reeves, L. Krinner, M. Stewart, A. Pazmiño, and D. Schneble
Physical Review A 92, 023628 (2015)• Evidence for a Quantum-to-Classical Transition in a Pair of Coupled
Quantum Rotors
B. Gadway, J. Reeves, L. Krinner, and D. Schneble
Physical Review Letters 110, 190401 (2013)

6

Chapter 2Theoretical and Experimental
Basics
The techniques for creating and characterizing Bose-Einstein condensates havebeen well-established for two decades [130]; however the details are often
dependent on the specific application. In the first section, we will briefly
discuss the experimental apparatus and the scheme for creating a BEC in
our laboratory and will then outline experimental techniques that are usedin the remainder of the dissertation. A more detailed description can befound in [88, 89, 131–133] and in the original publication describing our apparatus [134].

2.1

BEC in the Schneble Laboratory

In this section we will outline the tools needed for achieving a Bose-Einsteincondensates, and give a brief description of our specific apparatus.
A quantum degenerate gas has a temperature so low that the thermal
deBroglie wavlength λdB of the constituent particles becomes comparable tothe inter-particle spacing. The figure of merit we will seek to increase is thephase space density [135]:
P SD = n × λ3dB = n



2π~2
mkB T

3/2
,

with the density n, temperature T, mass of the atom m, the reduced Planck’s
constant ~, and the Boltzmann constant kB . The phase transition to a BEC
occurs when P SD ≥ ζ(3/2) ≈ 2.61 [6].

7In general, the position space distribution function of N classical, trapped,
non-interacting particles is given by a Boltzmann factor divided by a normalization factor (which equals the partition function of a trapped particle divided
by the free particle partition function) (β = 1/kB T ) [1]:Z ∞n (~r) = N × exp (−V (~r) β)

−1d ~r exp (−V (~r) β)3−∞N= 0 exp (−V (~r) β) ,Z1where V (~r) is the trapping potential (see later in this chapter). It confines
the atoms to a finite region of space and can be created using e. g. magnetic
fields or light. The potential will usually be a harmonic oscillator potential, in
which case the maximum density is given by N/Z10 . Alternatively, the averagedensity can also be approximated by the expectation value of the radial extent
of the individual atoms [132] (~r = (x, y, z)):
Z
h
i
p
N
d3~rexp −βV (~r) x2 + y 2 + z 3 .
hri = 0Z1
For particles in a three-dimensional harmonic potential the maximum density evaluates toN ωx ωy ωz (mβ)3/2
√
.
nHO =
8π 3/2p
For a linear potential of the form V (~r) = V0 ax2 + by 2 + cz 2 , the maximum
density is√
N V03 β 3 abc.
nlin =
64π

2.1.1

Laser Cooling and Magneto Optical Trapping

The journey towards quantum degeneracy usually starts with a room temperature alkali vapor in an ultra-high vacuum chamber which is subsequently
cooled most of the way to quantum degeneracy by using laser cooling [136,137]
in a magneto-optical trap (MOT) [138, 139] with subsequent molasses-cooling[140–142]. Laser cooling and trapping techniques [143] exploit the mechanical effects of light on neutral atoms at frequencies close to resonance andcan yield atomic samples with typical temperatures in the range of tens of
micro-Kelvins. Details about our magneto-optical trap can be found in [131].
For a brief description of the principle, let us first consider a two-level atomwith two possible internal states |ei and |gi with a transition of energy Ee,g =8

~ωe,g and linewidth Γ. For simplicity, we will assume that the particle canonly move in one dimension. Consider two counter-propagating laser beams,
propagating in the (against the) direction of motion of the atom. The atom canabsorb light (photons) from each laser beam. Upon absorption of a resonant
photon, from one of the laser beams with frequency ωL = ωe,g , the atom recoilswith the photon momentum ~kr = ~2π/λ = ~ωe,g /c (with c the speed of light)
in the same direction of the propagation direction of the laser beam, i. e. itchanges its momentum by that amount. Changing the frequency of the lightaway from transition frequency ωL = ωe,g + δ (detuning δ) then implies thatabsorption events happen less frequently. We will assume that δ < 0, i. e. thatthe laser beams are red-detuned. If an atom happens to be traveling towards
one of the beams, it perceives a blue shift, towards resonance, of the frequencyof that beam (due to the Doppler-effect), and a red shift of the frequency ofthe other beam. Hence the atom will scatter more strongly (more often) fromthe beam that it moves towards, and less photons from the beam it moves
away from. Hence it will receive, on average, more recoil kicks in the direction
opposite to its motion.
The absorption is followed by isotropic reemission, which means that on
average (averaging over hundreds or thousands of scattering events), the reemission does not change the momentum of the atom. Furthermore, the emitted
photons are blue-shifted with respect to the absorbed ones (in the laboratory
frame). This means that the energy of the emitted radiation is higher thanthe energy of the absorbed radiation, where the difference is provided by the
kinetic energy of the atom: kinetic energy is removed from the system by transferring it into the radiation field. If we consider motion in either direction andtake into account both laser beams, one can show [143] that an effective linear
friction force (F~ ∝ −~v ) arises for atoms which are slow enough. Note that
this effect is reversed (anti-friction) for blue detuning δ > 0, the emitted radiation being red-shifted with respect to the absorbed radiation in this case.
The friction is generalized to three dimensions by employing three mutuallyorthogonal pairs of laser beams.
One can also spatially vary the transition frequency ωe,g , for example byusing a quadrupole magnetic field (see also section 2.1.2 and Eq. 2.1) when
there are magnetic sublevels of the excited state |ei present. The quadrupole
field varies the level spacing of excited and ground state due to differential
Zeeman shifts (see section 2.2.4), and in a similar fashion as above, this spatial
variation of the resonance frequency together with counter-propagating laser
beams can create an effective restoring force F~ ∝ −~x towards the origin of thequadrupole magnetic field (again averaging over many absorption/reemission

9

cycles).
With three mutually orthogonal beam pairs, these ingredients thus create
the magneto-optical trap (MOT). The friction and restoring forces are present
everywhere within the capture range in real- and momentum-space, dependingon the laser intensity, detuning, profile, and magnetic field gradient. In ourdensity-limited (high atom number) MOT, the temperature is around 1 mK
(significantly higher than the Doppler-temperature TD = ~Γ/2kB = 146 µK[144] due to re-scattering of photons), with the number of atoms close to 1010
atoms [131], and densities in the 1011 cm−3 range (the diameter of the atomic
cloud loaded into our MOT is several mm).
When the magnetic field is switched to zero in a MOT, a configuration
called optical molasses is achieved. The temperature one can reach when not
requiring a restoring force is considerably lower; sub-Doppler cooling mechanisms [141,142] that arise from optical pumping between the AC-Stark-shifted
Zeeman sublevels of |gi in the polarization structure of the light field, finallygive temperatures on the order of several 10 µK [131] in the last stage of ourlaser-cooling sequence.

2.1.2

Magnetic Trapping

We eventually need a way to spatially constrain our atoms in the absence ofphoton-scattering events in order to attain the low temperatures (∼ 100 nK)
and high densities (∼ 1014 cm−3 ) necessary for our experiments. In the following, we will give a brief summary of the magnetic trapping techniques used in
our apparatus. A detailed description of their implementation can be foundin [132].
We first transfer the atoms from the MOT/molasses into a quadrupole
magnetic trap [145]. A quadrupole magnetic field has the form
 
−x
b  
~
(2.1)
BQ = × −y ,22zwhere b is the gradient along the axial direction. The Hamiltonian for a sta~ = µB gF mF B, where gF is thetionary atom in a magnetic field is Ĥ = −~µ̂ · B
hyperfine Landé g factor (we assume the atom is in the ground state), F is thetotal magnetic quantum number, mF is the projection of the total angular moˆ~ and µB is the Bohr magneton. If we
mentum F~ along the quantization axis B,
assume that the internal state of the atom is fixed (i. e. the quantization axis
can follow the direction of the local magnetic field direction adiabatically), the10Hamiltonian for an atom moving adiabatically in a spatially varying magnetic
field isp~ˆ 2
~ (~r) ,
+µ B
(2.2)
Ĥ =
2m
~ r) = B~ Q and µ = mF gF µB [132]. Hence, if mF and gF have identical
where B(~signs, we have created a trap for an atom in such a hyperfine state (quadrupole
magnetic trap). These types of atoms are called low-field seekers. For the
52 S1/2 ground state of 87 Rb, low-field seekers are |F = 1, mF = −1i, |2, 1i
and |2, 2i.
A major drawback for quadrupole magnetic trap is that it can only holdatoms down to a certain minimum temperature. The assumption made initially that the atoms stay in the same hyperfine state (including mF ), becomes
easily violated for atoms moving close to the trap center due to nonadiabaticity. At the trap origin, the field vanishes and there is no direction for thequantization axis. A particle that passes near the trap center would experience a (near) sudden change in quantization direction (exceeding the Lamorfrequency), likely causing it to flip the sign of mF , its total angular momentum
z-projection, which in turn transforms a trapped into an anti-trapped particle (Majorana loss). These losses scale as ΓM ∝ (b/T )2 [146], and prohibit
lowering the temperature below 100 µK (for typical trap parameters). There
are several strategies to circumvent this problem, for example Ioffe-Pritchard
traps [147,148] with a non-zero field minimum, plugging the hole with a repulsive laser beam [17, 149, 150] as well as time orbiting the field zero faster than
the atoms can follow (TOP-trap) [146]. This latter strategy is the approachtaken by our group.
A time-orbiting-potential trap (TOP-trap) is created by starting with aquadrupole field as described above, and adding a rotating bias field of theformsin(ωt)~ rot = B0 × cos(ωt) .
B
0
The time-averaged trapping-potential created in a TOP is then [132, 146]
2
2
z2
mωTOP
ρ2 mωTOP+
V (ρ, z)TOP ≈ µB0 +162pwhere µ = µB mF gF and ωTOP = µ × b2 /mB0 . The field zero moves rapidly
around the center of the trap, and its trajectory is typically called ‘circle ofdeath’ due to atoms undergoing Majorana loss there. The radius of the circle ofdeath is rCoD = 2B0 /b. Majorana losses can hence be suppressed if ω  ωTOPand kB T  µB0 .

11

2.1.3

Optical Trapping

An important requirement for the experiments discussed in this dissertationis that we need to trap atoms independent of their internal state at very low
temperatures and at a constant magnetic field. In order to achieve this, weemploy another means of trapping the atoms, namely the optical dipole force
created by far-off resonant beams of light. A trap formed using this effect isusually called FORT (far-off resonant trap) or ODT (optical dipole trap) [151].
An atom interacting with a static electric field will have an induced electric
dipole moment (polarization) proportional to the field strength.
~,d~ = αEwith α the complex polarizability and hence its energy will be [151]2
1
1  ~ ~~ .
U = − Re d · E = − Re (α) E
22This result still holds for oscillating electric fields, at least as long as resonant effects can be neglected. Resonant effects include the absorption of light
if the frequency of light is close to the transition frequency between two energy
levels of the atom, at which point dissipative forces, as described for the caseof the MOT and the molasses become significant.
Below the atomic resonance, one can reasonably assume that the polarization follows the applied electric field instantaneously (red detuning δ < 0,lowers energy, Re(α) > 0) while above resonance the polarization has a πphase angle with respect to the incoming electric field (blue detuning δ > 0,increases energy, Re(α) < 0). We can write1
U (t) = − Re (α) |E0 |2 sin2 (ωL t),
2with ωL the frequency of the light. Taking the time-average yields
I
1
Udipole = − Re(α)|E0 |2 = −Re(α) ,4
0 cso that we get a potential landscape that is proportional to the intensity of
the light at that point in space. This is the most basic form for describing
the optical dipole force of light on atoms. While it usually remains true thatthe dipole potential will be directly proportional to the intensity of the light,
the polarizability will generally depend on frequency and the internal state of
an atom, as well as the polarization of the light used, which will be discussed
12

in more detail in section 2.2.5. While we assumed that we are far away fromresonance, the equation above also holds closer to resonance (U ∝ 1/δ), withthe added caveat that spontaneous scattering processes occur more frequently(∝ 1/δ 2 ), thus heating the atoms more quickly.
A configuration of two Gaussian red-detuned laser beams crossing at aright angle yields an optical potential of the form

2x2 2z2
U (~r) ≈ Uy × exp − 2 − 2
wx
wz


2
2y
2z2
+ Ux × exp − 2 − 2 ,wywzwith wx and wy and wz being the beam radii in the x and y and z directions
respectively, thus creating an approximately harmonic trap around the minimum. Notice that the (usually very weak) change in potential of each beamalong the propagation direction was neglected in the above expression, which
breaks down in the case of very tightly focused beams (w ∼ λ) which quickly
diverge.

2.1.4

Evaporative Cooling

Unlike magneto-optical traps, magnetic and optical traps with their conservative potential can by themselves not provide an increase of the phase-spacedensity. In our magneto-optical trap, we achieve a PSD of ∼ 5 × 10−7 [132].
To achieve the necessary increase by six to seven orders of magnitude, we employ the technique of evaporative cooling [130, 152]. For evaporative cooling
one first removes the hottest fraction of atoms from the trap, and then allows
subsequent rethermalization and repeats this process many times. If one removes a low enough percentage of the atoms in the trap, the PSD increases.
Therefore, at least in principle, slower evaporation is better, where the limit
is set by inelastic collisions and collisions with the background gas.
Evaporation in the quadrupole/TOP trap is achieved using a ‘RF-knife’.
An ‘RF-knife’ uses oscillating magnetic fields (1 MHz to 100 MHz) to resonantly change the internal state of an atom from a trappable (low field seeking)to an untrappable (high field seeking) state, similar to the circle of death in
the TOP-trap but at finite field magnitude (for a more detailed description
see also the later section 2.2.4). The transition frequency in this case depends on the magnitude of the magnetic field and is the difference of energies
EB = mF gF µB Bz = mF gF µB |B|:
ωRF = |B(~r)|µB |gF (mF − m0F )| /~ .
13

The field grows away from the trap minimum, and hence one can choose an
oscillating field only resonant with magnetic fields corresponding to the highest
potential energies of the atoms. This establishes evaporation by expelling only
the most energetic atoms, with a new effective trap depth set by the expulsion
energy. One can slowly reduce the frequency of the radiation to gradually lower
the effective depth of the trap, thus lowering the temperature (and number)of the atoms in the trap, with a dramatic increase in the density and the PSD.
Optical traps naturally have a much lower total depth than magnetic traps(of order 1 − 10µK), such that one can simply lower the laser power of theindividual trapping beams, and thus gradually lower the trap depth and again
evaporate off the hottest atoms [153]. A drawback of this procedure is thatit also lowers the confinement and rethermalization rate, but for the crossedbeam configuration this effect dos not compromise feasibility of achieving BEC[154].

2.1.5

Achieving BEC

(b)

(a)

Figure 2.1: (a) Schematic of the apparatus, with the full-glass vapor cell (athigh pressure p ∼ 10−9 mbar), the transporter coils and science cell at ultrahigh vacuum (UHV, p ∼ 10−12 mbar. MOT-beams (solid dashed, red), repump
beam (translucent) and pumping beam (solid black, see section 3.3) added to
the MOT region. All MOT beams are retro-reflected. (b) Close-up of thescience cell (translucent) with the TOP coils (orange) and the ODT beams(red) in between the large coils used for MOT and magnetic trapping (gray).
Both graphics (a) and (b) reproduced from [134] with kind permission from D.
Pertot and with reproduction permission kindly granted by IOP Publishing.
c IOP Publishing. All rights reserved. The red and black beams in (a) (atthe position of the MOT) were added by the author.
14

We finally have all the ingredients necessary for creating an ultracold gas.
We will now outline a typical sample preparation for the specific case of ourexperiment.
We use the atomic species Rubidium-87 (87 Rb) which has the ground state
5 2 S1/2 with possible hyperfine states |F = {1, 2}, mF i, and a first excited state
52 P1/2 (F = {1, 2}, D1 -transition) and a second excited state 52 P3/2 (F ={0, 1, 2, 3}, D2 -transition), where our experiment uses D2 -light exclusively.
The resonant wavelength and natural linewidth of the D2 -transition are λ0 =780.2 nm and Γ = 2π × 6.065 MHz respectively [144] (see also Fig. 2.2).
Figure 2.1 shows a schematic of our apparatus [89,134]. Rubidium is loaded
from a low background vapor pressure (p87 Rb ∼ 10−9 mbar, p0 < 10−10 mbar)
into a MOT, consisting of three pairs of counter-propagating beams, red detuned from the D2 , F = 2 → F0 = 3-transition (cycling light) by 3.5 × Γ =18.7 MHz. An additional laser beam resonant with the D2 , F = 1 → F0 = 2transition (repump light) illuminates the atoms from one direction to avoid
buildup in the inactive F = 1 states, which are accessible in the rare eventof absorbing cycling light and transitioning to F 0 = 2 instead of F 0 = 3. The
vanishingly low probability of this occurrence on an event by event basis isdwarfed by the large number of cycling light absorption events, causing all theatoms to be optically pumped into F = 1 on a timescale of < 1 ms (the scattering rate into the off-resonant F 0 = 2 state is still on the order of 104 − 105 per
second).
After loading approximately 1010 atoms into the magneto-optical trap ofsize ∼ 0.5 cm at approximately 1 mK (P SD ≈ 10−7 . . . 10−8 ), we turn off themagnetic gradient field and hold the cloud for a short while at zero magnetic
field gradient (and zero field) while detuning the cycling light progressively
more, utilizing sub-Doppler cooling to lower the temperature of the atoms
to < 50 µK (P SD ≈ 10−5 . . . 10−6 ). We subsequently optically pump theatoms into the F = 1 state by turning off the repump-light and additionally
illuminating the light with ‘depump’ light (D2 , F = 2 → F 0 = 2). We thenturn on the quadrupole magnetic trap (b = 350 G/cm, gradient in the axial
direction) to trap about 1/3 of the initially loaded atoms (N ≈ 109 , T ≈0.5 mK, P SD ≈ 10−8 ), and transport the magnetically trapped atoms out ofthe vapor cell (p ∼ 10−9 mbar, lifetime τ ∼ 1 s) into a low pressure region(p < 10−11 mbar, lifetime τ > 100 s), where we can evaporate slowly enoughto allow an increase in phase-space density, losing about 30% to 50% of thetrapped atoms on the way out of the chamber (transport takes about 0.5 s toleave the chamber).
We then first evaporate in the quadrupole magnetic trap using a the aforementioned RF-knife technique. We use a frequency sweep from 80 MHz to
15

2

3

F=3
F=2
F=1
F=0

F=2

, σ--po90.1 nml--pol

D1, λ=795.6 nm

SDL, 7.1 nm, σ5 S1/2ΔEHFS=
6.8 GHz1

D2, λ=780.2 nm

ΔEHFS,D1=
0.8 GHz0
SDL, 79

2

mF
0

Cycling

ΔEHFS,D2≈
0.5 GHz

ODT, 1064 nm, π-pol

52P1/2-1

Repump52P3/2-2

Depump

-3

F=1

F=2
F=1

Figure 2.2: Level Scheme of Rubidium-87, with the ground state 52 S1/2 and thetwo first excited states 52 P1/2 and 52 P3/2 (split by spin-orbit coupling). The
remaining degeneracies are broken first by the hyperfine-coupling (interaction
with nuclear magnetic moment) and then by magnetic fields/Zeeman shifts if
present. The main spectroscopic lines of Rubidium are the transitions fromits ground state into either excited state, which are called the D1 and D2lines respectively. The main transitions used in the MOT and molasses phase(not including information about the mF levels and polarizations involved)
are drawn in solid (repumper, depumper and cycling light). Far off resonant
optical trapping is done red detuned from the D1 -line, several tens of THz
at λ = 1064 nm (short dashed line). The state-selective potential is createdusing σ − polarization at 790 nm (long dashed lines), which yields no potential
for |2, 0i-atoms and an effectively blue detuned (or repulsive) potential for
the |1, −1i-atoms. The wavy solid arrow indicates a magnetic dipole allowedtransition and the wavy dashed arrow indicates a magnetic dipole forbidden
transition.
10 MHz in approximately 10 s, after which Majorana-losses stop the efficientevaporation from the quadrupole magnetic trap (T ≈ 50 µK, N ≈ 108 , P SD ≈10−4 ). Finally we add the aforementioned rotating magnetic field to create√theTOP-trap (BT OP = 18G, trapping frequencies ωr ≈ 2π × 25 Hz = ωax / 8),16

and continue evaporating (either in the TOP magnetic trap or in the ODT)
to finally reach BEC at a temperature of Tc = ~ω̄N 1/3 /kB ∼ 300 nK, whereN ∼ 106 is the remaining atom number at the transition and ω̄ ∼ 2π × 60 Hz
is the geometric mean of the trapping frequencies [135].
horizontal beam

m

+
-1 1

Ca

vertical beam

OL 2

a
er

&
T1
D
O
G

IM

SDOL

z
x

y

Scienceλ/4 Wave-Plate

m
er
Ca

-1 1
+

Mirror

a

IM

G

O

D

T2

&

Cell

OL 1

Dichroic Mirror

Figure 2.3: Sketch of the optical schematic. Atoms (disk) are in a crossed
beam optical dipole trap and (solid lines with arrows) inside the UHV sciencecell (glass cuvette). Imaging (large translucent beam) is done along either
ODT-direction and is separated from the ODT beams using a dichroic mirror. When atoms are in an image, they cast a shadow in the illumination
onto the camera. Atoms are then recorded as absorption signals on a PIXISCCD camera. Optical lattices along the ODT1/ODT2 directions are createdby varying the amount that is retro-reflected using a pair of acousto-opticalmodulators [134]. Along the vertical direction (dashed green line) we use anear-resonant laser beam (∼ 790 nm) and a λ/4-wave retarder to create stateselective optical lattices (SDOL). The lattice depth is controlled only by theforward beam since the beam is fully retro-reflected. Coordinate system is thesame as in Fig. 2.1

17

2.2

Condensates in Optical Lattices

In this section we will discuss the basic techniques that we use after preparing
an optically trapped ultracold atomic cloud, including optical lattices, hyperfine state manipulation and optical imaging.

2.2.1

Optical BEC, Detection and Optical Schematic

From a BEC (or thermal cloud) in the magnetic trap we now typically transfer into the optical trap, where we are in a convenient position to conduct theexperiments discussed later in this dissertation. The optical schematic of themain apparatus is shown in Figs. 2.1 and 2.3. We trap the atoms in the intersection of two perpendicular, horizontal beams (at wavelength λ = 1064 nm)
creating a crossed optical dipole trap in which the final evaporation usually
happens (beam waist w = 135 µm, beam power P ∼ 3 W (trapping andtransfer from TOP-trap) and P ∼ 1 W (after forced evaporation)).
After concluding experiments, we quickly (∼ 1 µs) turn off all trapping
potentials and let the atoms fall for ≈ 15 ms. The cloud is usually too small
to be imaged initially (∼ 10 µm), but by dropping the atoms, we allow themto expand in time-of-flight (TOF), with the resulting atomic distribution after
time of flight representing (approximately) the momentum space distribution
at the time of release. We use resonant cycling light (D2 , F = 2 → F 0 = 3) toilluminate the atoms, which then cast a shadow onto a remote-triggered 16-bit
CCD camera (Princeton PIXIS 1024B). A typical detection cycle consists ofilluminating the atoms (a) (see Fig. 2.4 (a)), waiting until the atoms leave theimage area, then illuminating again, which creates a reference image (b) (seeFig. 2.4 (b)) and last taking a dark image without illumination (c) (not shownin Fig. 2.4).
A standard absorption image/optical density is created following the recipe(see Figs. 2.4 and 2.5) [130]apix − cpix,
ODpix = −lnbpix − cpix
where the number of atoms in any given pixel is given by N = A ∗ ODpix
where the factor A depends on pixel size, resonant scattering cross section
given polarization, bias field direction (and technically hyperfine state), andapix are the individual pixels of image (a) (and similar for (b) and (c)). In ourcase A ≈ 125 .

18

60k

40k

20k

Pixel Counts

(a)

(b)
0

Figure 2.4: Typical raw camera images for (a) the illumination of the F = 2- atoms and (b) the reference illumination without any atoms present. Color
scale gives the counts registered on each camera pixel, which is roughly comparable to the number of photons per pixel. The red circle indicates an absorption signal which signifies the presence of atoms. The dark image without
illumination is not shown here.
Stern-Gerlach separation
In order to separately image the population in the different magnetic Zeeman
sublevels of F = 1 and F = 2 we employ a technique called Stern-Gerlach
separation (as mentioned in Fig. 2.5) [130, 155]. As we have already seen in
Eq. 2.2 the energy of a particle in a magnetic field depends on its internal
state. If we apply a spatially dependent magnetic field (i. e. a magnetic field~ (~r) = b0 x~ex we subject the atoms to a
gradient) of the approximate form B
force which is dependent on their internal state:
F~ (mF ) = −µB gF mF b0~ex .
This force can be used to spatially separate different magnetic Zeeman sublevels at the beginning of time-of-flight.
In our experiment we use the same coils which also generate the quadrupole
magnetic trap to Stern-Gerlach separate different magnetic sublevels. We firstphysically move the coils in the negative x-direction (directions are defined in
Figs. 2.1 and 2.3) and slowly (over the course of 1 ms) rotate the externallyapplied bias-field from the z-direction into the x-direction. We then turn on
the current through the coils to generate a magnetic field gradient of b0 ∼19

Figure 2.5: Typical absorption image generated from Fig. 2.4 by OD =−ln [(a − dark) / (b − dark)] (dark images not shown in Fig. 2.4) to generate the lower panel of this figure (F = 2 atoms). The same is done for the
F = 1 atoms (raw image not shown in Fig. 2.4). Most of the inhomogeneities
and fringes are eliminated using this method, however some residual fringing
remains, which becomes important for lower atom numbers (two examples arecircled). The removal of these fringes will be discussed in chapter 4. The
hyperfine state is fully resolved using Stern-Gerlach separation of different
magnetic moments and imaging F = 1 - atoms and F = 2 - atoms in different
illuminations.
100 G/cm, which points approximately in the x direction (i. e. the radial
direction of the coil assembly). The field gradient is applied for ∼ 5 ms which
imparts momentum onto the atoms, which can then spatially separate during
the remainder of time-of-flight (∼ 10 ms). This separates magnetic species ofdifferent magnetic moments. In the case of 87 Rb the Landé g-factor is of equalmagnitude for F = 1 and F = 2 but of opposite sign. We can separate allstates by first imaging F = 2 and then imaging F = 1 as described earlier inthis section.

