Ultracold Collisions
and Fundamental Physics
with StrontiumbySebastian Blatt
Mag. rer. nat., Leopold-Franzens-Universität Innsbruck. 2005

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirement for the degree ofDoctor of Philosophy
Department of Physics

2011This thesis entitled:

Ultracold Collisions and Fundamental Physics with Strontium
written by Sebastian Blatthas been approved for the Department of Physics

Jun YeDeborah~

The final copy of this thesis has been examined by the signatories, and we find that boththe content and the form meet acceptable presentation standards of scholarly work in theabove mentioned discipline.

Blatt, Sebastian (Ph. D., Physics)
Ultracold Collisions and Fundamental Physics with Strontium
Thesis directed by Professor Adjoint Jun Ye

The success of strontium-based optical lattice clocks in the last five years has led to a
recommendation by BIPM of strontium as a future standard of frequency and time. Dueto the excellent agreement between three international labs, the strontium optical clocktransition is the best agreed-upon optical frequency to date. We use the international opticalclock data to limit present-day drift of fundamental constants and their coupling to theambient gravitational potential. Strontium lattice clocks are still making rapid progress andpromise a large signal-to-noise improvement over single-ion-based frequency standards by
employing O(104 ) atoms. Reaching quantum-projection-noise limited measurement requires
a careful study and control of the many-body interactions in the system. We measureinteractions between ultracold fermions at the 10−17 level and relate them to s-wave collisions
due to a loss of indistinguishability during the spectroscopic process. This new understandingof the many-body effects will increase the precision of current optical lattice clock systems
and can lead to the accuracy level that has so far been pioneered only in single particle(trapped ion) systems. A second generation strontium system is used to control ultracold
interactions in an otherwise ideal gas of bosonic 88 Sr via the optical Feshbach resonance effect.
These new measurement and control capabilities pave the way to reach atomic shot-noise
limited optical clock performance without detrimental effects from large atom numbers.

für Irina

Acknowledgments

I joined the strontium gang in November of 2004 and have stayed with them through adiploma thesis and all of graduate school. I would like to thank both Jun and Rudi Grimm
for making that first year possible. During my first visit to the lab, I was awed that anybody
could know and understand the function of every little piece in that huge pile of lasers, optics,
electronics, cables, instruments, and vacuum hardware. I was amazed that almost half ofthe lab was supposed to be for a single experiment, and really wanted to learn how to build
something like that myself.
Luckily, it turned out that the Sr experiment was large enough to require many people
to actually run it and I learned something from each of them. Working with a lot of smart
people in really cramped spaces until the early morning hours let me appreciate the value of
teamwork. Without being able to rely on everyone knowing their part of the experiment very
well and keeping it running, none of the work over the years would have been possible. Jun’s
enthusiasm for physics and his willingness to consider even our craziest ideas still amazes
me. I would like to thank Jun for showing us that combining superior technical skill withbroad interest and enthusiasm leads to many new ideas and great results. The importance of
free pizza, a coffee machine, and large amounts of grilled meats cannot be overstated either.
Even when I first joined, Marty Boyd and Andrew Ludlow let me play with all of theexpensive stuff right away. Knowing that I was handling the only laser diode left in thelab and that it cost more than I earned in four months was a great motivator. Over manyvinights of taking spectra and eating cheap pizza, we became friends and I gradually learned
more about what we were doing. Tetsuya Ido infected me with his enthusiasm for electronics
and taught me how to design them myself. Seth Foreman showed me all the black magic
involved in operating an octave-spanning frequency comb and also managed to improve myping-pong skills when everything was actually working. Tanya Zelevinsky showed me how tothink differently about many aspects of the experiment and I learned a lot from that. I would
like to thank Gretchen Campbell for teaching me that working harder than everyone else
results in getting much more done. I learn something new every time I talk to Jan Thomsen
about the experiment; I am still amazed at how well dry ice enhances the vacuum lifetime.
Since Mike Martin took over the frequency comb and spectroscopy laser side of things, I
have learned a lot by simply asking him what he is currently doing. Marcio Miranda’s
good-natured humor in the face of breaking lasers has kept me going more than once. Matt
Swallows taught me not to believe anything my grandmother says and how important it is toalways back up everything with facts. Mike Bishof took over the control software for Sr2 andmade it better. Thanks to Yige Lin for quietly improving many aspects of the experiment.
Building Sr3 would not have been possible without Travis Nicholson, Loctite, and Gamma
Ray. Ben Bloom has been doing a great job in taking over the Sr3 software and control
hardware development. Jason Williams actually knows what he is talking about when hespeaks about Feshbach resonances and both Ben and him have been willing to take over theBlue Beast, without which I would not have been able to write this.
Over the years, I have had many pleasant discussions and scientific interactions with the
YeLabs/Jin crew: Adela Marian, Aleksandra Foltynowicz-Matyba, Amodsen Chotia, Arman
Cingöz, Avi Pe’er, Ben Stuhl, Benjamin Lev, Brian Neyenhuis, Brian Sawyer, Craig Benko,
Dajun Wang, Darren & Eric Hudson, David Balslev-Clausen, Dylan Yost, Flavio Cruz,
Florian Adler, Jason Jones, Josh Zirbel, Kang-Kuen Ni, Kevin Cossel, Kevin Holman, Kevin
Moll, Lisheng Chen, Mark Notcutt, Mark Yeo, Matt Hummon, Matt Stowe, Mike Thorpe,
Piotr Maslowski, Silke Ospelkaus-Schwarzer, Steven Moses, Thomas Schibli, Thomas ZanonWillette, Ticijana Ban, Tom Allison, Travis Briles, Xueren Huang, and Yong Xia. The labvii
would not be the same place without Jan Hall and Ma Long-Sheng; it is always a great
experience to find out just how much there still is to learn about physics.
I would still be working on building Sr2 instead of having done an experiment with Sr3,
were it not for the exceptional technical support available at JILA. Especially during thebuilding phase, there was not a day when I was not running to the electronics shop, themachine shop, the purchasing department, or the computer guys with some problem thatwould have taken the next two to three days to figure out by myself. Thank you all for making
these experiments work, they would not be done yet without you: Terry Brown, James
Fung-a-Fat, Mike Whitmore, Carl Sauer, David Tegart, Paul Beckingham, Blaine Horner,
Tracy Keep, Hans Green, Tom Foote, David Alchenberger, Kim Hagen, Todd Asnicar, Ariel
Paul, JR Raith, Mike Paige, Pete Ruprecht, Jim McKown, Joel Frahm, Alan Dunwell, Dave
Errickson, Brian Lynch, and Jeff Sauter. Thanks also to Diane Moreland for organizing
everything from travel to room reservations.
Clock comparisons against clocks at NIST Time & Frequency would not have been possible
without our setup in Scott Diddams’ lab. Thanks to Loree Kaleth for getting me access to
NIST and to Scott and Leo Hollberg for helping me to mess with all the oscillators anddistribution amplifiers. Thanks to Tom Parker, Steve Jefferts, and Tom Heavner for making
me aware of the intricacies of rf frequency transfer and having me characterize the frequency
chain between our lab and NIST.
We would have had a hard time understanding many theoretical aspects of our experiments
without the help of Ana Maria Rey, Paul Julienne, Chris Greene, Murray Holland, Dominic
Meiser, Sergey Porsev, and Peter Zoller.
Thanks to many other friends I stayed (more or less) sane all through grad school: Amanda
Carpenter, Dominique Wagner & Clemens Bruschek, Doug Holbrook, Grzegorz Miecznik,
Helga & Peter Rabl, Jeff Long, Johannes Traxl, Jonas Pologe, Kirk Ullmann, Laura White,
Lisa Kaser, Maria & Art Richmond, Nate Sizemore, Paul Hultgren, Rajiv Bhat, Robert Ito,
Sandra Tkotz & Jan Neuber, Suzanne Whitby & Martin Graus, and Thomas Monz.
I would like to thank my parents for their constant support and encouragement and making

viii
all of this possible. I would not be here without you. Thanks also to my brothers, Florian &Christian, my grandparents Gerda, Edith & Otto, and my parents-in-law, Bernadette andHubert, for all they have done to support my wife and myself over the years.
My wife Irina has the greatest patience of all.
Thank you for being there and especially for being there when the going gets tough.

Contents

Abstractiii

Acknowledgmentsv

List of Tables

xiii

List of Figures

xv

Introduction

1

1. The 87 Sr Frequency Standard

5

1.1. Lattice clock overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2. Current 87 Sr error budget . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2. Laser Spectroscopy of Lattice-Trapped Atoms

15

2.1. Carrier and Sideband Transitions . . . . . . . . . . . . . . . . . . . . . . . .

172.2. Spectra for a harmonic trap . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.3. 1D Optical Lattice Potential . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.4. Tunneling in the WKB approximation . . . . . . . . . . . . . . . . . . . . .

23

2.5. Lattice Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

2.6. Adiabatic Wannier-Stark ladder . . . . . . . . . . . . . . . . . . . . . . . . .

32

2.7. Single-site potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

x
2.8. Sideband spectra in a one-dimensional optical lattice . . . . . . . . . . . . .

37

2.9. Carrier spectroscopy and effective misalignment . . . . . . . . . . . . . . . .

42

3. Ultracold Collisions in Lattice Spectroscopy

49

3.1. Experimental procedure and results . . . . . . . . . . . . . . . . . . . . . . .

51

3.2. Spin model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

3.3. Spin model fit to experimental data . . . . . . . . . . . . . . . . . . . . . . .

63

3.4. Two particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

3.5. Interaction-dominated regime . . . . . . . . . . . . . . . . . . . . . . . . . .

68

4. Variation of Fundamental Constants

73

4.1. Sensitivity constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

4.1.1. Sensitivity of the 133 Cs clock transition . . . . . . . . . . . . . . . . .

77

4.1.2. Sensitivity of optical clock transitions . . . . . . . . . . . . . . . . . .

78

4.1.3. Sensitivity of 87 Sr absolute frequency measurements . . . . . . . . . .

78

4.2. Global absolute frequency record . . . . . . . . . . . . . . . . . . . . . . . .

79

4.3. Linear Drifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.4. Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

4.4.1. Gravitational Potential Variation on Earth . . . . . . . . . . . . . . .

85

4.4.2. Gravitational Redshift . . . . . . . . . . . . . . . . . . . . . . . . . .

89

4.4.3. Gravitational Coupling . . . . . . . . . . . . . . . . . . . . . . . . . .

90

5. Optical Feshbach Resonance in 88 Sr

95

5.1. Complex scattering length, S-matrix, collision cross sections & rates . . . . .
5.2. OFR as an isolated decaying Feshbach resonance99. . . . . . . . . . . . . . . 101

5.3. Photoassociative Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4. Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.4.1. Vacuum system & magnetic coils . . . . . . . . . . . . . . . . . . . . 107
5.4.2. Blue laser & repumps . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

xi
5.4.3. Red lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.4.4. Dipole trap & photoassociation lasers . . . . . . . . . . . . . . . . . . 124
5.4.5. Dipole trap geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.5. Trap potential calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.6. Modeling the phase-space distribution . . . . . . . . . . . . . . . . . . . . . . 133
5.7. Measuring the phase-space distribution . . . . . . . . . . . . . . . . . . . . . 135
5.8. Inelastic loss as relative momentum knife . . . . . . . . . . . . . . . . . . . . 136
5.8.1. Single-particle and relative momentum distributions . . . . . . . . . . 136
5.8.2. Effect of relative-momentum dependent loss . . . . . . . . . . . . . . 138
5.9. Inelastic loss from a Gaussian distribution . . . . . . . . . . . . . . . . . . . 142
5.10. Results for small optical length . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.11. Elastic contributions to the cross section . . . . . . . . . . . . . . . . . . . . 144
5.12. Coupled-channels calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6. Summary & Outlook

153

A. Absorption imaging of trapped particles

159

A.1. Beer’s law and classical scattering cross section . . . . . . . . . . . . . . . . 159
A.2. Polarization, selection rules, and atomic response function . . . . . . . . . . 160
A.2.1. Atomic matrix element . . . . . . . . . . . . . . . . . . . . . . . . . . 161
A.2.2. Atomic response and laser spectrum . . . . . . . . . . . . . . . . . . . 164
A.2.3. Scattering cross section . . . . . . . . . . . . . . . . . . . . . . . . . . 165
A.3. Atomic saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
A.4. Time evolution of sample with probe pulse . . . . . . . . . . . . . . . . . . . 167
A.4.1. Photon rescattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
A.4.2. Doppler shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
A.4.3. Cloud displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
A.4.4. Radiation trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

xii
B. Monte-Carlo Collision Simulation

173

B.1. Bird’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
B.2. Inelastic collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
B.3. Elastic collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
C. Related Publications

183

Bibliography

187

List of Tables

1.1. Most recent 87 Sr error budget . . . . . . . . . . . . . . . . . . . . . . . . . .

124.1. Sensitivity of optical clock transitions to variation in α . . . . . . . . . . . .

79

4.2. Peak-top-peak variation in gravitational potential from different sources . . .

86

List of Figures

1.1. Electronic level diagram of strontium . . . . . . . . . . . . . . . . . . . . . .

6

1.2. Sketch of vacuum system and geometry . . . . . . . . . . . . . . . . . . . . .

7

1.3. 1D optical lattice and probe beam . . . . . . . . . . . . . . . . . . . . . . . .

8

1.4. Optical clock comparison layout . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.1. Example absorption spectrum of a harmonically bound particle . . . . . . .

21

2.2. WKB tunneling rate in an optical lattice . . . . . . . . . . . . . . . . . . . .

24

2.3. Band structure of the one-dimensional optical lattice . . . . . . . . . . . . .

26

2.4. Quasimomentum-resolved band structure and width . . . . . . . . . . . . . .

29

2.5. Comparison of Wannier and Mathieu functions . . . . . . . . . . . . . . . . .

31

2.6. Adiabatic band structure with respect to transverse coordinate. . . . . . . .

35

2.7. Experimental sideband spectra . . . . . . . . . . . . . . . . . . . . . . . . . .

38

2.8. Carrier Rabi flopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

2.9. Carrier line shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

2.10. Single-site Rabi frequency inhomogeneity . . . . . . . . . . . . . . . . . . . .

47

3.1. Clock laser stabilization scheme . . . . . . . . . . . . . . . . . . . . . . . . .

52

3.2. Clock frequency shift with varying sample density . . . . . . . . . . . . . . .

54

3.3. Thermally averaged spin model results (Figure from Ref. [47]) . . . . . . . .

64

3.4. Two-particle spin model spectroscopy . . . . . . . . . . . . . . . . . . . . . .

67

xvi
3.5. Two-particle Hamiltonian in dressed state picture . . . . . . . . . . . . . . .

69

4.1. Global absolute frequency measurements of the 87 Sr optical lattice clock . . .

81

4.2. Linear weighted least-squares analysis of optical clock fractional frequency drifts 84
4.3. Solar gravitational potential on Earth . . . . . . . . . . . . . . . . . . . . . .

87

4.4. Gravitational coupling constants . . . . . . . . . . . . . . . . . . . . . . . . .

93

5.1. Magnetic Feshbach resonance vs. Optical Feshbach resonance97. . . . . . . .

5.2. Molecular potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.3. PA loss spectra for small ℓopt . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.4. Sr3 vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.5. Blue laser schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.6. Red laser schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.7. Dipole trap laser schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.8. Experimental setup and geometry . . . . . . . . . . . . . . . . . . . . . . . . 129
5.9. Conditional relative momentum distribution . . . . . . . . . . . . . . . . . . 139
5.10. Effect of relative-momentum dependent loss . . . . . . . . . . . . . . . . . . 141
5.11. Photoassociative spectroscopy for n = −2 . . . . . . . . . . . . . . . . . . . . 145
5.12. Elastic contribution to the scattering cross section . . . . . . . . . . . . . . . 147
5.13. Coupled-channels calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 150
A.1. Cross section dilution from probe laser spectrum . . . . . . . . . . . . . . . . 165
B.1. Simulation of inelastic loss at constant cross section . . . . . . . . . . . . . . 178
B.2. Simulation of cross-dimensional thermalization . . . . . . . . . . . . . . . . . 181

Introduction

recision spectroscopy of electronic transitions in atomic systems has led to many tech-Pnological advances over the last decades. Atomic transition frequencies have become

the most accurately measured physical quantities, which has led to the redefinition of thesecond in terms of the 133 Cs hyperfine transition frequency. Development of more accurate
atomic frequency standards based on optical transitions has sped up greatly with the invention of the femtosecond optical frequency comb. These lasers emit a coherent train of laser
pulses at radio frequency repetition rates which provides a coherent link between the opticaland radio frequency domains. Absolute frequency measurements of optical transition frequencies with these devices has become routine over the last decade. Most optical standards
are based on narrow intercombination lines available in two-valence-electron systems such as
neutral alkaline earth atoms. This work focuses on experiments related to such an opticalfrequency standard using neutral 87 Sr trapped in an optical lattice.
The optical lattice clock entered the scientific scene in 2005 with work on 87 Sr by Hidetoshi
Katori’s group at the university of Tokyo [1], our group at JILA [2], and Pierre Lemonde’s
group at LNE-SYRTE in Paris [3]. Since then, several other groups have realized opticallattice based frequency standards such as fermionic 171 Yb [4] and bosonic 88 Sr [5, 6]. Theirwidespread use in laboratories across the world has made optical lattice clocks very wellcharacterized precision measurement systems. The 87 Sr standard in particular has been
accepted as a secondary frequency standard by the Bureau International des Poids et Mésures

2(BIPM). The main reason for this choice is the excellent global agreement on the absolutevalue of the optical transition frequency in the Sr lattice clock. In fact, the 1 S0 -3 P0 transition
frequency in neutral 87 Sr at 429, 228, 004, 229, 873.65(37) Hz [7] is the best agreed-upon
optical frequency there is [8].
An optical lattice clock consists of ultracold atoms trapped in an optical lattice at a
magic wavelength where the trapping potential is matched for both clock states. The atoms
are interrogated using a highly frequency-stable spectroscopy laser tuned to a narrow clocktransition. The spectroscopy laser is frequency-stabilized to the transition frequency ontimescales of a few seconds by recreating a new sample and repeating the experiment roughly
once per second. This approach combines the advantages of trapped ion clocks with the large
signal-to-noise (S/N) achieved in an atomic beam based frequency standard by creating anarray of identical microtraps that are interrogated concurrently.
Optical lattice clock systems have already exceeded ion clocks in terms of measurement
precision, although their accuracies have not yet reached the levels demonstrated in NIST’s
ion-trap frequency standards. Current lattice clock accuracies are limited by two effects.
The first is due to the influence of the room temperature blackbody radiation (BBR), which
produces an AC Stark shift of the clock transition. This purely single particle effect limits
all state-of-the-art optical and rf frequency standards, and its uncertainty increases withdecreasing clock transition frequency. Uncertainty in the BBR shift will be reduced withtechnical advances in controlling the sample environment. The second limiting effect arises
from clock frequency shifts related to the atomic density. Understanding and control of theseshifts is much more interesting and challenging. The main conceptual challenge remains aclear understanding of the intricacies of the underlying many-body system dynamics. Characterizing and controlling these dynamics will allow even larger S/N gains and should result
in lattice clock systems with unprecedented precision and accuracy.

This thesis is split into three parts. We will describe the experimental setup of the 87 Srstandard only briefly; by now, there are many excellent reviews on how such a standard is3
built. We will provide references to the relevant PhD theses and papers. An important aspectin understanding the spectroscopic process of atoms trapped in a deep optical lattice are theirmotional degrees of freedom. We will investigate the resulting spectroscopic features in detail.
The spectroscopic process is complicated by the fact that O(10) atoms occupy a typical latticesite. Even though 87 Sr is fermionic, the atoms will experience interactions because of theirnecessarily different motional state. We relate these fundamental motional inhomogeneities
to an inhomogeneous excitation process which allows fermionic s-wave interactions. These
interactions modify the spectroscopic lineshapes and introduce small but important frequency
shifts at the 10−16 level. A detailed understanding of the many-body interactions at thislevel leads to the counterintuitive idea of suppressing the interactions by increasing theirmagnitude. As a demonstration of the high level of understanding gained since 2005, we will
use the unprecedented agreement between all of the groups working on 87 Sr to investigate
physics beyond the standard model. What can we say about cosmological problems like thevariation of fundamental constants? Finally, we will investigate whether it is possible tomanipulate the interactions in an alkaline-earth based many-body system using laser light.
Magnetic-field-induced scattering resonances do not exist in the spinless ground state ofalkaline earth atoms. We experimentally demonstrate that the Optical Feshbach Resonanceeffect can be useful in manipulating the interactions in these systems.
The work presented here is a summary and extension of the work presented in Refs. [8–10]. At this point, a complete survey of the experimental setup and all important effects
influencing the operation of an optical lattice clock has grown far beyond the scope of asingle thesis or review article. Related publications and theses about our system and otheroptical lattice clocks are listed in Appendix C.