2.2.2

Optical Lattices and Lattice Calibration Using Talbot Kicking

In experiments with ultracold atomic gases, one of the most important tools is
the optical lattice, a potential landscape created by retro-reflection of a single
frequency (linewidth less than 1 MHz) laser beam, typically with a TEM-00
20

Gaussian spatial mode, onto itself. Around the center of a Gaussian laser
beam, the electromagnetic field is well approximated by plane electromagnetic
waves of amplitude E0 (here the laser beam is propagating in the x-direction):~E(x,t) = Re ~ε E0 ei(kr x−ωt)√E~r (x, t) = R × Re ~ε E0 ei(−kr x−ωt) ,where kr = 2π/λ = ω/c is the wave-vector of the light, R is the reflectivity ofthe mirror reflecting the laser beam/plane wave, ω the frequency of the lightand ~ε the polarization of the electromagnetic wave. We have seen before that
the
ε0 c
I0 =
|E0 |2 ,2which is constant in space for a (time-averaged) plane wave.
The intensity, and thus the optical potential, of the interfering plane waves~
E(x, t) and E~r (x, t) (assuming that R < 1) is then√U ∝ I0 1 + R + 2 R cos (2kr x) ,so that the maximum depth of the interference pattern (i. e. for R = 1) isthen 4 × I0 .
We have now created a perfectly periodic, perfectly sinusoidal potential for
our ultracold atoms. The sinusoid spatial shape is especially nice, since one caneasily calculate the band-structure (dispersion relation) of such a potential.
In our experiment we have three pairs of counter-propagating beams. In thetwo horizontal directions, we take the optical trapping beams and double-pass(i. e. ‘there and back’) them through a pair of acousto-optical modulators, enabling us to independently turn on and off (with R ≈ 0.15) the retro-reflectedbeam and control its intensity see Fig. 2.3 and [89,134]. The third direction isa ‘state-selective’ optical lattice. Only the forward propagating beam is controlled and the backward propagating beam is created by full retro-reflection
using a mirror. Details will be described in section 2.2.5.
The determination of the exact depth of this lattice potential is of major
experimental importance. Thus far, we only know it up to a proportionality
constant determined by reflectivity and polarizability of the atoms at a certain
wavelength and polarization, each of which can be difficult to know precisely
in a given setup (at least at the precise position of the atoms). We will now
discuss a method to circumvent needing to know these parameters precisely
by directly inferring the lattice depth seen by the atoms using a method called
Kapitza-Dirac diffraction [57, 156, 157].
21

-6

0.9

-4

-2

p/pr
0

2

4

6

0.45
OD

0

Figure 2.6: Typical absorption image resulting from a Kapitza-Dirac diffraction (KDD) pulse from a lattice of depth V0 = 30(1) Er . Pulse length is 10 µs
and the time of flight was 17.4 ms.
Starting with a BEC, we pulse on the lattice potential for a short time (τ ∼10 µs, < 1 µs rise time) and then immediately turn off all trapping potentials,
initiating time of flight, and then finally detect the momentum distribution of
the atoms. The atoms diffract from the optical potential, creating a regulardiffraction pattern with wave-packets centered at momenta n × 2 × ~kr (kr =
2π/λ, n ∈ Z).
We show the result of a typical experimental run in Fig. 2.6, where we see
clearly separated momentum orders at integer multiples of 2 pr (pr = ~kr ),with the lattice depth encoded in the relative populations of the individual
momentum orders.
The Hamiltonian which governs this system is (neglecting interactions)Ĥ = p̂2 /2m + V0 sin2 (kr x). Since a BEC is a wavefunction which is essentially a purely zero-momentum matter wave, the dynamics significantly simplify to a discrete set of coupled equations between the amplitudes cn (t) of
the n-th momentum order (n = {0, ±1, ±2, . . .} corresponding to p/2~kr ={0, ±2, ±4, . . .}). The resulting differential equations are [157, 158]:
iċn =V0
4Er n2cn +(cn−1 + 2cn + cn+1 ) ,~4~

where Er = ~2 kr2 /2m is the recoil energy.
22

In the limit of short pulses (much shorter than either the inverse recoil
frequency Er /~ = ~kr2 /2m or the inverse of the total potential depth V0 ), there
exists an analytical solution for the population of the nth diffracted peak as afunction of the potential depth of the lattice as Pn = Jn2 (V0 tpulse / ~) [157,158];
however this eventually breaks down once the potential becomes too shalloow
and longer pulse times are needed.
We can find numerical solutions of the differential equations [157] for agiven finite pulse time and a range of potential depths to find the best match
between simulated population and observed population extracted from absorption images.
We have now described a way to reliably calibrate intermediate and deep
lattice depths (V0 > 1Er ) using short (t < 1 µs) and intermediate length(t ∼ 10 µs) pulses, but the population in the non-zero orders becomes verysmall for small lattice depths. For a lattice depth of V0 = 0.1Er , the relative
population in the nonzero orders is on the order of 10−4 , which is beyond thedetection limit of our apparatus. It will later be necessary to quantify lattice
depths at this level.
Thus far we have only applied one lattice pulse to the atoms. We cantry to amplify the population in the diffracted orders by pulsing more thanonce. As is discussed in [159], this situation generally results in essentially localizing in discretized momentum-space, if the time between KD-pulses is notcarefully chosen. However, we can employ a technique called Talbot-resonant
kicking [160,161], where the pulses are spaced by the inverse of the two-photon
recoil frequency t2kr = h/4Er = 68 µs. For small lattice depths, the population
diffracted in the first diffracted order oscillates sinusoidally with a small amplitude when applying a single pulse of variable length. Using pulses shorter
than the oscillation frequency (t2kr ) results in, (initially quadratical) growthin the diffracted population (see Fig. 2.7). The population grows beyond theamplitude of the oscillations, allowing for a much more accurate determinationof small lattice depth.
We again can directly simulate the equations governing the time evolutionXt − tj V0
4Er n2cn +Box(cn−1 + 2cn + cn+1 ) ,ic˙n =~t4~w
j
now with the Box-functions of width tw = 10 µs and spaced by tj −tj+1 = 68 µs
turning the lattice potential on and off.
In Fig. 2.7 we show a numerical computation of this for two different smalllattice depths. We see that using a long pulse does not indefinitely increase
the diffracted population, but rather results in the aforementioned oscillations
23

Population in ±2pr [%]0

Time [μs]
50

100

1

.5

0
0

5
Number of Kicks

10

Figure 2.7: Simulated relative population of the first diffracted order for Talbot
resonant lattice pulses on 87 Rb for V0 = 0.1 Er (disks) and V0 = 0.2 Er(circles) for a pulse time of τ = 10 µs and a pulse spacing of T = 68 µs. The
dashed (dashed-dotted) line, labeled by the upper axis, is the population in thediffracted orders when a single pulse of varying length is used at potential depthof 0.1 Er (0.2 Er ). The dashed line only is magnified five-fold for visibility.
Oscillations continue at the level indicated in the figure.
of the population. In contrast the resonantly pulsed lattice diffraction grows
quadratically, greatly amplifying the diffracted orders for more accurate gauging of lattice suppression for state-selective lattices.
This effect is limited by the eventual real-space separation of momentum
space-orders after about 10 kicks for the ±2pr and 0pr orders. Using thismethod we can reliably set an upper bound of lattice depth of V0 = 0.2 Er .

2.2.3

Hyperfine-State Rabi Oscillations

Among the most important abilities in our laboratory is the arbitrary manipulation of the internal hyperfine-ground state. As was already mentioned, we
use 87 Rb in its ground state, which has F = 1 and F = 2 possible hyperfine
ground states, with the initially prepared hyperfine state |F = 1, mF = −1i.
We have seen in section 2.1.4 that the hyperfine state can be changed using
oscillatory magnetic fields that couple the magnetic dipole transition between
ground-states. We will elaborate further on this experimental technique here.
We first consider a magnetic two-level system in an external bias magnetic
field Bz (along a field quantization axis, here z) with states |↑i and |↓i. The
24

system will later be subject to another, oscillating magnetic field Brot sin (ωrot t).
A magnetic two-level system is governed by the dipole Hamiltonian (we apply
a rotating field):~ =µĤ = µ~ · B(t)~ · (Bz~ez + ~erot (t)Brot ) .
After transforming into a co-rotating frame (rotating wave transformation)
we obtain the typical Rabi-Hamiltonian [2, 162]:
ĤRabi =~(δσz + Re (ωR ) σx + Im (ωR ) σz ) ,
2

(2.3)

where δ = (µBz )/~ − ωrot and ωR = µBrot /~. From here on out we will assume
that ωR is real, an assumption which can typically be made if only one field ofunchanging strength is used. Assuming that the initial state is |↓i the resulting
time-evolution of the population in the |↑i state is:
q

ωR2
2
2
2
P↑ (t) = 2
sinωR + δ t/2 .
(2.4)
ωR + δ 2
This equation is visualized in Fig. 2.8 as a function of detuning δ. In
Fig. 2.8 we can see that for short pulses (compared to the Rabi-frequency
ωR ) the width of the spectrum (vary detuning and keep pulse time fixed) islimited by the pulse time, while the width for large detunings is limited by
the Rabi-frequency ωR . On the other hand, for constant detuning and varying
time evolution we get the well known Rabi-oscillation. On resonance (δ = 0)
we get full population transfer to the |↑i state with periodicity
ωR , while offp2
resonance we get partial transfer with shorter periodicity ωR + δ 2 .
This treatment for |↓i and |↑i generalizes for any magnetic-dipole allowed
pair of states in the (weak-field) rubidium ground state (e. g. |1, −1i ↔ |1, 0i
and |2, 0i ↔ |1, ±1i are allowed, while |1, −1i ↔ |2, 1i is dipole forbidden,
see also Fig. 2.2).
Experimentally the oscillatory field is realized using either a single loopof copper wire mounted directly beneath the science-glass cell (for transitions
within the F = 1 and F = 2 states, f ∼ 10 MHz) or using a microwavewaveguide antenna (for transitions between F = 1 and F = 2) which irradiates
the atoms with radiofrequency radiation at 6.8 GHz (see also Fig. 5.1 (a)).
Transitions between these states can be detected by separating atoms in
different hyperfine states by their magnetic moment by using Stern-Gerlach
separation (see section 2.2.1). This horizontally separates the internal stateswith different magnetic moments, which means that the states |1, −1i ↔ |2, 1i,|1, 0i ↔ |2, 0i and |1, 1i ↔ |2, −1i are still overlapped, see Fig. 2.5, where25

Population |↑〉

1
(a)

(b)

(c)

(d)

.5

0
-5

0

5

-5

0 5
-5 0 5
Detuning δ [ωR]

-5

0

5

Figure 2.8: Rabi Spectra: Population transfer to the |↑i state when startingin the |↓i state as a function of detuning from resonance. Pulse areas aret × ωR = π × {0.1, 0.5, 1, 10.5} for (a), (b), (c), and (d). respectively. Dashedline in (d) is the Lorentzian envelope function ωR2 / (ωR2 + δ 2 ).
for example |2, 0i and |1, 0i show up in the same position horizontally andare only separated vertically. We circumvent this by first imaging only the
F = 2 atoms (which disperses them and pushes them out of the imaging
region) using resonant cycling light (D2 , F = 2 → F 0 = 3, cycling light).
Then we utilize the ability of our camera to shift illuminated pixels of theimaging region into a dark (unused) region of the chip (light to this region isblocked by a razor blade), and image the F = 1 atoms by optically pumping
them (D2 , F = 1 → F 0 = 2, repumping light) into F = 2 and then imaging
them the same way approximately 3 ms later. Then both illuminations are
shifted up again futher into the blocked off region of the chip, and create thereference image (illumination without atoms) another 3 ms later. We repeat
this procedure without illumination of the camera to get the ‘dark’ picture forthe three images created before, thus creating absorption images which allow
a full reconstruction of the hyperfine ground state distribution.
These techniques can be combined to measure a Rabi-resonance curve.
If we start with a fully polarized sample (all atoms in one hyperfine state(say |↓i), the resulting time evolution of the Hamiltonian Eq. 2.3 is a Rabi
oscillation of the form given in Eq. 2.4, which, for constant detuning δ, isjust a sinusoidal oscillation of the population between |↓i and |↑i, while forconstant pulse time and varying detuning we get a sinc-squared response,
typically referred to as Rabi spectrum. Figure 2.9 shows such a Rabi spectrum
at low field (approximately 5 G) obtained by irradiating |1, −1i atoms with26

microwave radiation to transition to |2, 0i. The extremely narrow line-widthshows the good control of magnetic field. A a more in depth discussion follows
in chapter 5.

Population

.3

.15

0
-1

0
δ [kHz]1

Figure 2.9: Rabi-resonance spectrum data for a Rabi-oscillation between |2, 0i(initial state) and |1, −1i. The pulse time here is 1.8 ms and the solid curve
is a fit to the data with a Rabi-frequency of 114(2) Hz. The offset from zerofrequency is due to a slight shift of the levels due to the optical trap shifting|2, 0i and |1, −1i very slightly differently.

2.2.4

Hyperfine Structure in Magnetic Fields

In the above section we mention the resonance condition. For alkali-metals intheir ground state, we can find the resonant oscillation frequency of the individual allowed hyperfine pairs at finite magnetic field by calculating the magnetic
field dependent energy of the involved hyperfine states. The eigenenergiesof the respective hyperfine states only depends on one externally determined~ . We will assume for simvariable: the magitude of the magnetic field B = B
plicity that field is only applied in the z-direction (B ≡ Bz ). The eigenenergiesof the hyperfine-ground states of an alkali atom can then be found using theBreit-Rabi formula [144, 163]:∆EHFS+ gI µB (mI + mJ )B
4I + r
2∆EHFS
4(mI + mJ )x±1++ x2 ,
2
2I + 1E (J = 1/2, I, mJ , mI , B) = −

27

(2.5)

Figure 2.10: Breit-Rabi diagram for the 87 Rb 52 S1/2 ground state showing theenergy dependence of the eigenstates of the Hamiltonian Eq. 2.5 on the external
applied magnetic field. In the low field regime the F quantum number is agood quantum number (eigenstate of the Hamiltonian), while in the high fieldregime the z-projection of mJ (and mI ) are good quantum numbers.
with x = µB B (gJ − gI ) /∆EHFS and ∆EHFS the hyperfine splitting between
the ground states (F = 1 and F = 2 with projection quantum number mF
arbitrary) at zero external field. This formula assumes, that the magnetic
fields involved are much smaller than the effective magnetic fields associatedwith spin-orbit coupling, i. e. that fine-structure splitting (for rubidium thisis the energy difference between 52 P3/2 and 52 P1/2 ) is small compared to the
Zeeman energies of the levels in a magnetic field (fulfilled for DC magnetic~
fields accessible in a laboratory). This ensures, that spin-orbit coupling (L~ to the total angular momentum J)~ is not influenced by the magnetic
and S
2~ and Jz are good quantum numbers. Note that this formula
field, and thus |J|is only valid for the ground states of alkali atoms. In the case of excited states(e. g. 52 P3/2 or 52 P1/2 ) the full Hamiltonian for the hyperfine energies has tobe solved numerically [144, 164].

2.2.5

State-Selective Trapping Potential

A core requirement in our laboratory and especially for the experiments described in chapter 7 is the ability to create a state-selective optical lattice,i. e. a lattice that depends on the hyperfine state which the atoms are in.

28

V/Er

100
0
(a)

Γsc, pk [1/s]

-100

6
3
(b)
0
770

780

790
λ [nm]

800

Figure 2.11: (a) State selective optical potential depth and (b) peak spontaneous scattering rates (b) as a function of wavelength calculated for thestates |2, 0i (dashed) and |1, −1i (solid), with the tune-out wavelength (λ ∼790.1 nm) marked by the vertical dashed line for an intensity of 0.38 kW/cm2 .
The lattice potential depth here is V|1,−1i = 30Er and the scattering rate isΓsc, pk = 4.3 sec−1 for |1, −1i and Γsc, pk = 3.4 sec−1 for |2, 0i. This is theconfiguration and state-pair used in chapter 7.
State-selective optical lattices have seen a lot of experimental efforts, including our group [92–96,165–172]. A detailed description of state-selective opticalpotentials can be found in [88, 89, 173].
The basic idea behind a state selective optical lattice is that red-detuned
light (light-frequency below the closest resonance) attracts atoms into higher
intensity, while blue-detuned light (light-frequency above the closest resonance) repels atoms from regions of high intensity. When one now considers
light of a frequency in the middle between two resonances, there must be acrossover point, where the atoms experience no potential (tune-out point). In
general of course the tune out point will depend on the details of the excitedstates (i. e. how strong the coupling to the respective states is), but moreinterestingly, the tune out point also has to depend on internal state of theatom and polarization of the light (see also Fig. 2.2). We can imagine a pair29

of states where selection rules would allow for one state to couple to boththe blue-detuned and the red-detuned excited state, while the other state is
only allowed to couple to the blue-detuned excited state, thus creating a situation where one of the states is ‘tuned out’, while the other state experiences
effectively a blue-detuned optical potential.
For an atom with arbitrary excited state structure the optical potentialfrom the AC-stark shift of a given ground state |ii can be calculated using second order perturbation theory (the first order vanishes due to the absence of anelectric dipole moment in neutral atoms), where the interaction-Hamiltonian~of the light is given by the dipole operator [151] HI = ~r · E:

∆Ei =

~
X hl|d~ · E|ii
Ei − El

l6=i

2

.

We will now consider a situation as for 87 Rb at near infrared wavelengths(approximately λ = 700 nm to λ = 1400 nm) where only the D1 (λ ∼ 795 nm)
and D2 (λ ∼ 780 nm) lines are close by, and contributions from other states
become negligible since they are so far off-resonant. The optical potential dueto the laser induced coupling of the 52 S1/2 state to the 52 P1/2 and 52 P3/2 statesthen becomes [89] (neglecting counter-rotating terms):


li 2
li 2
2
X
X
cq
cq
3πc  ΓD1
2ΓD
 I (~r) ,U (~r, i, q) =+ 32
32
ωD1 l∈P ω − ωli
ωD2 l∈P ω − ωli
1/2

3/2

where ΓD1,2 is the scattering rate of the D1,2 transition, q denotes the polarization of the light (which can be {σ − , π, σ + }, cliq are the Clebsch-Gordon
coefficients which encode the coupling strength between two states at a given
polarization and the transition frequencies ωD1,2 ≈ ωli .
One of the great drawbacks of state-selective potentials is that they typically are still close enough to resonance that residual scattering is an issue.
The scattering rate can be calculated by knowing Γsc = Γ0 U/~∆ and applying
it term-wise to the above equation, which yields [89]:


li 2
li 2
2
2
2
X
X
c
c
Γ
2Γ
3πc  D1qq
D2
 I (~r) .
Γsc (~r, i, q) =
2 +
3
3
2
ωD1 l∈P (ω − ωli )ωD2 l∈P (ω − ωli )2
1/2

3/2

In Fig.2.11 we show the resulting potential for the state-pair |1, −1i ↔ |2, 0i
which will be used in chapter 7 at an intensity of 380 W/cm230

2.2.6

Bose-Hubbard Model and Interacting Bosons

As discussed in chapter 7, we start our experiments with an atomic crystal,
which is generated by ramping an optical lattice into the strongly interacting
regime. To better understand this, we first discuss the Hamiltonian for a single
atom in an optical lattice:Ĥ = −

−~2 ∇2+ V0 sin2 (kr x) + sin2 (kr y) + sin2 (kr z) ,2m

where kr = 2π/λ is the recoil wavevector of the lattice potential and λ is the
wavelength of the lattice light. We can solve this Hamiltonian by invoking the
Bloch theorem [174, 175] for particles in periodic media which states that the
eigen-wave-functions of a single particle in a periodic potential landscape has
the property:
φq (~r) = ei~q·~r u (~r) ,where ~q is the quasi-momentum (or crystal-momentum) and u (~r) has theperiodicity of the lattice (with {a, b, c} ∈ Z and {j, l, m} ∈ Z and ~r =(x, y, z)):


λ
λ
λ
u (~r) = u ~r + a ~ex + b ~ey + c ~ez
2
2
2
X
=
αj, l, m exp [2ikr (jx + ly + mz)]
j, l, mThe eigenfunctions φq (~r) to the above Hamiltonian are called Bloch-waves,and because of the above equation, the quasi-momentum is periodic with period 2kr . All solutions for the above Hamiltonian are thus contained in theinterval {−kr , +kr }, this interval is called the first Brillouin-zone. The eigenvalues form quasi-continuous ‘bands’ called Bloch bands, as is illustrated in
the equations above and in Fig. 2.12 (for clarity only in one dimension).
The Bloch-wave-basis is fully delocalized in real space, but for the following
discussion it is usually helpful to define a complementary, maximally localizedbasis called the Wannier states. The Wannier-states are restricted to a single
Bloch-band B, and can be computed as follows:X
wB (~r) = N
α (~q) φq, B ,q~ ∈ {BZ1 }where the summation is restricted to the first Brillouin-zone and the coefficients α (~q) have magnitude one and are chosen such that x2 expectation
31

Energy [Er]

6

3

0
-1

0
Quasi-momentum [kr]1

Figure 2.12: Lowest two Bloch-bands for a one-dimensional lattice of depth
V0 = 3Er plotted as a function of quasi-momentum. Energy is in units of
Er = ~2 kr2 /2m.

Density [a-1]value of |wB (~r)|2 is minimized (i. e. such that the Wannier-state is maximally localized), and N normalizes the Wannier-state. For very deep lattices,
the Wannier-states become approximately harmonic-oscillator ground states,with harmonic oscillator frequency pgiven by the harmonic approximation at
the bottom of a lattice well: ω0 = 2 V0 /Er Er where Er = ~2 kr2 /2m.

2

1

0
-1

0
Position [a]1Figure 2.13: Comparison of the probability-density of Wannier-functions(short dashed), Bloch-functions (solid, normalized to one atom per unit cell),and the harmonic oscillator ground state for the harmonic approximation of
the bottom of a lattice well (dash-dotted). Density is given in units of inverse
lattice spacing.

32For more than one atom (we are starting with a BEC), the full manybody Hamiltonian of contact-interacting atoms in a external potential V (~r)
is [6, 135]:

 2 2
Z
~∇
3
†
+ V (~r) Ψ̂ (~r)
Ĥ = d ~rΨ̂ (~r) −
2m
(2.6)
ZZ
1 4π~2 asc
3
3 0 †
†
0
0
0
+
d ~rd ~r Ψ̂ (~r) Ψ̂ (~r ) δ (~r − ~r ) Ψ̂ (~r) Ψ̂ (~r ) ,2 m
where Ψ (~r) and Ψ† (~r) are the bosonic field annihilation and creation operators at position ~r, asc is the scattering-length which defines the strength ofthe contact interaction. The second term of Eq. 2.6 interactions come fromthe Born-approximation for atom-atom scattering, when we assume the low
energy or s-wave limit [135]. We hence get a delta-shaped pseudopotential,
4π~2 asc δ (~r) /m, for the inter-particle interactions (contact interactions). We
will assume that the atoms reside in a simple cubic lattice landscape of theform:V (~r) = Vex (~r) + Vl (~r) = Vex (~r) + V0 sin2 (kr x) + sin2 (kr y) + sin2 (kr z) .
Here, Vex is a weak external trapping potential which localizes the sample toa finite region in space, as is typical in experiments.
If the lattice is deeper than all other energy-scales, such as temperatureand interactions (typically already fulfilled for potentials deeper than 3Er ),then we can make a tight-binding approximation in the lowest Bloch-band,
where we assume that particles in the Wannier states are mostly contained
within single lattice sites (harmonic approximation) with the Wannier-statescorresponding to the lowest Bloch-band w0 . We can then expand the bosonicfield operators in the Wannier-basis:XΨ̂ (~r) =âj w0 (~r − ~rj ) ,jwhere ~ri is the minimum of the ith lattice site and summation i runs over
all lattice sites (one two or three-dimensional). The operators âj and â†j are
the bosonic annihilation and creation operators which annihilate or create an
atom on site j (in the Wannier-state).
Inserting this expansion into the main Hamiltonian, allows us to carry out
the integrations, and we arrive at:
X †X
UXn̂i (n̂i − 1) +n̂i (εi − µ) .
ĤBH = Jâi âj +2 i
ihi, ji33

This is the famous Bose-Hubbard Hamiltonian [60], with the tunneling strength
J and the interaction parameter U . The last term is the mean-field of themodel, with the site-dependent energy-offset arising from the weak external
confinement εi ≈ Vex (~ri ). The summation hi, ji is over nearest neighbors only,with next-nearest neighbor tunnelings and higher order tunnelings neglected.
The tunneling strength J is calculated as follows:

 2 2
Z
~∇
3
?
+ Vl (~r) w0 (~r − ~ri+1 ) ,J := Ji = d ~rw0 (~r − ~ri ) −2m
where i + 1 simply denotes any nearest neighboring site to ~ri . The onsite
interaction parameter U is given by:Z
4πasc ~2U=
d3~r |w0 (~r)|4 .
m
We will restrict the discussion to the repulsive case of asc > 0. The BoseHubbard model has two competing terms, where the first term is a tunneling
term, which tends to delocalize the atoms, and the second term is an interaction term which penalizes two atoms siting on the same site. The interactions
tend to localize atoms on a site, since motion or tunneling often requires atoms
to move onto already occupied sites.
For temperatures kB T  ~J this competition causes a quantum phasetransition [59]. The superfluid to Mott-insulator transition, where in the superfluid phase particles are completely delocalized and are coherent with respect to each other (i. e. phase stable so that they can interfere), while in the
Mott-insulator phase atoms are localized on a site and are completely phaseincoherent (particles are in a product state). The transition happens whenzJ ≈ U where z is the coordination number or number of nearest neighbors(six for the simple cubic lattice). In our system, the temperature after ramp-up
of the lattice is around 4 nK [93] giving kB T /~U ≈ 0.1.
In the superfluid limit, the state of the system is:X  † N
|ΨiSF ∝âj
|0i,jwhere |0i is the vacuum-state or the empty lattice, while in the Mott-insulator
limit, the state is (for simplicity written here at exactly one particle per site)a product state of the form:
Y †|ΨiMI ∝âj |0i ,j34

where as was already mentioned we have one particle per site which is phaseincoherent with the neighboring site.
The coherence manifests itself in the 3D Bose-Hubbard model in time-offlight in ultracold atomic gases as sharp peaks in momentum space at momenta{a2kr , b2kr , c2kr } (where {a, b, c} ∈ Z. A beautiful representation of thiseffect can be seen e. g. in Fig. 5 of reference [134], which is also an example
for the superfluid-to-Mott-insulator transition of the Bose-Hubbard model fora cubic 3D optical lattice in our apparatus. The phase in which the atoms are
predominantly in is however a function of mean field (or average number ofparticles per site), where particles are more likely to be localized if exactly one
particle per site is in the lattice. Phase diagrams can be found for example inFigs. 1 and 2 of reference [176] and a mean-field phase diagram can be foundfor example in Fig. 6.2 of [90]. If a weak harmonic confinement is present
at fixed J and U , the particles experience differing local densities and hence
trace out a vertical line (a line along varying mean field and constant J/U )in the phase diagram (and hence experience differing occupation numbers persite). This is studied directly for example in [177] and also in appendix B.2.
In our experiment we typically load between 3 × 104 and 2 × 105 atoms into
lattices of depths up to 40 Er depth (periodicity is either 532 nm or 395 nm).
This results in interaction parameters of ∼ 1 kHz and tunneling strengths J/hbetween 1 Hz and 200 Hz (depending on lattice depth), which are occupied by
up to 7 atoms per lattice site in the center for lattice periodicities of 532 nm
and approximately 2 × 105 atoms (see appendix B.2).

35

Chapter 3
Experimental Upgrades
Since I started in the group of Prof. Schneble we were able to implement several
upgrades to the apparatus and the sequences for generating BECs (as discussed
in chapter 2). As a result of these changes, we are now able to operate morestably, more reliably, and faster on a daily basis. These improvements turned
out to be crucial for the measurements presented in chapter 7. The results
depend on rapid data-taking to avoid the detrimental effects of drifts. We will
present the changes and improvements in the following chapter.

3.1

Laser Cooling and Trapping

Creating the cycling light for the MOT/Molasses stage of the experiment requires a large amount of D2 -cycling light, on the order of 100-500 mW, thelarger the better typically. The light is generated using a Toptica DL-pro-780
master laser generating ∼ 30 mW of 780 nm light with linewidth < 500 kHz.
This light is amplified using a Toptica BoosTA laser amplifier which originallyoutput ∼ 900 mW. Frequency-shifting using an AOM-double pass arrangement(∼ 50% efficiency) and fiber coupling (∼ 50% efficiency) left about 150 mW
to 200 mW available for use in the MOT/molasses phase of the experiment.
We exchanged the amplifier chip with a 2 W chip (#TA -0780-2000-5),which can be driven at up to 1.5 W output given the BoosTA’s control electronics and laser head. This increased the output to 350 − 400 mW of lightavailable for the MOT, thereby significantly increasing the number of lasercooled atoms (fluorescence as measured in section 3.3 increases by 50% for
a 200 mW baseline and by 100% for a 150 mW baseline) while not greatly
influencing the temperature at this (density-limited) stage. After transfer into
the magnetic trap, the higher atom number allows for faster evaporation since36

greater density increases the rethermalization rate (details below).
Finally we mention that the locking signal for the repump laser (for frequency stabilization of the laser to the D2 F = 1 → F 0 = 2 transition), whichused to be generated using polarization spectroscopy in a Rubidium vapor cell,
is now generated by standard saturation-absorption spectroscopy (SAS) using
a lock-in technique. The lock-point specific to the repumper transition drifts
for polarization spectroscopy, but is much more stable for SAS, thus improving the stability of the laser locking, which now typically stays locked to thetransition for the entire day (as opposed to several hours in the past).

3.2

Rubidium Source

The rubidium in our chamber originates from sublimation of a 1 g sample ofa natural isotope mixture of bare rubidium metal inside the vacuum chamber.
The rubidium was introduced into the vacuum chamber (in 2005/6 when theapparatus was first built) in a sealed glass vial behind an angle gate valve.
The vial was broken after the vacuum chamber was evacuated. The vapor
pressure of rubidium at room temperature is ≈ 3 × 10−7 mbar, which would
mean that an atomic cloud would have a lifetime of less than a hundredth of asecond, which is too short to transport it into the ultra-high vacuum part of ourapparatus after transfer from the MOT into the magnetic trap. To control therubidium vapor pressure, we have the rubidium valved off in a separate part ofour vacuum chamber. Daily operation then traditionally consists of opening
this valve for several hours (1-2 hours) a day until the pressure becomes toohigh (i. e. lifetime in the vapor cell too short), and then closing the gate-valve
until the next day. Since the initial/final atom number strongly depends on
the pressure in the cell (size of the MOT depends on pressure and transport
loss depends exponentially on pressure), this means that the initial BEC size
tends to drift over the course of a day. We were able to alleviate this problem
considerably, with a full solution planned in the future.
The strategy is to simply cool the part of the chamber containing thevial with the rubidium metal, in order to lower the vapor pressure enough tohave stable operation with the angle valve open permanently. When locallycooling the chamber, we will assume that the cool part emits a certain flowof rubidium which, in a flow equilibrium with the ion-pump, leads to a stablepressure below the room-temperature vapor pressure.
For a better understanding of the situation, we will develop a simple model
to motivate how vapor pressure at the Rb-source corresponds to the pressure
seen by the atoms in the MOT (for more details about vacuum, cf. [178]).
37

Figure 3.1: (a) Schematic of the temperature control described in the maintext. The Rb-source in our experiment is rubidium metal in a glass vial (vialbroken in the vacuum). The vial sits on top of a CF-blank flange inside of an
approximately 10 cm long bellows. Temperature readings are taken using aplatinum resistance thermometer glued to the CF-blank flange. The bottomof the flange is cooled using a cold finger contacted to a thermoelectric cooler.
The waste-heat of the cooler on the hot side is carried away to a large reservoir(bucket) using water pumped by a standard aquarium pump. A flow meterswitches off the power to the cooler in case of no or too little water flow. (b)Photograph of the setup in (a). (c) Photograph of the current source for thecooler, the Watlow PID controller, and the water reservoir, which contains theflow switch and the pump. (d) Planned future upgrade to the temperature
regulation of the Rb-source for directly cooling elongated bellows.
A schematic which visualizes the parts used in the calculation and their connections is shown in 3.2. We start by approximating our vacuum chamberas consisting of two pipes in the molecular flow regime which are connectedto each other while the ‘free’ end of each pipe is held at a constant pressure.
The first pipe, IP1 (rubidium source), has a diameter d = 2 cm and lengthl = 25 cm and the second pipe, IP2 (MOT-chamber), has diameter d = 7 cm
and length l = 70 cm. The pipes are connected to each other and the ‘free’ endof the IP1 is connected to a volume at rubidium vapor pressure pRb at varying38

Rb-vapor
IP1, d1, l1

10-8 mbar
to 10-7 mbar

Q1
Cooling
assembly

MOT, pc

Q2

IP2, d2, l2

Ion Pump
10-10 mbar

Figure 3.2: Schematic to visualize connections between vacuum parts which
need to be considered to estimate flow-equilibrium. The source for the Rbvapor is cooled, which adjusts the Rb-vapor pressure originating from the
Rb-source. Two connected pipes link the Rb-source to the ion-pump, with the
MOT approximately at the mating point of the pipes.
temperature T while the ‘free’ end of IP2 is connected to a pump, which we
assume to be held at a pressure of pp = 10−10 mbar. We can simply demandthat the throughput Q = C ∗ ∆p (where C is the conductance) of each pipe isidentical to get the pressure at key positions in the system. The conductanceof a pipe in the molecular flow regime (i. e. pressures below 10−3 mbar, whichis always fulfilled for our system) is given by
Cmol =

vrms πd3,lwhere vrms is the thermal rms velocity (of rubidium) at a given temperature
and atomic weight, d is the pipe diameter and l is the pipe length. The
conductances are then C1, 2 = {2.5, 37.7} l/s (liters per second, where weneglect any variation in conductance due to temperature). We equate thethroughputs and solve for the pressure at the connection pc (which is theapproximate location of the atoms in the MOT-loading position)pc =C2 pp + C1 pRb.
C1 + C2

For rubidium at room temperature (pRb = 3 × 10−7 mbar), the pressure at theconnection point is pc ≈ 2 × 10−8 mbar, which is about a factor of 10 to 20
too high compared to the desired pressure of approximately 10−9 mbar.
Lowering the temperature by 5 K, 10 K and 20 K respectively, the pressure39

at the connection becomes
pc = {8, 5, 1} × 10−9 mbar.
Lengthening the first pipe to 50 cm results in
pc = {4, 3, 0.7} × 10−9 mbar,
and lengthening the first pipe to 100 cm lowers the pressure to
pc = {2, 1, 0.4} × 10−9 mbar.
A thermo-electric cooler (Peltier element) can easily achieve this temperature change and is compact enough to be added onto the current, very constrained setup, when using a compact water cooling block for removal of thewaste-heat. The setup is shown in Fig. 3.1 (a) and (b). We use a standard TEC-127-06 Peltier cooler with a switchable current of ∼ 2 A, controlled
by a Watlow temperature controller (PM6C1CA-AAAABAA) via an IXISCPC1709J solid state relay. The waste-heat is carried away from the thermoelectric cooler using a copper block [179] cooled by flowing water. The water is
supplied from a large bucket (∼ 20 l) via a standard aquarium pump. A flow
meter switches off the main power supply in case the water flow is impeded.
Photographs and schematics of the setup are provided in Fig. 3.1.
After opening the gate valve when using a cooled Rb-source, we observe adecrease of lifetime of magnetically trapped atoms (which indicates an increasein pressure) in the region of the main vapor cell vacuum chamber (see Fig. 3.3
and Fig. 2.1). The increase crosses the critical value of lifetime (∼ 1 s) afterroughly 100 days (or three months). The slow increase is a function of themain chamber slowly equilibrating to the presence of an additional gas load
(Rb-vapor) which slowly coats the walls of the MOT-chamber. For a morefine-grained observation time, the pressure can drop/rise on days where thelaboratory temperature is slightly cooler or warmer, due to the walls of thevacuum vessel pumping a little more or less efficiently. The slow decrease of
the pump performance also factors into the exact time evolution of this system.
The revised schematic already gives a marked improvement over the situation
before, where the valve to the Rb-source had to be opened and closed daily;
now instead the valve has to be adjusted once every two to three months. The
drift in background pressure can easily be compensated by e. g. changing the
MOT-loading time slightly, changing one or two evaporation parameters orchanging the power of light provided during MOT loading.
In the following we will discuss a straightforward adjustment that can bemade in the future, where one would have to adjust the rubidium-source on40τ-1 [s-1]

.5

.4

.3
0

5
15
10
Time after valve open [days]

Figure 3.3: Measured (inverse) lifetime τ −1 of magnetically trapped Rb-atoms
after laser cooling. Time measured after the gate-valve was opened. The
slow linear increase exceeds the maximum inverse lifetime (∼ 1 s−1 ) afterroughly 100 days, with no saturation visible in the finite observation timeshown. Temperature of the Cf-blank-flange (see Fig. 3.1) was stabilized to∼ 0◦ C.
the order of only once a year (see Fig. 3.1 (d)). The current cooling setup has
several problems. First, the cooling element freezes over at the temperaturesneeded to maintain a low enough pressure, which eventually (over a timespanof several weeks) causes the control loop to be unable to reach the setpoint
because the accumulating ice lowers the effective cooling power. Second, thedevice is essentially at the limit of what the cooler can feasibly do even without
ice buildup. As seen in Fig. 3.3, the pressure at the MOT-cloud slowly creeps
higher, and eventually reaches a point where its lifetime becomes too low.
This is due to the pressure slowly equilibrating after the valve is opened, withno saturation observable in the finite time we measured. The final pressure
is a function of temperature and differential pumping between Rb-source andthe ion pump. Additionally, the water reservoir is not quite large enough todissipate all the heat easily, so it heats up from approximately room temperature to approximately 10 − 15◦ C above it, which further reduces the cooling
capacity. Finally cooling the CF-blank flange underneath the Rb-vial is notefficient, since most of the sample sits at a much higher temperature. It would
hence be much more effective to cool the vacuum bellows directly.
A suggestion for an update is shown in Fig. 3.1 (d) where instead of coolingthe bottom of the CF-blank flange, one could in principle use a much longer
tube of approximately 50-100 cm length, which houses the Rb-vial at its end,41

giving more flexibility and spatial access. One would now cool the bellows
directly using a copper clamp, with minimal clamping force to cool a largepart of the vacuum bellows. The cooling capacity can also easily be increased
further by attaching the water to filtered tap water from the main water line,
which is typically about 5 to 10 degrees colder than room temperature, as
opposed to 10 degrees warmer than room temperature. The resulting decreaseof the overall sample temperature along with the increased differential pumping due to the longer spatial separation between sample and main chamber
would allow for much easier and therefore much more reliable control over the
background pressure inside the main chamber. Since the vacuum system willneed to be opened up in the future because of a slowly dying ion-pump, we
can easily realize the aforementioned changes.
Finally, upon opening the vacuum chamber, one can use enriched 87 Rb
instead of natural abundance Rb (natural abundance of 87 Rb in rubidium
metal is roughly 28% [180]), which immediately would lower any unwanted
background pressure significantly, thus lowering transport losses and increasing
MOT size.

3.3

Optical Pumping

It is noted in section 2.1.5 that after the finishing the MOT/molasses phase,
we optically pump the atoms from F = 2 to F = 1 and then capture only the|1, −1i state in the magnetic trap. This lowers the atom number to essentially
30% of that in the initial MOT. It turns out that in our apparatus we caneasily implement a more selective optical pumping into the |1, −1i state (seeFig. 3.4). We now describe this improved scheme.
The level-structure and transitions involved are shown in Fig. 3.5. In theregion of the MOT, the ambient field (Earth’s magnetic field and stray fields
in the laboratory) is zeroed for sub-Doppler cooling in the optical molasses
using three single coils along the x, y and z directions. By switching off oneof these coils during the optical pumping step, we provide a clear quantization axis/direction. If we send the pumping beam along this direction with apolarization of σ − we preferentially (ideally even fully) pump into the |1, −1istate, while the overall performance is unaffected. For this purpose, we simply
rerouted the laser-beam to a new direction, and built an electronic switch forthe coil current.
To determine the atom-number reliably, we first load the MOT, capture
the atoms in the magnetic trap and then record the fluorescence signal which
is observed after turning off the magnetic trap and turning the MOT back on
42

Figure 3.4: (a) Optical schematic for pumping the molasses-cooled atoms from
the MOT. Light from a fiber is first linearly polarized by a polarizing beam
splitter cube, expanded using a pair of lenses and finally circularly polarizedusing a λ/4 waveplate before illuminating the atoms. External magnetic field(earths magnetic field and stray fields in the laboratory) is provided by turning
off the ambient magnetic field compensation coil in the propagation directionof the beam. (b) Photograph of the experiment with the individual parts
labeled, showing directly the spatial position and size of the ambient fieldcompensation coil.
(observation time ∼ 100 ms). The short observation time ensures that the
MOT is not loaded during the observation. A direct comparison between the
new and old performance is thwarted by the fact that the path of the light
had to be permanently changed, and that the absolute atom number in the
MOT (as detected via fluorescence with a photodiode) drifts throughout theday. However, we can compare the performance of the new optical pumping
scheme to a scheme similar to the old one where we have no dedicated pumping
beam, but simply turn of the repumping beam after molasses for 10 ms before
catching the atoms in the magnetic trap. This repumps the atoms into F=1
due to spurious absorption events into F 0 = 2, which is the baseline we will
43

-3

-2

-1

mF
0

1

2

3

F=3
F=2
F=1
F=0

σ +-poπ-polσ− -poll

52P3/2F=252S1/2
F=1
-2

-1

0
mF

1

2

Figure 3.5: Level diagram for optical pumping. σ + -polarized light (blue, solid)
distributes population into more positive Zeeman-sublevels, which can thenspontaneously decay (gray dashed) into F = 1 (on average mF does not change
for spontaneous decay). σ − -polarized light (red, solid), distributes population
into more negative Zeeman-sublevels. Spurious π and σ + polarization ensuresthat most of the population ends up in F = 1.
compare to. We then vary the waveplate angle and compare the fluorescenceto the baseline measurement of regular optical pumping without a bias fieldto get the relative increase in captured atoms.
Assuming perfect conditions, we would expect the atom number to drop
virtually to zero at a certain angle, indicating that we have perfect σ + - polarization with all population ending up in the |1, 1i state (henceforth thiswaveplate angle is π/2). At π/4 and 3π/4 waveplate angle, we then get linear
polarization, which in this coordinate frame is an equal superposition of σ +and σ − polarization, so all F = 1 hyperfine states are thus similarly likely to
be occupied, and the fluorescence increase is close to unity. At zero and π
waveplate angle we expect near perfect σ − polarization. Therefore we expect
essentially all atoms to end up in |1, −1i (the state which we want to trap),
resulting in an enhancement factor of about three.
Realistically, we cannot get a perfectly polarized sample after just one stepof optical pumping, because our initial state after the molasses phase is a44

Relative Number

2

1

0
0
λ/4-Waveplate angle

πFigure 3.6: Fluorescence signal for the new optical pumping scheme divided
by fluorescence for the old scheme (using optical pumping of molasses only
by turning off repumping light). The fluorescence is proportional to atom
number captured and is plotted as a function of the angle of the λ/4-waveplate
immediately before the beam enters the chamber (see Fig. 3.4). Solid line is asine function fitted to the data as a guide to the eye. Horizontal dashed line
is at unity and vertical dashed lines are at π/4 and 3π/4.
perfect mix of all hyperfine states of F = 1 and F = 2. Let us consider again aperfectly σ − polarized beam. Starting with a finite fraction of atoms residingin the |2, 2i state, these atoms can absorb a photon from the optical pumping
beam and will transition to the excited hyperfine state |20 , 1i, which can decay
to the states |1, {0, 1}i and to the states |2, {0, 1, 2}i. Since the atom cannot
absorb if it is in any |1, mF i state, a decay to F = 1 means that this atom islost for magnetic trapping.
Figure 3.4 shows the result of conducting this experiment, where we change
the waveplate angle and compare to a scenario where we pump by turning
off the repump beam. The above reasoning explains why the fluorescenceincrease does not drop to zero for a waveplate angle of π/2 and why thefluorescence increase is only roughly two and not three: a significant portionof the atoms (presumably about one third) ends up in the |1, {0, 1}i statesfor σ − polarization of the light, which results in an increase to two-thirds of
all atoms which can be optically-trapped (from initially one third). Similarly
about one sixth of all atoms land in the |1, −1i state if we use σ + polarization,
which results in an enhancement factor of roughly one half.
The heating due to this process is negligible, since the atoms scatter on
average less than 3 photons before ending up in an F = 1 and therefore becom-

45ing dark. The temperature imparted by a photon recoil is only about 200 nK
where the temperature after molasses is on the order of ∼ 10 µK. While thepumping efficiency is not 100%, the number of atoms has still approximately
doubled at negligible additional heating, thus increasing evaporation speed
later in the sequence.
In the future, one could imagine a second step of optical pumping, where
after the first step of optical pumping is finished, one uses a new σ − polarized
repumping beam which counter-propagates with the pumping beam to put theatoms back to F = 2, and then adds a second identical optical pumping step
to increase the number of optically-trappable atoms even further. A counter
propagating beam is likely required to not impart too much linear momentum
and displacement during the pumping phase. While this procedure should,
in principle, increase the number of atoms even further, it cannot be easily
realized in our current apparatus due to geometric constraints. It should be
noted too that only a smaller marginal number gain would result at the costof a much higher complexity.
A caveat of the optical pumping method described in this section is thatfor the case of perfect σ − - polarization, a small fraction of the population
will get trapped in the state |2, −2i due to not being able to absorb any light(off-resonant absorption to the |3, −3i-state is irrelevant since it can only
decay back to |2, −2i, see also Fig. 3.5). Any population left in this state will
be quickly expelled when turning on the magnetic trap due to |2, −2i being
magnetically anti-trapped, resulting in a loss on the order of 20% to 30% of the
MOT loaded atom number. In a realistic experimental setup the polarization
will not be perfectly σ − due to small relative misalignments of the direction of
the beam and the magnetic field or due to the initial polarization of the beam
being not perfectly linear (extinction ratio for the unwanted polarization of
typical polarizing beam splitters is on the order of 10−3 [181]). At a scattering
rate of > 106 per second for the main polarization, even a fraction 10−3 in adifferent polarization will empty the |2, −2i - state in less than 1 ms.

3.4

Quadrupole Magnetic Coils

The re-design and exchange of our quadrupole-trap magnetic coils, which now
also allow accessing a Feshbach resonance at ∼ 1 kG [35, 182] when usedin Helmholtz-configuration, will be detailed in a dissertation written by ourcolleague A. Pazmiño. The maximum axial field gradient changed from b ≈350 G/cm to b ≈ 480 G/cm.

46

3.5

Evaporation Speed Increase

Because we start with a denser and hotter cloud (adiabaticity of compression
from 350 G/cm to 480 G/cm is assumed), the evaporation can be sped up. As
is discussed in [131] for the case of quadrupole potential, an increase of the trap
gradient by η results in an increase of the trap density by η and an increase in
the temperature of η 2/3 . In the case of the TOP trap, an increase in gradientby η results in an increase in the trap frequencies by η (all else unchanged).
Then the density increases by η 3/2 and the temperature increases by η. In [132]it is shown that elastic scattering
is proportional to density times square root√of temperature (Γel ∝ n T ). Since in our case η ≈ 1.4, the evaporation speed
can then be increased by a factor of η 4/3 ≈ 1.5 in the quadrupole trap andby a factor of η 2 ≈ 1.9 in case of the TOP trap. Furthermore, due to opticalpumping and more power in the cycling beam, we now have approximately 3 to
4 times as many atoms as initially in the magnetic trap, which correspondingly
allows the evaporation to be sped up by approximately that same factor.
We note that once the atoms are cool enough to be loaded into the optical
dipole trap (∼ 2 µK), the trapping frequencies of the optical dipole trap are
ωi ≈ 2π × 80 Hz, and their geometric mean does not go lower than ω̄ ≈2π × 60 Hz for typical evaporation, while the geometric mean of the TOP-trap
frequencies is, even with the new coils, only ω̄ ≈ 2π × 50 Hz (changed fromω̄ ≈ 2π × 35 Hz). Therefore, our long-established strategy of loading the ODT
as early as possible (limited by its ∼ 5 µK depth) remains unchanged.
In historical sequences, the TOP evaporation was continued essentiallyto the point of BEC in the TOP trap, with evaporation in the optical trap
essentially only compensating for the heating which results from the transfer
process between TOP and ODT, including three full stages of evaporation in
the TOP, all taking on the order of 10 s. With the changes outlined above, wetransfer from the quadrupole trap to the TOP-trap at roughly 50 µK and weneed only one stage of evaporative cooling in the TOP (now only taking ∼ 5 s)
before we can transfer into the ODT, where evaporation is concluded in ≈ 2 s.
Overall, the changes to the trap loading and evaporation sequence result in avery significant overall reduction of the cycle time for BEC production from90 s to 20 s.

3.6

Summary and Current Performance

Another result of the above upgrades is the increased stability of daily operation. In Fig. 3.7 we show the current atom number stability during a typical
47

old
new

MOT load

MT load &
move

14 s
3.5 s

(0.5+4.5) s(0.3+2.2) s

MT evap.
(QP &TOP)(15+37) s(4+4) simage &
move &
misc.
(3+5+1) s(1+1+1)sODT load& evap.
(1+3.5) s(1+2) sTable 3.1: Comparison of old and new sequences, detailing the duration for
production of a BEC. Old sequences achieve BEC in approximately 85 seconds
while new sequences achieve BEC in approximately 20 seconds. Miscellaneous(Misc.) is the time between sequences, the second move step returns thetransporter to its original position.