Chapter1

The 87Sr Frequency Standard
n this chapter, we will give a brief summary of the operation of a 87 Sr optical latticeIclock. This overview summarizes many important experimental procedures and tools

and detailed descriptions can be found in the papers and theses cited. Also see the list oftheses and papers in Appendix C, especially Refs. [11–14].

1.1. Lattice clock overview
Preparing the atomic sample. Strontium is an alkaline earth, a two-electron system withcorresponding singlet and triplet states, as shown in Fig. 1.1. The main transition is the
30 MHz wide 5s2 1 S0 -5s6p 1 P1 transition at 461 nm which is used in all current experiments
to load a magneto-optical trap (blue MOT) from a Zeeman-slowed atomic beam from aneffusive oven. A sketch of the vacuum chamber with beam directions is shown in Fig. 1.2. In
this way, depending on the Sr isotope used, 106 − 108 atoms are trapped and cooled to mK
temperatures. The MOT transition is not completely closed and roughly one in 105 atoms
leaks to the metastable 5s5p 3 P2 via 5s4d 1 D2 . Since the 3 P2 state has a magnetic dipole
moment, those atoms are not lost from the magneto-optical trap, but remain trapped in
the quadrupole field [15–17]. To prepare a ground state sample, a repumping scheme must
be used. There are several excited triplet states that will decay back to 1 S0 via 5s5p 3 P1
and all of them have been used. In our lab, two lasers on the 5s5p 3 P0 − 5s6s 3 S1 and6
5s5p 3 P2 − 5s6s 3 S1 are continuously interacting with the atoms while they are cycling on
1

S0 -1 P1 in the blue MOT [11, 18].

Figure 1.1.: Electronic level diagram of strontium, excluding hyperfine structure. Note thatthe clock transition 1 S0 -3 P0 only exists in 87 Sr due to the hyperfine interaction. In thebosonic isotopes, is has to be induced by a magnetic field.

After the blue MOT has been loaded with atoms for several hundred ms, the magnetic
field gradient is reduced drastically, the blue light is switched off and atoms are loaded into
a second stage magneto-optical trap (red MOT) operating on the 7.5 kHz wide 5s2 1 S0 −5s5p 3 P1 transition at 689 nm. The narrow linewidth allows cooling down to the recoil limitof several hundred nK [18, 19], and in the presence of hyperfine structure, the red MOT
dynamics require a second laser [20]. Typically, the atoms are cooled to 1 − 3 µK after about
200 ms of cooling at the end of the red MOT stage.
Magic wavelength optical lattice. The remaining atoms are then transferred into
an optical lattice formed by a vertically oriented, retro-reflected laser beam, as sketchedin Fig. 1.3. The individual microtraps are the pancake-shaped high-intensity regions of
the standing wave. The atoms are further cooled with Doppler cooling in the transverse
directions as well as sideband cooling along the lattice axis. Typically, 103 -104 atoms remain

7

Figure 1.2.: Sketch of the vacuum system. The atoms emerge from an effusive oven at∼575 ◦ C and are collimated with two retroreflected transverse cooling beams. The resulting
atomic beam is slowed in a Zeeman slower and the beam enters the vacuum chamber. Three
retroreflected beams at 461 nm form a MOT at the chamber center. The magnetic field
gradient is reduced and three retroreflected red beams at 689 nm form another MOT.
The red MOT is overlapped with the retroreflected lattice beam entering from the topof the chamber. The probe beam at 698 nm is overlapped with the lattice, but is not
retroreflected. Typical lattice lifetimes are ∼1 s.

8at temperatures of 1 µK (2 µK) along (transverse to) the lattice axis. The lattice wavelengthis set to the magic wavelength at 813.4208(5) nm for the 5s5p 1 S0 − 5s5p 3 P0 clock transitionin 87 Sr [21], where the AC Stark shift for the ground 1 S0 and excited 3 P0 clock states are equal
to first order. The clock transition at 698 nm is allowed only in the fermionic isotope throughhyperfine mixing of 3 P1 into the 3 P0 state [22], but can be induced by a large magnetic fieldin bosonic alkaline earths as well [5, 6, 23, 24].

Figure 1.3.: The optical lattice is formed by a retroreflected beam at 813 nm. The retroreflector is a curved dichroic mirror that transmits the probe light at 698 nm to avoid
standing wave effects in the spectroscopy.
Spectroscopy. In the one-dimensional magic wavelength optical lattice, spectroscopy is
performed by copropagating the spectroscopy laser along the lattice axis (see Fig. 1.3). In thisway, spectroscopic information is obtained in the Lamb-Dicke regime, where motional effects
only appear in vibrational sidebands, well separated from the electronic carrier transition [25,26]. The clock laser frequency is then stabilized to the clock transition frequency. On shorttime scales, the clock laser obtains its frequency stability from a lock to a high-quality-factor
optical cavity [27] (see Fig. 1.4) since information from the atomic sample only arrives at rate
given by the experimental cycle time of ∼ 1 s. The spectroscopy laser typically interacts withthe atoms for 80 ms, resulting in a duty cycle of ∼10%. The stabilized clock laser references

9
an octave spanning optical frequency comb [13].
Absolute frequency measurement. The optical frequency comb serves as a distribution center that makes the clock laser frequency stability available at other optical andmicrowave frequencies (see Fig. 1.4). To measure the absolute frequency of our Sr standard,
we use a stabilized optical fiber link between our lab and the NIST Time and Frequency division [28]. The absolute frequency reference is provided by a hydrogen maser referenced to theprimary Cs standard. The maser references an RF oscillator, which is used to modulate theamplitude of a telecom laser. The laser light is transferred to our lab, where the amplitude
modulation frequency is detected and compared against the optical frequency comb. In thisway, the absolute frequency of the Sr clock transition can be measured.
Direct optical comparisons. The quality of the RF frequency transfer method is limited
by how well the hydrogen maser can be referenced to the primary Cs standard (NIST F-1).
To compare other optical clocks at NIST to Sr, we use a direct optical frequency transfer
method [28, 29]. Another telecom laser is phase-locked directly to the frequency comb in ourlab. The laser light is then directly transferred via the optical fiber link. At NIST, the lightis beat against another octave-spanning optical frequency comb which serves as the opticalfrequency distribution center at NIST [30]. In this way, we measure frequency ratios between
Sr and other optical standards with high precision [12, 31].

1.2. Current 87Sr error budget
Any frequency standard needs to be evaluated carefully for systematic effects that influence
its accuracy. Using the direct optical-to-optical comparison scheme outlined in the previous
section, systematic effects can be studied by modulating one clock’s parameters between two
settings while continuously comparing against the other clock.
An example of a systematic effect is the optical lattice wavelength. The sensitivity of
the clock transition frequency to the optical lattice wavelength in the vicinity of the magic
wavelength can be tested in this way, and a conservative estimate of the lattice wavelength

10Figure 1.4.: Spectroscopic information from the atoms at the center of the vacuum chamberis used to tune the spectroscopy laser on timescales of the experimental duty cycle. The
laser itself is stabilized to a highly mechanically-stable optical cavity on short timescales.
The spectroscopy laser stabilizes a self-referenced optical frequency comb. The comb
references other lasers, such as a telecom laser that is used to transmit the spectroscopy
laser’s stability to NIST for optical clock comparisons. The optical clock layout at NIST
is similar and another frequency comb is used to compare the telecom laser’s optical phaseto the respective optical clock. The optical length of all fiber links is stabilized.

11stability allows putting an uncertainty on the final result. In principle, such an uncertainty
could also be arrived at by calculating the sensitivity of the spectroscopy to variability inthe lattice wavelength. As far as possible, it is important not to rely on such estimates since
one can never be sure that the calculation has taken all important effects into account. For
this reason, all systematic effects should be experimentally tested. Ideally, every parameter
contributing to the experimental result should be varied individually. Evaluating a fulluncertainty budget thus requires many iterations of the optical clock experiment and anychange to the experimental layout requires the reevaluation of systematic effects.
In Tab. 1.1, we show the uncertainty budget from the last full evaluation of the JILA87

Sr clock against the primary US frequency standard, NIST F-1 . Ref. [7] includes a very

thorough investigation of all systematic effects that influence such a measurement. The
paper is a good example of the metrological procedure required to claim uncertainties below
the Hz level in an optical frequency measurement. A full discussion is beyond the scope of
this introduction, but we highlight the effects that limit state-of-the-art optical frequency
standards below.
The first two entries of Tab. 1.1 describe shifts of the clock transition frequency with respect
to the lattice intensity. The third entry describes the AC Stark shift of the clock transition
frequency with respect to the ambient room-temperature blackbody radiation (BBR). The
next two entries describe sensitivity of the clock transition with respect to the magnetic field.
The seventh entry characterizes the shift of the clock transition with respect to the atomic
density in the optical lattice. The next entry is a conservative estimate of spectroscopiclineshape modification due to imperfect nuclear-spin polarization. The servo error estimatebounds the effect of possible integrator offsets in the digital servo that stabilizes the clock
laser to the atomic transition. Due to the tight confinement along the optical lattice axis,
the second order Doppler effect is very small.
The total 87 Sr optical clock systematic uncertainty is much smaller than the uncertainty
in the calibration of the intermediate hydrogen maser used to compare Sr versus the Cs
fountain, since the Sr error budget was measured in a direct optical-to-optical comparison

12

Correction

Uncertainty(10−16 )(10−16 )Lattice Stark (scalar/tensor)

−6.5

0.5

Lattice hyperpolarizability

0.1

0.1

BBR Stark

54.0

1.0

AC Stark (probe)

0.15

0.1

1st order Zeeman

0.2

0.2

2nd order Zeeman

0.36

0.04

Density

3.8

0.5Line pulling00.2

Servo error

00.5

2nd order Doppler0≪ 0.01

Sr systematics total

52.11

1.36

Maser calibration−4393.7

8.5

Gravitational shift

12.5

1.0

Total

−4329.1

8.66

Contributor

νSr − ν073.65 Hz0.37 Hz

Table 1.1.: 87 Sr error budget from Ref. [7].

13against the NIST Ca optical clock. The large discrepancy between the quality of rf frequency
measurements and direct optical comparisons is a strong indicator of why a future redefinitionof the second in terms of an optical standard is being pursued.
Another effect that will become more and more important as remote optical frequency
standards are compared is the gravitational red shift. Our lab at JILA and the NIST Time& Frequency division are separated by 3.5 km and a height difference of 11.3(2) m between
the Sr clock and the Cs fountain has been estimated by GPS receivers in each building. The
height difference by itself introduces a frequency correction on the 10−15 level. However, theuncertainty in the red shift correction not only includes the mere height difference but also an
upper limit on the “transverse” variation in the gravitational field from the nearby mountains.
The gravitational potential also fluctuates in time and these effects will become more andmore important as optical clock comparisons become more accurate or are performed overlonger distances. The red shift uncertainty here includes an estimate of the gravitational
isosurface variation between JILA and NIST at the 10 cm level using the National Geodetic
Survey markers next to either lab [7].
The largest systematic effects related to the atomic system are the AC Stark shifts inducedby the room-temperature BBR and transition frequency shifts when the atomic density isvaried. Those two shifts are also the conceptually most worrying and interesting effects. In
Cs fountains, these effects appear at much larger magnitude, but with the increase in opticalclock accuracy, all state-of-the-art optical standards are becoming limited by the same effects.
The BBR correction was first introduced by W. Itano [32] and since then, atomic frequency standards have been defined at zero temperature. The correction here is based ontheoretical calculations of the clock state polarizabilities at the BBR wavelengths [33]. Halfof its uncertainty comes from insufficient knowledge of these polarizabilities, the other half is
experimental. Even though the temperature of the vacuum chamber is measured in multiple
spots and rolling corrections are applied, the metallic vacuum chamber is not a black body.
For these reasons, the BBR shift in optical clocks has received much attention in the last few
years [34, 35]. Work on measuring the shift experimentally is under way in many labs, but

14
any such measurement requires a specialized setup [36] and there are no experimental results
so far. Nevertheless, the BBR is a pure single-particle effect and will be understood andcontrolled with a technical solution involving a temperature-controlled environment similar
to the ones employed in Cs fountain clocks.
The next largest uncertainty are clock frequency shifts related to the atomic density. These
effects are intellectually much more challenging and interesting since they can potentiallycompromise both precision and accuracy. For this reason, recent work in our lab has focusedon understanding and controlling these shifts. The density shift is an intrinsically many-bodyeffect arising from atomic interactions. Its appearance is especially surprising since the atoms
are at temperatures of ∼1 µK and s-wave interactions should be suppressed by the Pauli
exclusion principle. The density shift will be discussed in detail in Chapter 3. To understandits origin, however, we need to understand the spectroscopic process in considerable detail.

Chapter

2

Laser Spectroscopy of
Lattice-Trapped Atoms
aser spectroscopy of tightly confined atoms has been investigated in the context of ion

Ltraps for more than three decades [25, 26]. The key advantage of trapping particles

for laser interrogation is signal-to-noise (S/N) gain by extending the coherent light-atominteraction time. However, the atom needs to be tightly trapped with respect to the interrogating wavelength to suppress contamination of the signal by the atomic motion. In
addition, the trapping potential needs to be the same for ground and excited state coupled by
the spectroscopy light to avoid coupling the motional and spectroscopic degrees of freedom.
The same advantageous spectroscopic conditions as for ion traps can now be achieved forneutral atoms by trapping them in a tight optical lattice. The theoretical description of thespectroscopic process still applies, but has to be modified to account for the conditions in
optical lattices.
The second, and much more important, difference between ion trap and optical lattice
spectroscopy is that spectroscopy in optical lattices allows interrogating many sites of thelattice simultaneously. The lattice can act as an array of identical microtraps and thus the√
S/N is in principle enhanced by a factor of N , where N is the overall number of atoms
interrogated. If, additionally, the particles in different lattice sites can be entangled, one can16hope to increase the spectroscopic S/N by another factor proportional to√N.

For these reasons, high-resolution spectroscopy in optical lattices has been investigatedactively over the last decade, mostly based on the “magic wavelength lattice clock” proposal [37, 38] building on parallel ideas for applications in cavity QED experiments [39].
First 87 Sr spectroscopy results were available shortly thereafter [1, 40, 41]. The first systematic investigation of high-resolution 87 Sr spectroscopy for application in an optical atomic
clock was published in 2006 [2], followed by similar results from the Paris [3] and Tokyo [42]groups. Soon after, we used the same system to achieve the highest quality factor in any
kind of coherent spectroscopy [43].
To achieve higher resolutions and to make use of the full enhancement factor due to N ,the spectroscopic process must be investigated in deeper detail. The main obstacle to highresolution spectroscopy was overcome by using the magic wavelength lattice [21], and theParis group verified that higher order polarizability contributions are small [44]. Next, theeffect of the hyperfine interaction on spectroscopy was investigated in detail [22], leading tospin-polarized operation of the lattice clock used in all groups today [7, 42, 45].
Soon after, it became obvious that the number of atoms trapped in an individual latticesite is an important factor in spectroscopy. A significant systematic shift of the opticalfrequency with number of atoms per lattice site was first observed in direct optical clock
comparisons [31]. These comparisons still used simultaneous spin polarization to mF = ±9/2,
but the density shift persisted even with a single spin species [46]. The dominant role ofRabi frequency inhomogeneity due to populating different transverse motional states was
described qualitatively as due to inhomogeneity-induced s-wave interactions [9], which led
to several theoretical models trying to describe the underlying effect [47–50].
The importance of understanding the interactions in a many-body system for clock spectroscopy using more than one particle was highlighted recently by demonstrating that density
shifts can be suppressed if the interparticle interactions are increased by placing the particles
in a two-dimensional optical lattice [51].
In this Chapter, we will introduce the framework for understanding the optical lattice

17
spectroscopy and introduce optical sideband spectroscopy as an important tool to understandthe system parameters. With these methods, we will proceed to model collisions between
ultracold fermions and understand the density shift as arising from system inhomogeneities.
The framework for understanding spectroscopy of trapped particles builds on the results
available from laser spectroscopy of ions trapped in RF traps. The main ideas have beendeveloped early on and have been described in many theses from our lab [11, 12, 14] andthe Paris group [52–54]. The main parameter to understand spectroscopy of tightly boundparticles is the Lamb-Dicke parameter [26]p/νtrap ,η = νrec(2.1)

pgiven by the ratio of probe light recoil frequency νrec= h/(2mλ2p ) to trapping frequency

νtrap , where λp is the probe light wavelength and m is the mass of the particle. In the limitη ≪ 1, the particle’s motional wave function has a small extent with respect to the probing
wavelength and the probing process will not change the motional state of the particle. This
limit is called the Lamb-Dicke regime and the process is very similar to what happens inMössbauer spectroscopy: the particle is so tightly bound that the recoil from absorption andreemission of a probe photon gets absorbed by the crystal lattice (the optical lattice here).

2.1. Carrier and Sideband Transitions
The response of a particle to the probing light in the dipole approximation is given by the
Rabi frequencyΩ ∝ hψf |eikp ·x |ψi i,(2.2)

between initial and final motional state |ψi i and |ψf i, and we assumed the probe to be aplane wave with wave vector kp = 2π/λp k̂p , and x is the position operator of the particle.
The Lamb-Dicke approximation consists of expanding the matrix element in orders of kp ·hxi:Ω ≡ Ω0 [hψf |ψi i + ikp · hψf |x|ψi i + . . .)] .

(2.3)

18
If we assume three-dimensional harmonic confinement with trap frequencies (νx ,νy ,νz ) andinitial (final) trap state |ni = |nx , ny , nz i (|mi = |mx , my , mz i), this expansion can besummed up analytically and reduces to the well-known expression [25, 26, 55]YΩm←n = Ω0j∈{x,y,z}= Ω0†hmj |eiηj (âj +âj ) |nj iYe−ηj2 /2j∈{x,y,z}

s

n<
∆n
j !
(iηj )∆nj Ln< j (ηj2 ),<j(nj + ∆nj )!

(2.4)

where âj (â†j ) are bosonic annihilation (creation) operators, n<
j = min(nj , mj ), ∆nj =|nj − mj |, Lαn is a generalized Laguerre polynomial, and the ηj are the per-axis Lamb-Dicke
parameters√pηj ≡ (k̂p · x̂j )kp aj / 2 = (k̂p · x̂j )νrec/νj = (k̂p · x̂j )for oscillator length 2πaj ≡

p

s

h
,
2mλ2p νj(2.5)

h/(mνj ), and trap axis direction x̂j . In most of the following,

we are interested in the carrier transitionΩc (n) ≡ Ωn←n = Ω0Y2e−ηj /2 Lnj (ηj2 ),

(2.6)

j

where the motional state is unchanged. However, the first blue (red) sideband, where one
motional quantum is added (removed) is also of interest (change in quantum number assumed
along ẑ here)iηz L1nz (ηz2 )Ωbsb (n) ≡ Ω(nx ,ny ,nz +1)←n = Ωc (n) √nz + 1 Lnz (ηz2 )iηz L1nz −1 (ηz2 )n >0,Ωrsb (n) ≡ Ω(nx ,ny ,nz −1)←n z= Ωc (n) √nz Lnz (ηz2 )(2.7)

In the Lamb-Dicke regime, we can expand the Laguerre polynomials in orders of ηj to simplify
the expressionsΩc (n) = Ω0YjRemarks2e−ηj /2 [1 − nj ηj2 ]√hnz 2 iΩrsb (n) = Ωc (n)iηz nz 1 + ηz2√nz + 1 2Ωbsb (n) = Ωc (n)iηz nz + 1 1 +
ηz .
2

(2.8)

19
• The red sideband does not exist if the particle is in the ground state initially, consistent
with the original expression in terms of âj and â†j .
• The first order sideband Rabi frequencies Ωrsb and Ωbsb are 90◦ out of phase with thecarrier frequency Ωc .

• The atomic response always depends on the initial motional state |ni of the particle
and decreases with increasing |n|.
• The atomic response distributes among all target states |mi such that |Ω0 |−2
Q P
Q−ikp ·x†|mj ihmj |eikp ·x |nj i = j hnj |nj i = 1.
jmj hnj |eP

m |Ωm←n |2.2. Spectra for a harmonic trap
Using the results from the last Section, we can discuss the absorption spectrum of a harmonically trapped particle. The spectrum will be dominated by the carrier transition at the basefrequency. The first sideband transitions are detuned from the carrier by the energy added or
removed due to the extra motional quantum (or phonon). Here, the extra detuning is givenby +νz (−νz ) for the blue (red) sideband. If νz is larger than the power-broadened carrier
linewidth Ωc and the natural linewidth of the carrier transition, the system is said to be in
the resolved sideband regime. The first blue and red sideband transitions produce spectralfeatures that are well separated from the carrier transition and addressable by tuning thespectroscopy laser to the corresponding detuning ±νz from the carrier.
In the following, we will limit our discussion to completely coherent population dynamics,
and neglect all decoherence processes. In this regime, we can describe the response of an ensemble of particles as mixture of single particles. In the absence of decoherence, each particle
responds to the spectroscopy light according to its carrier and sideband Rabi frequencies. In
the resolved sideband regime, the relative phases of carrier and sideband Rabi frequencies

2

=

20
can also be neglected and the excited state population at time t can be written asp|Ωrsb (nz )|2t2pe (nz , ∆, t) =sin|Ωrsb (nz )|2 + (∆ + νz )2|Ωrsb (nz )|2 + (∆ + νz )22p|Ωc (nz )|2t2|Ωc (nz )|2 + ∆2
+
sin
2
2
|Ωc (nz )| + ∆
2


2
p
t
|Ωbsb (nz )|222|Ωbsb (nz )| + (∆ − νz )+sin|Ωbsb (nz )|2 + (∆ − νz )2
2

(2.9)

For a thermal mixture in a truncated harmonic oscillator, we find the thermally averaged
excited state population
Pe (Tz , ∆, t) =Nz
1 − qz X
q nz pe (nz , ∆, t),
1 − qz1+Nz n =0 z

(2.10)

zz
with Boltzmann factor qz = exp(− khν) and maximal quantum number Nz .
B Tz

An example spectrum of pe with respect to ∆ for a particle in the nz = 1 state of a harmonic
trap probed along the ẑ direction is shown in Fig. 2.1(a). First order blue and red motional
sidebands appear detuned by the trap frequency from the central carrier transition. We
have assumed negligible transition linewidth, resulting in the sinc2 sidelobes around carrierand sidebands. Note that the sidebands are only suppressed with respect to the carrier
because the sideband Rabi frequencies are attenuated by ηz , and Ω2bsb /Ω2rsb ≃ 1 + 1/nz . In acompletely homogeneous system, the Rabi flopping on the sidebands has the same contrast
as the carrier.
In panel (b) of the same figure, we added a slight inhomogeneity to the system by assuminga thermal distribution truncated at Nz = 6 and plot the thermally averaged excited state
population for Tz = 3 µK. Note that the inhomogeneity reduces the initial rise of the
sidebands, with respect to the carrier. The difference in the excited state fraction dynamics
is small as long as the exposure time is short. This can be seen in panels (c) and (d)which compare the carrier Rabi flopping for the two cases (a) and (b). The Rabi frequency
inhomogeneity in (b) and (d) causes decay and revival of the Rabi flopping contrast in
panel (d). Even with the added inhomogeneity, we can thus always find an exposure time –experimental conditions permitting – where the carrier or the sidebands have full contrast.

1.0

1.0

0.8

0.8

0.6

0.6
Pe

Pe

21

0.4
0.2

0.2

0.0
-100

-50

0
50
Detuning HkHzL

0.0
-100

100

(a) Pure state nz = 1.

Pe

0.4

-50

0
50
Detuning HkHzL

100(b) Thermal average at Tz = 3 µK, Nz = 6.

1.0
0.8
0.6
0.4
0.2
0.0
0

20

40

60

80

60

80

Exposure time HmsLPe(c) Pure state nz = 1.

1.0
0.8
0.6
0.4
0.2
0.0
0

20

40
Exposure time HmsL(d) Thermal average at Tz = 3 µK, Nz = 6.

Figure 2.1.: Example absorption spectrum of a harmonically bound particle. The trap
frequency νz = 80 kHz and the Lamb-Dicke parameter ηz = 0.3, and the harmonicallybound particle is probed purely along z. Natural linewidth is neglected and Ω0 = 1 kHz.
2

The exposure time t is set to t−1 = Ω0 e−ηz /2 to produce a π pulse on the carrier for nz = 0
in panels (a) and (b). Panels (c) and (d) show coherent carrier Rabi flopping over 80 ms
corresponding to the situations in (a) and (b).

22
The situation shown in panels (b) and (d) corresponds to typical parameters used in the
optical lattice clock experiment. The simulated data, however, does not at all look like what
is seen in the experiment. The maximal carrier contrast is 90% and the sidebands neverreach the excited state fraction observed in the carrier. The imbalance between red and blue
sideband is always present as well. Clearly, there is more inhomogeneity in the experimentthan we have assumed so far.

2.3. 1D Optical Lattice Potential
To find a more realistic description of the inhomogeneity of the vertical one-dimensionallattice system, we need to look at the trapping potential more carefully. In the following, we
assume a one-dimensional magic-wavelength optical lattice formed by a retro-reflected laser
beam. The crucial role of atomic polarizability, the effects pertaining to lattice polarizationand magnetic field, and the effect of nuclear structure on spectroscopy will be completely
omitted. For more information on these effects refer to the detailed explanations in Refs. [11].
For our purposes, imagine a conservative trapping potential that is exactly the same forboth clock states. In this Section, we would like to exhibit the effect of the motional degrees
of freedom on spectroscopy in the context of Eq. 2.2. The material here summarizes andextends previous discussions in Refs. [9, 11, 14, 51–54, 56].
In this spirit, we approximate the optical lattice potential around its focal point as
2

2

U (x) = U (x, y, z) = −U0 e−2r /w0 cos2 κz,(2.11)

where w0 is the Gaussian waist of the retro-reflected beam, and κ = 2π/λ is the wavevectorof the beam for the magic wavelength λ = 813.43 nm. The trap depth U0 is given by thetime-averaged laser power P and the dynamic AC polarizability at the lattice wavelengthα(λ) as
U0 =

4P
α(λ).
πǫ0 cw02(2.12)

Experimentally, P , w0 , and α(λ) are hard to know accurately enough. Instead, knowledge
about the trap geometry is usually inferred from measured trap frequencies under the as-

23
sumption of the overall form of Eq. 2.11. To understand this relation clearly, the potential
has to be understood in more detail.
If we neglect gravity for now, U has the form of a periodic potential and is cylindricallysymmetric around z. Atoms are trapped in the anti-nodes of the periodic potential andare thus confined longitudinally on the order of λ/2. The transverse confinement along r
is much weaker, since the waist w0 of an optical beam is typically much larger than λ.
We conclude that the aspect ratio (∝ w0 /λ) of a microtrap will be large. Even if we hadallowed the Gaussian beam waist to vary with z, the shape of adjacent microtraps would bevery similar, as long as only pancakes within the Rayleigh range zR = πw02 /λ are populated.
Since the number of similar pancakes ∝ zR /λ ∝ (w0 /λ)2 , it scales as the aspect ratio squared.
From these simple arguments, we conclude that it is reasonable to think of atoms trappedin identical pancake-shaped microtraps. For each microtrap, we can calculate longitudinal
and transverse trapping frequencies which can then be compared to the spectroscopicallymeasured results.

2.4. Tunneling in the WKB approximation
A complication to the simple picture above is that the potential is periodic in z which allows
tunneling between different microtraps. The tunneling rate gives each microtrap level a
finite energy width. For shallow lattices, tunneling can dominate the system dynamics andwe cannot speak of isolated microtraps anymore. As a first estimate, we will try to use the
WKB approximation to describe the tunneling rates in fairly deep lattices that are typicalfor optical clock experiments.
In the semi-classical WKB approximation, the tunneling rate is given by a collision attempt
rate with the potential wall and a tunneling amplitude that exponentially decays with thewall height and width (see e.g. Ref. [57]). For U (z) = U0 (1 − cos2 κz), we find a tunneling
rate
Erec
ΓWKB =~p" r#
Z π−ξ0 r
E/ErecEU0
exp −2− cos2 ξ ,dξ 1 −
ξ0
Erec ξ0
U0(2.13)

24
with lattice recoil energy Erec ≡ h2 /(2mλ2 ) ≃ h × 3.47 kHz and atomic mass m = 87 amu.
We use the dimensionless excursion ξ = κz, and the classical turning point is given bypξ0 = arccos 1 − E/U0 with ξ0 ∈ [0, π/2]. Numerical results for a lattice depth U0 = 120Erec

are shown as the solid red curve in Fig. 2.2. This trap depth is a typical value for our opticalclock experiments and would correspond to a longitudinal trap frequency of ∼75 kHz. The
tunneling lifetimes predicted from 1/ΓWKB are basically infinite unless the particle is very
highly excited. For reference, the black dashed line indicates a tunneling rate of 1 s−1 for87

Sr in the magic wavelength lattice.

æ

æ

æ

Tunneling rate HErec ÑL

æ

0.01

æ
æ
æ

10-6

æ

10-10
10-14
10-18
0

20

40

60

80

100

120

EErecFigure 2.2.: WKB approximation to the tunneling rate in a cos2 lattice (red solid curve)
versus full band structure calculation (blue circles) in a 120 recoil deep lattice. The black
dashed line indicates a tunneling rate of 1 s−1 for the 87 Sr magic wavelength lattice. The
WKB approximation fails to describe tunneling in the infinite lattice because only two
adjacent sites were included.

The blue circles in Fig. 2.2 show the tunneling rates calculated using the full lattice band
structure for an infinite lattice. Even for the relatively deep lattice used here, the WKB
approximation clearly underestimates the tunneling rate in a periodic potential with manysites. Including the periodicity correctly is hard, since the transverse and longitudinal potentials are multiplied instead of added in the Schrödinger equation and thus do not allow
separable solutions.

25

2.5. Lattice Band Structure
To treat tunneling to all orders, we need to think of the potential as an array of longitudinal lattices, one for each transverse motional quantum state. This picture is supportedby the following argument: because the microtrap aspect ratio is large (typically ≫ 50),
we can think of the transverse motional dynamics as much slower than the longitudinal dynamics. This separation of time scales means that tunneling happens instantaneously on
the transverse time scale. For the same reason, there are also many more transverse thanlongitudinal quantum states occupied at a given temperature. This means that we can thinkof the transverse degrees of freedom as a classical parameter that parametrizes the quantum
mechanical degree of freedom along the longitudinal direction. For this reason, tunneling
events that change the transverse quantum number are not important since such a change
will not influence the system dynamics drastically.
To understand the longitudinal tunneling dynamics, we consider r as a classical parameterthat simply scales the trap depth and focus on describing the longitudinal problem. A
detailed derivation [14] shows that the one-dimensional longitudinal Schrödinger equation
reduces to the Mathieu equationd2 φ+ (a − 2q cos 2ξ)φ = 0,dξ 2(2.14)

for the longitudinal spatial wave function φ(ξ), with ξ = κz. The Mathieu parameters
U0 (r)
,
4Erec
E
− 2q,
a=
Erec
q=

(2.15)

describe the kinetic energy E and the trap depth U0 (r). The Mathieu equation only has
stable solutions for certain combinations of a and q and stability diagrams can be foundin standard mathematical works [58], works on ion traps [59], or in the context of band
theory [60]. A plot of a + 2q versus 4q is shown in Fig. 2.3 and directly shows the evolutionof the energy structure from isolated harmonic oscillator levels to broad energy bands withdecreasing trap depth from infinity.

26

140
120

EErec

100
80
60
40
20
0
0

20

40

60

80

100

120

140

U0 ErecFigure 2.3.: Band structure of the infinite one-dimensional optical lattice without gravity
plotted as allowed kinetic energy E versus trap depth U0 in recoil units. Bands with odd(even) band index n are shown in blue (red). The dashed line indicates the lattice depth
E = U0 .

27
A spatial wavefunction basis (Wannier basis) that is localized at the position of eachmicrotrap can be constructed and for small band width and band index n, the Wannier
wave functions for lattice site m can be written in terms of the Mathieu equation’s standard
solutions [14]whereΘm (ξ) =r2Θm (ξ) cen (ξ),π

(2.16)
1 ξ ∈ [mπ, (m + 1)π)(2.17)

wnm (ξ) =0 elseis the supporting function of the m-th lattice site. In this approximation, matrix elementsin the Wannier functions can be easily calculated, for example to estimate lattice heating
rates [14]. However, the approximation is based on completely isolated sites and is applicable
only for low band index n in a deep lattice. To estimate, for instance, the effects of tunneling,
we need to use a numerical method to generate more realistic Wannier states that also apply
to shallower lattices. A useful method is based on representing the Schrödinger equation in
momentum space, since the potential term will then couple only two specific momenta to
the particle momentum. We rewrite the periodic Schrödinger equation as1
H=~2 k̂ 2 U0
+ [1 − cos 2κẑ],2m
2(2.18)In a plane wave basis hx|ki ∝ eikz , the periodicity in the potential transforms into coupling
only two other momenta to a given momentum k:

 2 2
U0
U0
~k
+
|ki − (|k + 2κi + |k − 2κi).
H|ki =
2m
2
4(2.19)The periodicity of the potential leads to periodic Bloch waves |n, qi as the eigenfunctions of
this operator. Here, the dimensionless quasimomentum q is limited to the first Brillouin zoneκq ∈ (−κ, κ] and n indexes the band structure as above. We let q be a fixed value within
the first Brillouin zone, and transfer Eq. 2.18 into a plane wave basis parametrized by q. We
1

Many references describe the application of Bloch’s theorem and the Wannier basis to optical lattices.
This summary is based on [56, 61–64].

28
can omit odd multiples of κ in the basis, since they are not coupled by Eq. 2.19. We lethξ|j, qi ∝ ei(2j+q)ξ ,(2.20)

with integer j, and we truncate the basis such that −J ≤ j ≤ J. This choice is equivalent
to treating a system of 2J + 1 lattice sites with periodic boundary conditions. As above, we
use the dimensionless position ξ ≡ κz. In this basis, the Hamiltonian matrix (in units of therecoil energy) is sparse and has tridiagonal structure [56, 62](−2J + q)2
− u4




u
u
2


(−2J + 2 + q) − 4
−4




.
.
u
.. ..


−
4




u
.
.
.

,
.
.
.
H(q) = I2J+1 + 
.
.
.

2


.
.


.. ..
− u4






u
u
2

−4 
− 4 (2J − 2 + q)


u
2
−4
(2J + q)(2.21)
where u ≡ U0 /Erec and I2J+1 is the (2J +1)-dimensional identity matrix. Diagonalizing H(q)numerically gives a set of (2J + 1) real q-dependent eigenvector coefficients cnj (q) which arenormalized toJX[cnj (q)]2 = 1,

(2.22)

j=−J

as well as the q-dependent eigenenergies En (q). Here, n indexes the band structure and theeigenvectors are sorted according to their eigenvalue. The band structure for u = 120 isshown in Fig. 2.4(a). There are seven trapped bands, and all but the two highest are very
flat, indicating well-separated microtraps.
To quantify the tunneling rates with respect to trap depth, we calculate the band widths
as the difference between the q = 0 and the q = 1 energy structure in Fig. 2.4(b). The
tunneling rate is given by the inverse of the band width. For the magic wavelength lattice,
the recoil frequency is h/(2mλ2 ) ≃ 3.47 kHz, so that a bandwidth of 2.9 × 10−4 Erec would
correspond to a tunneling rate of 1 s−1 . Even at typical trap depths of 120Erec , tunneling

29

10

140
120

0.1
Band width  Erec

EErec

100
80
60
40

0.001

10-5

20
0
-1.0

-0.5

0.0
qΚ

0.5

1.0

(a) Band structure for U0 = 120Erec . The dashedline indicates U0 .

10-7

0

20

40

60

80

100 120 140

U0 Erec(b) Band widths as a function of U0 . The dashedline indicates a tunneling rate of 1 s−1 at Erec =h × 3.47 kHz.

Figure 2.4.: Quasimomentum-resolved band structure calculations from diagonalization of
Eq. 2.21 with J = 30.

30
can be significant if the particles are in higher longitudinal bands or are highly excited in
the transverse direction (reducing the effective lattice depth).
It is instructive to also calculate the tunneling rates from a Wannier function perspective.
The Bloch vectors in the truncated plane wave basis used here areJ

1 X nhξ|n, qi = √cj (q)ei(2j+q)ξ ,π j=−J(2.23)

which are normalized over [−π, π] ∋ ξ. If we choose a sufficiently dense set of quasimomenta
qk ∈ (−1, 1], the Wannier function for lattice site m can be represented as1
wnm (ξ) = √

N
X

N k=1

e−iqπm hξ|n, qi.

(2.24)The prefactor makes the resulting Wannier function site-normalized.
Figure 2.5 shows the site-localized Wannier functions (solid blue), the Mathieu function
approximation (dotted red), and the harmonic oscillator approximation (dotted green) for all
trapped bands in a 120Erec lattice. The agreement between Wannier function and Mathieufunction is almost perfect within a lattice site, even for the high-lying bands with finite
width. The harmonic oscillator approximation shows discrepancies even in the n = 2 band.
The Mathieu functions are periodic in the lattice spacing and the Wannier functions are
localized in each site. However, the localization of the Wannier functions is so good thatthe cut-off Mathieu function approximation in Eq. 2.16 is almost perfect (see right handside plots). Although the harmonic oscillator functions could be improved by simple firstorder perturbation theory in the anharmonic z 4 term, we recommend the cut-off Mathieufunctions for any site-local calculation at this lattice depth since they are readily available
in mathematical packages.2
In the tight-binding approximation [64], the tunneling dynamics can be described by particles hopping from site to site and the band width is determined by a single tunneling matrix
elementZ ∞dx [wnm (x)]∗ Hwnm+1 (x).
Jn ≃ −−∞

2

For instance, as the MathieuC function in Mathematica.

(2.25)

1.0
0.5
0.0
-0.5
-1.0

Ψ0 HxL2

Ψ0 HxL

31

0.2

0.4

ΞΠ

ΞΠ

0.0

10-4
10-9
10-14
10-19
10-24
10-29
-2.0 -1.5 -1.0 -0.5 0.0

-0.2

0.2

0.4

ΞΠ

ΞΠ

0.0

10-5
10-10
10-15
10-20
10-25
-2.0 -1.5 -1.0 -0.5 0.0

-0.4

-0.2

0.2

0.4

ΞΠ

ΞΠ

0.0

1
0.001
10-6
10-9
10-12
10-15
10-18
-2.0 -1.5 -1.0 -0.5 0.0

Ψ3 HxL2

1.0
0.5
0.0
-0.5
-1.0
-0.4

-0.2

0.2

0.4

1.0
0.5
0.0
-0.5
-1.0

1.5

2.0

0.5

1.0

1.5

2.0

0.5

1.0

1.5

2.0

0.5

1.0

1.5

2.0

0.5

1.0

1.5

2.0

0.5

1.0

1.5

2.0

0.5

1.0

1.5

2.0

0.01
10-5
10-8
10-11
-0.4

-0.2

0.0

0.2

-2.0 -1.5 -1.0 -0.5 0.0

0.4

ΞΠ

ΞΠ
1

1.0
0.5
0.0
-0.5
-1.0

0.01
Ψ5 HxL2

Ψ5 HxL1.0ΞΠ

Ψ4 HxL2

Ψ4 HxL

ΞΠ

10-4
10-6
10-8

-0.4

-0.2

0.0

0.2

10-10
-2.0 -1.5 -1.0 -0.5 0.0

0.4

ΞΠ

ΞΠ

1.0
0.5
0.0
-0.5
-1.0

0.1
Ψ6 HxL2

Ψ6 HxL

0.5

1

1.0
0.5
0.0
-0.5
-1.0

Ψ2 HxL2

Ψ2 HxL

0.0

1.0
0.5
0.0
-0.5
-1.0
-0.4

Ψ3 HxL

-0.2

Ψ1 HxL2

Ψ1 HxL

-0.4

10-5
10-11
10-17
10-23
10-29
10-35
-2.0 -1.5 -1.0 -0.5 0.0

0.001
10-5
10-7

-0.4

-0.2

0.0
ΞΠ

0.2

0.4

-2.0 -1.5 -1.0 -0.5 0.0ΞΠ

Figure 2.5.: Wannier functions (solid blue), periodic Mathieu functions (dotted red), andharmonic oscillator wave functions (dotted green) for the 120Erec lattice and all trappedbands n = 0 to 6.

32
Within the validity of the tight-binding approximation, the relation between J and the band
width (or tunneling rate) is given by
4Jn = En (q = 1) − En (q = 0),(2.26)

and from Eqs. 2.23 one can show that calculation of J in the numerical eigenvalues reduces
to a simple integral over the complex unit circle
N

1 X
Jn = −En (qk )eiπqk ≃ −
N k=1

Z 1

−1

dqEn (q)e

iπq

= −2

Z 1

dqEn (q) cos πq.