82

N×10-4

N×10-44

4

0
0

Run number

300

0
0

500

1000
Run number

1500

Figure 3.7: Typical BEC number stability. Data taken on two separate days,
the second day starts at run number 1595. Total number drifts slowly throughout the day; small adjustments are made using MOT-loading time. Zero atomruns are the reference images described in chapter 4. Inset: Typical numberstability in 2011, before the changes were made to stabilize number. Total
atom-count drifts quickly, with higher variance than in the main figure.
day of taking data for a publication, compared to a typical day of taking data
for a publication in the past. We see that the stability of produced BEC sizes
is much improved, and drifts are also decreased and easily compensated for.
Finalizing the changes suggested in 3.2 should get rid of drifts experienced
during a typical day, with only minor adjustments needed in e. g. MOT loading time, MOT beam powers or evaporation time to set atom-number to thedesired (or previous day’s) value.

48

Chapter 4
Image Analysis
4.1

Backgrounds and Fringes

As is outlined in section 2.2.1, we use coherent (spatial and temporal), resonant
light to image the shadow cast by the atoms in the science cell onto a CCD chip.
In a raw image, any sort of imperfection, such as a speck of dust, can cause
circular fringes in the image plane. Also close-to-plane-parallel surfaces in the
beam path will cause ‘etaloning’, resulting in linear fringes. Typically mostof these fringes are easily taken care of in standard absorption imaging [130]by taking three total images: (a) illuminate the CCD with atoms present, (b)illuminate the CCD without atoms present and (c) take a dark image without
illumination. Then the absorption of light by the cloud of atoms can easily becalculated as (a-c)/(b-c) (see also section 2.2.1).
This, of course, assumes that the fringes are stable in time. To the bestof any experimenter’s ability, this can be nearly achieved by building opticalsystems which are maximally stable and not prone to any sort of vibration (useoptical benches, short posts, non-adjustable optics, water-cooled CCD camera
without fan) and which are unaffected by air currents and accumulating dust.
One can further reduce the impact of time-dependence by taking signal andreference images as close in time as possible. However, regardless of the caretaken to eliminate any problems, the combination of temporal separation,
acoustic noise and air currents (to some small residual degree) will cause aslight movement of the fringes. The actual relative movement of the fringes is
hard to predict a priori, and one could in principle have many fringes moving(seemingly) arbitrarily and independently of each other, with the maximum
complexity of the problem (i. e. parameters which need to be monitored)
limited only by the number of pixels on the camera (106 ).

49

As we will see in this chapter, the situation is fortunately often not quite
so dire, and only a limited set of ‘typical’ fringe motions is actually relevant.
We will pick out the set of typical fringes using the technique of principal
component analysis.

4.2

Reference Fringes

The technique described here is based on work conducted in Hamburg, Amsterdam and Shanghai [183–186]. We start with a set of empty absorption
images ({Rj } = (a − c)/(b − c)) called reference images, obtained by simply
not loading the MOT before running the sequence as normal. As stated previously, slow accumulation of dust over the course of the experiments has tobe factored in, so the optimal set of fringes may be time dependent. In ourcase this was done by gathering these reference images interleaved with datataking (typically every fifth image) instead of before/after data taking. This
allows the basis of reference images to include any changes that happen overthe course of taking the data.

4.3

Principal Component Analysis

The tool to analyze the large amount of images and find or filter out the fringes
that occur is principal component analysis (PCA) [187]. PCA is typicallynconducted on a data set {Dj } where
P each data point Dj ∈ R is a real n-vector
and the dataset has zero mean j Dj = 0. PCA then finds the eigenvectorsof the covariance matrix of the full data set A = [D1 , D2 , D3 , · · · ] [187]:
C = A · AT ,ordered by magnitude of the corresponding eigenvalue (the largest eigenvalue
giving the first principal component). Each principal component Pi (in ourcase representing a fringe pattern) lives in the same vector space Rn as eachdata point Dj (a typical image containing fringes and the absorption signal
from the atoms). The first principal component
P1 is the direction in thisP
space that has the greatest variance, i. e. j (P1 · Dj )2 is maximized, so that
there is no other vector P0 along which greater scatter is observed. The second
principal component is the direction orthogonal to the first principal component along which the biggest scatter occurs. The third principal component
is the direction orthogonal to the first and second principal component along
which the biggest scatter occurs. In other words, PCA yields an orthogonal50

(and, if normalized an orthonormal) basis, which is only complete when thedataset has more (linearly independent) entries than the dimensionality of thevector space of Dj .
Similar to the use of PCA in reconstructing faces in the ‘eigenface’ method[188], here it is used to identify the most important (read: most variance)features from a data set (read: BEC-absorption images with fringes). PCA
thus allows for the reconstruction of a given data point (new or present in thecurrent set of data) using only the few most important principal components.
In our case we can easily cast the two-dimensional reference images Rj withjj
pixels Rx,
y into vector form Rk where, e. g. k = x + dimx × y, and dimx isthe numberof pixels in the x coordinate. We then subtract the mean image
P
R = j Rj /N (N is the total number of images) from each Rj to generate
the data set {rj } = {Rj − R} on which we can perform PCA yielding thenormalized principal components pi .
A new image with atoms, RAtom , will be processed by first subtracting themean image, R, to give rAtom = RAtom −R, and by then calculating the overlapof that subtracted image with the first few (typically n = 5 − 20) normalized
principal components. The processed image is thenRAtom, proc = rAtom −nX(pi · rAtom ) × pi .

i=1

We take reference images (Rj ) throughout typical data-taking (‘training
set’ of images). For the final analysis, we compute the principal components
and mean from all reference images, and we then analyze (process) all atom
images with the mean and principal components computed from these reference images. One should note that each image is also shifted to have zero mean
individually, i. e. the set which is used to create the principal components is
actually 0
rj = {rj − 1 hrj i} ,which means that for the data matrix each column-vector and each row-vector
have zero mean. Consequently, an image inside the ‘training set’ of reference
images can be recreated as follows:X
Rj = R + 1 hrj i +pi (pi · r0j ) .
i

On the other hand, new images that are not part of the ‘training set’ canonly be recreated approximately, unless the number of training-images (andhence the number of non-vanishing principal components) exceeds the numberof pixels, which is a prohibitively large set of images (≈ 105 ).
51

For new images to be processed, we will typically only subtract the meanof the reference fringes (R) and the overlap with the principal components
multiplied by the respective principal component. While one should technicallystill zero the mean of the new image which we want to process (i. e. subtract
1 hrAtom i) this is complicated by presence of actual atoms, which shift the
expectation value of the mean (hrAtom i) of the new image away from zero. A
better approach is to zero the optical density in an empty region close to or
surrounding the atoms.

4.4

Discussion of Errors

With the implementation thus far, we make an (albeit very minimal) error
introduced by the small residual overlap between the principal components
and the actual atom signal. To understand this, we can imagine an image ascreated as follows:
RAtom = 0 + SN + FR + A.
An ideal image would have optical density 0 everywhere. A real image on topof that would then have shot noise, SN, (from the shot noise from photons
counted per pixel) as well as the aforementioned fringes, FR, and finally thesignal of the atoms we are actually looking for: A.
The effective dimensionality of the shot-noise images is the number of imaged pixels, but it does not induce any systematic shift. The shot noise oneach pixel is (approximately) proportional to the inverse square root of thep−1
counts at the pixel ( Npix ), which for our 16-bit camera is on the orderof ≈ 0.01/pix=
ˆ ± 1 . . . 10 atoms/pix. With enough images to average over(> 100), the SN part contributes mostly to late principal components (i. e.
beyond the 20th principal component). The overlap of new realizations of
SN with the used principal components is non-zero but very small, with an
expectation value of zero, so that this contributes no net systematic error.
The A signal, on the other hand, can have a systematic nonzero overlap
with the principal components. We should expect the overlap to be extremely
small, simply because the vector space associated with a typical image is so
extremely large (approximately 105 basis vectors) compared to the few principal components ∼ 10 used, so we expect overlap by coincidence to be limited
to 10−4 (in other words, it is unlikely that the atomic distribution looks like afringe). Furthermore the principal components typically have non-zero components all over the image, while the atom signal is typically localized.
In practice, the overlap is on the order of < 0.1 atoms/pix for 104 atoms
and < 1 atom/pix for 105 atoms, most of which is a shift of the mean, which52can be easily subtracted off in later image processing of the region immediately
neighboring the atoms (for comparison, the residual fringing on the other handis on the order of 2 atoms/pix or larger). If large atom numbers (N > 5 × 104 )are used, one however needs to worry about this residual overlap, which will
briefly discussed in the next section.
Masking the Atomic Signal
In this section we discuss how the removal of fringes can be achieved withhight fidelity even if large atomic clouds are in the image by masking out theatomic signal. This is not used for the analysis in chapter 7, so the reader may
wish to initially skip to section 4.5.
If large atom numbers are required, one should ‘mask-off’ the region/regions which contains or will contain the atoms (with the masked area being
rather a little too small than too large, i. e. just the size of the cloud rather
than twice the size of the cloud). An example of a mask is shown in Fig. 4.2(b). An implementation of the masking is as follows. The mask m is an imagesized array with 0 in the ‘mask region’ and 1 everywhere else, and is simply
multiplied pixel by pixel onto the mean subtracted image rm
Atom = m × rAtom .
However, when the same is done with the principal components pm
i = m × pi ,this introduces an error since the masked principal components pm
i are nowneither normalized nor orthogonal to each other. Luckily we can mitigate thisfact by following a procedure similar to the one outlined in [185], and leastsquares-fit the masked principal components to the masked mean subtracted
image:
!
!#
"
X
X
cm
cm
0
0
m
m m
m
m
m m
m
Min
rAtom − p1 −ci pi · rAtom − p
1 −
ci pi
,
Npix
Npix
i
i
p
The constant component c0 / Npix 1m allows for mean-centering the image,which needs to be included now since it would otherwise interfere with the
optimization.
We willp
consider the constant offset as the ‘zeroth’ principal component(p0 := 1m / Npix ). We can then write:


2
X
m 
Min  rmcm.
Atom −i pi
i=0

We can easily calculate an analytic solution to this problem given a finite
set of principal components rather than numerically optimizing each time a53

new picture comes in. To do this we first take the partial derivative of theabove equation for a single coefficient cl (Latin summation is over principal
components and Greek summation is over pixels):∂ X m X m m 2(rα −pα,i ci ) = 0
∂cm
l
α
i=0
X
X
∂ X 0
m
m m m
(rα − 2rαm cmcm
i pα,i +i ci0 pα,i pα,i0 ) = 0
m
∂cl α
i
i,i0
which yields:
X

m m
pm
α,l pα,i ci =

X

X

X

rαm pmα,lαα,iBl,i cmi =rαm pmα,l .

αi

The matrix B is the product of the matrices P C T · P C, and P C has columnspm
i which are the principal components and the number of columns is thenumber of principal components used. B is a square, symmetric matrix whose
number of columns is equal to the number of principal components used. The
component weights for reconstruction are then simply
X−1 m mBl,ncm
pα,l rα .
l =n,αThis can be written in vectorial form also, as
−1· P C T · rm .
~c m = P C T · P CThis makes it a little easier to understand what the final result of the optimization is: If we assume that we have started out with unmasked principal
components pi , we predicted that the result for the weights of the individual principal components should be ci = pi · r. Now if we consider that the
principal components before masking were orthonormal, then the B matrix is
simply the identity B = 1, so our initial heuristic of simply computing component vector overlap is confirmed here independently. The picture of leastsquares optimization generalizes this to a basis which is not orthonormal. In
cases where the components are not normalized, the matrix B is still diagonalwith entries that are the square of the normalization for each principal component, and if the components are neither orthogonal nor normal, the matrix
B becomes off-diagonal, thus compensating for the non-orthogonality.
54

The processed image is then found using the new principal component
weights cm
i along with the unmasked principal components:
X
RAtom, proc = rAtom −cm
i pi ,
i=0

as opposed to the original principal component weights ci = pi · r.
For the analysis done in chapter 7 we will proceed without masking, since
the involved atom numbers are suitably small (less than 2 × 104 atoms per
picture) so that the error introduced by the atoms is negligible. In the future
the masking strategy can be very easily adapted as described here.

4.5

Results

In this section we show the results of implementing the method above on aset of ∼ 300 empty images. Figure 4.1 (a) shows a replica of Fig. 2.5 (on an
expanded scale), with the most severe artifacts (i. e. dead pixels at the bottom
left and low illumination at the top and left margins) masked off. All images(and all empty images) will be masked like this before any further analysis, inorder to prevent any skew from the large deviations in these regions.
We can clearly see that both F = 1 and F = 2 images still have artifacts
in them, on the order of OD ∼ ±0.05 (where the shot noise floor should beof order ∆SN OD ∼ ±0.01). The main portion of these artifacts is relatively
constant from illumination to illumination, and is due to very small residual
drift of the imaging power during an imaging sequence and also due to someimaging light bleeding over into the ‘dark’ section of the camera chip (maskingof the CCD chip described in section 2.2.1).
It is clearly seen that the main portion of these artifacts
P is easily subtracted
out in our case by just subtracting the mean (R = j Rj /N ) of the empty
images (see Fig. 4.1 (b)). It is important to interleave the empty images withthe actual data and to have the sequence for taking an empty image deviate
only minimally from the sequence of generating actual data. An example forthis would be shuttering off the MOT (cycling) laser or turning off the gradient
field during the MOT loading phase and then proceeding with the sequence as
one would normally. This ensures that any drifts that may happen between
images or during/before images remain constant for the empty images. In
other words, it would be a bad idea to quickly take many several hundred
images before and after data taking, since the resulting set of empty images
might not have the same characteristic features as the actual data-images.
The minimum number of images needed is on the order of ∼ 100, which
55

Figure 4.1: The image shown in (a) is a recreated version of Fig. 2.5 in chapter
2, with all other images being a processed version of (a). (a) Optical density
images for a typical iteration of the experiment showing both F = 1 and F = 2
images (∼ 3 × 104 atoms in total). The temporal distance between reference
and F = 2 illumination is double compared to F = 1, hence the deviations
are bigger for F = 2. (b) Optical density image after subtracting the mean
empty image removing most of the artifacts. (c,d) Optical density image after
subtracting the mean image and the first ten principal components where theoverlaps calculated with masking (c) and without masking (d) the atom signal.
The very faint signal at |1, 0i (circled in (d)) corresponds to about 500 atoms.
suppresses the per pixel shot noise enough to not dominate until later principal
components (30 or later).
In Fig. 4.1 (b) (especially for F = 2 atoms) we can still see artifacts thatare not constant due to the relatively long time interval (6 ms) between illumination and reference illumination. F = 1 also still has artifacts, but they
are comparable to the shot-noise level. In Figs. 4.1 (c) and (d) we see theresult of subtracting the first 10 principal components with and without theatom-masking procedure described in section 4.4. In both cases, the artifacts
still present in (b) are mostly removed, at least to a level well below the shotnoise. The remaining artifacts seen in Fig. 4.1 (c) and (d) F = 2 are probably
56due to imaging the atoms either slightly out-of-focus or slightly off resonancecreating the tiny ripples seen between the atom-clouds.

Figure 4.2: (a) Difference between the images Fig. 4.1 (c) and (d) at a magnified scale (×100 for F = 1 and ×20 for F = 2), showing the error induced
by not masking at small atom-numbers is typically negligible (shot noise is at
the 0.01 level). (b) Masks used for the F = 1 and F = 2 atoms respectively
laid over Fig. 4.1 (c).
The magnified difference between including the masking procedure and notincluding the masking procedure is shown in 4.2 (a), indicating that for smallenough atom numbers, masking is not needed. The masks used to block theatomic signal are displayed in Fig. 4.2 (b).
The first two principal components (not the mean empty image and not theconstant image, which might be considered −1st and 0th principal component)
for F = 1 and F = 2 are displayed in Fig. 4.3 (a) and (b), showing fringe
patterns of different shapes, generally distributed over most of the image. Somefringe patterns, for example F = 1 top right corner, repeat in a similar fashion,
which means that they not only occur sometimes, but they potentially also
shift in time slightly. Figure 4.3 (c) and (d) shows the real part of the Fouriertransform of those two principal components, showing a clear structure. Higher
principal components (20 and higher) show no structure either in real space orin Fourier space, which indicates that most of the later principal components
serve to emulate certain realizations of shot-noise, which is of course impossible
to predict. Therefore higher principal components should in fact be excluded,
since otherwise the data is ‘over-fitted’. In the case that even smaller signals
need to be detected, but shot noise is a problem, averaging over different
realizations of the shot noise by averaging images can lower the shot noise of
the averaged image further.
Figure 4.4 illustrates the above point by displaying the average absolute57

Figure 4.3: First two principal compnents ((a) and (b)) and the real part oftheir Fourier transform ((c) and (d)).

Figure 4.4: Mean absolute value of the corrected overlap between masked
principal component i and image RAtom,M , averaged over ∼ 200 images for
F = 1 (a) and F = 2 (b) atoms. Overlap eventually drops off into a constantnoise floor, indicating that principal components after the 20th are simply
recreating shot-noise patterns, and should hence be disregarded. The zerothprincipal component is the constant offset.

58

value of the principal component weight for each principal component up to100, when averaged over ∼ 200 different absorption images with atoms. The
component weights quickly decay to a level of 0.01. A conservative approachthen would take only the first 5 principal components, and a more aggressive
approach would use all the way up to ∼ 20 principal components; any morethan this is only fitting to shot noise. The number of principal componentsthat need to be used will of course depend on the specific imaging system andthe environment it is placed in, but can in general be evaluated by repeating
the analysis done in Fig. 4.4.
The finite average value of 0.01 to which the overlap decays to for highprincipal components, can be explained as follows. We want to calculate theoverlap of an arbitrary principal component with shot noise. We know the
first and second moments of the shot noise (i. e. the mean and variance) to be(N is the number of pixels)
1 X
SNα = 0N α
1 X
2=
SNα2 := σSN≈ 10−4 .
N α

hSNi =
SN2

We want to know the result of pi · SN, and since the principal components arenormalized and mean centered, we also know their first and second moments:
1 X
pα,i = 0
N α
1 X 2
1
pα,i =:= σp2 i ≈ 5 × 10−6 .
=N αN

hpi i =
p2i

An estimate for the overlap can be made when assuming that late principal
components are also realizations of shot-noise with that are merely differently
normalized:
pi · SN =X

pα,i SNαα

σpi X
SNα SNα0 .
≈σSN αThe last line calculates the correlation of two independent realizations ofshot noise, which is zero on average. The expectation value of its variance59can be calculated as [189] (assuming the shot-noise is approximately Gaussianwith zero mean)
!
1X
σ4
σ4
var
SNα SNα+β = SN − SN
≈ 5 × 10−14
N α
N
N2
!
q
X
4
4
≈ 5 × 10−2 .
− σSNσcorr := σSNα SNα+β = N σSNα

The overlap can finally be estimated to bepi · SN ≈σpi
σcorr ≈ 0.01 ,σSN

as is also seen in Fig. 4.4.

60

Chapter 5
In-Situ Magnetometry
For our experiments with spontaneous emission discussed in chapter 7, a finecontrol over the magnetic field, down to the level of 100 µG, is important.
However, magnetic fields can be hard to control in the usually congested apparatus used to create a degenerate quantum gas. Here we address this issue by
showing that the atomic cloud used in experiments can itself serve as a probe
to determine the magnetic field. This chapter is an extended version of ourpublication In situ magnetometry for experiments with atomic quantum gases,
Review of Scientific Instruments 89, 013108 (2018) [129], with co-authors M.
Stewart, A. Pazmiño, and D. Schneble. Most passages and figures are quoted
verbatim from the publication, and the author of the dissertation gratefully
acknowledges the contributions from his co-authors. The author of this dissertation and D. Schneble conceived of the experiment. The author took andanalyzed the data with contributions from M. Stewart and A. Pazmiño. The
author made the numerical simulations for Fig. 5.4. The author and D.
Schneble wrote the manuscript with help from M. Stewart and A. Pazmiño.

5.1

Introduction

Experiments with ultracold atomic quantum gases [135] often call for the manipulation and control of the atoms’ spin degree-of-freedom, including workwith spinor condensates [190] or homonuclear atomic mixtures in state-selective optical potentials [92–96, 165–167] where a control of Zeeman energies toa fraction of the chemical potential (typically on the order of one kilohertz or
one milligauss), may be required. With fluctuations and slow drifts of ambient laboratory magnetic fields on the order of several to tens of milligauss,achieving such a degree of control over an extended amount of time requires61dedicated field-stabilization techniques. However, in a multi-purpose BEC machine, this may be challenging given geometric constraints that can interferewith shielding or with placing magnetic-field probes sufficiently close to theatomic cloud, which is often subject to short-range, drifting stray fields fromnearby vacuum hardware or optomechanical mounts. To address this problem,
we have developed a simple method for direct monitoring of the magnetic fieldat the exact position of the atomic cloud, by employing the cloud as its own
field probe, in a way that does not interfere with its originally intended use.
The idea is that hyperfine ground-state Zeeman sublevels that are not usedin an experimental run can be employed for a rapid, concurrent sampling ofRabi resonances, in the same run, thus making it possible to record and ‘tagon’ field information to standard absorption images, which can be used bothfor slow feedback control or for stable-field postselection. We emphasize thatour pulsed, single-shot method, which features an accuracy of tens of microgauss and has an effective bandwidth of one kilohertz, is not meant to competewith state-of-the-art atomic magnetometers [191–196]; rather, its distinguishing feature is that it can be implemented without additional hardware andindependently of geometric constraints, while featuring a performance that iscompetitive with that of advanced techniques for field stabilization in a dedicated apparatus [197, 198]. It can, at least in principle, be used over a wide
range of magnetic fields, starting in the tens of milligauss range.
This chapter is structured as follows. Section 5.2 presents the principle andimplementation of our method. Section 5.3 discusses the expected measurement accuracy as well as an experimental test based on a tagged measurementof slow Rabi oscillations on a magnetic-field sensitive transition. Section 5.4 describes an application featuring the precise characterization of a state-selective
optical lattice potential via microwave spectroscopy [96].

5.2

Method and Implementation

5.2.1

Principle of Operation

The principle of the method is illustrated in Fig. 5.1 for the S1/2 (F = 1, 2)
hyperfine ground states of 87 Rb, which are split by 6.8 GHz. The atomic
sample is located in an externally applied bias field B0 along z leading toa differential Zeeman shift δz /2π = 0.7 MHz/G × B0 between neighboring|F, mF i states. Starting with all atoms in the state |ai ≡ |F = 1, mF = −1i, asequence of microwave pulses i distributes population to |2, 0i, |2, −1i, |2, −2i(i=1,2,3), and then via |2, 0i to |1, 1i (i = 4) and further to |2, 1i and |2, 2i62

(i = 5, 6). To ensure isolated addressing of each transition, the detunings δiand Rabi couplings Ωi are chosen to be small compared to δz (by three orders ofmagnitude in the example discussed below), and the ordering of the individual
pulses is chosen to avoid spurious addressing of near degenerate single photontransitions: |2, −1i ↔ |1, 0i ≈ |2, 0i ↔ |1, −1i and |2, 1i ↔ |1, 0i ≈ |2, 0i ↔
|1, 1i. Other transitions are near degenerate but magnetic dipole forbidden,|∆mF | > 1.

Figure 5.1: In situ magnetometry scheme. (a) A Bose-Einstein condensate of87
Rb atoms in a bias field B0 is subjected to a series of microwave pulses that
distribute population over the |F, mF i ground state manifold, depending on
the exact value of the field. (b) Relevant states for the 6-pulse sequence, the|1, −1i (red) and |2, −2i (blue) states are used for the measurement of Fig. 5.3.
(c) Outline of a typical experimental run (gray) with the magnetic field tagging
added in (white). (d) Rabi resonance for Ωi τi = 0.94π, choice of detunings
δi = 0.82Ωi (at B0 ), and effects of magnetic-field changes on the transfer
probabilities pi (for identical Ωi ) from which the field is then reconstructed.
The pulse parameters are adjusted such that the final populations PF =2,mF
expected at B0 are comparable, and that their sensitivity to small deviations63

1

δB from B0 is maximal, cf. Fig. 5.1 (d). Assuming all δi > 0 at B0 , thepopulations change away from the nominal field B0 is negative for i = 1, 2, 3
and positive for i = 4, 5, 6, with each transition shifted by a different amount.
The change in the set of final populations then allows for an unequivocal andprecise reconstruction of B = B0 + δB. In quantitative terms, the transition
probabilities pi for the individual pulses can be calculated from the relative
final-state populations PF,mF = N (F, mF )/N as
p1 = P1,1 + P2,0 + P2,1 + P2,2
p2 = P2,−1 /(P1,−1 + P2,−1 + P2,−2 )p3 = P2,−2 /(P1,−1 + P2,−2 )p4 = (P1,1 + P2,1 + P2,2 )/(P1,1 + P2,0 + P2,1 + P2,2 )p5 = P2,1 /(P1,1 + P2,1 + P2,2 )p6 = P2,2 /(P1,1 + P2,2 )Each pi is related to the magnitude of the magnetic field B via (see also
chapter 2 section 2.2.3)
 2
τi
Ωi
(5.1)
sin2 (Ω̃i ),
pi =
2
Ω̃i
where Ω̃i = (Ω2i + δi2 )1/2 and δi = δi (B) is the modified detuning of the ith
pulse from the ith addressed resonance. Assuming that the Rabi couplings Ωi
are known from an independent calibration, the magnetic field B can then be
extracted by fitting ~(δi + ωi ) = E(Fi , mFi ; B) − E(Fi0 , mFi0 ; B), where ωi isthe microwave frequency for the ith pulse, and where
E(F, mF ; B) = −~∆ ~∆ p±
1 + mF x + x2 + gI µB mF B
8
2

(5.2)

is the Breit-Rabi [163] energy of the levels involved in the transition (see also
chapter 2 section 2.2.4), where the +(-) sign holds for F=2(1), x = (gI −gs )µB B/∆, with gs the g-factor of the electron, ∆ = 2π × 6.834... GHz andgI = −9.951... × 10−4 for 87 Rb [144].