(2.27)

02.6. Adiabatic Wannier-Stark ladder
So far, we have neglected gravity in the description of the optical lattice potential andespecially in the models used to describe the tunneling rates. Suppression of tunneling in
optical lattice clocks has been a concern from the very beginning. To prevent the atoms fromtunneling while the clock laser is interacting, two steps were taken. As we have seen in thelast section, the lattices for optical clocks are typically fairly deep (> 100Erec ). In addition,
the lattice is oriented along gravity, which breaks the symmetry between lattice sites.
This effect is fairly simple to understand: the energy shift between lattice sites due togravity is given by∆g ≡mg≃ 0.79 Erec ,κ

(2.28)

for gravitational acceleration g ≃ 9.81 m/s2 . If this energy splitting is larger than the lattice
bandwidth (without gravity), energy levels in neighboring sites are no longer degenerate andthe band width becomes suppressed. Each band n separates into individual states in isolated
lattice sites. Since the acceleration is uniform along the lattice, the new energy states areevenly split by ∆g and this structure is called a Wannier-Stark ladder.
Tunneling from a higher-lying state in the Wannier-Stark ladder to a lower-lying state
becomes exponentially suppressed with the ratio of ∆g over the original bandwidth Γn . If
the mismatch becomes large enough, such that ∆g becomes on the order of the interband
splitting, resonant tunneling can be allowed from a lower-lying Wannier-Stark ladder to a33higher lying one. As we will see, this is not an issue in deep optical lattices and we only haveto consider tunneling within an isolated ladder.
The new eigenstates that describe particles in isolated sites can be easily obtained within
the tight-binding approximation. The Wannier-Stark states are given by superpositions of
the original Wannier states [64]M
1 X m mψnℓ (ξ) = √zn wn (ξ),
M m=−M(2.29)

where ℓ is the new Wannier-Stark site index. The new energy levels can be obtained bydiagonalizing the tridiagonal Hamiltonian matrix in the Wannier states
V−M −Jn−Jn V−M +1 −Jn... ...


−J
n




.
.
.
′
,
..
..
..
Hn = εn I2M +1 + ... ...


−Jn







−Jn VM −1 −Jn −Jn VM
R
where Vm = m∆g /Erec and εn = dx[w0n (x)]∗ Hw0n (x).

(2.30)

However, calculating the tunneling lifetimes of the new Wannier-Stark states is hard. A

numerically involved method is based on calculating the poles of a scattering matrix [63].
None of the work available treats the transverse potential, and tunneling in the vertical
one-dimensional optical lattice remains hard to calculate accurately.
Following the arguments in Section 2.3, we will treat the transverse coordinate as a classical parameter that adiabatically follows the longitudinal dynamics since any longitudinaltunneling event is much faster than the transverse oscillation time. For these reasons, we
model the full potential as an “adiabatic” Wannier-Stark ladder and disallow any change inr during a tunneling event.
A plot of the adiabatic band structure is shown in Fig. 2.6(a), where the infinite latticeenergies and their band widths are shown as a function of r/w0 , where the radial trap profile

34
2

2

is modeled by replacing U0 7→ U0 e−2r /w0 . None of the lower-lying bands become very wide,
unless we go to large radii r ∼ w0 . For transverse temperatures T , the thermally averaged
radius is given by
h
i
2U0 2
−1


3
dρρ
exp
−
ρ
kB T
0
T
kB T
U0
2
2
h
i =
ρrms ≡ hρ i ≃ R ∞≃ 3.0 ×
2U0 2
2U0
µK
Erec
dρρ
exp
−
ρ
kB T
0
R∞

(2.31)

with ρ ≡ r/w0 , and kB × 1 µK ≃ 6.0 Erec . Typical transverse temperatures are 2 − 4 µK,
resulting in ρrms ≃ 0.2 − 0.3. Longitudinal temperatures can be as low as 1 µK, when
the sample is laser-cooled in the optical lattice. In this regime, the lowest bands are notsignificantly broadened, but their energy is strongly dependent on ρ.
The longitudinal Wannier states are thus parametrized by ρ and we can find a tunnelingrate for each band and each transverse position. A thermal average over ρ will give anorder-of-magnitude estimate for the tunneling rate. Since we do not have a full model forthe tunneling, we give upper and lower bounds on the thermally averaged tunneling ratefrom the following considerations.
An upper bound on the tunneling rate is given by the lattice band structure in the absenceof gravity. Fig. 2.6(b) shows the radially varying tunneling rate without gravity as solid
colored lines. A lower bound on the tunneling rate is given by the WKB approximation
Eq. (2.13), since it completely neglects the effect of many sites, but handles the shape cos2
shape of the longitudinal potential. The results from the WKB calculation are shown as thedotted colored lines in Fig. 2.6(b). The dashed black line indicates ∆g . A full model for thetunneling rate should smoothly interpolate between the WKB expression (valid for Γ ≪ ∆g )and the lattice structure without gravity (valid for Γ ≫ ∆g ).
A thermal average over the radial coordinate is shown in panel (c). Note that the first threelongitudinal bands have a bandwidth that is always much smaller than ∆g , indicating thattunneling is negligible even in a thermal sample. For typical lattice depths and longitudinaltemperatures of 1 µK, more than 99% of the population resides in the lowest longitudinalband. We conclude that tunneling is not an issue in a vertical one-dimensional lattice attemperatures of a few µK.

35
0
1
-20
0.01

10-4

GErecEErec

-40
-60
-80

10-6

-100

10-8

-120
0.0

0.2

0.4

0.6

0.8
rw0

1.0

1.2

10-10

1.4

0.0

0.2

0.4

0.6

0.8
rw0

1.0

1.2

1.4

(a) Radial energy structure and band widths with-(b) Radial band widths without gravity (solidout gravity. The dashed line indicates the adia-lines) and WKB tunneling rate (dotted lines).

batic longitudinal lattice depth as the transverse

The dashed line indicates the gravitational en-

coordinate r varies.

ergy shift ≃ 0.79Erec between lattice sites for a
fully vertical lattice.

1

XG\Erec

0.01

10-4

10-6

10-8

10-10

0

2

4

6

8

10

Transverse temperature HΜ KL(c) Thermal average over radial direction of panel(b). The full trap depth is U0 /kB ≃ 20 µK.

Figure 2.6.: Adiabatic longitudinal band structure in a transverse Gaussian beam potential
22

U0 7→ U0 e−2r /w0 , with U0 = 120Erec . The tunneling rate in the Wannier-Stark lattice
should interpolate between the band structure without gravity and the WKB tunnelingaround the gravitational splitting between lattice sites.

36
Note that transitions between neighboring Wannier-Stark states can also be driven by
the spectroscopy laser [56], but that the amplitude of these transitions becomes negligible
already at lattice depths of a few recoil. Recent experimental work shows control of tunnelingdynamics in a Wannier-Stark ladder by probing these transitions, but at lattice depths of≤ 4Erec [65].

2.7. Single-site potential
As shown in the last section, we can ignore tunneling dynamics in the vertical one-dimensionallattice. However, the lattice band structure is still important, since the energies of the
Wannier-Stark states are given by the lattice without gravity. In this Section, we will develop
a simplified model of the single-site potential to finally describe the optical sideband spectra
we observed.
We approximate the lattice potential Eq. 2.11 around an antinode of the cos2 term up tosecond order in r and fourth order in z
U (z, r) ≃ U0


2 2 κ4 4 2κ2 2 2−1 + κ z + 2 r − z − 2 z r ,
w0
3
w0
2 2

(2.32)

where the gravitational shift is absorbed into an offset of the z coordinate. Treating thequartic distortion and the r-z coupling term in first order perturbation theory for harmonic
oscillator states |ni gives an energy spectrum
En /h = νz (nz +1/2)+νr (nx +ny +1)−

νr
νrec 2(nz +nz +1)−νrec (nx +ny +1)(nz +1/2), (2.33)
2νzwith recoil frequency νrec = Erec /h and longitudinal and transverse trap frequenciesνz = 2νrec
s
νr =

r

U0
Erec

U0
.
mπ 2 w02(2.34)

Measuring both trap frequencies determines the trap parameters U0 and w0 completely. The

37
number of states in the trap is approximately given by Nz Nr2 , with
r
U0
U0
Nz ≃
=
,
hνz
4Erec
(2.35)
νz
Nr ≃ Nz .
νrTypical values in our experiment are νz ≃ 80 kHz and νr ≃ 450 Hz, giving Nz (Nr ) ≃6(1000). As shown in the last Section, the longitudinal trap frequency changes with the radial
coordinate. Here, this relation is encapsulated in the r2 z 2 coupling term. The frequency gap
between longitudinal states is
γz (n) ≡ (Enx ,ny ,nz +1 − Enx ,ny ,nz )/h = νz − νrec (nz + 1) − νrecνr(nx + ny + 1).
νz(2.36)

We will use the single-site approximation in the next Section to describe the spectralfeatures in absorption spectroscopy of particles trapped in the one-dimensional optical lattice.

2.8. Sideband spectra in a one-dimensional opticallattice
We finally arrived at the important quantity γz (n) that describes the spectroscopic featuresin a one-dimensional optical lattice and can model the equivalent of the harmonic confinement
features in Fig. 2.1. Two example absorption spectra are shown in Fig. 2.7.
The first spectrum (red squares) represents a sample loaded directly from the 1 S0 -3 P1
MOT into the optical lattice without any further cooling. The second spectrum (blue circles)
represents the same situation, but with longitudinal sideband cooling and transverse Doppler
cooling applied to the sample before spectroscopy. Several features are evident. As in Fig. 2.1,
there is a central sharp carrier transition as well as a red and a blue first order sideband.
The relative height of the blue and red sideband is an indication of the temperature along
the probe direction (see Section 2.2). However, the sidebands are not symmetric, but slope
towards the carrier, while maintaining a sharper edge that faces away from the carrier.
In this Section, we will build a detailed model of the spectroscopic features in Fig. 2.7 toextract important system parameters such as longitudinal and transverse temperatures by

38

0.5

Excited state fraction

Carrier
0.4
0.3

Blue sb.

0.2

Red sb.

0.1
0

−80

−60

−40

−20
0
20
Detuning (kHz)

40

60

80

Figure 2.7.: Example absorption spectra in the one-dimensional optical lattice probed along
the lattice axis with a Rabi pulse time of 80 ms. Three features are evident: a central
carrier transition, where the longitudinal motional state nz is unchanged, and a red (blue)
sideband where nz is reduced (increased) by one. The data points in red represent a sample loaded from the 1 S0 -3 P1 MOT. Data points in blue show the change in the spectralfeatures after the sample is cooled in-trap using longitudinal sideband cooling and transverse Doppler cooling. Fits to the data (solid lines) are based on the single-site expansionof the lattice potential.

39
fit. Based on the model, we will also be able to explain spectroscopy on the carrier transition
in detail.
The sideband features can be well understood in the single-site potential picture derived
in the previous Section. From Eq. 2.36, we see that the first-order longitudinal sidebands
nz → nz ± 1 will appear at different detunings from the carrier for each nz . In addition, eachlongitudinal sideband depends on its radial state through the r2 z 2 coupling term. Even if only
the nz = 0 state is populated, the sideband spectrum will not be Lorentzian as in a harmonic
trap, but will be smeared out towards the carrier since γz (n) decreases with increasing nr .
The sharp sideband edge indicates the highest γz (n) at the lowest n in the sample. The
position of the sharp edge thus lets us determine γz (nx = ny = nz = 0) = νz − νrec precisely.
Since the sidebands are still well separated from the carrier, we will derive a lineshape expression valid for the blue sideband first. Similar to Eq. 2.10, we assume a thermal distribution with transverse temperature Tr and find the thermally averaged excited state population
for the nz -component of the ℓ-th order longitudinal sideband asPeℓ (nz , Tr , ∆, t) =

1 − qr1 − qr1+Nr

2 X
Nr X
Nr

nx =0 ny =0

qrnx qrny ×(2.37)2pt|Ωℓ (n)|2|Ωℓ (n)|2 + [∆ − ℓγz (n)]2 ,sin22|Ωℓ (n)| + [∆ − ℓγz (n)]2with radial Boltzmann factor qr ≡ exp(−hνr /kB Tr ). Since there are many Rabi frequencies Ωℓ (n) contributing to the sum, we simplify the problem by cycle-averaging the timedependent Rabi oscillations and set sin2 (·) 7→ 1/2.
To allow for experimental imperfections, we allow for a slight angular mismatch ∆θ between the probe pointing k̂p and the lattice direction ẑ and for this discussion, we choose
the x̂ and ŷ axes in the radially-isotropic harmonic confinement such thatk̂p ≃ kp [ẑ +∆θ(x̂ + ŷ)].
2(2.38)The transverse extent of the probe beam is large and can be approximated by a cylindricallysymmetric function with respect to the net probe direction k̂p . The extent of the probe
beam wp is generally chosen to be much larger than the trap waist w0 . The resulting Rabi

40
frequency from Eq. 2.2 isΩ(n) = Ω0 hn|eikp ·x [1 + O(a2r /wp2 )]|ni,with transverse oscillator length 2πar ≡p(2.39)

h/(mνr ). For typical parameters, a2r /wp2 ≃ 10−4 ,and we conclude that the transverse shape of a cylindrically symmetric probe beam withlarge cross section cannot influence the spectroscopic results. The Lamb-Dicke parameters
become
1
ηz =
λp

r

h
2mνz

∆θ
ηx = ηy = ηr =
2λp

r

h
.
2mνr

(2.40)

Since the mismatch ∆θ is assumed to be small, we can approximate the Laguerre polynomials
in the Rabi frequency as in Eq. 2.8 and find a carrier Rabi frequency#"Y2Ωc (n) ≃ Ω0e−ηj /2 (1 − nz ηz2 )[1 − (nx + ny )ηr2 ](2.41)

jand blue (red) first-order longitudinal sideband Rabi frequencies√ 
nz Ωbsb (n) = Ωc (n)iηz nz 1 + ηz2 ,2
√nz + 1 2Ωrsb (n) = Ωc (n)iηz nz + 1 1 +ηz .
2(2.42)

Note that with the symmetric misalignment choice ηx = ηy ≡ ηr and that all transverse
motional quantum numbers appear as the combination nx + ny in Eq. 2.38. If we introduce
a radial quantum number nr and a corresponding degeneracy factor nr + 1, we can collapse
one of the sums in the equation and find the time averaged and radially averaged excitedstate population for the carrier ℓ = 0 and blue (red) sideband ℓ = +1 (−1):
2 X

Nr
1
(nr + 1)qrnr Ωc (nz , nr )2 fℓ (nz )1 − qrℓhPe (nz , Tr , ∆)it ≃,2 1 − qr1+Nr n =0 Ωc (nz , nr )2 fℓ (nz ) + [∆ − ℓγz (nz , nr )]2(2.43)

r

where we neglected fourth order terms in ηz inηz2 (nz + 1) ℓ = −1fℓ (nz ) = 1
ℓ=0





η 2 nℓ = +1z z(2.44)

41
On the shallow sideband slope, (∆/γz ≪ 1), the sum can be simplified further by realizingthat its main contribution arises from the term where ∆ = ℓγz (nz , nr ), which leads to a
relationship
nr + 1 ≃ −νz[∆ − γ̃z (nz )],νrec νr(2.45)

between radial quantum number and base longitudinal frequency gap γ̃z (nz ) ≡ νz − νrec (nz +1). If we approximate the radial sum by this main term, we find a simple functional formnz(∆) ≡ hPebsb (nz , Tr , ∆)it ∝
σbsb



∆
1−
γ̃z (nz )e−α[1−∆/γ̃z (nz )] Θ[γ̃z (nz ) − ∆],(2.46)

with α ≡ [γ̃z (nz )/νrec ](hνz /kB Tr ). The applicability of this approximation is ensured by
cutting off the the line shape at ∆ = γ̃z (nz ) with the Heaviside function Θ. For the relevant
parameters, this lineshape can be approximately area-normalized to three significant figureswith a prefactor α2 /γ̃z (nz ).
The longitudinal trap anharmonicity produces a blue sideband for each nz at a different
detuning. Assuming a thermal distribution in z with temperature Tz , we can describe the full
blue sideband shape as a Boltzmann-weighted superposition of area-normalized components
σbsb (∆) ≡Nz
Xe−Enz /kB Tz

!−1 N
z
X

nz
(∆),
e−Enz /kB Tz σbsb

(2.47)

nz =0

nz =0

where the longitudinal energy Enz /h = νz (nz + 1/2) − νrec /2(n2z + nz + 1) includes the quartic
distortion of the longitudinal trap as an approximation to the full lattice band structure in
Section 2.5. The red sideband lineshape is derived analogously.
A nonparametric method to extract the longitudinal temperature Tz from sideband spectra
can be derived by comparing the integrated areas underneath the first order blue and red
sidebands. Regardless of the details of the component lineshapes σ nz , the only difference
between the red and blue sidebands is that the Boltzmann weights are shifted according to
nz 7→ nz + 1. Since nz = 0 does not contribute to the red sideband, the ratio of sideband
areas isPNz −En /kB Tz
Re zd∆ σrsb (∆)
e−E0 /kB Tz
R= 1 − P Nz.
= PnNzz=1−Enz /kB Tz−Enz /kB Tz
d∆ σbsb (∆)
e
e
n =0
n =0
z

z

(2.48)

42
After extracting νz from the sharp sideband edge, and given a calibration measurement of νrand νz for a particular lattice laser power and waist, we can calculate the energy structure Enz
as shown previously. With this information, Eq. 2.48 can be solved numerically for Tz . Notethat comparing the sideband peak heights is generally not accurate since power-broadeningof the underlying individual components is dependent on the motional state through Ω(n).
The fits (solid lines) in Fig. 2.7 are based on fitting the sideband lineshapes using Eq. 2.47.
nz(∆) does not use the main-term approxiHowever, the underlying functional form for σbsb

mation in Eq. 2.46. For most applications, Eq. 2.46 is sufficient and easy to implement in a
fitting routine to extract the transverse temperature Tr from longitudinal spectra after fixing
Tz to the number extracted from the sideband area ratio. Note that the sideband lineshapeis not very sensitive to the value of ηr2 and thus ∆θ because the main term approximation
describes the overall shape fairly accurately and ηr2 appears only in Ωc (nz , nr ), which cancels
in the main term.

2.9. Carrier spectroscopy and effective misalignment
We have seen in the last Section that the sideband shapes are fairly insensitive to a possible
effective misalignment ∆θ between the probe beam and the lattice axis. However, it is clearthat ∆θ would introduce Rabi frequency inhomogeneity through Eq. 2.41.
Describing the carrier spectroscopy for non-zero ∆θ leads to simpler formulas if we rotate
the coordinate system such that the inhomogeneity is along a particular trap axis, say
kp ≃ kp (ẑ + ∆θx̂),
such that ηx = ∆θ
λpq

(2.49)h, and ηy = 0. For a particular nz , we retain the time dependence in2mνr

Eq. 2.43, set ℓ = 0, and let Nr → ∞:Pec (nz , ∆, t) =(1 − qx )

∞
X

nx =0

qxnx



p
Ωc (nz , nx )2t2
22Ωc (nz , nx ) + ∆.
sinΩc (nz , nx )2 + ∆2
2

(2.50)

This expression can be directly fit to carrier lineshapes and Rabi flopping data. However,for our purposes, it is instructive to expand the Rabi frequencies in terms of ηx2 to exhibit a43
simple dependence on the misalignment ∆θ. Care must be taken with expanding the Rabi
frequencies but leaving the sinusoidal behavior unchanged, and we findpΩc (nz , nx )2
t
2
2
2
sin
Ωc (nz , nx ) + ∆Ωc (nz , nx )2 + ∆2
2


 


2
p
∆
t
Ωc (nz )222
222≃ 1 − 2nx ηxsin
nx η x.
Ωc (nz ) + ∆ 1 −22
22Ωc (nz ) + ∆Ωc (nz ) + ∆2(2.51)

To describe carrier Rabi flopping in a simple fashion adequate for a fitting routine, we set∆ = 0. In this case, we obtain
Pec (nz , t) =∞X

(1 − qx )qxnx sin2 [φ(nz )(1 − ηx2 nx )/2]

nx =0

1 1 − qx qx cos[φ(nz )(1 − ηx2 )] − cos φ(nz )
= +
,
2
2
1 + qx2 − 2qx cos φ(nz )ηx2

(2.52)with2

2

φ(nz ) ≡ Ω0 e−ηx /2 e−ηz /2 Lnz (ηz2 )t.

(2.53)

If more than one longitudinal state nz is occupied, a Boltzmann-weighted superposition of
Eq. 2.52 is used as before. For zero misalignment ηx → 0, Eq. 2.52 reduces to completely coherent Rabi flopping sin2 φ(nz )/2. In the presence of misalignment, the Rabi fringes dephase
to 1/2 because the nx dependence introduces many different Rabi frequencies. With higher
transverse temperature Tr , more transverse states nx will contribute, and the Rabi frequency
inhomogeneity increases. Larger misalignment angle ∆θ increases the inhomogeneity in asimilar way, by introducing a larger change in Rabi frequency for the same nx . Note thatthe dephasing of the oscillations is a completely coherent process and that rephasing of theoscillations still occurs. However, because many nx contribute, the revivals will take many
seconds.
Experimentally, we obtain the excited state fraction by first measuring the populationin the ground state by recording fluorescence on 1 S0 -1 P1 . This process is destructive andremoves the ground state atoms from the lattice. After the measurement pulse, the excitedstate population is repumped back to the ground state via 3 P1 . These atoms are then measured again by fluorescence detection. The experimentally measured excited state fraction
is given by the ratio of excited state fluorescence counts over the sum of ground and excited44
state counts. The repump efficiency β ≤ 1 can vary, which causes dephasing of the Rabi
flopping to β/(β + 1) instead of 1/2.
Figure 2.8 shows the effect of transverse temperature and misalignment angle on the carrier
Rabi flopping. The top panel shows the excited state fraction as a function of Rabi pulse
time for sample temperatures of 1 µK (blue circles) and 3 µK (red squares), as determinedfrom sideband spectra and time-of-flight expansion. The Rabi oscillations are visible forabout 10 cycles in the cold sample, but decay quickly after 5-6 cycles in the hotter sample.
The data is fit with Eq. 2.53 and we obtain Ω0 = 59 Hz and 76 Hz, respectively. Both fits
result in a misalignment angle ∆θ ≃ 10 mrad.
In the lower panel, ∆θ is increased by misaligning the probe beam with respect to thelattice axis and the effect on the visibility is even more drastic than when increasing thetemperature. Fits with Eq. 2.53 give Ω0 = 55 Hz and ∆θ = 10 mrad for the least misaligned
case (blue circles). A fit to the red squares results in Ω0 = 56 Hz and ∆θ = 17 mrad,consistent with a geometrical estimate of ∆θ. For the largest misalignment (green triangles),the small ηx2 approximation used in deriving Eq. 2.53 starts to break down and the fit
looks much worse. However, the fit still predicts ∆θ = 40 mrad, again consistent withthe geometrical estimate.
Each data point in the figure is obtained by scanning out a full carrier spectrum for a given
pulse time and measuring the central fringe height. An example spectrum fit with Eq. 2.50
is shown in Fig. 2.9 for a probe time of 1.7 ms. In this case, the temperature is allowed tovary when fixing the misalignment to 10 mrad from Fig. 2.8. The resulting temperatures
Tz = Tr = 2.1(2) µK agrees well with the sideband method and time-of-flight expansion.
In our experimental apparatus, even in the experimentally best aligned case, the carrier
spectroscopy data is consistent with a misalignment angle ∆θ ≃ 10 mrad. From geometricalestimates, the alignment precision should be much be better than that. The angle ∆θ should
be interpreted as an effective misalignment that is due to the geometrical overlap between
the probe wave vector and the lattice shape. Both probe and lattice beams pass through thesame optical elements after being combined on a dichroic beamsplitter. The main suspect for45

1

Excited state fraction

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

0

10

20

30

40
50
Pulse time (ms)

60

70

80

0

10

20

30

40
50
Pulse time (ms)

60

70

80

1

Excited state fraction

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Figure 2.8.: Carrier Rabi flopping as the central peak excited state fraction in a lineshapescan for varying Rabi pulse time. The top panel shows the faster dephasing with anincrease in temperature from 1 µK (blue circles) to 3 µK (red squares). The bottom panel
shows similar behavior with an intentionally misaligned probe beam, where the blue circles
correspond to the most well aligned case (see text).

46

0.7

Excited state fraction

0.6
0.5
0.4
0.3
0.2
0.1
0

−1.5

−1

−0.5

0
0.5
Detuning (kHz)

1

1.5

2

Figure 2.9.: Carrier line shape for the best-aligned case. The data (blue circles) are fit with
Eq. 2.50 and the temperature is allowed to vary while fixing ∆θ = 10 mrad.
introducing the effective angular mismatch are aberrations caused by the vacuum viewportsthat both the probe and lattice beams traverse.
As we have seen above, the main result of both effective misalignment and temperature is
to increase the inhomogeneity in Rabi frequencies. Note, however, that the inhomogeneity
discussed here is Rabi frequency inhomogeneity due to variation of the atomic response
to a perfectly homogeneous optical field over the small extent of a single lattice site. To
characterize these kinds of effects in a source-independent way, we define a single-site mean
Rabi frequency Ω̄ and an rms spread ∆Ω byXΩ̄ =w(nx )w(nz )Ω(nx , nz ),nx ,nz∆Ω =sX

nx ,nz(2.54)

w(nx )w(nz )Ω(nx , nz )2 − Ω̄2 ,using the appropriately normalized Boltzmann weights w. The ratio ∆Ω/Ω̄ characterizes the
single-site inhomogeneity in a dimensionless way and we plot results for temperature andmisalignment induced inhomogeneity in Fig. 2.10. Typical values for this experiment are47
0.05-0.4 for the misalignment angles determined from the data in Fig. 2.8.

40∆Ω/Ω̄ Contours

+1.0
+0.9

1
0.5

∆θ(mrad)

+0.8

3

+0.7

1.5

0.2

+0.6

0.7

20 0.06

+0.5
0.3

+0.4

0.1

10
0.01

∆Ω/Ω̄

30

+0.3
+0.2

0.03

+0.1
1

2

3
T (µK)

4

5

6

0.0

Figure 2.10.: Contour plot of the single-site Rabi frequency inhomogeneity parameter∆Ω/Ω̄ when assuming an effective misalignment ∆θ along x and a homogeneous sample
temperature Tz = Tr = T . The dashed contours indicate points where Ω̄ becomes smaller
than ∆Ω, because the transverse Laguerre polynomials in the Rabi frequency expression
can become negative for sufficiently large nx .

To summarize: we have shown how optical spectroscopy of carrier and motional sideband
transitions in a vertical one-dimensional optical lattice can be used to extract importantsystem parameters. We have determined the full potential from measurement of sideband
frequencies, extracted information about the kinetic energy in all directions, and found a
way to characterize the single-site Rabi frequency inhomogeneity.

48
The inhomogeneity characterization in terms of ∆Ω/Ω̄ is useful because it allows a general discussion of inhomogeneity-induced effects. In particular, we will extensively use theinhomogeneity parameter to characterize inhomogeneity-induced optical clock shifts in thenext Chapter.

Chapter

3

Ultracold Collisions in LatticeSpectroscopy
e have seen in the previous Chapter that spectroscopy of many particles can enhanceW

the signal-to-noise of the measurement by using many atoms simultaneously tomeasure a spectroscopic feature. A major limitation to making the particles evolve equally
under the influence of the spectroscopy laser is that their response depends on the individual
atom’s motional state. This simple fact is in principle unavoidable if using fermionic particles
since their motional state cannot be the same if there is more than one particle per site.
The main reason for using fermionic particles for clock spectroscopy is that interactions
between identical particles (they only differ in their spatial wavefunction) are suppressed by
the Pauli exclusion principle. The suggestion of building an optical clock using Fermions arose
because one of the main limitations to the Cs frequency standard are interaction-induced
clock frequency shifts during the Ramsey dark time. In a Cs fountain clock, the atoms areprepared in a MOT and then are launched upwards through a microwave cavity where thefirst Ramsey pulse takes place. The atomic cloud slows under the influence of gravity andfalls back down through the same microwave cavity and a second Ramsey pulse interacts
with the atoms. The problem is that that the Cs atoms interact during this whole sequence.
In contrast, in an optical lattice clock, the atoms are confined to the sites of an optical lattice

50
and their motional state is an eigenstate of the trapping potential. The thought is then that
the Pauli exclusion principle should prevent the egregious clock frequency shifts that limit
the Cs fountain clock accuracy.
With the increasing performance of the Sr optical lattice clock, in particular after reaching
accuracies below 10−15 , our experiments have shown that the above picture is too simple.
When the single-site occupation number is varied by loading more or less atoms into theoptical lattice, we have found that the optical clock frequency experiences a shift proportional
to the atomic density.
How can we reconcile this experimental fact with the Pauli exclusion principle? Rememberthat the Pauli exclusion principle only holds for indistinguishable particles. The Sr atoms
in our experiments are optically pumped into a single spin state and their electronic state isprepared in either 1 S0 or 3 P0 . This means that the particles are initially indistinguishable
in their internal degrees of freedom. However, because their external degrees of freedom are
distinguishable (they are in different motional states), they react to a perfectly homogeneous
spectroscopic excitation in an inhomogeneous way. If we consider two atoms in different
motional states, their Rabi frequency will be slightly different. This difference makes theirevolution on the Bloch sphere different and the atoms slowly become distinguishable as the
spectroscopic excitation progresses. As soon as they are not completely indistinguishable
anymore, they are allowed to interact. This interrogation-induced distinguishability is fundamental to the spectroscopic process and will lead to density-dependent clock frequency
shifts.
Our experimental data [9, 46] has led to multiple theoretical models that try to relate theunderlying interactions to the spectroscopic results [47–50]. We will present an introduction to the many-body model developed by A. Rey and co-workers [47], since its generality
allows specialization to all other published models. The understanding gained from applying the model to explain the experimental data has led to rapid progress in understanding
and controlling the density shifts in both our Sr system and NIST’s Yb lattice clock [4],although the underlying interaction mechanics are very dependent on the atomic species

51
and the particulars of the system under consideration. The model has been generalized tobosonic atoms [66] and predicts the paradoxical fact that density shifts can be suppressedby increasing the interaction energy in a two-dimensional optical lattice. This suppression
has also been experimentally verified [51, 67] and work on measuring more of the importantinteraction parameters is in progress.
Here, we will limit the discussion to the results obtained in a one-dimensional opticallattice and present a general introduction to modeling the loss of indistinguishability and its
influence on spectroscopic lineshapes in Rabi spectroscopy.

3.1. Experimental procedure and results
For this experiment, all atoms were initially prepared in the same nuclear spin state and theexcited electronic state 3 P0 . The remaining ground state atoms were removed from the latticeby scattering blue light on the 1 S0 -3 P1 transition. This preselection via the spectroscopy light
results in a slightly more homogeneous sample by removing atoms with very different Rabi
frequencies.
The atoms are prepared in either one of the mF = ±9/2 magnetic sublevels in the presenceof a small bias magnetic field, resulting in a splitting of about 250 Hz between the ±9/2transitions. The spectroscopy light propagates along the lattice axis and is linearly polarizedalong the magnetic field axis and the lattice polarization [22]. To initialize the stabilization
procedure, the polarization alternates between ±9/2 on every cycle and a spectroscopic tracethat contains both transitions is built up by scanning the detuning inwards from the outeredges of both spectra. Each half of the initial spectrum is fit with a model lineshape and its
center, maximal amplitude and FWHM are extracted.
The clock laser is then frequency-stabilized to the clock transition; the stabilization procedure is shown schematically in Fig. 3.1. The sample is polarized to ±9/2 alternating every
two experimental cycles. Two points on each component lineshape are measured by detuning the clock laser by η FWHM from the component center. The choice of η determines

52
the final ground state fraction Pg targeted during each interrogation. Four values of Pg are
obtained at the four lock points indicated by the green circles in Fig. 3.1 for η = 0.5. The
four values are fed into two separate digital servos that estimate the component line centers.
The estimate for line width and peak excited state fraction are assumed to be constant. The
servos adjust the detuning steps such that all four lock points have equal Pg when measured.
The clock transition frequency is then defined by the average of the two component line
center estimates. In this way, the atomic response stabilizes the noisy clock laser to the atomic
reference and makes the clock frequency first-order insensitive against any mF -antisymmetricsystematic effects such as a varying bias field.

Figure 3.1.: The clock laser is stabilized to a spectroscopic lineshape by alternatively probing at a negative or positive detuning and comparing the final ground state fraction. The
sample is alternately polarized into mF = ±9/2 every two cycles. A separate servo foreach case equalizes the final ground state fraction. A virtual clock transition frequency isdefined by the average of both servoed line centers.

As we discussed in the introduction to this Chapter, the density-related clock frequency
shifts considered here arise from interactions between the atoms during the spectroscopic
pulse. The shifts are quite small (10−16 level) and a lot of averaging is necessary to determine
their magnitude accurately. The shifts were measured by modulating the atomic density

53
every 30 s for an experimental cycle time of ∼1 s which corresponds to a locking duty cycleof ∼4 s. The (virtual) Sr clock frequency was continuously compared against the NIST Ca
standard over the optical fiber link [31, 46] and the difference between consecutive segmentsof low and high density clock frequency data was averaged over long time scales.

By varying η, the locking point on the lineshape was varied. When scanning η in fineenough steps, the measurement determines the differential line shape asymmetry between
low and high density settings. It is important to note that the value of mF is assumed tohave no influence on the component lineshapes. Both clock states have no electronic spin
(J = 0) and the only difference between different magnetic sublevels is the orientation of
the nuclear spin with respect to the bias field. The collision process is determined by theinteraction of the electron shells. Because of the full outer electronic shells in the spinlessstates, the coupling between electronic spin and nuclear spin is extremely weak and it is very
unlikely that a collision event induces a nuclear spin flip. For these reasons, the value of mF
should have no influence on the scattering process and all component lineshapes look thesame.

The experiment was performed over a range of locking points η and for two sample temperatures. Each setting of η required at least a full day of continuous averaging against
the Ca standard. The resulting clock frequency shift data [46] at the reference density
ρ0 = 1 × 1011 cm−3 is shown in Fig. 3.2. We note two features: (1) the magnitude of theclock frequency shift decreases with locking points closer to the peak of the line profile; (2)the magnitude of the shift increases with increasing temperature.

In the remainder of this Chapter, we will develop a model that describes the spectroscopic
response of N particles in the presence of interactions. Using the N -particle model, we
can then make predictions about the relative influence of interactions and Rabi frequency
inhomogeneity on the lineshapes.

Clock frequency shift @ ρ0 (Hz)54

2
T = 3µK
T = 1µK

1.5
1
0.5
0
−0.5
0.1

0.2

0.3
0.4
0.5
0.6
0.7
Final ground state fraction

0.8

0.9

Figure 3.2.: Clock frequency shift [46] as a function of locking point on the spectroscopiclineshape when varying the single-site density. The shift decreases when locking closer tothe top of the peak and the shift increases for higher sample temperature. The data arescaled to the reference density ρ0 = 1 × 1011 cm−3 .

55

3.2. Spin model
In this Section, we will derive an N -particle pseudo-spin model that can be used to describe the spectroscopic process within a single lattice site. The derivation here is basedon the theory by A. Rey [47], which can be extended to bosonic atoms [66] and has beensuccessfully used to model the s-wave collisional dynamics in our system for both one- andtwo-dimensional optical lattices [51, 67]. More recently, the theory has also been adapted todescribe p-wave collisions in a Yb optical lattice clock [68].
For nuclear-spin-polarized fermionic atoms in a single-site, the general description requiresa fully many-body Hamiltonian [47]


X Z
1 2
3
†
H=
d r Ψ̂α −∇ + V (r) Ψ̂α2mα∈{e,g}
Z
Z
h
i
~Ω
~ωeg
3
d r (ρ̂e − ρ̂g ) −d3 r Ψ̂†e e−i(ωL t−k·r) Ψ̂g + h.c.
+
2
2
Z
+ u−
d3 r ρ̂e ρ̂g ,
eg

(3.1)

that includes both ground (g) and excited (e) electronic states. Here, the field operator Ψ̂†αcreates a fermion in electronic state α, and the density operator ρα ≡ Ψ̂†α Ψ̂α measures theatomic density in state α. The first term describes the kinetic energy within the single-site
potential V , which we assume to be the same for both electronic states. The second and thirdterms describe the energy splitting ~ωeg and the bare Rabi frequency coupling Ω0 between
the clock states with laser frequency ωL . Here, Ω0 is defined as the Rabi frequency in theabsence of any motional effects. The final term introduces the fermionic s-wave interactions
2 −between ground and excited state atoms via u−eg = 4π~ aeg /m. The singlet scattering lengtha−eg describes the energy shift of two atoms in an antisymmetric superposition of electronic
states. For bosonic atoms, more interaction terms would need to be included [66].
To specialize H to the situation at hand, we expand the field operators in terms of fermioniccreation (annihilation) operators c†α,n (ĉα,n ) that create (destroy) a particle in electronic state
α and motional state n. We find
Ψ̂α =

X
n

ĉα,n φn (r),

(3.2)

56
using motional state eigenfunctions φn . In Chapter 2, we found that the longitudinal motional wavefunctions are well described by the cut-off Mathieu functions and that the transverse degrees of freedom can be modeled as a degenerate two-dimensional harmonic oscillator.
We assume separable wave functions along the trap axes x̂i and writeYφn (r) =ψni (xi ),

(3.3)

i

An important limitation to the applicability of this operator expansion is that we haveassumed particles to be in particular motional eigenstates. The eigenenergy picture is notnecessarily valid when considering spectroscopic time scales shorter than typical trap oscillation periods. Here, a typical spectroscopy time scale is 80 ms and a typical longitudinal(transverse) oscillation period is 12 µs (2 ms). For time scales ∼2 ms, we cannot consider theatoms to be in a transverse motional eigenstate and wavepackets of many motional states
would have to be considered. In this regime, any inhomogeneity-induced collisions should
also be suppressed which can be explained by a local picture. The atoms are effectively
confined to a volume determined by their velocity and the spectroscopy pulse time. If thisvolume is significantly smaller than the trapping volume, the excitation process becomes
more homogeneous. No collisions can occur since the atoms cannot travel far enough toencounter collision partners that have been excited in a slightly different way. Here, we will
focus on long pulse times, use motional eigenstates and neglect to describe such effects. However, these effects would become important when trying to describe Ramsey spectroscopy
with short pulse times such that interactions only have to be considered during the Ramsey
dark time.
Using the operator expansion, we find a simplified Hamiltonian under the rotating wave
approximation in a rotating frame at the laser frequency ωL
X ~Ωn
X
X

En ĉ†α,n ĉα,n −ĉ†e,n ĉe,n +H = − ~δĉ†g,n ĉe,n + h.c.
2
n
α,n
n
X
An1 ,n2 ,n3 ,n4 ĉ†e,n1 ĉe,n2 ĉ†g,n3 ĉg,n4 ,
+ u−
eg

(3.4)

n1 ,n2 ,n3 ,n4with laser detuning δ ≡ ωL − ωeg , and motional-state dependent Rabi frequency Ωn as
in Eq. 2.6. The interaction term is proportional to the number of ground and excited state57
atoms and the probability amplitude of a given collision process is given by the wave function
overlap between the motional states involved
An1 ,n2 ,n3 ,n4 ≡

YZ ∞
i

−∞

dxi ψn∗ 1 (xi )ψn2i (xi )ψn∗ 3 (xi )ψn4i (xi ).
i

i

(3.5)

Since the probability amplitude is an overlap integral over different motional states, we
can immediately conclude that the dominant contributions will arise from terms where thenj are as similar as possible. However, there are still many contributing terms, and it isnecessary to simplify the sum considerably. The complexity of the collision term for N ≥ 2
is the main limitation in modeling the 1D optical lattice. However, a detailed discussionof the assumptions made will lead to a simpler model that contains all other density shiftmodels in the literature as limiting cases [9, 48, 50].
If we require strict energy conservation for elastic collisions, the energy of the createsparticles 1 and 3 has to be equal to the destroyed particles 2 and 4. Thus the motional state
indices nj in the interaction term are constrained by
En1 + En3 = En2 + En4 .

(3.6)

We consider the collision process as a simultaneous motional index conversion
n2 7→ n1 ,
4

(3.7)

3

n 7→ n ,and would like to limit the allowed index mappings to simplify the collision term. We makethe following assumptions(i) Although the anharmonicity along the transverse directions is small for typical temperatures, the full trap potential is anharmonic along each axis (see Section 2.7). The
anharmonicity suppresses collision processes that conserve energy by changing two
quantum by forbidding simultaneous change of two quantum numbers within index
vector nj to conserve energy. In other words, processes, where one particle reduces
its motional quantum number and the other particle increases its motional quantum

58
number according to(n1k = n2k + m ∧ n4k = n3k − m) ∀ k, m

(3.8)

are forbidden.
(ii) We assume that the degeneracy in the transverse degrees of freedom is lifted and that
there are no accidental degeneracies such that
En 6= Em ∀ n, m.