5.2.2

Experimental Implementation

Our experiments are performed in a magnetic transporter apparatus [134] (seealso chapter 2 section 2.1.5), with an optically trapped condensate of N ∼1

For δB  B0 , fluctuations perpendicular to z can be neglected, since they are quadratically suppressed.

64

1 × 105 atoms in the |ai ≡ |1, −1i ground state. At the end of an experimentalrun (which usually contains steps for the manipulation of the motional and/orinternal state of the atoms), the atoms are released, given about 1 ms to
expand (to avoid interaction effects), then subjected to the magnetometry
pulse sequence described above, and subsequently detected using absorption
imaging. For the determination of the state populations PF,mF we use SternGerlach separation (see chapter 2 section 2.2.1). In addition, to distinguish
the F = 1, 2 states with |mF | = 1 (note that the gF factors in 87 Rb have thesame magnitude), absorption imaging of the F = 2 states is first performed
using resonant F = 2 → F 0 = 3 light, which disperses the F=2 atoms while
the F = 1 atoms continue their free fall. After optical pumping of the F = 1
atoms to F = 2 (using F = 1 → F 0 = 2 light) these atoms are then imaged as
well.
Several considerations determine the optimum choice of parameters for themagnetometry pulse sequence. Maximizing the magnetic-field sensitivity of thepi (see Eq. 5.1) for a fixed coupling Ωi yields optimum detunings δi ≈ 0.58Ωi(at B0 ) and pulse durations τi ≈ 1.24πΩ−1(the pulse area should be keptibelow 3π/2 in order to avoid sidelobes as high as the main lobe in the Rabi
spectrum). Additional minimization of the sensitivity to possible fluctuationsof Ωi (with microwave amplifiers typically specified only to within 1 dB) modifies these conditions to δi ≈ 0.82Ωi and τi ≈ 0.94πΩ−1
i , respectively. Ideally,
the chosen coupling strength Ωi of each transition should be proportional tothe differential magnetic moment ∆µi = ∂B E(Fi , mFi ; B)−∂B E(Fi0 , mFi0 ; B) of
the two states involved in the transition. Furthermore, the expected range δBof fluctuations around B0 sets the optimum choice of Ωi through δB ∼ ~Ωi /µB ,and, in turn, the accuracy of the measurement goes down with increasing Ωi .
In our experiment, we can comfortably realize kHz-range microwave couplingson all transitions (which are independently calibrated from sampling single
Rabi resonances).
To demonstrate our method, we applied a bias field of 5.9 G using a pairof Helmholtz coils with 10 ppm current stabilization. Figure 5.2 shows theresults of a typical short-time measurement of the magnetic field along thebias field direction, using an AC-line trigger to start the pulse sequence (seeappendix C.1). The dominant contribution to field fluctuations around B0 isseen to be ambient AC-line noise with an amplitude around 1 mG, containing
the first few harmonics of 60 Hz. From here on, we compensate for this byfeeding forward the sign-reversed fit function onto an identical secondary coilof a single winding. The subtraction of the fit results in residual fluctuations
up to ±0.4 mG, without apparent phase relationship with the AC-line.

65Figure 5.2: Measurement of magnetic-field fluctuations (at B0 = 5.9 G), referenced to an AC-line trigger with a variable delay. (a) Reconstructed field noise,
as a function of time after an AC-line trigger. The solid line is a fit functiona cos(ωAC t+φ1 )+b cos(3ωAC t+φ3 )+c cos(5ωAC t+φ5 )t, with ωAC = 2π×60 Hz.
(b) Residual field variation after subtracting the fit function.

5.3

Characterization of Performance

5.3.1

Slow Rabi Cycling

We characterize the remaining fluctuations further, and in particular determine whether they represent the actual magnetic field in a time interval
close to the measurement. For this purpose we implement slow Rabi cycling (at B0 = 9.045 G) on the maximally magnetic-field sensitive transition |ai ≡ |1, −1i ↔ |bi ≡ |2, −2i, with a differential magnetic moment of∆µ3 = 2π × 2.1 kHz/mG. This measurement is performed by varying the coupling time of the oscillation and then recording the number of atoms in |bi.
To accommodate the Rabi cycling measurement, we choose a truncated pulse
sequence in which the population in |ai is subsequently distributed over fivetransitions instead of six. We note that this experiment is an example for themode of operation depicted in Fig. 5.1(a), in which a ‘measurement’ (of the
Rabi cycling) is followed by a magnetic field ‘tag’. Magnetic-field fluctuations
will lead to a rapid dephasing of the Rabi oscillation. However, using the field
tag, the effect of (slow) magnetic-field fluctuations on the oscillation can beeliminated.
For a well-resolved, single-cycle oscillation, the instability of the detuning

66Figure 5.3: Slow Rabi cycling between |ai = |1, −1i and |bi = |2, −2i, withmagnetic-field reconstruction based on a 5-pulse sequence. (a) Observed timedependence of the transferred population after eliminating AC-line fluctuations
as demonstrated in Fig. 5.2. The large shot-to-shot scatter is due to residual
field fluctuations. (b) Data points post-selected to be within a 100 µG-window.
A clear oscillation is recovered, that only dephases after the first cycle. Theline is the expected oscillation. (c) Population at a constant time of 400 µs
vs. the measured field tag. The solid line is a Rabi resonance fit with thepulse time and Rabi frequency fixed to expectation. (d) Scaled population vs.
scaled time. Gray points are original data scaled by average detuning. The
shaded line is a simulation, with B0 known to within 55 µG, and Ω known towithin 1 dB (see text).
should not exceed about one tenth of the Rabi frequency. Here we chooseΩ = 2π × 0.61(3) kHz, at an average detuning of δ = 2π × 0.44(3) kHz.
We see that the raw data resulting from multiple repetitions of the Rabi
oscillation experiment has large associated scatter due to the long term drifts
and shot-to-shot jitter of the magnetic field. To demonstrate the effect of the
field tag, we plot the oscillation both as a function of inferred detuning (ata fixed duration) and time (at a fixed detuning). The results are shown in67

Fig. 5.3 (b,c). In addition, we also plot all data, as scaled population pΩ̃2 /Ω2vs. scaled time tΩ̃. Clearly, the field tagging leads to a marked improvementof the oscillation contrast.
The next section, 5.3.2, will give the details of a simulation of the exact
behavior of the field reconstruction. For the given example, and for the parameters of the five-pulse sequence used, we expect the reconstructed fields toscatter around the true magnetic field value with a 55 µG standard-deviation.
The simulation and data agree very well, with a slight deviation at late times,
potentially due to imperfect cancellation of the AC-line or higher-frequency
noise that is uncorrelated with the AC-line.
In our measurements, the high degree of correlation between the transferred
population and the detected magnetic field further confirms that the residualfluctuations occur on a scale that is long compared to the duration of the Rabi
cycle preceding the field measurement (cf. Fig. 5.3). We note that on longtime scales, the observed magnetic field drifts are typically on the order of one
to several milligauss, over the course of one hour.

5.3.2

Theoretical Accuracy & Operation Range

For the Rabi oscillation measurements described in section 5.3.1, the parameters of the magnetometry pulse sequence i = (1, 2, 4, 5, 6) were Ωi /2π =(2.3, 1.6, 2.6, 2.0, 2.7) kHz, τi = (150, 150, 150, 200, 120) µs and δi /2π =(1.8, 2.8, 2.0, 2.1, 3.4) kHz, which yielded an inferred accuracy of 55 µG. To
estimate the ultimate resolution and limits of our magnetometer for optimal
parameters (see section 5.2.2), we perform a Monte-Carlo simulation, using a
six-transition sequence. We start with a set of fixed (true) fields Btr drawn
from a Gaussian distribution around B0 that are supposed to be reconstructed.
The number of atoms transferred in the ith pulse at fixed pi is drawn from abinomial distribution, while the transfer probabilities pi themselves are subject
to uniformly distributed fluctuations of τi (±2 µs), Ωi (±1 dB), δi (±2π×7 Hz)
and the instantaneous magnetic field during each individual pulse due to uncanceled residual fluctuations (±100 µG). The Rabi frequencies are Ωi /2π =(0.9, 1.9, 3.1, 1.2, 1.9, 3.1) kHz and the optimized detunings and pulse areas
are δi = 0.82Ωi and τi Ωi = 0.94π as mentioned earlier.
Results of the simulation are shown in Fig. 5.4. For the optimum pulse parameters, the reconstruction of Btr is accurate to within a standard deviationof 25 µG. A reconstruction is consistently possible within a ±500 µG window
around B0 , if outliers with large fit uncertainties are removed. For larger distances from B0 , the default detunings δi can be readjusted, or alternatively,
larger Rabi couplings can be used, at the (inversely proportional) expense of68

Figure 5.4: Simulated field reconstruction (over 104 runs). (a) Reconstruction
error vs. fit uncertainty Be , with convergence in the shaded area Be < 80 µG.
(b) Reconstruction error vs. distance of Brc from B0 , after discarding fitswith Be > 80 µG. Proper convergence is obtained in a ±500 µG window. (c)Histogram of reconstruction errors for the data in the gray shaded areas (a,b).
The solid curve is a Gaussian with a σ of 25 µG.
the accuracy of the field reconstruction. The results of the simulation confirm
that most of the apparent remaining fluctuations in Fig. 5.2 are actual fluctuations of the ambient magnetic field, at least to within the reconstruction
uncertainty (±100 µG for the pulse parameters chosen in that experiment).

5.4

Application: Spectroscopy of State - Selective Optical Lattices

A number of experimental applications involve the use of homonuclear mixtures of alkali atoms in state-selective optical lattice potentials [92–96,165,166,
199] , which rely on the existence of a differential Zeeman shift between thestates involved. In certain cases, a highly stable separation between a deeply
lattice-bound state and a less deeply bound or free state may be desired, suchas when the states are subject to coherent coupling [124, 200, 201], requiring
precise control of both the lattice depth and the magnetic field.
Figure 5.5 (a) shows an experimental configuration in which we preparedan ‘untrapped’ ensemble of atoms in state |bi = |2, 0i, coupled to a state |ri =|1, −1i that is confined to the sites of a deep, blue-detuned lattice potential
with a zero-point energy shift hνho /2 = h×20(1) kHz, generated with circularly
polarized light from a titanium-sapphire laser (see section 2.2.5 for details).
To stabilize the magnetic field, we utilize post-selection down to the 100 Hz

69

Figure 5.5: Microwave spectroscopy of a free-to-bound transition in a stateselective optical lattice potential (wavelength 790.10(2) nm, σ − polarization).
(a) Population is transferred from the untrapped state |bi = |2, 0i to theconfined state |ri = |1, −1i. The gray lines indicate the magnetometry sequence following the transfer. (b) Bound-state population after a 400 µs longpulse with Ω = 2π × 450(1) Hz and variable detuning, and after accounting
for magnetic-field fluctuations. The sequence of spectra was taken at regularintervals over the course of one hour.
level based on the magnetic-field tagging described above, using parameters
similar to those in Fig. 5.4. The optical intensity I is stabilized to ∼1% usinga photodiode and a PID regulation circuit, yielding a transition
frequency√that should be stable to within about 100 Hz (since ωho ∝ I). However,this does not eliminate the possibility of slow drifts of the lattice depth (suchas due to temperature induced birefringence or small wavelength changes ofthe laser) in the course of an experiment, as can be seen in Fig. 5.5 (b). To
address these issues, the precise resonance condition can now be monitored
throughout data taking, using our method. The range of the drift is several
hundreds of Hz. We emphasize that the spectroscopic precision necessary forthis kind of experiment would not be attainable without canceling AC-line
induced magnetic field noise and compensating the shot-to-shot fluctuations
using the magnetic field tag.

70

5.5

Conclusions

In conclusion, we have demonstrated a simple method for in-situ monitoring of magnetic fields in quantum gas experiments with alkali atoms, with ademonstrated accuracy of 55 µG, an inferred accuracy of 25 µG for optimized
parameters, and a time resolution of 1 ms. This is a very notable improvement
over magnetic field stability in the past which was on the order of 5 mG ofdrift per day and up 10 mG due to AC-fluctuations of the magnetic field. As
already seen for the examples above, the magnetometry pulse sequence canbe tagged onto experiments that potentially involve several hyperfine states.
In principle, the number of transitions used for magnetometry can be reduced
down to two, as long as they move differently for a change of the magnetic
field (this can be achieved by having two detunings of opposite sign or differential magnetic moments of opposite sign). For example, the method could
work using only transitions 1 and 4 of Fig. 5.1 (b). Using a smaller√ number oftransitions generally degrades the accuracy (here by a factor ∼ 3 when all
pulse parameters are left constant, compared to using six transitions), but it
increases the measurement bandwidth (here by a factor of 3), which could be
an important independent consideration for certain applications.
Thus far, we have only described the use of this method as a scalar magnetometer (in order to be able to ignore fluctuations in perpendicular directions).
It should also be possible to access fluctuations of the ambient field in morethan one spatial direction, if the bias field is rotated during the magnetometry
pulse sequence (with two transitions used per direction). This can become
important if one wants to use this method for stable field post-selection at low
fields.
Finally, for comparison to other magnetometry
techniques, a sensitivity√
may be specified as [202]√η = ∆Bmin T , i.e. as the minimum detectablechange in field ∆Bmin = 2 ln 2σ ∼60 µG multiplied by the square root of thecycle time. Since typical field fluctuations in laboratories usually stem fromAC-mains or are very low frequency (such as fluctuations of Earth’s magnetic
field), synchronizing the experiment to the AC-line can yield one measurementin the effective√ integration time of 1 ms. In this case,3an effective sensitivityη ∼ 300 pT/ Hz (in a measurement volume of 10 µm ) can be reached.
In comparison, other stabilization methods typically use fluxgate magnetometers [198] and large volume Helmholtz coils (1 m3 ). The bandwidth of
the fluxgate magnetometers is on the order of 1 − 5 kHz [203].

71

Chapter 6
Review of the Weisskopf-Wigner
Model
The description of spontaneous emission of photons from an excited atom,
as described by Weisskopf and Wigner [100] on the basis of Dirac’s radiation
theory [204], is a fundamental concept in quantum electro-dynamics. In thischapter we will give a brief overview of the Weisskopf-Wigner model in order
to discuss results used in chapter 7.

e

Ee

bk

Γ

g

0

Figure 6.1: Spontaneous Emission. A two-level atom initially prepared in the
excited state |ei at energy Ee undergoes decay to the ground state |gi at rateΓ under emission of a photon |bk i at wavenumber k and momentum ~k.

6.1

Electric Dipole Hamiltonian

We want to consider the interaction of a single atom with the quantized electromagnetic vacuum. To that end, we will first start with the Hamiltonian of72

a particle in a classical monochromatic electromagnetic field [2]2~p~ˆ − q A
+ V (~rˆ)Ĥ =2m~ is the vector
where q is the charge of the particle, and V is the potential, and A
potential. Note that we have neglected the energy in the electromagnetic fieldin the above equation. In our case, we consider the electron in the electricfield of a nucleus, so q = −e, where e is the elementary charge. Expanding thesquare yields~ + eA~ · p~ˆ + e2 A~2ep~ˆ · A
Ĥ = Ĥ0 +2m
 
where Ĥ0 = p~ˆ2 /2m + V ~rˆ is the Hamiltonian without field. We will neglect
the last term for since it is negligible for small fields, and we will choose the~ = 0, transverse waves) so that we can write:
Coulomb gauge (∇ · A
e ~ ˆĤ = Ĥ0 + A· p~.
m
We make the following ansatz for the vector potential:


~
~
~
A = A0 cos k · ~r − ωt
~ 0 sin (ωt)≈A
where ω is the frequency of the electromagnetic field and we have made thedipole approximation (~k · ~r ≈ 0, i. e. the spatial variation of the electric field
is negligible over the size of the atom) in the second line of the last equation.
The electric field at the place of the atom is then
~ = iA~ 0 ω sin(ωt) = ~ε E0 sin(ωt),
E
where E0 is the magnitude of the electric field and ~ε is its polarization. We
arrive at:e  ˆ
~ε · p~ E0 sin(ωt) = Ĥ0 + Ĥ 0 .
Ĥ = Ĥ0 +mωhi
Using the relation p~ˆ = m ~rˆ, Ĥ0 /i~, we can evaluate the transition matrix
elements of Ĥ 0 to
0
Ĥj,k
= −i

ωj,k
~ˆ
hj|~ε · (−e)~rˆ|kiE0 sin(ωt) ≈ −ihj|~ε · d|kiE
0 sin(ωt)ωˆ
where d~ = −e~rˆ is the dipole operator, and we have assumed that the radiation
used is near resonant with the energy difference between state |ji and state
|ki.
73

6.2

Field Quantization and Weisskopf–Wigner
Hamiltonian

Now we quantize the electromagnetic field [102, 205], which, in the dipole
approximation, yields the following expression for the electric field at frequency
ω and with polarization ~ε:
r

~ω 
ˆ
b̂ω ~εeiωt + b̂†ω ~εe−iωt ,
E~ω =
2ε0 V
where V is the volume of space and ε0 is the vacuum permittivity and wehave again used the dipole approximation in the second line. The operator b̂†ω(b̂ω ) creates (annihilates) a single excitation (photon) of the electromagnetic
field at frequency ω with the polarization ~ε. The energy of the quantizedelectromagnetic field at of frequency ω is Ĥfield = ~ω(n̂ + 1/2) where n̂ is thenumber of excitations (i. e. photons) in the field. We can simply insert thequantized form of the electric field into the above Hamiltonian. If we assume
the unperturbed Hamiltonian Ĥ0 to be that of a two-level atom with groundstate |gi, excited state |ei, and excited state energy ~ωeg , we arrive at the
Hamiltonian of a two level atom in the quantized electromagnetic vacuum:[102, 205]:


X
1
†
Ĥ ≈ ~ωeg |eihe| +~ωk b̂k, s b̂k, s +
2
~k, s
!
r
X
~ωk
ˆ
†
~ b̂ |gihe| + h. c. ,−i~ε~ · hg|d|ei+~k, s2ε0 V k, s~k, s~ε~k,s is the polarization vector of the light, ωk is the frequency of the lightand s is one of the two possible polarization states (e. g. linear horizontal
and linear vertical). Since we do not externally apply a field in this case,but are rather considering the quantized zero-point fields of the quantumelectrodynamic vacuum, we have to sum over all modes which are allowedin the volume V in our physical space. The operator b̂~†k, s creates a photon ofpolarization s and wavevector ~k. Note that the time-dependence of the electricfields e−iωt was shifted into the states rather than the Hamiltonian (see also
Eq. 6.1).

74

6.3

Decay Rate and Lamb Shift

To solve the dynamics of the Weisskopf-Wigner model, we first make an ansatz
for the wavefuction, by assuming the system is in a superposition state ofbeing either a not decayed atom (|ei) and no photons in the radiation field ora decayed atom (|gi) with one photon in the radiation field [206]:X|Ψi = A(t)e−iωeg t |e, 0i +eiωk t B~k, s (t)|g, 1~k, s i ,
(6.1)
~k, s

which can be inserted into the Schrödinger equation i~∂t |Ψi = Ĥ|Ψi to get
the set of coupled equations:X?
g~k,∂t A(t) = ie−i(ωk −ωeg )t B~k, ss~k, s∂t B~k, s = ig~k, s ei(ωk −ωeg )t A(t)r~ωk~ˆ~ε~k, s · hg|d|ei.
g~k, s = i
2ε0 VFormal integration of the second equation can then be inserted into the firstdifferential equation, which yields a decoupled integro-differential equation:
Z t
X
2
0
∂t A(t) = −g~k, sdt0 e−i(ωk −ωeg )(t−t ) A(t0 ) .
(6.2)
~k, s

0

At this point no formal/closed solution can be found; however, with certain assumptions, one can arrive at good approximate expression for A(t).
The most famous is the Markovian or Weisskopf-Wigner approximation, forwhich we identify two timescales for dynamics in the system: (1) phase evolution (∂t arg(A(t)), and (2) amplitude evolution (∂t |A(t)|). If the amplitude
evolution is much slower than the phase evolution, which in this case implies
ω0  |gk |2 , then we can approximately evaluate the integral in the integrodifferential equation as (A(t) ≈ A(t0 )) :
Z t
Z ∞
0
0 −i(ωk −ωeg )(t−t0 )0dt e
A(t ) ≈ A(t)dt0 e−i(ωk −ωeg )(t−t )00With the above approximation Eq. 6.2 is now reduced to an ordinary firstorder differential equation, with the only challenge being the evaluation of the75

following integral (we have taken the continuum limit, which transforms thesummation over momentum states to an integration over momentum states):
Z ∞
XZ
2
Γ
0
3~
dt0 e−i(ωk −ωeg )(t−t ) := − iδL .
d k g~k,s20s
A detailed version of how to evaluate this integral is shown for 3D in [102]or [206]. We thus arrive at the differential equation:Γ∂t A(t) ≈ −A(t)− iδL2where the real part of the integral represents the spontaneous decay rate andthe imaginary part of the integral represents the Lamb shift. The Lamb shift,i. e. the change of the resonance frequency of the atom induced by the zero
point electromagnetic field oscillations, is similar to the shift in resonancefrequency of a classical pendulum when introducing a damping term. The
spontaneous emission rate Γ evaluates to [102, 205]:
Γ=

3
ωeg
~ˆ 2
d|ei
,
hg|
3πε0 ~c3

and the Lamb shift evaluates to [206]
Z ∞

ωk3
Γ
dωk
P
.
δL =
3
2πωeg
ωk − ωeg
0Here P denotes the Cauchy principal value of an integral, and me is the electronmass. The integral diverges, a problem which can be solved e. g. by massrenormalization of the electron, as done by Bethe [207].
The Markovian approximation breaks down whenever the excited stateenergy is not much larger than the inverse lifetime (linewidth Γ). However,for a typical atomic optical decay, the linewidth is many orders of magnitude
smaller than the transition frequency (Γ ∼ 2π×107 Hz vs. ωeg ∼ 2π×1015 Hz),so that the Markovian approximation is well satisfied. As we will see, this is
not the case for decay in photonic bandgap materials.

76

Chapter 7
Spontaneous Emission of Matter
Waves from a Tunable Open
Quantum System
This chapter is an extended version of our publication Spontaneous Emission
in a Matter-Wave Open Quantum System, arXiv:1712.07791 (2017) [208], andits revision Spontaneous emission of matter waves from a tunable open quantum system, Nature 559, 589 (2018) [209], with co-authors M. Stewart, A.
Pazmiño, J. Kwon, and D. Schneble. Most passages and figures are quoted
verbatim from the publication, and the author of this dissertation gratefully
acknowledges the contributions from his co-authors. Additional sections andfigures will be marked with an asterisk (*) above the section heading or figure caption respectively. D. Schneble, the author of this dissertation, and M.
Stewart conceived the experiment. The author took the measurements withassistance from A. Pazmiño and J. Kwon. The author analyzed the data with
contributions from M. Stewart. Numerical simulations were performed by theauthor. D. Schneble supervised the project. The results were discussed andinterpreted by all co-authors. The manuscript was written by the author and
D. Schneble with contributions from A. Pazmiño, J. Kwon and M. Stewart.

7.1

Introduction

The decay of an excited atom undergoing spontaneous photon emission into
the fluctuating quantum-electrodynamic vacuum provides an emblematic example for the dynamics of an open quantum system. Recent experiments
have demonstrated that the gapped photon dispersion in periodic structures77

can give rise to novel spontaneous decay behavior including the formation ofbound states [116, 121, 122]. So far, these effects have been restricted to theoptical domain. Here, we experimentally demonstrate similar behavior in asystem of elementary emitters, realized with ultracold atoms in an optical lattice, that decay by emitting matter-wave, rather than optical, radiation into
free space. By controlling vacuum coupling and excitation energy, we directly
observe exponential and partly reversible, non-Markovian dynamics and detect a tunable bound state containing evanescent matter waves for emission at
negative excitation energies. Our system provides a flexible platform for theemulation of open-system quantum-electrodynamics and studies of dissipative
many-body physics with ultracold atoms [124–126].
The Weisskopf-Wigner model of spontaneous emission [100,205] is a central
concept in quantum optics [102], describing how an excited atom can decay
to its ground state due to coupling to zero-point oscillations of the electromagnetic vacuum. It simultaneously represents one of the first open quantum
systems discussed in the literature, and it is an area of research that has recently seen a resurgence of intense theoretical effort [99, 127, 210, 211]. In its
usual Markovian formulation, the model makes the assumption that the decay
proceeds on a much slower time scale than the optical period, which leads to amemoryless, exponential decay of the excited-state amplitude and to an associated Lamb-shift of the transition frequency. For free-space photon emission,
the Markovian approximation is generally fulfilled to high accuracy.
On the other hand, modifications to the mode density of the vacuum canchange the features of spontaneous decay. This was first recognized in the
1940s [101] and again decades later [106] in the development of cavity quantum electrodynamics [212–214], where the decay can be altered to the extreme
point of coherent vacuum Rabi oscillations when the spectrum is restricted toa single mode. Between these two limits lies the regime of a vacuum witha bounded continuous spectrum, in which a strong modification of spontaneous decay behavior occurs close to the boundary. An example is given by
photonic crystals (also called photonic bandgap materials) [104, 105], wherea periodic spatial modulation of the refractive index gives rise to a gapped
dispersion relation. For emission close to a bandgap, the Markovian approximation can no longer be made, and novel features appear including oscillatory
decay dynamics for energies above the band edge and the formation of atomphoton bound states below [215]. Over the past two decades, experiments on
spontaneous emission in photonic bandgap materials, including the microwave
domain, have observed some of these effects, specifically, modified spontaneousemission rates [109, 113] and Lamb shifts [117], as well as spectral signatures
for non-exponential decay [116]. Very recently, experiments have probed the78

long-predicted bound state [103, 216], using both transmon qubits coupled
to corrugated microwave guides [122], and atoms in photonic-crystal waveguides [121] with the prospect of engineering systems with optical long-range
interactions [217].
Here, we realize an atom-optical analog [124–126] of emission in a onedimensional photonic bandgap material, where the singularity in the modedensity near the edge of the continuum leads to particularly strong deviations from Markovian behavior. In our system of matter-wave emitters, thefree tunability of the excitation energy and decay strength allows for a systematic exploration of the emergence of non-Markovian dynamics, including
partial reversibility and the formation of a matter-wave bound state which canbe directly detected. Importantly, the close spacing of emitters gives rise tocollectively enhanced dynamics beyond the Weisskopf-Wigner model.