(3.9)

Under this assumption, the motional quantum numbers n uniquely determine theenergy of a state.
(iii) We assume that only the full motional quantum number vector can be exchanged
during a collision. Processes that only swap the motional state along one axis are
forbidden.
Within these assumptions, the only allowed collision processes aren2 = n 1 ∧ n4 = n3 ,(3.10)

n2 = n 3 ∧ n4 = n1 ,which conserve the number of particles occupying each mode n.
Since we consider fermionic particles, initially, there is only particle in each occupied mode.
Under our assumptions, this fact does not change during the spectroscopy process whichallows us to parametrize the Hamiltonian by a fixed set of motional states. The population
dynamics can then be calculated within that set of modes and a final description of a thermal
sample can be obtained by thermal averaging over the motional modes considered. We
conclude that the drastic simplifications in the collision term are necessary to separate themotional dynamics from the electronic degrees of freedom in the presence of interactions.
The interaction term becomes
u−
eg

X

n6=m


An,n,m,m ĉ†e,n ĉe,n ĉ†g,m ĉg,m + An,m,m,n ĉ†e,n ĉe,m ĉ†g,m ĉg,n .

(3.11)

59
Using the symmetry of A and defining the per-mode number operator N̂α,n ≡ ĉ†α,n ĉα,n , wecan simplify this term to


X
u−
eg
†
†
†
†
An,n,m,m N̂e,n N̂g,m + N̂e,m N̂g,n + ĉe,n ĉe,m ĉg,m ĉg,n + ĉe,m ĉe,n ĉg,n ĉg,m . (3.12)
2 n6=m
We introduce a per-mode pseudo-spin operator S n with components
1
Sn1 ≡ (ĉ†e,n ĉg,n + ĉ†g,n ĉe,n ),2
1
Sn2 ≡ (ĉ†e,n ĉg,n − ĉ†g,n ĉe,n ),2i
1
1Sn3 ≡ (ĉ†e,n ĉe,n − ĉ†g,n ĉg,n ) = (N̂e,n − N̂g,n ).
22(3.13)

Noting that N̂e,n + N̂g,n = 1 and using the fermionic anticommutator relations ĉj,n ĉ†k,m +ĉ†k,m ĉj,n = δj,k δn,m we obtain
1
1
1
3
Sn3 Sm− 1 = (N̂e,n − N̂g,n )(N̂e,m − N̂g,m ) − (N̂e,n + N̂g,n )(N̂e,m + N̂g,m )4441= − (N̂e,n N̂g,m + N̂g,n N̂e,m ),2
1
1
2= − (ĉ†e,n ĉe,m ĉ†g,m ĉg,n + ĉ†e,m ĉe,n ĉ†g,n ĉg,m ).
Sn1 Sm+ Sn2 Sm2

(3.14)

This allows us to write the interaction term as a spin-spin interaction in the electronic
pseudo-spins
−u−
eg

X

n6=m

An,n,m,m (S n · S m − 1/4).

(3.15)

Since the motional quantum number distribution is conserved, the kinetic energy term only
introduces a constant energy shift
X

En ĉ†α,n ĉα,n =

α,nX

En (N̂e,n + N̂g,n ) =

n

X

En ,

(3.16)nand can be omitted.
We finally find a simplified N -particle pseudo-spin Hamiltonian that is parametrized by
the initial motional state configuration {n1 , . . . , nN } under the collision term assumptions
given in the above derivation. The Hamiltonian is
H/~ = −δ

X
n

Sn3 −

X
n

Ωn Sn1 −X

n6=m

Un,m (S n · S n − 1/4),(3.17)

60with Un,m ≡ u−eg An,n,m,m /~. Note that this Hamiltonian is basically a Heisenberg Hamiltonian used to model spin dynamics in a solid state system:
H = −gµB B ·

X
i

1 X
Si − J
Si · Sj .
2(3.18)

hi,ji

Here, the set of occupied motional modes represents the sites of the Heisenberg lattice.
The system here has more inhomogeneity than Eq. 3.18, since the “external field” B is
site-dependent. In addition, the interaction term J becomes site-dependent and we allow
longer-range interactions than between nearest neighbors. However, we can still define ametric of nearness between the motional modes, since the interaction term Un,m decays
Q
quickly according to j |nj − mj |−1 [51]. Hamiltonians of the general form of Eq. 3.17 havebeen extensively studied in condensed matter physics to calculate the magnetization in solidstate materials under the influence of external and internal fields. For instance, a classic
paper by Holstein and Primakoff [69] discusses the eigenvalues and solutions of an even moregeneral Hamiltonian than Eq. 3.17 for a large number of sites. The Hamiltonian Eq. 3.17
is also very similar for the one used in describing spectroscopy of nitrogen-vacancy color
centers in diamond that couple to a background spin bath [70]. For our purposes, we cansimulate the dynamics of Eq. 3.17 directly for small N , since typical atom numbers per site
are small. For larger samples, a solid-state type approximation method might become more
appropriate.
We can gain a simple understanding of the pseudo-spin dynamics produced by H by letting
the Rabi frequencies Ωn 7→ Ω be homogeneous and neglecting the interactions Un,m 7→ 0. If
we introduce a total spin operatorS≡
we findX

Sn,(3.19)nH/~ = −δS 3 − ΩS 1 ≡ −V · S,

(3.20)

with V ≡ (Ω, 0, δ)⊤ . In the Heisenberg picture, the expectation values of the total spin
followhṠi = −ih[S, H/~]i.

(3.21)

61
Using the commutation relations for the components of angular momentum operators [S k , S ℓ ] =P
i~ m ǫkℓm S m [71], we find the equations of motion for an N -particle Bloch vectorhṠi = −V × hSi(3.22)

which describes a simple rotation of hSi around axis V . The rotation conserves the lengthof the Bloch vector |hSi| = N/2.
The same result is obtained when considering the Hamiltonian in Eq. 3.20 as a sum of N
identical single particle Hamiltonians. Each single-particle pseudospin evolves in the same
way within its own Bloch sphere and their sum evolves in the same way. If we allow a Rabi
frequency inhomogeneity again, each particle will evolve within its on Bloch sphere according
to its Rabi frequency Ωn . Each individual evolution is norm-conserving, but the ensembleof Bloch vectors dephases and rephases according to the distribution of Rabi frequencies.
Summing the individual Bloch vectors produces a total Bloch vector that exhibits periodic
changes in length while it rotates on the total Bloch sphere. These inhomogeneous dynamics
are exactly what we derived in Sec. 2.9 and can be qualitatively characterized by the mean
i1/2
h
PN
PN
2
−1
−1
Ω̄ ≡ Nof thej=1 (Ωnj − Ω̄)
j=1 Ωnj and standard deviation ∆Ω ≡ (N − 1)
Rabi frequency distribution. Note that the Rabi frequency distribution moments here are

not defined with respect to a thermal distribution, but with respect to the set of initially
populated motional modes under consideration.
If we limit ourselves again to homogeneous Rabi frequencies and homogeneous interactions,another interesting limit of the dynamics can be explored. We let
H/~ = −V · S − U S · S = −(V − U S) · S.

(3.23)The interaction thus acts like an additional external field that depends on the current spinstate. If we replace the self-interaction term with the solution for U = 0, we obtainH/~ ≃ −V · S − U (V × hSi) · S.

(3.24)In the Heisenberg picture, this iterative approximation leads tohṠi = −V × hSi − U (V × hSi) × hSi,

(3.25)

62
which is known as the Landau-Lifshitz equation for the magnetization of a solid under the
influence of an effective magnetic field V in the presence of damping (proportional to U ).
This equation was first introduced phenomenologically and includes the simplest nonlinear
term that leads to damping of the magnetization [72]. The Landau-Lifshitz equation is
widely used to model the ferromagnetic response of solids [73, 74]. We conclude that our
problem at hand is very similar to what happens in an isolated magnetized domain of a solidwhen a magnetic field is applied.
Including inhomogeneous interactions makes algebraic calculations significantly more difficult, but for low N , the spin Hamiltonian is still relatively easy to simulate using numericalmatrix calculations. Matrix representations of the spin operators are easily obtained bytensor products1
1
Snj k = I2 ⊗ · · · ⊗ I2 ⊗ σ j ⊗ I2 ⊗ · · · ⊗ I2
|
|
{z
} 2
{z
}
k−1

j

S =

N
X

N −k

(3.26)

Snj k

k=1

where I2 is the 2-dimensional identity matrix and σ j is the j-th Pauli matrix. Using these 2N dimensional matrices to represent H makes it immediately obvious that simulating the full
evolution of more than a few particles with inhomogeneous Rabi frequencies and interactions
is hard. Note that the Hamiltonian should only be used with a pure pseudo-spin polarized
initial stateN|ψi0 = ⊗N
j=1 |gi or ⊗j=1 |ei,

(3.27)

which guarantees the assumptions about motional mode populations that led to the spin
Hamiltonian. In the spin-matrix representation, the polarized states correspond to the first1

Kronecker products of matrices can be calculated with kron in Matlab and KroneckerProduct in Mathematica, for example. More optimized object-oriented representations of tensor products can be foundin specialized toolkits such as the Matlab Quantum Optics toolbox (http://qwiki.stanford.edu/
index.php/Quantum_Optics_Toolbox) or C++QED (http://www.uibk.ac.at/th-physik/qo/research/cppqed.html)

63
and last entry in the state vector. The propagator is a simple matrix exponential
U (t − t0 ) = exp[−iH/~(t − t0 )](3.28)and the excited state fraction is given by
N
I Ni
2 2
N
= N −1 [U (t − t0)ψ0 ]† [S 3 + I2N ][U (t − t0)ψ0 ]2

Pe ≡ N −1 hNe i = N −1 hS 3 +(3.29)

Thermal averaging of Pe is done by averaging results for different sets of initial motionalQ−Enj /kB T
modes {nj } according to their Boltzmann factor N. To model a realistic samj=1 eple, we also have to average the results of calculations over a distribution of N over theoccupied lattice sites.

3.3. Spin model fit to experimental data
The experimental data in Fig. 3.2 was fit with the N -particle spin model derived in the last
Section by A. Rey and coworkers [47]. The result of thermal averaging for T = 1 µK (3 µK)
is shown in Fig. 3.3 as the pink (blue) shaded region. The uncertainty in the fit is mostly
due to allowing a 10% variation in the pulse area. Each data point consists of at least one
full day of averaging against the NIST Ca clock. A conservative estimate of the clock laser
intensity variation on long time scales results in an uncertainty of about 10% on the π-pulse
condition. For the fits here, the singlet scattering length is set to a−eg = 200 a0 , which isconsistent with the unitarity limit given by the thermal wavelengthq
λT≡ ~/ 2πmSr kB (T + Tzp )2π(3.30)≃ 221 a0 @ T = 3 µK

where Tzp ≃ 3.5 µK corresponds to the ground state energy of the single-site potential.
The inset of Fig. 3.3 shows the 1 µK Rabi flopping data from Fig. 2.8 and the dotted purple
line corresponds to the fit from the same figure using ∆Ω/Ω̄ = 0.05. The solid blue curve
is a fit using the thermally averaged N -particle model. To match the data when including

64

Figure 3.3.: (Figure from Ref. [47]). The shaded regions indicate fits of the experimental
data in Fig. 3.2 with the thermally averaged N -particle spin model. The singlet scattering
length is set to the unitarity limit a−eg = 200 a0 and the uncertainty region results fromallowing the π-pulse condition to vary by 10%. The inset also shows a fit of the Rabi
flopping data in Fig. 2.8 with the full N -particle spin model. The dashed black curve is−the result for setting a−
eg = 0 and the solid blue curve is the fit for aeg = 200 a0 whenusing ∆Ω/Ω̄ = 0.15. The dotted purple line results when extracting ∆Ω/Ω̄ = 0.05 fromthe theory in Sec. 2.9 and setting a−
eg = 0.

65
interactions, the Rabi frequency inhomogeneity has to be increased to ∆Ω/Ω̄ = 0.15. For
comparison, the black dashed curve shows the N -particle model with ∆Ω/Ω̄ = 0.15 when
the interactions are neglected. Note that all three fits agree almost perfectly during the first
Rabi cycle, because ∆Ω/Ω̄ is still small enough to take several cycles to develop a significant
effect.
We conclude that it is necessary to include interactions to fit carrier Rabi flopping in thepresence of Rabi frequency inhomogeneity. Note that the agreement between the data and thetheoretical model is fair. However, the large amount of averaging necessary to describe thespectroscopy makes it hard to make general statements. In particular, predicting the behaviorof the thermally averaged N -particle model from simulations of the 2-particle Hamiltonian
are not reliable. Predicting a zero crossing (or no zero crossing) from the two-particle model
as a function of locking point is too simplistic.
Even though the optical lattice spectroscopy has been made as clean as possible, thelarge number of degrees of freedom still requires a full model. Performing optical lattice
spectroscopy for optical frequency standards still requires a precise measurement of thedensity shift. This important systematic cannot be neglected, but we will show that thedensity shift can be suppressed by engineering the trapping potential.

3.4. Two particles
Although the two-particle model does not easily generalize to N particles, it is instructive
to consider the dynamics for N = 2 since several parameter regimes become obvious. The
matrix representation of H for N = 2 is




0 
 −2δ −Ω2 −Ω11−Ω2 U12 −U12 −Ω1 
H/~ = 

2 −Ω −U
U
−Ω
 1
12
12
2


0
−Ω1 −Ω2 2δ

(3.31)

66
where the subscripts j indicate motional modes nj and we have used Un,m = Um,n . The
matrix is written in the tensor product basis |g1 g2 i, |g1 e2 i, |e1 g2 i, |e2 e2 i. We see that theinteractions only manifest in the e − g subspace. The tensor basis is the most convenient fornumerical calculations, but for our purposes, H becomes much more informative in a singlet
and triplet basis of symmetrized and antisymmetrized pseudo-spin superpositions. The basis
transformation is defined by


0
0
1

0
0
0

P =
√
0 1/ 2 1/√2√√
0 1/ 2 −1/ 2



0

1

,
0


0

(3.32)

which rotates the e − g subspace into a symmetric and antisymmetric superposition |±i ≡√(|egi ± |gei)/ 2 and rearranges the Hamiltonian such that the triplet states |eei, |ggi, |+i(symmetric under particle exchange) form the top left block.


√
√
0
−Ω̄/ 2 ∆Ω/ 2 
 −δ

√ 
√
 0
2
−∆Ω/
2
δ
−
Ω̄/


H2 ≡ ~−1 P HP −1 = ,−Ω̄/√2 −Ω̄/√2

0
0




√
√
∆Ω/ 2 −∆Ω/ 20
U12(3.33)

where we have defined Ω̄ ≡ (Ω1 + Ω2 )/2 and ∆Ω ≡ (Ω1 − Ω2 )/2. The antisymmetric (underparticle exchange) singlet state |−i is the only state that interacts, as shown by U12 in thebottom right corner of H2 . This Hamiltonian was first considered in the context of opticalclock density shifts in Ref. [48]. As we have seen, it is not general enough to describe thermal
averaging over motional modes and sites with N > 2. As we will see, it is nevertheless animportant conceptual tool to understand the system dynamics.
From Eq. 3.33, we can immediately see that if the system is initially prepared in a pseudospin polarized state |eei or |ggi, the induced dynamics will be constrained to the noninteracting triplet manifold unless there is a Rabi frequency inhomogeneity ∆Ω that couples
to the interacting singlet. Unless there is non-zero Rabi frequency inhomogeneity, the interaction cannot influence the lineshape.

67
To simulate the influence of nonzero inhomogeneity and interactions for nonzero inhomogeneity and interactions, it is useful to rescale H2 by Ω̄. Figure 3.4(a) shows the total spin
dynamics for ∆Ω/Ω̄ = 0.4 and u = U12 /Ω̄ = 0.4 when δ/Ω̄ is scanned across the resonance.
The system is prepared in the |eei state at the south pole of the Bloch sphere and the evolution of the Bloch vector from t = 0 to t = π/Ω̄ is shown as a colored trace. The trace colors
are interpolated from red δ/Ω̄ = −2 (red detuning) to blue δ/Ω̄ = 2 (blue detuning). The
light green trace indicates δ = 0 and we see that even for two particles, the individual Blochvector dephasing can be significant. Panel (b) shows the resulting spectroscopic line shapefor t = π/Ω̄: the maximal excited state fraction is attenuated and the spectrum becomesskewed. Switching the sign of u mirrors the lineshape around δ = 0.
1.0

1.0
0.5

0.5
XS 3 \

0.0XS 3 \

0.0

-0.5

-0.5
-1.0
1.0

1.0
0.5

0.5
0.0
XS 2 \ -0.5

0.0
-0.5 XS 1 \-1.0(a) Total spin expectation value hSi on Bloch
sphere for detuning δ/Ω̄ ∈ (−2, 2). Red (blue)

-6

-4

-2

0

2

4

6

∆W

-1.0

(b) Spectrum of hS 3 i versus detuning for the same
conditions as in panel (a) at Ω̄t = π.

traces indicate the evolution of hSi for Ω̄t ≤ π
and negative (positive) values of δ.

Figure 3.4.: Two-particle spin model for ∆Ω/Ω̄ = u = 0.4, where hS 3 i = −1(+1) indicatesthat both particles are in the ground (excited) state.

The lineshape skewing is very dependent on the interplay of ∆Ω/Ω̄, u, and the pulse time.
The position of the locking points on the central fringe adds to the complexity. This complex
interplay makes it difficult to formulate general descriptions that do not require numerical68
simulation. The spectroscopic lineshape parameter-dependence becomes even more complex
when considering a Ramsey scheme. Ramsey pulses that are short enough such that interactions can be neglected require a separate description not covered by either the full spin-modelor the simpler two-particle model in this Section. In the presence of interactions, a Ramsey
sequence can easily be generated by chaining the spin-model propagators (Eq. 3.28). However, even long dark times do not simplify the resulting lineshape parameter dependence
significantly.
A full description of the spectroscopic lineshape in the presence of interactions remains
a difficult problem as long as all the parameters in H are on similar orders of magnitude.
We will show that the description becomes much simpler when the interaction parameter u
dominates over all other energy scales in the Hamiltonian.

3.5. Interaction-dominated regime
Another instructive way of looking at the two Hamiltonian is a dressed basis at the mean
Rabi frequency Ω̄. The dressed eigenstates can be obtained by diagonalizing H2 when letting∆Ω 7→ 0. As discussed in the last Section, the triplet manifold becomes decoupled from the√singlet state and the triplets are split by twice the effective Rabi frequency Ω̄2 + δ 2 . If welabel the dressed states by their original triplet labels, we find the situation schematicallyshown in Fig. 3.5(a). When admitting a small inhomogeneity again, the spin-polarized
dressed states |eei and |ggi can couple to the singlet via ∆Ω. The singlet |−i is split off
from |+i by the interaction energy U = U12 . Since all energy scales are on similar ordersof magnitude, the ∆Ω-induced dynamics become a detuning-dependent interference pattern
between all allowed transitions in the energy diagram. The interference mainly occurs within
the central fringe of the spectroscopic lineshape. This complex interplay is exactly why thelineshape and thus the density shift is so parameter-dependent.
The situation simplifies drastically if we allow the interaction energy U to become large
with respect to Ω̄ and ∆Ω. In the small detuning regime δ ≪ U , the singlet becomes

69(a) Dressed states and coupling when the inter-(b) When U is the largest energy scale, the dressed

action U ≪ Ω̄, ∆Ω is a perturbation. Transfer

triplets cannot couple to the singlet and U does

between the Ω̄-dressed triplets and the singlet

not influence the lineshape.

state in the perturbative regime resulting in a
modified lineshape.

1.0

XS 3 \

0.5

0.0

-0.5

-1.0
-10

-5

0

5

10

15

∆W

(c) Same as panel (b), but if we detune far enough,(d) Example spectrum for N = 2, u = 8, ∆Ω/Ω̄ =resonant transfer with ∆Ω between one of the

0.4. The interaction separates into its on spec-

triplets and the singlet results in a spectroscopic

tral feature and does not perturb the carrierfeature when δ = U .

spectrum.

Figure 3.5.: Dressed triplet states for N = 2 with mean Rabi frequency Ω̄ and singlet statethat is separated by the interaction energy U . Panel (a) shows the regime where U isa perturbation on the Rabi dynamics. Panel (b) indicates that a large interaction will
suppress transfer to the singlet, and thus suppress effects of U on the lineshape. Panel (c)shows that if U ≫ Ω̄, a detuning of δ = U will produce resonant transfer to the singlet
and result in a single well-separated interaction sideband. An example spectrum for case(c) is shown in panel (d).