7.2

Introduction to the System

The experimental configuration is shown in Fig. 7.1(a). Using a deep threedimensional optical lattice with state-selectivity along one axis, we prepare asparse array of atoms confined to sites that are embedded in a system of isolated tubes acting as one-dimensional waveguides (see section 7.6 for details).
An atom’s internal state (|ri, red) is coherently coupled to a second, unconfined internal state (|bi, blue) using an oscillatory magnetic field. Each sitethus acts as a two-level matter-wave emitter, with harmonic-oscillator groundstate occupational levels |gi (empty) and |ei (full), supporting both the emission (for |ei → |gi) or absorption (for |gi → |ei) of a |bi atom. The excitationenergy of the emitter, which is given by the detuning ∆ of the coherent coupling from the atomic resonance, is converted into kinetic energy for atomic
motion along the axis of the waveguide.
One of the main features of each matter-wave emitter is its ability to undergo spontaneous decay as understood by Weisskopf and Wigner. Assuming
no lattice potential, the driven atom performs simple Rabi oscillations between
two internal states |ri and |bi described by the HamiltonianĤ = (~Ω/2)eiδt r̂b̂† + H.c.,(7.1)

where Ω denotes the strength and δ the detuning of the coupling from thetransition. The tight confinement of just one of the states (say |ri) strongly
couples the atom’s internal and motional degrees of freedom, producing azero-point energy shift ε̄0 = ~ω0 /2  ~Ω in the potential, as well as a kineticenergy shift εk = ~2 k 2 /2m for motion of the free |bi state at ~k momentum.
79

(a)

(b)

ii: Rotating frame

i: Bare statesiii: State-selective OL

b

Ωk

6.8
F=2, m =0 GHz
F

87

εk

Δ

δ

Rb

5 2S ½Ω0r
F=1, mF=-1

0

0

(c)
e

εk

Δ

0 k/kr 1 0 kπ/a 1

bk

Γ(Ω)

g

0

0
PBG

Figure 7.1:

Realization of matter-wave emitters. (a) Experimental configuration: An
occupied site of an optical lattice embedded in a single-mode matter waveguide acts as anelementary emitter of a single atom; adjacent empty lattice sites act as absorbers. The
bottom panel shows a momentum distribution in the waveguide, observed after releaseand free expansion, where ~kr = ~2π/λz (with λz = 790.1 nm) is the recoil momentum.
(b) Emission mechanism: (i) Bare internal-state pair |ri and |bi in 87 Rb; (ii) state pair in aframe co-rotating with a 6.8 GHz microwave field (variable detuning δ and coupling strengthΩ), and (iii) in the co-rotating frame after applying the state-selective potential of the lattice(detuning shifted to ∆ = δ + ε0 /2, where 0 = h × 40 kHz). The microwave couples (withΩk = Ωγk ) the trapped |ri state to free |bk i states with momentum k and kinetic energy
εk . (c) The filled (empty) potential well can be viewed as the excited (ground) state |ei(|gi) of a matter-wave emitter. The emission of atoms in this scenario bears close similarity
to the emission of photons in photonic bandgap (PBG) materials, both featuring quadratic
dispersions and energetically forbidden regions.

80As a consequence, the detuning and strength of the coupling are shifted to∆k = δ +(ε̄0 −εk )/~ and Ωk = Ωγk , respectively, with γk = hk|ψe i denoting theoverlap of the external wavefunctions. Integration over all possible momenta
k then yields [126]
X
Ĥ =
(7.2)
~gk ei∆k t |gihe|b̂†k + H.c. ,kwith gk = Ωk /2 i.e. the standard Weisskopf-Wigner Hamiltonian describing
spontaneous emission into a vacuum of modes (k, εk ) (see Fig. 7.1(b)). In
contrast to optical emission in free space, the dispersion relation εk is quadratic
as in a photonic crystal, cf. Fig. 7.1(c). In such crystals, the emission energy
relative to the edge of the continuum may be adjusted through the crystal’sband structure; in our system, the excitation energy ~∆ ≡ ~∆k=0 itself istunable including the case ∆ < 0. Importantly, the tunability also includes
the vacuum coupling gk , which is set by Ω.

7.3

Sample Preparation

In the experiment we use 87 Rb atoms in the hyperfine ground states |ri =|F = 1, mF = −1i and |bi = |2, 0i (the fact that |ri lies below |bi is inconsequential in the rotating frame). The atoms are confined to a two-dimensional
array of ∼ 103 isolated lattice tubes spaced at 532 nm, each with a radial confinement of ω⊥ = 2π × 26 kHz and a residual axial confinement of
ωz = 2π × 97 Hz that quantizes the mode spectrum for released |bi atoms inthe z direction (see Fig. 7.1), but is inconsequential for times much shorter
than τz = 2π/ωz ∼ 10 ms. A state-selective lattice with period λz /2 = 395 nm
and harmonic-oscillator frequency ω0 = 2π × 40(1) kHz strongly confines the|ri atoms along the tube axis (the harmonic oscillator frequency is the approximate
harmonic potential at the minimum of each potential well ω0 =√2 sEr,790.1 nm , with s the lattice depth in recoil energies Er,790.1 nm ).
Starting with an optically-trapped Bose-Einstein condensate [134], we firstcreate an atomic Mott insulator of |ri atoms by simultaneously ramping up
all three optical lattices, over a time of 90 ms, to depths of {40Er,1064 nm ,40Er,1064 nm , 30Er,790.1 nm } in the {x, y, z} directions, where Er,λ = h2 /2mλ2 isthe respective recoil energy. This procedure results in a deeply-confined, Mottinsulating sample, while also shifting the minimum of the tube potentials by∼ 10 µm with respect to the sample. The shift (which is comparable to thesample size and corresponds to the change of the gravitational sag during theramp [134] after the atoms are effectively pinned by the z-lattice) leaves the81resonance condition for individual atoms unaffected since |ri and |bi experience
the same tube potential.
After loading the atoms into the lattice, we transfer a fraction of ∼ 0.82 toan intermediate state |2, 1i, using a two-photon RF-pulse (microwave radiation
at 6.8 GHz and radiofrequency radiation at ∼ 3 MHz) of 0.95 ms duration.
These atoms are finally removed using resonant light on the D2 cycling transition (F = 2 → F 0 = 3). Using the |2, 1i state has the advantage that there
is no first order shift in the energy difference between |1, −1i and |2, 1i due
to either the magnetic field or the state selective optical potential. Also, collisional shifts [177] of doubly (triply) occupied sites are ∼ 50(100) Hz for ourlattice parameters, much less than the spectral width of the pulse. This yields
a sample of 2.8(2) × 104 |ri atoms with an average site occupation hni i . 0.5
in the tubes. Having thus created an initial state of matter-wave emitters, wethen switch on a 6.8 GHz microwave field of variable coupling strength Ω anddetuning ∆.

7.4

Markovian Dynamics

A common scenario considered in the Wigner-Weisskopf model is emission
deep into the continuum, such that the decay dynamics are much slower than
the time scale set by the excited-state energy (or the elevation above theband edge in the case of a photonic crystal). This allows for a Markovian
treatment and results in exponential decay of the excited state. Following
Fermi’s Golden Rule, the decay width Γ is the product of the mode density ρand the square of a matrix element Hge which, for optical decay, is the productof the electric dipole moment and the zero-point field of the resonant mode.
For our √system, an analogous analysis [126] (valid for √Ω/∆  1) leads to2
Γ = Ωk̄ / ω0 ∆, containing the 1D mode density ρ ∝ 1/ ∆ and Hge ∝ ~Ωk̄ ,pwhere k̄ = 2m∆/~ labels the resonant mode.
Because of the residual axial tube confinement, ωz , all measurements aretaken for Ω/ωz > 1 and associated time scales shorter than τz = 2π/ωz ∼10 ms. The measured |ri population is shown in Fig. 7.2 for parameters in the(quasi-)Markovian regime (Ω/∆)2  1 as a function of time (a) and detuning(b); the data in (a) are for Ω/∆ ≈ 0.4 at ∆ ≈ 2π×2 kHz (with Γ = 2π×72 Hz).
After a variable coupling time t, we observe an irreversible, exponential decay
in agreement with the expectation; however, the measured population does
not decay to zero but instead saturates at a finite value. We qualitatively
explain this discrepancy by taking into account that an excited emitter is not
isolated but part of a (mostly) ground-state array that enables reabsorption,
82

1

(a)
.6

10
20
Δ/2π [kHz]2
4
Evolution time [ms]

(c)

0
-2
Δ/2π [kHz]

(d)

20

30

0
-1

(e)

_

δL/2π [kHz]

2
k/kr

.5
0

0

0

30

1
arb. u.

k/kr

(b)
.6

0

2

Population

Population

1

-2

-2
1.5

1.5
3.7
Ω/2π [kHz]3.7
Ω/2π [kHz]Figure 7.2:

Markovian Regime. (a) Time evolution of the lattice population forΩ = 2π × 0.74(5) kHz and ∆ = 2π × 1.9(3) kHz. Each point is the average of at least 3 measurements, bars are the standard error of the mean. The red line is a phenomenological exponential decay curve with a fitted rate of 2π × 94(3) Hz and an offset of 0.503(4). In contrast,
the light gray lines represent the Markovian approximation (dashed, Γ = 2π × 72(12) Hz)
and the exact analytical solution for an isolated emitter [126] (solid). (b) Lattice population
as a function of ∆ for t = 0.4 ms and Ω = 2π × 1.5(1) kHz. The solid line is the Markovian
expectation with the overall decay width Γ scaled by 0.61(1). (c) Detected momentum distribution of |bi atoms versus ∆ for parameters as in (b). The dashed line is the single-particle
dispersion; data for small positive and for negative detunings are outside the Markovian
regime. (d) Raw TOF data for extracting the energy shift at ∆ = 2π × 6.0(3) kHz. (f)Measured shifts δ̄L = ∆k̄ − ∆ in the regime Ω/∆ < 1, for Ω̃t = 1.24 and averaged over
∆ = 2π × 1, 2, 4, 6 kHz. The data are extracted from the second moment (maximum) of
the momentum distribution, shown using blue squares (red circles). The blue solid and red
dashed lines are quadratic fits, the gray dotted line represents δL (Ω).

83

in analogy to an optically thick medium. In the Weisskopf-Wigner formalism,
the array is modeled by introducing site-dependent phases and projectors,
resulting in [124]XĤ =~gk ei(∆k t−kzj ) |gj ihej |b̂†k + H.c .
(7.3)
k,j

Following the Master-equation based treatment in [125], we expect excitations
to be transferred between neighbors on a timescale t ∼ 1/Γ = 2 ms, leading
to a slowdown of the decay in qualitative agreement with Fig. 7.2 (a) (see also
section 7.7). Moreover, the emitted atoms cannot escape in our system forlong times, which motivates the formation of a steady state as t approaches
τz . Here, additional dephasing effects that are not captured in this model
may arise from collisions between emitted atoms (scattering between modes
in each tube). At early times t < 1/Γ, with still weak reabsorption, the decay
at fixed time τ , cf. Fig. 7.2 (b), approximately displays the expected detuning
dependence of the Weisskopf-Wigner model, exp[−Γ(∆)τ ] ∼ 1 − Γ(∆)τ (albeit
with a downward rescaling of the actual value of Γ at the finite pulse timeτ = 0.4 ms used, with reabsorption effects already apparent).
We next characterize the momentum distribution of the emitted atoms.
For this purpose we apply a 0.4 ms long coupling pulse and then observe
the location of the |bi atoms after 15 ms of free fall, using state-selective
absorptive imaging (see section 7.6 and chapter 2 section 2.2.1). Based on
the Markovian approximation, isotropic emission with wave-packets centerednear the resonant momentum k̄(∆) is expected. Figure 7.2 (c) shows the observed momentum distribution as a function of ∆; the emission clearly traces
the parabolic dispersion. Moreover, the spectral width σk of the separated
wavepackets decreases with detuning, in qualitative agreement with the expectation (σk ∝ 1/∆ for large detunings [126]; a quantitative comparison iscompromised by the finite time of flight). The ‘intensity’ of the emitted matterwave pulse strongly depends on the detuning as already seen in Fig. 7.2 (b).
The standard Markovian treatment of the Weisskopf-Wigner model yields
a Lamb shift of the ground and excited states as a unitary coupling to the vacuum. An analogous analysis for our system [124,126] yields a shift δL = Ω2 /ω0of the excited-state energy, to (∆ − δL ). We measure the momentum distribution for variable Ω at several values of ∆ and then calculate the mean kinetic
energy of the wave-packets both from the second moment of the momentum
distributions and from the location of their fitted maxima (see 7.6 for details).
To facilitate comparison with the model, the data are taken for a constanteffective pulse area Ω̃t, where Ω̃ = (Ω/ω0 )1/3 Ω, and t ≤ 1 ms to mitigate propagation effects. The results for the (quasi-)Markovian regime (Ω/∆)2  1 are84

shown in Fig. 7.2 (d, e) as a function of Ω. The extracted shift has the sign
and approximate quadratic dependence of δL , but is a factor of roughly three
too large. We caution that, while this alone could point toward the existenceof collective enhancement, there is no indication for superradiance [124, 218]
from the decay data (which is consistent with the fact that there is no overall
population inversion in the array).

7.5

Non-Markovian Dynamics

Our system readily allows for the study of spontaneous emission outside theMarkovian regime, as already visible in Fig. 7.2 (c). In particular, the diverging
one-dimensional mode density near εk = 0 greatly enhances the effects of theedge of the continuum. For emission at low excitation energy, ∆/Ω  1, we
expect dynamics reminiscent of a two-level system, with damping provided by
low-energy modes. Results of measurements at ∆ = 0 are shown in Fig.7.3 (a).
We observe oscillations similar to the predictions of our isolated-emitter model[126] (which now features a finite offset, in contrast to positive detunings) but
with higher frequency and less damping, suggesting that the dynamics arecoherently enhanced by low-energy modes, whose wavelengths can extend over
several emitters (see also section 7.7).
For emission below the continuum edge, i.e. for ∆ < 0, we expect theformation of a stationary bound state [124, 126], as illustrated in Fig. 7.3(c). For our one-dimensional system, this state consists of a partly excited
emitter dressed by an evanescent, approximately exponentially decaying matter wave,
p with a binding energy ~ωB ≈ ~∆ and localization length [126]ξ = 1/ 2m|ωB |/~ . To isolate the bound state, the coupling needs to beturned on slowly to prevent the additional population of freely propagating modes [126] representing a non-adiabatic, transient shedding of matterwaves. However, we first proceed as before by switching on the coupling, at∆ = −2π × 1.7 kHz. The lattice population, shown in Fig. 7.3 (b), showsa transient oscillation (with much lower amplitude than at the edge) settlingto an asymptotic value below unity (with an observable |bi population, cf.
Fig. 7.2 (c)). Remarkably, our single-emitter model [126] now closely fits thedata within the experimental uncertainties. Indeed, for the chosen parameters,ξ is less than half a lattice period, which should lead to a relative suppressionof long-range couplings.
To access the properties of the bound state, we first determine the fractionof |bi atoms by comparing the asymptotic lattice population for a sudden andfor an exponential turn-on of the coupling. The results shown in Fig. 7.3 (d)85Population

.5
0

(b)

1

.5
1
Evolution time [ms]

g

0

.5
1
Evolution time [ms]

(e)
1
2
-2
t

40

t

Ω

10

0

t

1

0
Ω

10

-1

t

0
0

.1
.2
Transferred pop.

x/a

-2

1

0

ψ(x)/ψ(0)Counts0Ω0Momentum k/kr

Atom number

Ω

(d)

80

arb. u.

(f)

Δ

2

k

0

Ω
e

0

2

k/kr
0 1ε

(c)

.75

k/kr

Population

(a)

1

-40
2

Figure 7.3: Non-Markovian dynamics and bound-state formation. (a) Time evolution of
the |ri population for ∆ = −2π × 0.1(3) kHz and Ω = 2π × 3.0(3) kHz. The gray line is ouranalytical model. (b) Same as in (a) but for ∆ = −2π × 1.7(3) kHz. The fit parameters of
the analytic model are ∆ = −2π×2.08(3) kHz and Ω = 2π×2.79(4) kHz. (c) Illustration of astationary bound state for negative excitation energies. (d) Asymptotic fraction of |bi atoms
after t = 2.6 ms, for Ω as in (a) and ∆ = −2π × 2.2(3) kHz. The top (bottom) histogram isfor a sudden (adiabatic) turn-on of the coupling, 50 experimental runs each. (e) Illustrationof separation of evanescent and propagating waves along the weakly confining tube axis. (f)Momentum distributions of |bi atoms for the two scenarios considered in (d). Open (filled)
circles are for the sudden (adiabatic) turn on of the coupling and their difference (triangles).
The solid line represents the square of the Fourier-transform of the analytic evanescentwavefunction [126], fitted to the adiabatic data (∆fit = 2π × 2.1(1) kHz). The center ofmass of the non-adiabatic data set is shifted by ps = 0.32(1) ~kr relative to the adiabatic
data set; and that of their difference (with a half width at half maximum of 0.57(1)~kr ) is
shifted by ps = 0.80(1) ~kr . The expectation for free atoms is ps = 0.83(7) ~kr . Lower inset:
Corresponding real-space evanescent wavefunction (blue, solid) and Wannier function of alattice-trapped |ri atom (dashed, red). Upper inset: Raw data for momentum distributionsof (e) before subtraction of spurious higher-band contribution (see section 7.6).

show that 7.1(2)% of the population are in the evanescent wave, with a total |bi
population of 12.7(2)%. The observed fraction of |bi atoms in the bound state86(55%) is close to the expectation [126] of 47% for the chosen parameters, withthe excess possibly stemming from residual non-adiabaticity of the ramping(we noticed an inconsistency within [126] for the total |bi fraction, but thisdoes not affect the relative |bi fraction in the bound state). Importantly,
the recorded momentum distribution of the bound evanescent fraction of |bi
atoms, see Fig. 7.3 (f), can be directly compared to the prediction of ouranalytical model [126] due to the absence of propagation effects. Fitting themodel to the data for the adiabatic ramp-up (with ∆ as the only fit parameter)
reproduces its experimental value to within the experimental uncertainty, andthe agreement between the model and the measured momentum distribution
is indeed excellent. The localization length obtained from the exact model isξ = 142(3) nm (we note that for the parameters chosen [126] ωB ≈ 1.3∆).
Finally, additional direct evidence for the non-adiabatically released partcan be obtained by comparing the adiabatic momentum distribution to thatrecorded after the rectangular pulse. Since our sample is prepared off-centerfrom the potential minimum along the tube direction (cf. Fig. 7.3 (e); seesection 7.6), the distribution of the shed |bi atoms can separate in momentum
space from the bound fraction. The pulse duration used in this experimentcorresponds to about a quarter oscillation period (2.6 ms) along the tube axis,
thus maximizing this differential effect. As seen in Fig. 7.3 (f), the (symmetric)
difference of the two momentum distributions is centered at a finite momentum
expected for atoms released with zero velocity at the beginning of the pulse.
This finding is consistent with the time evolution in Fig. 7.3 (b), which suggeststhat the release of the unbound fraction occurs within a short time (∼ 0.5 ms)
after the coupling is turned on. We note that, as a result of propagation in the
tubes for a quarter oscillation period, the width of the momentum distributionof the released atoms reflects that of their distribution in real space (vz = zωz ).
The extracted half width at half maximum of 5.47(9) µm is comparable to theThomas-Fermi size of the initial condensate.
Much of the present work has focused on basic properties arising from thetunability of our Wigner-Weisskopf system, including the formation of boundstates below the edge of the mode continuum. On the single-emitter level,
this provides a direct analogy to atomic decay near the bandgap of a photonic
crystal. We note that, in yet another context, the observed non-Markovian oscillatory dynamics also reproduces predictions for electron photo-detachmentfrom negative ions [219, 220]. The optical lattice geometry opens up various additional avenues of inquiry. For emission sufficiently above the continuum edge, these may include novel types of superradiance that dependon the degree of coherence of the lattice population (superfluid or Mottinsulating) [124, 125] and have no analog in optical systems. Moreover, con87

trolling the longitudinal waveguide level spacing should allow for studies of thetransition between the Dicke- and Tavis-Cummings-type models in quantum
optics [102, 124] (restricted to co-rotating terms) including their modificationin the non-Markovian regime. Unlike photons, the emitted atoms can directlyinteract with each other, which should give rise to additional, nonlinear effects
modifying the population dynamics. For negative energies, the bound state
lends itself to the realization of lattice models [124] with modified tunnelingand interactions. Superficially, the structure of the bound state resembles thatof a lattice polaron [93,221] (for which a phononic Lamb-shift has recently beenmeasured [170]), with massive vacuum excitations replacing massless Bogoliubov sound excitations. Rather than reducing transport, the bound state here
leads to an enhancement of mobility. The presence of tunneling with a tunable
range is, for example, of interest for studies of integrability and thermalizationin one-dimensional geometries.

7.6

Experimental Details

7.6.1

Experimental Procedures

In this section we outline the experimental procedures and details that we used
to take our experimental data.
Atom Detection
We suddenly switch off the microwave coupling Ω and measure the population
remaining in the lattice and access the momentum distribution of the released
atoms with state-selective absorptive imaging, using a combination of bandmapping and Stern-Gerlach separation in time-of-flight (see also chapter 2
section 2.2.1).
The state detection begins with a 500 µs bandmap step, during which alllattice potentials are ramped down to zero (the bandmap is done for technicalreasons, inhibiting the rapid transverse expansion out of the tube potentials).
Subsequently all remaining trapping potentials are turned off suddenly, andafter a ∼ 1 ms expansion time, a pulse sequence for magnetic-field characterization (see below) is applied. This is immediately followed by a 5 ms longmagnetic field gradient pulse to separate states of different magnetic moments.
Finally, following a total of 14.5 ms after release from the optical potential, a
200 µs long imaging pulse, using resonant light on the F = 2 → F 0 = 3 D2
cycling transition, is used to detect the F = 2 atoms, yielding a total effective
time-of-flight (TOF) of 14.6 ms for atoms in this state. The F = 1 atoms
88are repumped for 100 µs using resonant D2 light (F = 1 → F 0 = 2) afteran additional 2.7 ms, immediately followed by another 200 µs long imaging
pulse (D2 , F = 2 → F 0 = 3) to detect the repumped F = 1 atoms for a total
TOF of 17.6 ms. Since all magnetic moments are unique within F=1 and F=2
separately, we fully resolve the population in each individual hyperfine state|F, mF i. We note that, technically, the cloud centers of F = 1 and F = 2
along the imaging direction are still overlapped during the second imaging
pulse; however, the F = 2 atoms are pushed out of the field-of-view by thetime the F = 1 detection occurs.
Momentum-space Calibration
Our standard momentum calibration relies on Kapitza-Dirac diffraction (KDD)[157] (see also section 2.2.2) from the z lattice. For a more precise determination of emission momenta, we take into account residual propagation in the
tubes which slows the atomic motion. The tubes are created, after ramping
up the z lattice, by partial retro-reflection of the Gaussian beams of our optical trap (1/e2 radius of w = 135 µm) [134], which leads to an increase of
the optical confinement ωz /2π from 72(1) Hz to 97(1) Hz (with gravity alongz). The tubes are again ramped down within 500 µs after the microwave
pulse (together with the z lattice, for band-mapping purposes), followed by aswitch off of the optical trap. We numerically simulate the motion of atoms in
the tubes, by assuming that the release (with momentum ±k̄) occurs midway
through the pulse at the center of the 72 Hz trap, and then calculate the trajectories in the time-dependent optical potential until detection after 14.6 msof time-of-flight. We see that the calibration differs from the KDD results by−(0.5, 1, 6)% for pulse durations of (0.2, 0.4, 1) ms, with negligible differences
for shorter pulses. These corrections are included in Fig. 7.2.

7.6.2

Characterization of Magnetic Fields and Optical
Potentials

The quoted uncertainty of 300 Hz in the detuning ∆ has contributions fromboth differential Zeeman- and AC-Stark shifts, which are characterized as follows:
Magnetic Fields
All experiments are carried out at fields between 4.9998(1) G and 5.0002(1) G.
Here the error bar corresponds to an uncertainty in the bare level splitting
89between |ri and |bi of 70 Hz. The magnetic field for each iteration of theexperiment is monitored using a series of Rabi-pulses during time of flight[129], in which atoms in |1, −1i are redistributed to the |2, −2i and |2, −1istates. This method allows field reconstruction to within 100 µG (for details
see chapter 5). The inhomogeneity of the magnetic field across the sample ischaracterized via a Ramsey pulse sequence. It does not exceed 70 µG, whichcorresponds to 15 Hz rms variation of ∆ across the sample for the |ri, |bi state
pair.
Differential Optical Potential
The state-selective optical potential is created using a σ − -polarized laser beam(waist w0 = 230 µm) at λz = 790.10(2) nm. Polarization and wavelength areset (using a λ/4-waveplate and laser-controls) such that the lattice potential
seen by the |ri-atoms is maximized, while the |bi atoms experience zero potential. The change in lattice potential with wavelength is 0.12Er /0.01 nm, while
the maximum theoretically possible polarization change (σ − to π) changes thepotential by 0.01 Er (We note that the polarization is stable in the experiment). We characterize the state selectivity using a sequence of ten KapitzaDirac pulses (10 µs) spaced at the Talbot-resonance time (see also chapter 2
section 2.2.2) [160] τ = (4Er,790.1 /h)−1 = 68 µs, to ensure a suppression of the
optical potential for |bi atoms by a factor of better than 100 (V|bi /V|ri < 100),
consistent also with the remaining wavelength uncertainty of 0.02 nm.
A crucial part of the experiment is a reliable determination of the resonancecondition, or alternatively, the value of ∆. Since the creation of our matterwave emitter (in Fig. 7.1 (c)) starts with a detuned Rabi oscillation of two
hyperfine states, we first (post-) stabilize the magnetic field as discussed above
and in [129]. The resonance condition (‘excited state energy’) of the matterwave emitters is determined by the detuning of the hyperfine spin and the zero
point shifts to both states induced by the state-selective potential. Since we
can currently control the wavelength to 0.02 nm rms accuracy only, the overall
uncertainty is limited to a maximum of ±340 Hz rms, with the most extreme
fluctuations limited to 1 kHz. To address and monitor this issue, we bracket
each measurement (such as measuring the time evolution at a given detuning)by a resonance curve, an average of which is shown in Fig. 7.4 (see also Fig. 5.5).
The resonance curve is taken without any transverse lattices on and with a low
atom number in the optical trap (∼ 30, 000 atoms) and at only partial transfer(maximally 30 percent) to minimize systematic mean field/density shifts. The
residual systematic shift of the resonance condition due to mean-field/density
effects is estimated to be less than 100 Hz via direct simulation of resonance90

Population

curves using the 1D-time-dependent Gross-Pitaevskii equation (see appendix
A. The scatter of the center of resonance curves during a typical measurement
has a rms value of 0.3 kHz, which is the quoted uncertainty of the detuning,
and which also matches the expected wavelength reproducibility of 0.02 nm.

.2

.1

0
-3

0
Δ [kHz]3

Figure 7.4: Average spectrum of coupling stationary atoms into the z lattice(generated from a series of 17 spectra taken over a one-day period whose fitted
centers are shifted to zero). The coupling strength is Ω = 740(10) Hz and thepulse time is 400 µs, with the datapoints binned into 300 Hz wide bins. The
solid curve is a fit to the data with Ω as a free fit parameter, while the dashedcurve has no free parameters, and the effective coupling strength has been
calculated using the wavefunction overlap between free and trapped species of
γ0 = 0.72.
To characterize inhomogeneities of the state-selective optical potential, we
first precisely calibrate the Rabi coupling strength (for a range of couplingstrengths) in the absence of state-selective potentials. We then compare themaximum observed population transfer in a Rabi spectrum to the expected
maximum population transfer into the lattice. Based on the comparison, we
estimate an upper bound for the inhomogeneity of 300 Hz rms. We notethat the inhomogeneities of the trapping potential at 1064nm do not exceed
20 Hz rms across the sample. For comparison, the absolute magnitude of thespectroscopic shift of (|1, −1i ↔ |2, 0i) in the center of the optical trap does
not exceed 400 Hz.
(*)Future Stabilization of the State-Selective Lattice Laser
In future experiments we want to stabilize the laser wavelength directly usinga wavemeter which has higher accuracy and precision (four digits of precision91

after the decimal of the nanometer in are needed instead of two digits). We areadding a more detailed calculation here which specifies exactly the differential
shift in level splitting when the wavelength changes by 1 pm. For a wavelength
shift of 1 pm, at tune-out wavelength for the free species and 30 Er potential
depth for the trapped species, then the trapped species potential depth changes
by ±0.015 Er /pm (away from 30 Er while the potential depth of the freespecies changes from 0 to ±0.012 Er /pm. For the free state we can assume
that the potential is a weak perturbation: the potential is small comparedto the two-photon recoil energy E2r = 4Er , which is the approximate kinetic
energy required to modulate the wavefunction at the periodicity of the latticepotential. Using perturbation theory on a state that is flat on the scale of alattice wavelength, we find that the energy shift of this optical potential onto
the free state is simply the average value of the lattice potential, i. e. half thelattice potential:∆Efree = V0 sin2 (kr x) =0.006
ˆHzEr= h × 22pm
pm.
The confined species is tightly trapped, which allows us to approximate theground state energy of the√particle in the potential well using the harmonic
approximation: ω0 = 2Er s/~ (s = V0 /E√
r is the lattice depth in recoils),√
which has a ground state energy of 0 = Er s. The slope of this is Er /2 s,
and hence
Er
Er
Hz
ˆ
∆Etrapped = ∆s × √ =0.0014
=h×5
pm
pm
2 s
.
Hence the differential shift between the the two states is 17 Hz/pm. Thismeans that the current stabilization/accuracy of the wavelength to 0.02 nm
is not accurate enough to fix the level splitting to a 100 Hz level, which is
however the stability given by magnetic field uncertainties.
The laser light for the state-selective lattice potential is generated using a Coherent-899 solid-state laser, pumped by a Coherent Verdi-V10 diodepumped solid-state laser. Measurements using a higher resolution wavemeter(Burleigh WA-1500, accurate to 1pm in wavelength, built in He-Ne reference)show that the laser is indeed very stable in the short term (over minutes), due
to it being locked to the external reference cavity. Over the course of longer
periods (half hour to several hours) the laser can undergo large wavelengthchanges (up to 0.02 nm change, undetected by the wavemeter in the laboratory) possibly due to a large external perturbation of the laser cavity (soundand noise) or due to the regulatory electronics running out of range. The laser
92then resettles at that wavelength, and the change cannot be detected in ourcurrent setup.
In the future we plan to directly stabilize the laser either by buying a moreaccurate wavemeter and using, for example, the external laser scan control tofeed back slowly onto the wavelength. The laser can be scanned around its
current lock-point by ∼ ±10 GHz which is approximately 0.02 nm wavelengthchange, so the inputs provided by the laser should be sufficient to stabilize thelattice potential down to a 10 Hz accuracy given the correct wavemeter. In
the long run it would be best to replace our current Ti-Sapph laser (Coherent899) with a more modern Ti-Sapph laser, which can easily reach line-widthsof 10 kHz at drifts of less than 1 GHz per day, which is sufficient for our
experiments.

7.6.3

Background Subtraction

k/kr

2
0
-2
0

10
Δ/2π [kHz]20

30

Figure 7.5: Raw momentum spectrum showing a detuning-independent, diffuse
background of ∼ 103 atoms. Processed figure and details in Fig. 7.2 (c).
The sequence used to thin out the atomic sample leaves roughly 103 atoms
in the |bi-state before the microwave pulse is applied. This results in a diffuse
background in the momentum distributions, as illustrated in Fig. 7.5. We
remove this background by subtracting out reference data taken for zero pulse
time. The result is Fig. 7.2 (c) of the main text.
In the main text in Fig. 7.3 (f) we also mention subtracting backgrounddue to the very small fraction of higher band population. The first excited
band is expected at approximately 2π × 36(1) kHz or ±3~kr . For subtraction
we first fit three Gaussian peaks for the adiabatic ramp and one soft edge
box and two Gaussians to the sudden turn-on data, requiring identical atom
numbers in the soft edge box and the small Gaussian. The fitted functions
are subtracted from the data, which then yields the main plot in Fig. 7.3 (f).
93

The number of atoms emitted from higher bands is on the order of 510(30) for
the sudden turn on and 260(30) for the adiabatic ramp, which is less than 1
atom in a higher band per tube. The total number emitted from higher bands
changes with pulse area because the spurious emission from the higher bands
is in the Markovian regime.

7.6.4

Energy Shift Data0[ ] [ ]

[ ] [ ][ ][ ][ ][ ] [ ]

-1_

δL/2π [kHz]

1

[ ]
[ ]

[ ] [ ]

[ ] [ ]

(a)

[ ]

[ ]
[ ]
[ ]

(b)

[ ]

-2
1

3

1

3
Ω/2π[kHz]Ω/2π[kHz]

Figure 7.6: Raw data used to obtain the energy shift as shown in the text.
(a) Second moment of k and (b) half separation squared both subtracted by∆/2π. The detunings are ∆/2π = {1000, 2000, 4000, 6000} kHz for the disks(black), triangles (red), squares (green) and circles (blue) respectively. Points
in brackets correspond to the non-Markovian regime Ω/∆ > 1.
The main motivation for a precise momentum calibration as describedabove lies in the smallness of the energetic shift. Another challenge is the blurring of the distribution due to propagation effects for small coupling strengths,i.e. long pulses. We use two measures for the determination of the energy ofthe emitted wave-packets; the squared separation of the wavepacket-centers(extracted by fitting) and the second moment of the (centered) distribution.
The accuracy of the peak separation measure is limited by the fact that it
ignores the physical broadening of the momentum distribution at larger coupling strengths and shorter times, while the second moment is sensitive to a
blurring of the wavepackets during detection. The data obtained using bothmethods are shown in Fig. 7.6. In the non-Markovian regime Ω/∆ > 1 thepeaks become indistinguishable (apex of the parabola in Fig. 7.2 (c)) and ameaningful measure of a shift cannot be extracted with either method.

94

7.7

Numerical Simulation of Array Effects in
Weak Harmonic Trapping Potential
1

Population

1

(a)

1

/4

1

/16

.6

0

2

1
Population

(b)

4

0

10

20

(c)

.5

0

.5
Evolution time [ms]1

Figure 7.7: Simulated decay dynamics for a 1-site and 3-site model (withthe central site initially populated), and for ωz = 2π× 0.1 kHz (a,c) and
5 Hz (b). (a) Dynamics for ∆ = 2π × 1.9 kHz and Ω = 2π × 0.74 kHz.
(b) Long-time decay dynamics of the 1-site (black) and 3-site (red) models
for Ω = 2π × 0.74 kHz and ∆ = 2π × 1.9kHz. The dashed line shows the
population of the central, initially populated, site; the dotted line shows the
population of the neighboring sites. (c) Dynamics for ∆ = −2π × 0.1 kHz andΩ = 2π × 3 kHz.
As stated in the main text, the Hamiltonian of the multi-site array [124]predicts the (resonant) transfer of excitations between neighboring emitters[125], in some similarity to the mechanisms for radiation trapping in an optically thick sample. Instead of attempting to solve this Hamiltonian, here
we consider a simplistic model of an array of three emitters that is coupled
to a quantized mode structure reflecting the weak longitudinal harmonic con95

finement. We chose our simplistic three-site model, with a centrally occupied
site as the simplest spatially symmetric generalization of the isolated-emitter
situation, which also corresponds to an ‘average segment’ in our system withapproximately one empty neighbor on each side of a populated site.
We start from the Rabi-Hamiltonian (in the rotating wave approximation)and expand it to couple one or several sites (|ri in the main text) to manydifferent, weakly confined levels (|bi in the main text). This Hamiltonian is(for simplicity only shown for two sites, but readily expanded to n sites)


2δ̄1
0
Ωγ1,1
Ωγ1,2
···
 0

2δ̄2
Ωγ2,1
Ωγ2,2

~


0
(7.4)
Ĥ = Ωγ1,1 Ωγ2,1 −2∆ + ωz
2 Ωγ1,2 Ωγ2,2
0
−2∆
+
3ω
z


..
..
.
.
where δ̄i = mωz2 ri2 /(2~) is a site-dependent detuning (i.e. a site-dependent
offset due to the weak harmonic confinement ωz experienced by both, thelattice-trapped, and free atoms), and the γi,j are overlaps between final andinitial state wavefunctions (calculated numerically). We use modes up to a
fixed frequency (ωmax = 2π × 5kHz) and restrict ourselves to ∆ + 2Ω < ωmax .
The results of the simulation for the Markovian parameters of the main
text are shown in Fig. 7.7 (a) and (b). The simulated decay for an isolated
emitter reproduces the prediction of our analytical model [126] (small discrepancies arise from the fact that the latter neglects terms of order (∆/ω0 )2 and(Ω/ω0 )2 and higher, whereas our numerical simulation retains all orders). For
the three-site array, the presence of neighboring wells leads to a slowdown of
the decay. The origin for this is reabsorption of emitted population by the
initially empty neighbors as seen in Fig. 7.7 (b). For the array considered, theprocess also leads to the formation of a temporary plateau in the overall site
population, which may be related to the offset observed in the experiment.
However we caution that the long-time decay occurs in a regime not accessible
experimentally (we assumed a denser mode structure in order to extend thecontinuum approximation; see below), and also the dynamics may be different
in an optical lattice extending over the entire mode volume.
In Fig. 7.7 (c) we plot the dynamics in the extreme non-Markovian regime
at the edge, where the coupling strength is much larger than the excited stateenergy. In this case we see that the single emitter displays oscillatory dynamics
that quickly damps out and settles to a nonzero value, again in agreementwith the analytic theory. On the other hand, the three-emitter array shows
oscillatory dynamics that are of much greater amplitude and last longer. We
96interpret this as a coherent enhancement of the dynamics through tunnelingto nearest neighbors. The range of this tunneling diverges at the band edge
causing the marked difference between single emitter and three emitters.
From our simulations we gain additional insight into the effects of thequantized mode structure. As discussed in the main text, the mode structure
should act like a true continuum for short enough times where uncertainty
should ‘wash out’ the levels. The simulations provide a quantitative test forthis. The simplest comparison is for an isolated emitter in the Markovian limit.
We see that for early times, the Markovian prediction [126] quantitatively
agrees with the numerical solution, with a marked deviation (‘revival’, seeFig. 7.8) observable only at t∗ > 0.25(ωz /2π)−1 (we restrict data taking tot < t∗ = 2.6 ms in the experiments in all but one case.) This is independent of
the set harmonic trapping frequency or where the site is located in the array(center or off center). We note that similar results are also obtained if thecontinuum is discretized by assuming a periodic-box type potential, where therevival time depends on the length of the box. We furthermore note that oursimulation does not reproduce δL for long times.

Population

1

.5

0
0

3
Time [ms]6

Figure 7.8: (*) Simulated decay dynamics for a 1-site model, for ωz = 2π×0.1 kHz, ∆ = 2π × 1.9 kHz and Ω = 2π × 0.74 kHz. The dashed line is theprediction of the Markovian approximation. At a time corresponding to half
the harmonic oscillator period (5 ms), we see a clear revival due to the finite
size of the system.

97

7.8

(*) Comparison to Non-Adiabatic Diffraction

As a final note we want to briefly point out that this work is related to work
done in collaboration with J. Reeves [96] on non-adiabatic diffraction of matterwaves. In the experiment by Reeves et. al [96], atoms in the freely propagatingstate (without transverse confinement) experience a coupling to the trappedstate. While the experimental techniques developed in [96] work lay the foundation for this dissertation, we want to compare the conceptual differences
between [96] and this work.
The most important difference between this work and [96] is the fact thatthe initial state here is in the lattice, and is made up of localized atoms (essentially in Wannier states) which reside on lattice sites. As a result, the initialstate here is made up of contributions from a broad range of momenta, whichleads to coupling to many different states, and hence is essential to the possibility of simulating an open quantum system. In contrast, the matter wave
diffraction starts with an almost pure p = 0 state, and only a discrete set ofstates (p = n × 2~kr ) is ever populated, yielding effective few-level (four level)
single particle dynamics.
Furthermore this work is restricted to coupling strengths which are much
weaker than the two-photon recoil energy (E2r = 4~2 kr2 /2m ∼ h × 15 kHz,
which is the scale on which the coupling strength becomes large enough toeffectively hybridize the optical potential of the free state (flat) and the boundstate (lattice); the experiments in [96] were conducted at (0.5 . . . 1.0) E2r .

98

Chapter 8
Conclusion and Outlook
The main result in this dissertation is the realization of a matter-wave analog
to the Weisskopf-Wigner model in a structured one-dimensional medium. We
realize the analogue by preparing a sample of atoms in a single a hyperfine statethat are strongly localized along all three spatial directions. Subsequent weakcoupling to a hyperfine state which is approximately free along one spatial
direction allows the recreation of a one-dimensional analog of the WeisskopfWigner model, where in our system we have, potentially coupled, arrays ofartificial emitters rather than a single atom. Due to the exceptional tunabilityof our system, we can probe the model in unprecedented detail and see atransition from Markovian to non-Markovian time-evolution when the excitedstate energy becomes comparable to the coupling strength. In the extreme
case of negative excited state energies, we observe the formation of a boundstate, and characterize its localization length.
Since these experiments required the reliable detection of small atom numbers, an important technical prerequisite was the implementation of a fringeremoval technique to purge images of any residual artifacts which are not images of atoms. The method is implemented using empty ‘training’ images on
which principal component analysis is done in order to generate a set of ‘typical’ problem images which can then be subtracted from the actual absorption
images.
A major technical challenge was the control of magnetic fields in our apparatus to the level of ∼ 50 − 100 µG. We developed a post-selection methodthat allows conducting experiments on a hyperfine pseudo-spin pair, and uses
unused hyperfine levels to characterize the field shortly after the time of measurement. While this technique only allows for post-selection field reconstruction, the continuous monitoring of the field at exactly the position of the atoms
allows for a level of stability needed. The measurement only takes 0.5 ms to99

2 ms (depending on the needed accuracy) and can easily be implemented onany existing setup with hyperfine-state control.
In the future, we are interested in observing many-body effects of thematter-wave emitter system, such as superradiance [218] and especially its
extension into the non-Markovian regime, as well as the emergence of effective Bose-Hubbard/Ising Hamiltonians when the excited state energy becomes
negative, especially the possibility of directly tuning the tunneling range via
the bound state energy. We are planning to investigate transport phenomena
utilizing the bound state (e. g. Bloch oscillations) and want to study the precise influence of tuning the reservoir. The reservoir can be tuned by either
applying a periodic potential to the reservoir as well, thus creating a situation
where one can have an upper and a lower band edge. Another way of tuning
the reservoir is to tune the weak harmonic trapping potential such that it becomes noticeably quantized (or equivalently conduct experiments for a longer
time in the same weak confinement), thus realizing a Tavis-Cummings typemodel [124, 222].
Finally, we are interested in increasing the dimensionality for the stateselective potential in order to increase the effective dimensionality of the systemform 1D to 2D or 3D, enabling us to study in detail how e. g. the diverging
1D density of states affects the dynamics close to zero excitation energy, as
compared to the coherent dynamics of 2D and 3D geometries. We are excited
to read about recent new proposals for our and analog systems [223, 224].
Furthermore, dimensionality plays a big role in superradiance of the system,
since the density of the (matter) waves decreases in 2D and 3D away from theemitter, which is not the case in 1D.

100

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[220] M. Lewenstein and K. Rzążewski. Quantum anti-Zeno effect. PhysicalReview A, 61:022105, 2000.
[221] Martin Bruderer, Alexander Klein, Stephen R. Clark, and Dieter Jaksch.
Polaron physics in optical lattices. Physical Review A, 76:011605, 2007.
[222] Michael Tavis and Frederick W Cummings. Exact solution for an Nmolecule-radiation-field Hamiltonian. Physical Review, 170(2):379, 1968.
[223] A González-Tudela and JI Cirac. Non-Markovian Quantum Optics withThree-Dimensional State-Dependent Optical Lattices. arXiv preprint
arXiv:1806.02528, 2018.
[224] T Shi, Y-H Wu, A González-Tudela, and JI Cirac. Effective manybody Hamiltonians of qubit-photon bound states. arXiv preprint
arXiv:1806.02527, 2018.
[225] Russell Anderson. Nonequilibrium dynamics and relative phase evolutionof two-component Bose-Einstein condensates. PhD thesis, Swinburne
University, 2010.
[226] Anthony J Leggett, SDAFMGA Chakravarty, AT Dorsey, Matthew PA
Fisher, Anupam Garg, and W Zwerger. Dynamics of the dissipative
two-state system. Reviews of Modern Physics, 59(1):1, 1987.
[227] AM Kaufman, RP Anderson, Thomas M Hanna, Eite Tiesinga, Paul S
Julienne, and DS Hall. Radio-frequency dressing of multiple Feshbachresonances. Physical Review A, 80(5):050701, 2009.
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Theodor W Hänsch, and Immanuel Bloch. Entanglement interferometry for precision measurement of atomic scattering properties. PhysicalReview Letters, 92(16):160406, 2004.
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121[231] Embedded Micro. https://embeddedmicro.com/products/mojo-v3.
Accessed: 2018-07-06.

122Appendix A
Simulation of Mean-Field Shift
In this chapter we briefly summarize the simulation used to estimate theresidual mean field shift in chapter 7. When weakly coupling (compared toE2r = 4~2 kr2 /2m) a weakly trapped 3D-BEC to a lattice-trapped state, thelattice trapped state will experience significantly higher densities (comparedto the non-lattice trapped state) due to the compression into the lattice wells.
This changes the mean-field energy or interaction energy of the system, which
results in a density dependent shift of the resonance frequency. We estimate
this shift here directly using numerical simulation. We begin with the twocomponent, coupled, time dependent Gross Pitaevskii-equation (TDGPE) in
one dimension [135, 225]:

 2 2
~ ∂x
~∆
2
2
+ V1 (x) + g11 |ψ1 | + g12 |ψ2 | +ψ1i~∂t ψ1 = −
2m
2
~Ω
ψ2
+
2
 2 2

(A.1)
~ ∂x
~∆
2
2
i~∂t ψ2 = −+ V2 (x) + g22 |ψ2 | + g12 |ψ1 | −
ψ2
2m
2
~Ω
+
ψ1 .
2
Here, we have two species with wavefunctions ψ1 (x) and ψ2 (x) g11 and g22 are
the intra-species interaction parameters and g12 is the inter-species interaction
parameter. V{1, 2} (x) are the potentials for either species and ∆ and Ω are thedetuning and Rabi-frequency known from Rabi oscillations.
We numerically integrate Eqns. A.1 directly, starting with all populationin ψ1 . The potential V1 (x) is simply a harmonic trap centered at x = 0 with
ω = 2π × 72 Hz, and the potential V2 (x) = V1 (x) + 30Er sin (kr x)2 . The spatial
distribution ψ1 (x) has minimum energy (is in the ground state). The ratio123

Rof the interaction parameters g11 and the total atom number |ψ1 |2 (x)dx are
set such that initially g11 |ψ1 (0)|2 ≈ ~ × 2π × 750 Hz, which is approximately
the mean field energy of a 3 × 104 -atom BEC in a 72 Hz trap. We then time
evolve for 400 µs the using variable detuning and a Ω = 2π × 630 Hz couplingstrength used in the experiment to create a simulation of the lattice spectrum
seen in Fig. 7.4.
We simulate Eqns. A.1 for both, interactions turned to zero, and for interactions turned to the relative values of 87 Rb. The result is shown in Fig.
A.1. We can see that for the low numbers and low population transfer usedin our experiment the interaction shift is indeed negligible (less than 100 Hz)
and thus negligible for the experiments described in chapter 7.

Population

.30

.15

0
-2

0
Detuning Δ [kHz]2

Figure A.1: Spectrum obtained via simulation of coupled time dependent
Gross-Pitaevskii equation A.1. Coupling strength is Ω = 2π × 630 Hz, pulse
time is τ = 400 µs. The red squares are with interactions turned on, andthe black circles are with interactions turned off. The solid black line and thedashed red line are fits to the black circles and red squares respectively (fitfunction is Eq. 2.4). Relative shift of the two curve-centers is δ∆ = 2π×64 Hz.

124

Appendix B
Other Measurements
During my PhD, we have also conducted a few measurements which detailother aspects of our apparatus, but that do not quite fit the context of themain dissertation. We will briefly describe them here, since they may be
important for future experiments.

B.1

Controlling the Scattering Length: Interspecies Feshbach Resonances

In previous dissertations from our group [89–91] there was a focus also on
interacting Bose-Hubbard models (BHM [59]) and spin-boson models [200,201,
226]. An important feature here is the interspecies s-wave scattering length
which details the point-interaction of bosons.
Ultracold atomic bosons (such as 87 Rb) usually interact in such a way, thatit can be described using only an effective contact interactions. This is called
the s-wave scattering regime, where a single parameter (the s-wave scattering
length as ) fully describes the interaction of two particles, with the interactionenergy given byZI = dx4πas ~2 n(x)2 /m
where n(x) = |Ψ(x)|2 is the density of atoms in space and Ψ(x) is the manybody wavefunction of the atoms. The scattering length as is determined by
the far-field phase-shift that occurs when a low energy wavefunction scatterson a potential (e. g. a wavefunction scattering off a barrier, or one atom scattering off another atoms). A positive (negative) scattering length translates
to repulsive (attractive) effective interactions.