70
separated from the triplets by a large energy gap, as shown in Fig. 3.5(b). Transitions
between triplets and the singlet are in principle still allowed, but their amplitude becomesstrongly suppressed by the energy separation as long as |U | ≫ ∆Ω. In this regime, thecoupling between triplets and singlet can be treated perturbatively for small detuning. In
other words: the influence of U on the central fringe is suppressed.
However, if we increase the detuning until the triplet splitting matches the singlet energy,
resonant transfer between one of the stretched triplets and the singlet becomes allowed. Thissituation is shown in Fig. 3.5(c). The Rabi frequency inhomogeneity allows transitions tothe singlet, even if ∆Ω ≪ Ω̄, U . The effect on the spectrum is shown in panel (d), where
we let U/Ω̄ = 8 and ∆Ω/Ω̄ = 0.4. The central fringe is symmetric and not influenced. At
δ = U , a separated interaction sideband appears, corresponding to resonant Rabi flopping
between a stretched triplet and the singlet. In this way, larger interactions can suppress theinfluence of interactions on the relevant parts of the spectroscopic lineshape.
The interaction energy U can be strongly enhanced in a two-dimensional optical latticewhere the atoms are confined to the lowest motional state in two axes. The overlap integrals
contributing to U become less suppressed and U can become the dominating energy scale in
the Rabi flopping dynamics. A density shift suppression by more than an order of magnitude
has been shown recently [51] in a two-dimensional optical lattice, and separated interaction
sidebands have been observed [67].

These new developments show that the presence of atomic interactions does not necessarily
put a hard limit on the number of atoms that can be used to enhance the spectroscopic
signal-to-noise ratio. The combination of precision measurement and ultracold many-bodysystems will allow the future development more precise and also more accurate optical latticefrequency standards.
On the other hand, many new proposals for quantum simulation with alkaline earth atoms
will have to rely on manipulation of the internal and external atomic degrees of freedom withhigh fidelity. The measurement, control, and understanding of the density shift demonstrates

71
that combining insights from different fields yields a better understanding of all of them. In
this spirit, we will apply the high precision of Sr optical lattice clocks to study fundamental physics by using the clock data to put limits on present day variation of fundamental
constants in the next Chapter.

Chapter

4

Variation of Fundamental
ConstantsM

any landmark physical experiments have succeeded in finding new physical phenomena, particles, or forces by testing accepted theories such as Newton’s theory ofgravitation. The surprising results from such experiments have led to improved theories ofphysics, such as general relativity. Within the last century, general relativity has become
one of the most well-tested physical theories. One of the main driving forces behind highaccuracy tests of general relativity are theories beyond the standard model, because theirscope can be experimentally constrained by null results from high-accuracy tests.
Fundamental constants determine the relative scale of different physical forces. For instance, the fine-structure constant α is the coupling constant between electromagnetic fields
and matter, and also sets the energy scale of electronic transitions in atoms. Since the equations of physics are parametrized by fundamental constants such as α, a natural question
to ask is whether they have to have particular numerical values such as α = e2 /(4πǫ0 ~c) ≃1/137, with electron charge e, vacuum permittivity ǫ0 , Planck’s constant h = 2π~, and speedof light c.
Such questions arise in the course of trying to describe all physical forces in a unified way
or in modeling the physics of the early universe. The experimental search for variations

74
in fundamental constants has gained much momentum from an analysis of astronomical
data from quasar spectra which indicated a non-zero fractional variation δα/α ∼ 10−6 oncosmological timescales [75], although later analyses showed a much smaller variation [76].
Limits on variation of α on geological timescales have also been extracted from analyses of thenatural fission reactor at Oklo [77]. Present-day limits on variations of fundamental constants
can be obtained from long-term absolute frequency records of atomic clocks, since atomic
transition frequencies are sensitive to fundamental constants such as α. A full review of thissubject is beyond the scope of this introduction, but many reviews [78–80] and extensive
lecture notes are available [81].
In this Chapter, we will analyze the international frequency record obtained from absolute
frequency measurements of 87 Sr optical lattice clocks to put limits on present-day drifts in
α and the electron-proton mass ratio me /mp [30, 82–85]. We also check the same data fora violation of local position invariance by testing for a coupling of fundamental constants tothe solar gravitational potential [30, 86–89].

4.1. Sensitivity constants
The energetic structure of an atom is determined by the interaction between many fundamental particles: quarks form the nucleons and nucleons bond via nuclear forces. The
electromagnetic charge Ze of the resulting atomic nucleus determines how many electrons
can be bound into electronic shells orbiting the nucleus. Almost all chemistry can thenbe understood as resulting from the energetic structure of this cloud of electrons. Most of
atomic physics and chemistry is based on the fact that the outermost (valence) electrons
determine an atom’s behavior almost completely. However, their quantized energy structure
is of course dependent on the full underlying many-body system.
Modern atomic structure calculations can model the electron cloud ab-initio. The resulting
valence electron energy levels relate the wavelengths of spectroscopic transitions in atoms
to the underlying physics and especially the fundamental constants. By continual checks

75
against experimental data across many atomic species, these calculations have become very
accurate over the last decades. Because of their accuracy, one can extract the sensitivity of
the resulting electronic structure to the value of the underlying fundamental constants.
Assume that we are interested in a dipole-allowed optical transition. In atomic units,
the transition frequency can be expressed via the atomic unit of energy, the Hartree energy
Eh ≃ 27.2 eV which is roughly twice the energy required to dissociate an electron from a
proton. The resulting expression for the atomic unit of frequency
ωopt ∝ Eh /~ = α2

me c2
,
~

(4.1)

exhibits the energy scale of electronic transitions as the product of the Compton frequencyof the electron me c2 /~ and α2 .
The notation here is slightly modified from the review article by S. Lea [80] to conformbetter to the notation used in our paper [8]. There is no standard notation for the sensitivity
constants and functions mentioned here. The confusing use of the SI value of the Rydberg
cR∞ to represent variations in frequency units is avoided, especially when talking about
frequency variations in the Cs standard. Since it is the current definition of frequency, there
can be no frequency variation of the Cs standard. Clock comparisons are either performed
against the Cs standard to obtain absolute frequencies or they are not. If they are not, thenthe result of a clock comparison cannot be represented in SI frequency units. Instead, theresult of a clock comparison must be converted to a dimensionless fractional frequency ratio.
Any absolute frequency measurement is a comparison of a frequency ω to the Cs frequency
ωCs and the measurement also returns the dimensionless value ω/ωCs . Only with the definition 1 s ≡ 9, 192, 631, 770 × (2π/ωCs ) can such a result be converted to a frequency in SI
units, but we emphasize that in reality all such experiments measure dimensionless frequency
ratios.
A fractional variation in measured dimensionless fractional frequencies can be expressedas the logarithmic derivativeδ(ω/ωCs )= ∂t ln ω/ωCs = ∂t ln ω − ∂t ln ωCs .
ω/ωCs

(4.2)

76
We now write the proportionality factor in Eq. 4.1 as a dimensionless function F ({ηk }),where {ηk } denominates the set of fundamental constants ηk determining the atomic structure:ω = F ({ηk }) × (units)(4.3)

ωCs = FCs ({ηk }) × (units).
A dimensionless variation in these frequencies is then seen to separate into a variation of F
and a variation of the units used in the atomic structure calculation∂t ln ω = ∂t ln F ({ηk }) + ∂t ln (units)(4.4)∂t ln ωCs = ∂t ln FCs ({ηk }) + ∂t ln (units).
from which we see that the units in the atomic structure calculation do not matter whencomparing against the experimental data:δ(ω/ωCs )= ∂t ln ω − ∂t ln ωCs = ∂t ln F ({ηk }) − ∂t ln FCs ({ηk }).
ω/ωCs

(4.5)

Any measured fractional frequency variation can thus be directly compared to the sensitivity
calculations from the atomic structure calculations (usually done in atomic units).
The numerical variation in atomic structure calculations is fit with a simple model
F ({ηk }) = const ×

Y c
ηkk ,

(4.6)

k

and numerical constants ck are extracted for the transitions of interest. In this parametrization,∂t ln F ({ηk }) = ∂tXln ηkck =kXk

ck ∂t ln ηk ≡

X
k

ck

δηk
.
ηk

(4.7)

The fractional frequency variation of a species j compared against the Cs standard thus
becomes a simple linear combination of the fractional variation of fundamental constants
δ(ωj /ωCs ) X jδηk=(ck − cCs.
k )
ωj /ωCs
ηk
k

(4.8)

In this way, we can extract information about δηk /ηk from absolute frequency measurements
since the sensitivity constants ck are known.

77
Also note that any other atomic clock comparison producing dimensionless frequency ratios
can be checked for variation in fundamental constants viaδηkδ(ωj /ωi ) X j=(ck − cik ).
ωj /ωiηkk(4.9)

4.1.1. Sensitivity of the 133 Cs clock transition
Which fundamental constants are important for the 133 Cs standard based on a hyperfine
transition? Hyperfine transition energies are given by the nuclear magnetic momentµN = g N µp = g N

e~
,
2mp

(4.10)

with nuclear g-factor gN , nuclear magneton µp , and with proton rest mass mp . A typicalatomic energy scale for a hyperfine transition is thus
ωhfs ∝ α2meµN m e c2
= gN
a.u.,
µB h
mp

(4.11)

where µB = e~/(2me ) is the Bohr magneton, and a.u. indicates the atomic unit of frequency
Eh /~.
The Schmidt model relates gN to the proton g-factor across many species [78, 80, 90, 91].
If we assume that these relations hold even when the fundamental constants are varying,
and that the proton g-factor itself is constant, we can isolate the electron-proton mass ratio
me /mp as one of the contributors to the sensitivity function F .
The main contributor to F is the fine structure constant α. For transitions between groundstate hyperfine levels, the contribution can be written as α2 Frel (Zα), where Frel summarizes
relativistic corrections that scale with the nuclear charge Z [80, 87, 91]. Other contributions
are small, and we can writeωhfs = const × α2 Frel (Zα)

me
a.u.,
mp

(4.12)

78
such thatme∂t ln ωhfs = ∂t ln Frel (Zα) + ∂t ln+ ∂t ln a.u.
mp∂ ln Frel (Zα) δα δ(me /mp )= 2+++ ∂t ln a.u.
∂ ln ααme /mpδ(me /mp )δα+ cme /mp+ ∂t ln a.u.
≡ cααme /mp(4.13)For the clock transition in 133 Cs, we find [92]cCs
α ≃ 2 + 0.83

(4.14)

cCs
me /mp = 1,
where the integral numbers are exact from the analytical powers of α and me /mp given
above.

4.1.2. Sensitivity of optical clock transitions
For optical transitions, only the relativistic corrections related to Zα contribute significantly,
and we find
ωopt = const × Frel (Zα) a.u.,(4.15)

such that∂t ln ωopt =∂ ln Frel (Zα) δαδα+ ∂t ln a.u. ≡ cα
+ ∂t ln a.u.
∂ ln ααα(4.16)For the 87 Sr 1 S0 -3 P0 optical clock transition we find [93]cSr
α = 0.06,(4.17)

which is comparatively small as shown in Table 4.1.

4.1.3. Sensitivity of 87 Sr absolute frequency measurements
In summary, measurements of the absolute frequency of the 87 Sr clock transition are sensitive
to variation in two fundamental constants: the fine structure constant α and the electronproton mass ratio me /mp . To shorten the notation in the following, we will write µ ≡ me /mp ,79Species

Transition

Wavelength (nm) cα

87

1

S 0 - 3 P0

698

0.062

Sr

171

Yb

1

S 0 - 3 P0

578

0.314

199

Hg, 201 Hg

1

S 0 - 3 P0

266

0.813

Al+

1

S 0 - 3 P0

267

0.008

171

Yb+

2

S1/2 -2 D3/2

436

0.881

199

Hg+

2

S1/2 -2 D5/2

282

-3.19127

Table 4.1.: Sensitivity of high-Q optical transitions to variation in α, calculated from
Ref. [93].
not to be confused with magnetic moments. We findδ(ωSr /ωCs )
Cs δα
Cs δµ
= (cSr+ (cSrα − cα )µ − cµ )ωSr /ωCsαµδαδµ= (0.06 − 2 − 0.83)+ (0 − 1)
α
µ
δα δµ
− .
= −2.77αµ(4.18)

87
Because cSr
Sr are a clean measurement
α is small, the absolute frequency measurements ofof the 133 Cs sensitivity to both α and µ. The 87 Sr standard can thus also act as an anchor
in comparisons against more sensitive species.

4.2. Global absolute frequency record
In the previous Section, we have shown that absolute frequency measurements can be sensitive to possible variations in fundamental constants. In particular, we are interested inanalyzing the absolute frequency record of the 87 Sr optical lattice clock for such variations.
Figure 4.1 shows all published absolute frequency data from our lab (red), the Paris group(green), and the Tokyo group (blue). The data is referenced to ν0 = 429 228 004 229 800 Hz.
Vertical 1-σ error bars indicate the total uncertainty in each optical frequency measurement
and horizontal error bars show the time period over which the measurement took place.

80
The last four measurements agree within 1.7 Hz [7, 42, 45, 94], and the agreement between
the JILA and Paris data [7, 45, 94] is on the 1 × 10−15 level, which approaches the Cs limit
and makes the 87 Sr 1 S0 -3 P0 optical clock transition the best agreed-upon optical frequency
to date.
Improving the absolute frequency measurement much beyond the uncertainty achievedin our last measurement [7] will be difficult without significant improvements in the Cs
standard and the dissemination of derived RF frequencies via hydrogen masers. The lastmeasurement in the graph was performed by remote comparison (see Ch. 1) between the 87 Srstandard in our lab and the primary US frequency standard, NIST F-1, at NIST Time andFrequency in Boulder. The measurement resulted in over 48 h of consecutive data, takenover a period of 60 h and was stopped when both Cs and Sr standards were limited by theirrespective and simultaneous comparison against the intermediate hydrogen maser [7, 12].
Direct comparisons of optical clocks can be performed to much higher levels of accuracy [31]and promise to give much more stringent limits on the variation of fundamental constants [95].
The figure also shows a linear and a sinusoidal fit to the frequency data. The fits were
performed by minimizing the weighted sum of errors
χ2 =

X
k

[νk − f (tk , p)]2σν2k + σt2k [f ′ (tk , p)]2

(4.19)

over the corresponding parameter vector p. The points in the data set and their error bars aredenominated by (tk ±σtk , νk ±σνk ) and the sum is over the whole data set. Here f is the fitting
function to model the variation in the frequency record and f ′ indicates its partial derivativewith respect to the independent variable t. We used a Levenberg-Marquardt algorithm [96]modified by Lybanon’s method [97] to find the minimum of χ2 . Quoted parameter errors are
1-σ and were obtained from the Hessian matrix of χ2 evaluated at the minimum [98]. The
parameter uncertainties could be reduced slightly by explicitly including the measurement
R t +σinterval 2σtk as a time average (2σtk )−1 tkk−σttk dτ f (τ, p) ≡ F (tk , p) and fitting the frequencykrecord with F to avoid the significant difficulty of handling errors in the independent variable
explicitly and correctly [99].

81

90

νSr − ν0 (Hz)

80
70
JILA
Paris
Tokyo

60
50
1/05

7/05

1/06

7/06
Date

1/07

7/07

1/08

Figure 4.1.: Global absolute frequency measurements of the 87 Sr optical lattice clock against
the Cs standard from our group, the group at LNE-Syrte in Paris, and the University ofTokyo group. The data is shown with respect to ν0 = 429 228 004 229 800 Hz. A linear
fit to the data is shown as a dotted line and a sinusoidal fit is shown as a solid line. Its
peak to peak amplitude is indicated by the shaded gray area.

The last four high-accuracy data points determine the fits due to their small error bars.
These fits will be used in the following Sections to limit the variation of fundamental constants. The linear fit can be used to check for linear drifts in the fine structure constant andthe electron-proton mass ratio (see Sec. 4.3). The sinusoidal fit limits coupling of fundamental constants to the gravitational potential (see Sec. 4.4).

4.3. Linear Drifts
If we assume that fundamental constants have been different at some time in the early universe, the simplest model is that they are changing over time. With present day experiments,
we are limited to year-scale time frames. The simplest check we can do is to approximate a82
linear variation in time t around the beginning of the measurement period at t0 :ηk (t) ≃ ηk (t0 ) + (drift rate)(t − t0 )δηk(drift rate)(t − t0 ),=ηkηk (t0 )(4.20)and try to put limits on the fractional drift rate using experimental data. By using absolute
frequency measurements of optical standards, we will be able to extract drift rates for α andµ, as shown in Eq. 4.18. We will refer to the fractional drift rates as δα/α and δµ/µ forthe rest of this Section, since that is their standard notation. The distinction between thefractional drift rates and more general variations is important since we will allow nonlinear
variations in the next Section. Under the assumptions above, we can write Eq. 4.8 as
δ(ωj /ωCs )δα δµj(t − t0 ).
(4.21)−= (cα − 2.83)
ωj /ωCsαµ
By fitting a linear function to a frequency record such as in Fig. 4.1, we can extract afractional frequency drift rate xj and compare it against the coefficient in Eq. 4.21. However,
it is immediately apparent that we cannot extract information about both δα/α and δµ/µ
from a single absolute frequency record. Data from at least two different clock species needs
to be fit and the resulting set {xj } can then be orthogonalized. By including data from morethan two species, we obtain an overdetermined system of linear equations that is solved ina least-squares sense. The least-squares solution and the parameter covariance matrix are
obtained by solving the weighted normal equations in matrix form (see e.g. [100]).
To aid in the visualization of the procedure, the experimental fractional drift rates are typically plotted against the numerical sensitivity constants such as in the top panel of Fig. 4.2.
Here, we have included absolute frequency data from hydrogen 1S-2S spectroscopy [83, 84],
the Yb+ ion clock at PTB [85], the Hg+ ion clock at NIST [30] in addition to our drift rate
extracted from Fig. 4.1. Each species has a different sensitivity as indicated by the x-axis.
The Cs contribution to each species’ sensitivity is shown as a vertical dotted line. We extract
drift rates
δα/α = (−3.3 ± 3.0) × 10−16 /yr
δµ/µ = (1.6 ± 1.7) × 10−15 /yr,

(4.22)

83
which agree well with zero at the current level of accuracy.
A detailed visualization of the error in the above result is shown in the bottom panel ofFig. 4.2. Each species contributes a linear constraint
xj − σxj ≤ (cjα − 2.83)

δα δµ−≤ xj + σxj ,αµ(4.23)

where σxj is the 1-σ error bar from the drift rate fit to data as in Fig. 4.1. These linear
constraints define a region in the ( δα, δµ ) plane that contains the valid values of δα/α andα µ
δµ/µ for the measured drift rate xj ± σxj . Each region of validity is shown as a color-codedbar in the bottom panel of Fig. 4.2. The intersection of the regions of validity defines an
elliptical contour of the reduced χ2 (shown in white). The quadric shown here is the joint
confidence region for δα/α and δµ/µ. Its projection onto the axes defines the individual 1-σ
confidence intervals for each variable given in Eq. 4.22. Note that the eigenaxes of the error
quadric are not along the axes of the plot, indicating that the covariance between δα/α andδµ/µ is significant. The addition of the 87 Sr data only reduced the previous uncertainties
only by ∼15%. Nevertheless, it is important to check for frequency variations across a wide
variety of species and systems.
The utility of the bottom half of Fig. 4.2 becomes apparent when discussing the effect ofadding results from direct optical clock comparisons to the results shown here. The slopeof each confidence region is given by value of cα − cCs
α compared to unity (the Cs sensitivity
to δµ/µ). Steeper slopes than the constraint due to Hg+ will not be obtained with current
optical clock systems compared to Cs. However, when optical clocks are compared directly,
they will contribute constraints on δα/α only – as seen from Eqs. 4.9 and 4.16 – but with muchhigher precision. Adding such a result to the plot amounts to adding a completely verticalregion of validity to the plot. Even if the direct optical-to-optical results were only measured
with the precision of the absolute frequency measurements, they would immediately put
much tighter constraints on both δα/α and δµ/µ.
After the Sr studies were completed, preliminary data from the comparison of NIST’s Hg+
and Al+ standards became available. This addition puts the current limits on the drift of αDrift x (10−15 /yr)

84

0

Yb+

H
−5

δµ
δα
Fit to (cαj − cCsα ) α − µ = xjHg+δα
−16 /yrα = (−3.3 ± 3.0) × 10
δµ−15 /yrµ = (1.6 ± 1.7) × 102

δµ(10−15 /yr)
µ

Cs
Sr

3
4
5
Clock/Cs Sensitivity −(cα − cCs
α )

6

4
2
0
−2
−0.8

−0.6

−0.4
−0.2
δα
(10−15 /yr)
α

0

0.2

Figure 4.2.: Linear weighted least-squares analysis of optical clock fractional frequency
drifts x to extract drift rates δα/α and δµ/µ. The top panel shows fractional frequency
drift rates with respect to each experiments’ sensitivity to variation in α. Other species
included are results from hydrogen 1S-2S spectroscopy at MPQ [83], Yb+ ion clock datafrom PTB [85], and Hg+ ion clock data from NIST [30]. The bottom panel shows thelinear constraints contributed by each species as colored regions of validity in the (δα/α,δµ/µ) plane. Their overlap defines a joint 1-σ region of confidence, shown as a white error
ellipse. Its projections onto the axes are the 1-σ confidence intervals for δα/α and δµ/µ.