125

N×10-5

1.0
100

0.7

60

0.4
8.95

Scattering Length [ab]

140

9.00
9.05
9.10
Magnetic Field [G]Figure B.1: Interspecies Feshbach resonance. Total atom vs. magnetic field ofa mixture of |2, 0i and |1, 1i-atoms (46(7)% are |F = 2, mF = 0i), showing aloss signal at 9.0428(2) G (systematic uncertainty of up to 2 mG not included).
The solid line is a Lorentzian fit to the data of width 14(1) mG, signifying
the presence of an inelastic Feshbach resonance between the two states. The
dashed line (labeled by the dashed axis) is the expected modification of thereal part of the scattering length, centered at the center of the loss feature(other parameters taken from [227]).
Since the scattering phase-shift depends on the microscopic details of thepotential landscape between the atoms as their separation decreases, one cantune as using a so-called Feshbach resonance [34, 35]. Here the change inthe microscopic interaction at zero incident wave energy occurs due to boundmolecular states coming into resonance (i. e. when a molecular bound state
approaches zero binding energy). These molecular states can usually be tuned
relative to the free wavefunctions using magnetic fields.
Feshbach resonances can furthermore obtain imaginary character if different freely propagating states (different from the input states) are energeticallyallowed, causing two-body losses to become greatly enhanced. Typically Feshbach resonances are measured using losses (due to enhancement of two- or
three-body losses) but the real part of the s-wave scattering length can also
be measured directly, e. g. through interferometry [228].
In our apparatus we are specifically interested in inter-species Feshbachresonances, where in Fig. B.1 we observe a loss feature when two species arein our optical trap and we scan the magnetic field, indicating the presence ofan inelastic Feshbach resonance. The scattering length takes the form [227]:∆Ba(B) = aBG 1 −B − B0 − iγB /2126

Population Transfer

.10

(a)

.05

.00

Population Transfer

-200

.4

-100

0

(b)

.2

0
-40

-20
ν-ν0 [Hz]020

Figure B.2: Two-photon spectroscopy of a Mott-insulator. Relative population transfer to the |2, 1i-state from the |1, −1i-state for two-photon radiofrequency spectroscopy using a 100 ms pulse. The atoms sit in a 3-D deeply
confining optical lattice of spacing 532 nm and depth 30 Er,1064 (a) Result ofspectroscopy for 190(5) × 103 atoms, showing sites which are populated by up
to 7 atoms. (b) Result of spectroscopy for 5(1) × 103 atoms. A second peak isbarely visible.
where aBG is the background scattering length, B0 is the center of the Feshbachresonance, ∆B is the width (zero crossing if the resonance were fully elastic)and γB is the inelastic width. The interaction energy can then calculated by
the real part of the scattering length only, while the imaginary part induces
two-body losses.
We see a well resolved loss feature when approaching a field of 9.043(2) G(value measured in [227] is 9.0448(5) G), indicating the presence of the inelastic
Feshbach resonance and hence control also over the real part of the scattering
length and the interaction energies.

127

B.2

Measuring Mott-Shell Populations

Following an experiment conducted by [177] we became interested in the occupation number of our atomic Mott insulators. We prepared a Mott-insulating
sample of |1, −1i atoms by loading them into a deep (30Er,1064 depth) 3-D lattice of lattice spacing 532 nm. We then dial the magnetic field to 3.23 G where
the differential quadratic shift between |1, −1i and |2, 1i vanishes, allowing us
to use pulses of ∼ 1 Hz coupling strength and as long as 100 ms. This allows forhighly accurate spectroscopy which can resolve the shift in scattering length
between |1, −1i ↔ |1, −1i and |1, −1i ↔ |2, 1i (which is 100.4 ab vs. 97.7 ab(where ab is the bohr radius). We use microwave radiation at ∼ 6.8 GHz andradio-frequency radiation at ∼ 2 MHz, resulting in an effective two-photon
Rabi-Frequency of less than 5 Hz.
Results are shown in Fig. B.2 where we can clearly see populations of up to
7 atoms per site for high total atom numbers (on top of a thermal background)
whereas for low total atom numbers we can see at most 2 atoms per site.

128

Appendix C
Electronics
We regularly develop and maintain in-house built electronics, as part of dailylaboratory operation. Here we present a few important electronic schematics
which were built during my time in the group.

C.1

AC-line Trigger

The ac-line trigger (see Fig. C.1) is implemented with the help of the Cicero
and Atticus Software [229] and an Opal Kelly XEM3001 Spartan-3 FPGA
(field programmable gate array). We implement a simple Schmitt-trigger circuit (see appendix C.2) connected to the AC-mains, which generates a 0 − 5 V
trigger signal to synchronize with the AC line signal. The FPGA board is
programmed by Cicero/Atticus using the manual instructions to wait for the
next high signal at the re-trigger input pin (in this case pins JP2-46 (signal)and JP2-49 (ground)) immediately before starting any magnetic field sensitive
measurements. The FPGA is programmed to generate and distribute timing
information/triggers to the computer’s analog and digital cards based on aninternal crystal oscillator which creates a master clock at frequency 10 MHz,
from which slower frequencies are derived to time the experimental sequence,thus enabling the necessary wait for synchronization with the line after evaporation and before sensitive experiments.

C.2

Schmitt-Trigger Circuit

As mentioned in section C.1 we use a Schmitt-trigger-type circuit to derivea clock signal which is line-synchronous. The circuit is shown in Fig. C.2.
We use op-amps to create a standard Schmitt-trigger circuit. The input pin
129

USB

FPGA

PFIO

JP49

PC

S-TrigA-Cards

Retrig D-Cards

Figure C.1: Schematic of the setup for the line trigger. The computer communicates the details for the entire experimental sequence to the FPGA board
via USB. The FPGA board generates timing for the sequences at variable time
resolution. The timing is sent to the computer via the PFIO input. Before
magnetically sensitive experiments, the FPGA halts, and the digital outputs
enable the Schmitt-trigger circuit to go high at the start of the next AC-clock
cycle. The retrigger signal is sent to a dedicated pin on the FPGA which thenresumes the sequence.
for the FPGA is value sensing, not edge sensing, hence if the trigger signal isgiven directly to the FPGA, it would start when the Schmitt-trigger is high,and not (as is required for us) at the same phase during each AC-cycle. To
circumvent this problem, we first pass the trigger signal through a Q-flip-flop
which is clocked by the Schmitt trigger. The input to the Schmitt-trigger
circuit is a digital output from the PC, which goes high only when a retrigger
is requested, thus correctly communicating with the FPGA.

C.3

Stable Amplified Photo-diodes.

Accurately stabilizing optical intensities is very important for the experiments
in our laboratory. Following description in [230] we built temperature compensated photo-diode amplifiers which convert the photo-diode current into ameasurable voltage which is not sensitive to drifts in temperature, using theAD8304 integrated circuit and a standard FDS-100 photo-diode from Thorlabs. The output of the circuit is the logarithm of photo-diode current.

130

Vcc

.tran 0.1

V-

V+

R14
750

Supply+Supply-

15

-15

R5
33k

U1
LF365

U2
LF365

HD74LS00P
1 14 Vcc
2 13
3 12
4 11
5 10
6 9
1N5221B
7 8
D1AndLogic

R8

R9
1k

100k 1N5221B

R10
1k

R12
40k

R11
1k

R13
10k

D2

PCInput

15k

V+

R4

R3
1000k

R15
1250

V+

R2
15k

V-

SINE(0 30 60)

100

R6
15k

V+

ACMainsInput

R7

R1
150k

SN74F74N
1 14 Vcc
2 13
3 12
R16
4 11
100k
5 10 Vcc
6 9
7 8
To JP46
QFlipFlop

Figure C.2: Schmitt trigger circuit for line synchronization built using two
LF365 operation-amplifiers. R12 and R13 are actually one trim-pot for adjusting the trigger phase. R14 and R15 are also a trim-pot for adjusting the
Vcc drive voltage for the logic devices. The AndLogic device conditions the
direct output of the Schmitt-trigger op-amp to proper logic levels and the Qflip-flop blocks trigger signals until requested by the PC, digital control cards.
In order to incorporate this into a PI-stabilization loop we use an ADL5330-EVAL variable gain amplifier (VGA) which is then connected to a Minicircuits ZHL-1A power amplifier in order to drive an acousto-optic modulator(AOM). The RF-source which is amplified by the VGA is a Wenzel temperature stabilized crystal oscillator.

131

R2
1k

F ilt1

A D8304
1 14
F DS 100
Vcc
2 13
3 12
1 3
Vcc
B uf
4 11
2
B uf
10µ
5 10
PD
6 9
10µ
B uf 7 8R E F 02
P DA mp
1 8
1µ
2 7
B uf
3 6 Vcc
10µ
4 5
1n

15V

15

Ra

Rb

100k

30k

C filt
0.1n

Rc
30k

VoltageR ef

Figure C.3: Logarithmic temperature stabilized photo-diode amplifier, as used
for the optical dipole trap. The sensitivity and minimum detectable opticalintensity can be easily adjusted by changing resistors Ra, Rb and Rc (details
in the data-sheet of AD8394).

132

RF
SourceWenzel

Log

PI

ADL

ZHL-

PD

Cont.

5330

1A

Light

+1
AOM
0

ZX80DR230

Figure C.4: Schematic for PI-control of optical intensity using logarithmic
photo-diodes and variable gain amplifiers ADL5330.

133

Appendix D
Development of Smart Laboratory
Devices
It was mentioned in section C.1 that we use an FPGA to generate a variable
time base for synchronizing and timing our experiments. As a result of thiswe got interested in working more with FPGA devices and were able to findgood introductory projects from colleagues at Stony Brook.
An FPGA (Field Programmable Gate Array) is a highly customizable digital electronic device which has a collection of typically 104 to 106 logic cells
that can be interconnected arbitrarily to each other, and typically 102 in/out
pins that can be freely defined by the user. As a result one can build arbitrary complex logic functions on these devices, or implement highly parallelized
computations or data-acquisitions. FPGAs are a complementary device to microprocessor chips, which are very good and very fast at sequentially executing
one complex task, while an FPGA is typically more geared towards parallel
executions of many simpler tasks. The outstanding customizeability of FPGAs
comes at the penalty of typically slower clock speeds (∼ 108 Hz) compared tomodern day processors (∼ 109 Hz) and less ease of use for programming thedevice (FPGAs are typically still programmed closer to the actual machine
logic than microprocessors which offer a wide variety of high-level programming environments, such as Matlab, Mathematica, or Python).
I will outline two projects on two different FPGA platforms that showcase
how FPGAs can be used in an experimental physics laboratory, and thatshowcase the utility of these devices.

134

D.1

Introductory Project: FPGA Based Down
Counter for Ultrafast Experiments

Shift
3.3 V
ZX60-33LN
1 3
V+ 2 4
Amplifier x 2

RT_1
100

C1
0.1µ

RT_2
100

To_FPGA_fclk

Vpd
PD-Out 100 MHz

Figure D.1: Circuit used to condition photo-diode signal to reach logic levels
in order to be used as a clock input to the FPGA.
An early project we did with FPGAs is to implement a so-called countdown circuit. I will describe it here since it is a very easy introductory project
to familiarize oneself with FPGAs and the syntax of the programming language. Furthermore it perfectly showcases how easy it is on an FPGA togenerate different clock frequencies from a master clock, which is the basis forthe variable time-base used in the experiments (see section C.1).
The lab of T. Weihnacht utilizes a laser oscillator which generates ultrashort laser pulses (∼ 30 fs) at a repetition rate of ∼ 108 Hz, which need tobe amplified. Since other equipment (for example cameras) can usually only
sustain a repetition rate of ∼ 1 kHz, the amplifier is only operated at a thisrepetition rate, while all other pulses are discarded. The lowered repetition
rate of the amplifier and the other equipment has to be derived as an integer135multiple of the repetition rate of the laser-oscillator such that the amplifier
and the camera remain properly synchronized with the laser oscillator.

Figure D.2: Photographs of FPGA board (black) and photodiode conditioning
circuit.
The current way of deriving a lowered repetition rate clock is to triggera circuit made from IC-flip-flops and IC-decade counters using a photo-diode136

signal which records the output of the laser-oscillator. The circuit effectively
counts down the original frequency of the laser-oscillator and gives a trigger
only once every ∼ 105 -th pulse, thus synchronizing the amplification and theexperiment. In practice this requires careful soldering and patience when thecircuit does break, and once constructed the circuit can only count down by
one number, which cannot be easily changed.
It turns out that this task is relatively easy to implement using FPGA
logic. In turn this makes the circuit more easy to customize and more easy
to replace once broken, since the little soldering that is involved initially uses
components that are less prone to failure and the main FPGA board can beeasily bough off the shelf.

D.1.1

Basic Idea: Code snippet

FPGA devices normally have no internal mechanism for clocking or synchronization; clocking or synchronization is usually supplied externally by the user
or by a crystal oscillator on typical FPGA board which is directly connectedto the FPGA-chip. This is the basic idea which the final device is based on.
If we can condition the photo-diode signal which monitors the output powerof the laser-oscillator sufficiently to use it as a ‘clock’, the implementation of
the down-count circuit is straightforward. A ‘clock’ in this context is a signal which has a sharp rising edge which defines the start of each clock cycle.
Typically it also has a duty cycle of 50% so that a typical clock will be asquare-wave which is 0V half of the time and 3.3V% (or whatever the logic
levels of the device are) the other half of the time. The only important featureof a clock is the sharp rising edge
The central code for this looks as follows:
always @ ( posedge f c l k )//At each c l o c k −edge e x c e c u t e t h i s b l o c k// f c l k i s t h e photo−d i o d e s i g n a l// c o u n t e r _ s t o r e i s h a l f t h e number t h e// i n p u t c l o c k i s d i v i d e d by// s c l k i s t h e o ut pu t c l o c k
begin// B e g i n n i n g o f b l o c ki f ( rst_n == 1 ’ b0 )// R e s e t and r e a d i n new number
begin
c o u n t e r <=24’ h000000 ;c o u n t e r _ s t o r e <=c o u n t _ s e t ;end
e l s e i f ( c o u n t e r == c o u n t e r _ s t o r e )// I n v e r t o ut pu t when count i s r e a c h e d
begin
c o u n t e r <= 2 4 ’ h000000 ;s c l k <= ~ s c l k ;

137

end
else
// I n c r e m e n t c o u n t e r
begin
c o u n t e r <= c o u n t e r + 1 ’ b1 ;end
end

where the variable count_set is set by the user by interconnecting various
inputs pins to the FPGA, and the clock fclk is supplied externally by the user
to a special clock pin. In the current setup, input pins can be directly connected
to neighboring pins that are set permanently high or permanently low, which
in turn programs a 24-bit number. The number is only updated upon resetto guarantee reliable operation. The variable sclk is directly connected to an
output and can be used to trigger as it is logic level 3.3V-CMOS standard. All
that is left is to supply a constraints file which defines which pins the inputs
and outputs are connected to and a circuit which conditions the photo-diodesignal.

D.1.2

Hardware and Implementation

The hardware which this was implemented on was an Embedded Micro Mojo
V3 [231]. The full source code and constraints files are listed at the end of thischapter and the circuit which was used to condition the photo-diode signal isshown in Fig. D.1. A photograph of the devices is shown in Fig. D.2.

D.1.3

Entire Source CodeIn the following I recreate the entire source code for this project. First is thebase code which manages counting:module downcounter (input fclk ,i n p u t [ 2 3 : 0 ] count_set ,i n p u t rst_n ,ou tp u t s c l k);// Front ma tte r// Counter R e g i s t e r For DownCountingr e g [ 2 3 : 0 ] c o u n t e r = 2 4 ’ h000000 ;// Counter R e g i s t e r t o s t o r e count t o numberr e g [ 2 3 : 0 ] c o u n t e r _ s t o r e = 2 4 ’ h000FFF ;// H e l p e r R e g i s t e r For A s s i g n i n g I n t o Output
r e g h e l p e r = 1 ’ b0 ;assign sclk = helper ;// a s s i g n c n t = c o u n t e r ;always @ ( posedge f c l k )begin
i f ( rst_n == 1 ’ b0 )138beginc o u n t e r <=24’ h000000 ;c o u n t e r _ s t o r e <=c o u n t _ s e t ;end
e l s e i f ( c o u n t e r == c o u n t e r _ s t o r e )begin
c o u n t e r <= 2 4 ’ h000000 ;h e l p e r <= ~ h e l p e r ;
end
else
begin
c o u n t e r <= c o u n t e r + 1 ’ b1 ;end
end
endmodule

Second is the top or wrapper module which manages auxiliary variables:
module mojo_top (input fclk ,i n p u t [ 2 3 : 0 ] count_set ,i n p u t rst_n ,ou tp u t s c l k ,ou tp u t [ 2 3 : 0 ] h i g h s ,ou tp u t [ 2 3 : 0 ] lows ,ou tp u t l e d);
r e g h a l p = 1 ’ b0 ;a s s i g n l o w s = 2 4 ’ h000000 ;a s s i g n h i g h s = 2 4 ’hFFFFFF ;a s s i g n led = halp ;always @ ( posedge f c l k )begin
i f ( rst_n == 1 ’ b0 )begin
h a l p = 1 ’ b1 ;
end
else
begin
h a l p = 1 ’ b0 ;end
end
downcounter inst_downcount (. fclk ( fclk ) ,. count_set ( count_set ) ,. rst_n ( rst_n ) ,. sclk ( sclk ));endmodule

And this is the entire constraints file:NET " f c l k " LOC = P51 | IOSTANDARD = LVCMOS33 ;NET " s c l k " LOC = P24 | IOSTANDARD= LVCMOS33 ;NET " rst_n " LOC = P38 | IOSTANDARD = LVCMOS33 ;NET " count_set <23>" LOC = P57 | IOSTANDARD = LVTTL;NET " count_set <22>" LOC = P74 | IOSTANDARD = LVTTL;NET " count_set <21>" LOC = P78 | IOSTANDARD = LVTTL;NET " count_set <20>" LOC = P82 | IOSTANDARD = LVTTL;

139

NET " count_set <19>" LOC = P84 | IOSTANDARD = LVTTL;NET " count_set <18>" LOC = P92 | IOSTANDARD = LVTTL;NET " count_set <17>" LOC = P94 | IOSTANDARD = LVTTL;NET " count_set <16>" LOC = P99 | IOSTANDARD = LVTTL;NET " count_set <15>" LOC = P101 | IOSTANDARD = LVTTL;NET " count_set <14>" LOC = P111 | IOSTANDARD = LVTTL;NET " count_set <13>" LOC = P114 | IOSTANDARD = LVTTL;NET " count_set <12>" LOC = P118 | IOSTANDARD = LVTTL;NET " count_set <11>" LOC = P120 | IOSTANDARD = LVTTL;NET " count_set <10>" LOC = P123 | IOSTANDARD = LVTTL;NET " count_set <9>" LOC = P131 | IOSTANDARD = LVTTL;NET " count_set <8>" LOC = P133 | IOSTANDARD = LVTTL;NET " count_set <7>" LOC = P139 | IOSTANDARD = LVTTL;NET " count_set <6>" LOC = P141 | IOSTANDARD = LVTTL;NET " count_set <5>" LOC = P1 | IOSTANDARD = LVTTL;NET " count_set <4>" LOC = P5 | IOSTANDARD = LVTTL;NET " count_set <3>" LOC = P9 | IOSTANDARD = LVTTL;NET " count_set <2>" LOC = P11 | IOSTANDARD = LVTTL;NET " count_set <1>" LOC = P16 | IOSTANDARD = LVTTL;NET " count_set <0>" LOC = P21 | IOSTANDARD = LVTTL;NET " h i g h s <23>" LOC = P58 | IOSTANDARD = LVTTL;NET " h i g h s <22>" LOC = P75 | IOSTANDARD = LVTTL;NET " h i g h s <21>" LOC = P79 | IOSTANDARD = LVTTL;NET " h i g h s <20>" LOC = P83 | IOSTANDARD = LVTTL;NET " h i g h s <19>" LOC = P85 | IOSTANDARD = LVTTL;NET " h i g h s <18>" LOC = P93 | IOSTANDARD = LVTTL;NET " h i g h s <17>" LOC = P95 | IOSTANDARD = LVTTL;NET " h i g h s <16>" LOC = P100 | IOSTANDARD = LVTTL;NET " h i g h s <15>" LOC = P102 | IOSTANDARD = LVTTL;NET " h i g h s <14>" LOC = P112 | IOSTANDARD = LVTTL;NET " h i g h s <13>" LOC = P115 | IOSTANDARD = LVTTL;NET " h i g h s <12>" LOC = P119 | IOSTANDARD = LVTTL;NET " h i g h s <11>" LOC = P121 | IOSTANDARD = LVTTL;NET " h i g h s <10>" LOC = P124 | IOSTANDARD = LVTTL;NET " h i g h s <9>" LOC = P132 | IOSTANDARD = LVTTL;NET " h i g h s <8>" LOC = P134 | IOSTANDARD = LVTTL;NET " h i g h s <7>" LOC = P140 | IOSTANDARD = LVTTL;NET " h i g h s <6>" LOC = P142 | IOSTANDARD = LVTTL;NET " h i g h s <5>" LOC = P2 | IOSTANDARD = LVTTL;NET " h i g h s <4>" LOC = P6 | IOSTANDARD = LVTTL;NET " h i g h s <3>" LOC = P10 | IOSTANDARD = LVTTL;NET " h i g h s <2>" LOC = P12 | IOSTANDARD = LVTTL;NET " h i g h s <1>" LOC = P17 | IOSTANDARD = LVTTL;NET " h i g h s <0>" LOC = P22 | IOSTANDARD = LVTTL;NET " lows <23>" LOC = P66 | IOSTANDARD = LVTTL;NET " lows <22>" LOC = P67 | IOSTANDARD = LVTTL;NET " lows <21>" LOC = P80 | IOSTANDARD = LVTTL;NET " lows <20>" LOC = P81 | IOSTANDARD = LVTTL;NET " lows <19>" LOC = P87 | IOSTANDARD = LVTTL;NET " lows <18>" LOC = P88 | IOSTANDARD = LVTTL;NET " lows <17>" LOC = P97 | IOSTANDARD = LVTTL;NET " lows <16>" LOC = P98 | IOSTANDARD = LVTTL;NET " lows <15>" LOC = P104 | IOSTANDARD = LVTTL;NET " lows <14>" LOC = P105 | IOSTANDARD = LVTTL;NET " lows <13>" LOC = P116 | IOSTANDARD = LVTTL;NET " lows <12>" LOC = P117 | IOSTANDARD = LVTTL;#lows <11> and not a s s i g n e d , GND nearby , u s e GND;
#NET " lows <10>" LOC = P126 | IOSTANDARD = LVTTL;#Not s e t , u s e LED pin , s e r v e s d u a l p u r p o s e o f LED and low p i n ;NET " lows <9>" LOC = P127 | IOSTANDARD = LVTTL;

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NET " lows <8>" LOC = P137 | IOSTANDARD = LVTTL;NET " lows <7>" LOC = P138 | IOSTANDARD = LVTTL;NET " lows <6>" LOC = P143 | IOSTANDARD = LVTTL;NET " lows <5>" LOC = P144 | IOSTANDARD = LVTTL;NET " lows <4>" LOC = P7 | IOSTANDARD = LVTTL;NET " lows <3>" LOC = P8 | IOSTANDARD = LVTTL;NET " lows <2>" LOC = P14 | IOSTANDARD = LVTTL;NET " lows <1>" LOC = P15 | IOSTANDARD = LVTTL;NET " lows <0>" LOC = P26 | IOSTANDARD = LVTTL;#S e t lows <11> and lows <10> i n o r d e r#t o be a b l e t o s y n t h e s i z e t h e c i r c u i t .
NET " lows <11>" LOC = P29 | IOSTANDARD = LVTTL;NET " lows <10>" LOC = P30 | IOSTANDARD = LVTTL;NET " l e d " LOC = P126 | IOSTANDARD = LVTTL;The pinout is also visualized in Fig. D.3 for clarity, where the Figure represents
the layout of the board which was used.

Pin

P22 P17 P15 P12 P10P8P6P2 P144 P142 P140 P138 P134 P132 P127 P124

P24

H

H

L

H

H

L

H

H

L

H

H

L

H

H

L

H

sclk

0

1

L

2

3

L

4

5

L

6

7

L

8

9

L

10

fclk

Pin P26 P21 P16 P14 P11

P9

P7

P5

P1 P143 P141 P139 P137 P133 P131 P126 P123

P51

H/L
Bit
H/L
Bit

L

Pin P58 P67 P75 P79 P81 P83 P85 P88 P93 P95 P98 P100 P102 P105 P112 P115 P117 P119 P121

H/L H
Bit
H/L 23
Bit

L

H

H

L

H

H

L

H

H

L

H

H

L

H

H

L

H

H

L

22

21

L

20

19

L

18

17

L

16

15

L

14

13

L

12

11

Pin P57 P66 P74 P78 P80 P82 P84 P87 P92 P94 P97 P99 P101 P104 P111 P114 P116 P118 P120

Figure D.3: Pinout visualization for downcounter circuit, to visualize how to
set bits.

D.2

OutlookIn the future one can use more advanced FPGA devices like the Red Pitaya
which already has DA-chips and AD-chips installed for live signal processing
or fast PID stabilization of signals.

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