85
and µ at [95]δα/α = (−1.6 ± 2.3) × 10−17 /yr
δµ/µ = (−1.9 ± 4.0) × 10

−16

(4.24)/yr.

4.4. Gravity
From a heuristic standpoint, there are two simple ideas for the variation of fundamental
constants.
“Our universe is changing, and physics was different at different times.”“Our universe is changing, and physics is different in different places.”
We investigated the first idea in detail in the last section. What about the second idea? If
the physics is really different in spatially separated parts of the universe, what could be thecause? According to general relativity, space is defined by the distribution of masses. Since
we are interested in the variation of fundamental constants, we can immediately ask whether
they could be coupled to the mass distribution in the universe, which was of course also very
different in the early universe than it is now. If that were true, how would such a coupling
between fundamental constants and the gravitational potential show up in clock frequency
measurements?

4.4.1. Gravitational Potential Variation on Earth
For an earth-bound clock, changes in the ambient gravitational potential can come fromdifferent contributors. Here is a list of contributions sorted in terms of magnitude fromlargest to smallest:• Relative position of Earth with respect to the Sun.
• Relative position of Jupiter, Saturn, and Venus with respect to Earth.
• Relative position of the Moon with respect to the clock.

86
rmin (m)rmax (m)∆U/c2

1.99 × 1030

1.47 × 1011

1.52 × 1011

3.3 × 10−10

Jupiter 1.90 × 1027

5.89 × 1011

9.68 × 1011

9.4 × 10−13

Saturn 5.69 × 1026

1.20 × 1012

1.66 × 1012

9.9 × 10−14

4.87 × 1024

3.82 × 1010

2.61 × 1011

8.1 × 10−14

Moon 7.35 × 1022

3.63 × 108

4.06 × 108

1.5 × 10−14

truck 10410∞

7.43 × 10−25

graduate student 102

0.1∞

7.43 × 10−25

Body Mass (kg)
Sun

Venus

Table 4.2.: Peak-to-peak variation in ambient gravitational potential on Earth over one
cycle. Astronomical data from Ref. [101].
• Relative position of other planets to Earth• Gravitational forcing (Earth tides)• Local moving masses (trucks, trains, etc.)
It might seem surprising at first that the Moon contributes less than Saturn, but a quick orderof magnitude estimate can be found from the following argument. Consider the minimal andmaximal distances rmin and rmax of the corresponding body of mass m with respect to theclock. We then evaluate the peak-to-peak amplitude of the gravitational potential variation
over one cycle∆U
U (rmax ) − U (rmin )
Gm
=
=− 2
2
2
c
c
c



1
rmax

−

1
rmin



,

(4.25)

with gravitational constant G, and we obtain Tab. 4.2.
For comparison, we also estimate the contributions of Earth tides. Earth is not a rigid
body, and “gravitational forcing” (mostly due to the Moon) changes the local elevation withrespect to Earth’s center periodically. The largest contributions change the local gravitational
potential by several ten cm [102]. An upper limit on the gravitational potential variation
due to Earth tides is∆U/c2 = (1 m) × g/c2 ≃ 1.1 × 10−16 ,(4.26)

87

∆U

m⊙

U0

U(r)

aǫ
r

a
E a
Ω

Figure 4.3.: Solar gravitational potential on Earth. Earth (blue) orbits the Sun (red, massm⊙ ) on an elliptical orbit with ellipticity ǫ and semimajor axis a. The ellipticity is givenby the distance ǫa between the center of the ellipse and the two-body center of mass. The
radial distance between Sun and Earth is r and the orbit has a mean angular velocityΩ given by Kepler’s third law. The eccentric anomaly E is the angle between the major
axis and the orthogonal projection of Earth’s position onto a circle with radius a aroundthe center of the ellipse. Earth’s orbit is projected onto the solar gravitational potential
U (r) and the peak-to-peak gravitational potential variation ∆U as well as the mean solar
potential U0 are indicated.

with local gravitational acceleration g ≃ 9.81 m/s2 . In summary, the contribution of the Sun
to the ambient gravitational potential variation on Earth is three orders of magnitude larger
than all others combined.
Earth travels around the Sun on an elliptical orbit, and high-accuracy models of the
trajectory are available [103]. For our purposes, a simple approximation is sufficient. The
derivation presented here is based on methods summarized in Ref. [104]. Figure 4.3 shows
the geometry of the two-body Sun (red) and Earth (blue) system and indicates the geometricmeaning of all variables used in the following derivations.

88
The solar potential at radial distance r(t) isGm⊙ aU (r)/c = − 2
=−
ac r
2



Ωa
c

2

a
,
r

(4.27)

with Sun mass m⊙ , the orbit’s semimajor axis a ≃ 1 au = 1.496 × 1011 m, and Earth’s mean
angular velocity with respect to the Sun (from Kepler’s third law)rGm⊙Ω≃
≃ 2 × 10−7 s−1 .
3a(4.28)The Earth-Sun orbit r with ellipticity ǫ ≃ 0.0167 is found by solving Kepler’s equationΩt = E − ǫ sin E,(4.29)

which relates the eccentric anomaly
E ≡ arccos



1 − r/aǫ(4.30)

to the orbit’s elapsed phase since perihelion M ≡ Ωt, also called the mean anomaly. From
Eq. 4.29, we find that the inverse radial distance is proportional to the derivative of theeccentric anomaly with respect to the elapsed phase
−1

dE
a
dM
= (1 − ǫ cos E)−1 = ,
=
dM
dE
r

(4.31)

which can be expanded as a power series in the ellipticity [104]
∞
X
a
=1+2
Jn (nǫ) cos nM = 1 + ǫ cos M + O(ǫ2 ).
r
n=1(4.32)To leading order in ǫ, the dimensionless solar gravitational potential on Earth is thus
U (t)/c2 = −(Ωa/c)2 (1 + ǫ cos Ωt),(4.33)and its variation with respect to the mean potential is∆U (t)/c2 = −(Ωa/c)2 ǫ cos Ωt.

(4.34)

Note again that t = 0 indicates the perihelion, where Earth is closest to the Sun, and the solar
gravitational potential on Earth is most negative. As shown in Tab. 4.2, the peak-to-peakdimensionless variation is 2ǫ(Ωa/c)2 ≃ 3.3 × 10−10 .

89

4.4.2. Gravitational Redshift
One of the reasons for precision measurements using atomic clocks is the study of gravity
itself. General relativity predicts that light experiences a frequency shiftωωno gravity=√

1
1 − 2u(4.35)

directly given by the dimensionless ambient gravitational potential u ≡ U/c2 . Clocks tickfaster the deeper they are bound in a gravitational well. As we have seen in the previous
section, even the solar gravitational potential on Earth’s surface is neither temporally nor
spatially stable. As Earth moves around the sun, the potential experiences shifts on theorder of 3 × 10−10 , which means that an optical atomic clock at 429 THz base frequency runs
about 140 kHz differently in January (perihelion) than in July (aphelion).
This number is of course meaningless, because clock measurements always involve comparison against a reference, i.e. we need to compare a redshifted frequency ω ′ at u + δu tothe base frequency ω at u:ω′ − ωδu=+ O(δu2 ) ≃ δu.
ω
1 − 2u(4.36)

A clock comparison is thus directly sensitive to the differential gravitational potential.
The important point here is that the spatial separation between clocks matters. A difference in elevation close to Earth’s surface incurs a frequency shift of 1 × 10−16 m−1 . Our
absolute frequency measurements against the Cs standard at NIST had to take these effects
into account, as the elevation difference between the labs is (12.5 ± 1.0) m. In light of how
much the elevation in Earth’s potential matters for clock measurements, the agreement between our measurements and the Paris results at the 1 × 10−15 level becomes even more
impressive: Paris’ mean elevation above sea level is 35 m, Boulder’s is 1655 m, leading to aredshift of 1.77 × 10−13 due to Earth’s gravitational potential.
The agreement between JILA and Paris thus is two orders of magnitude better thantheir relative redshifts. This fact points to an important component of international clock
comparisons: they must be referenced to a common gravitational isosurface. For this purpose,90international clock comparisons (and the Global Positioning System) are referenced to amodel isosurface called the geoid. A good introduction to international timekeeping andthe geoid is Ref. [105]. Each of the 87 Sr absolute frequency measurements were performed
against local RF standards that were individually referenced to the geoid. In this way, theabsolute frequencies between Boulder, Paris, and Tokyo become comparable without having
to worry about the local gravitational potential up to the specified accuracy of the local RF
standard.

4.4.3. Gravitational Coupling
As a simple model, we will assume that a fundamental constant η couples to the ambient
gravitational potential U in a dimensionless fashion via a coupling constant kη , i.e.
δη∆U= kη 2 .
ηc(4.37)

Inserting this assumption into Eq. 4.9, we find that a clock comparison experiment should
be sensitive to such coupling, sinceδ(ωj /ωi ) X j∆U=(cη − ciη )kη 2 .
ωj /ωicη(4.38)

Using the orbital expansion from Sec. 4.4.1 and assuming an absolute frequency measurementof an optical clock (Eq. 4.16), we find a sinusoidal fractional frequency variationΩ2 a2
δ(ωj /ωCs )= −[(cjα − 2.83)kα − kµ ] 2 ǫ cos Ωt.
ωj /ωCsc(4.39)

Note that both frequency and phase of the sinusoidal variation are fixed by the orbital
mechanics. As in Sec. 4.3, we have a situation where the only free parameter in the fit is theamplitude of the effect. The sinusoidal fit in Fig. 4.1 gives an amplitude ofySr = (−1.9 ± 3.5) × 10−15 ,(4.40)

where the frequency of the fit was fixed at Ω = 1.991317 × 10−7 s−1 and the phase was
fixed such that an antinode of the sinusoid coincides with the 2005 perihelion on January

91
2, 01:00 (UTC). The fitted amplitude is negative, and the Sr data anticorrelates with thegravitational potential. The fit is again mostly determined by the last three high-accuracy
data points from the JILA and Paris groups.
As before, data from one species cannot determine limits on the individual coupling constants kη and we again need to orthogonalize similar data from multiple species. There iseven less data available for this kind of analysis than for the linear variation since a good
limit on the sinusoidal variation requires many data points over the course of a single oscillation period. The only other species that have been analyzed for annual variations are the
NIST Hg+ ion clock [30] and a NIST hydrogen maser array [89]. By correcting a sign error
in Ref. [30] when applying Eq. 2 in the subsequent paragraph (corrected in Ref. [106]), we
find the constraint from the Hg+ clock data as
yHg+ = (0.7 ± 1.2) × 10−15 .

(4.41)The sign of the constraint for the hydrogen maser data is derived from the averaged fit inFig. 3 of Ref. [89]:yH-maser = (0.03 ± 0.47) × 10−15 .

(4.42)

Note that the hydrogen maser is also sensitive to variation in the light (up or down) quark
mass mq divided by the strong interaction coupling constant ΛQCD [92]. We write the sensitivity constant for the H-maser as cq = 0.109 [92] and the corresponding coupling constantto the gravitational potential as kq .
We divide the measured amplitudes yj by the peak-to-peak gravitational variation u0 =2(Ωa/c)2 ǫ = 3.3 × 10−10 . Together with the sensitivity equations, the measured values yj /u0
and their uncertainties form a system of linear constraints
ySr /u0 = 2.77 kα + kµ ,yHg+ /u0 = 6.03 kα + kµ ,(4.43)

yH-maser /u0 = 0.83 kα + 0.109 kq .
We can again use weighted linear regression [100] to find the coefficients kη and their uncer-

92
tainties askα = (2.5 ± 3.1) × 10−6kµ = (−1.3 ± 1.7) × 10−5

(4.44)

kq = (−1.9 ± 2.7) × 10−5 .
To aid with the visualization of this procedure, we write Eq. 4.43 in the general form
yj /u0 = cjα kα + cjµ kµ + cjq kq ,(4.45)

yj= kα + djµ kµ + djq kq ,jc α u0(4.46)and divide by cα to obtain

where we defined new numerical prefactors dµ ≡ cjµ /cjα and djq ≡ cjq /cjα . We now interpret thisequation as a linear function in the variables dµ and dq , and interpret the experimental data
for the left hand side as measurements of this linear function with the unknown parameterskα , kµ , and kq .
The resulting graph is shown in Fig. 4.4. The linear function in dq and dµ defines aplane. Since there are only three data points, the linear regression is not strictly necessary,
but it immediately generalizes to the addition of more species and returns the correct error
estimates for the fit parameters. The value of the plane at dµ = 0, dq = 0 is kα and its
gradient along the dµ (dq ) axis is kµ (kq ).
The combination of maser data with optical clock data could be questioned, since hydrogenmasers are not frequency-stable and have long-term nonlinear drifts. However, the additionof the maser data only pivots the fitted plane around the Sr-Hg+ axis and influences neither
the value nor the error bar of kα and kµ .
In conclusion, we find that α, µ = me /mp , and mq /ΛQCD do not couple to the ambient
gravitational potential at the current level of accuracy (∼10−15 ). These results will continue
to improve with more data becoming available from direct optical clock comparisons. Experiments that test the weak equivalence principle also be analyzed to test for the gravitational
coupling discussed in this Section, but such analyses are much less direct [107, 108]. Recently,
a suggestion has been put forward that views atom interferometry as a clock experiment at

Norm. Amplitude (10−7)

93

Hg+
30
0

Sr

H-maser

−30
0

j
Sr
Hg+
H-maser

0.1
dq
cαj
2.77
6.03
0.83

kα
kα
kα
kα

0.2

0.4

+dµj kµ+0.36kµ+0.17kµ+dqj kq+0.13kq

0.2

0dµ= yj /(cαj u)= (−2.1 ± 3.2) × 10−6
= (3.5 ± 6.0) × 10−7
= (1 ± 17) × 10−7

Figure 4.4.: Graphical representation of linear regression to extract the gravitational coupling constants.

94
the atomic Compton frequency [109, 110], with correspondingly low fractional uncertainty.
This interpretation is much-disputed [111, 112] and clock experiments remain the most direct
way of investigating gravity [113].

Chapter5

Optical Feshbach Resonance in 88Sr

S

cattering resonances in ultracold atomic gases [114, 115] allow control of atomic interactions and have been used very successfully over the last decade to study many novel

phenomena. The underlying phenomenon is the magnetically controlled Feshbach resonance(MFR) which arises from coupling of the free particle states to a bound molecular state. Such
scattering resonances were studied first by Ugo Fano [116, 117] and Herman Feshbach [118].
MFR is schematically shown in Fig. 5.1(a). Different hyperfine structure states in theelectronic ground state manifold of an alkali metal atom lead to multiple molecular groundstate potentials. Their relative energies can be tuned by an external magnetic field. If
atoms are initially prepared in a given hyperfine structure state (the open channel), anda bound state in a different molecular ground state potential (the closed channel) becomesdegenerate with the free particle threshold, a scattering resonance arises, and the free particle
scattering phase shift can be modified. Because the bound molecular state is in the groundelectronic manifold, it is long-lived and the molecular coupling does not introduce loss. In
the zero collision energy limit, the scattering process is completely determined by a reals-wave scattering length.1 In the vicinity of the resonance, the scattering length diverges
1

A complete discussion of scattering theory relevant to collisions in ultracold atomic gases is beyond thescope of this work, but an excellent summary can be found in Ref. [119]. A good overview on quantum
degenerate gases is Ref. [120]. The main review paper for Feshbach resonances in ultracold gases isRef. [121].

96
and the achievable change in scattering length is only limited by secondary processes suchas three-body collisions [121].
These dispersive changes in scattering length have allowed exquisite control and led to
the realization of few-body quantum states [122, 123], and the study of strongly correlated
systems [124, 125]. These resonances have played a leading role in building atomic systems
that model simple condensed matter systems and have become the dominant experimental
tool to tune many-body interactions in atomic physics experiments [121].
With the success of ultracold gases of alkaline earth atoms in recent years for precision
measurements (see previous Chapters), many interesting proposals for quantum simulationof many-body systems have been put forward [126–128]. However, magnetically induced
Feshbach resonances do not exist in the spinless electronic ground state of these systems since
there are no molecular potentials available that can be strongly coupled by experimentallyfeasible magnetic fields. Instead, the coupling field has to be at optical frequencies and thecoupled molecular potential is an excited molecular state. The coupled molecular boundstate can now decay via spontaneous emission and the resulting decoherence rate is the mostimportant factor to consider in understanding the resulting Optical Feshbach Resonance(OFR) [121, 129, 130]. The experimental situation is shown in Fig. 5.1(b), which describes
a photoassociation (PA) experiment [131] where a PA laser couples free particles to a boundstate in a metastable molecular potential. On resonance, molecules are formed which candecay to free particles with high kinetic energy or deeply bound ground state molecules.
Once a molecule has formed, it is considered lost from the experiment. In a PA experiment,
the laser is tuned across the resonance and atomic loss spectra are measured. If the PA
laser is detuned from molecular resonance, the scattering phase shift can still be modified,just as for a magnetic Feshbach resonance. However, the molecular loss limits the achievable
scattering length modification. In a dressed state picture [see Fig. 5.1(c)], the OFR looks
very similar to the MFR. The closed channel can be tuned with respect to the open channel
by either a magnetic field or the laser detuning. The tuning changes the interatomic distance(the Condon point) at which the open and closed channels become resonantly coupled. The

97(a) Magnetic Feshbach resonance (MFR) between(b) Optical Feshbach resonance (OFR) between

molecular potentials in the electronic groundelectronic ground state potential (open chan-state.

nel) and metastable molecular potential (closed
channel that decays).

(c) Dressed potential for either magnetic or optical
Feshbach resonance. The Condon point RC indicates the interatomic separation around which
the resonant interaction is localized.

Figure 5.1.: Magnetic and optical Feshbach resonances arising from coupled molecularpotentials. The coupling is adjustable by either the magnitude of an external magnetic
field (MFR) or the detuning of a photoassociation laser (OFR). A resonance appears when
the free particle energy becomes resonant with a state in the closed channel. The only
conceptual difference is that the OFR closed channel introduces decay when a metastablemolecule is formed and subsequently dissociates.

98
only conceptual difference between OFR and MFR is that OFRs have intrinsic loss given by
the natural radiative decay of the metastable molecules and that the OFR strength can betuned independently by adjusting the PA laser intensity.
The OFR effect was first observed in alkali atoms [132–135] but was not found useful due tolarge inelastic losses. Much narrower optical intercombination lines are available in alkaline
earth atoms and are predicted to overcome this loss problem [136]. These predictions in
combination with our group’s study on narrow line photoassociation in 88 Sr [137] stimulated
work on the OFR effect in thermal and quantum degenerate gases of Yb [138, 139].
In our experiment, we worked with bosonic 88 Sr, which has a small background s-wave
scattering length of abg ≃ −1.4a0 [140, 141], with the Bohr radius a0 . The sample is trappedin an optical dipole trap at temperatures of several µK. In this regime, s-wave scattering
dominates, but the small abg makes the sample effectively non-interacting on experimental
timescales. In combination with the simple electronic structure, this makes 88 Sr an ideal
test bench for the OFR effect. For these reasons, we can specialize the general theory to
negligible background scattering which simplifies the treatment considerably.
We will first introduce the general theoretical background for describing the OFR effect inboth degenerate and thermal gases. Based on the general theory, we will develop the formalism used to describe a single, isolated OFR. Care is taken to make the formalism applicableto the regime where s-wave scattering dominates, but where particles collide at non-zerocollision energies. Extraction of the relevant OFR parameters from experimental data suchas PA loss spectra requires a careful modeling of the sample dynamics in the presence of interactions. While analytic treatments can highlight important limiting cases, a Monte-Carlosimulation of the sample is required to model the interaction dynamics accurately. Using
these methods we are equipped to model the sample dynamics under the influence of the
OFR effect in a universal way and predict the requirements for applying OFR in a quantum degenerate gas. Our results will be directly applicable to experiments with quantum
degenerate gases of Sr, which have become available in the last two years [142, 143].

99

5.1. Complex scattering length, S-matrix, collisioncross sections & rates
We introduce the necessary notation for describing the OFR effect, based on Refs. [119, 121,
130, 144]. We will use the energy-dependent complex scattering length definition adopted inRef. [145]α(k) ≡ a(k) − ib(k) ≡ −tan η00 (k)
k

(5.1)

to describe collisions for small but non-zero collision energy in the thermal regime wheres-wave scattering dominates, but the finite collision momentum matters. Here, a (−b) is thereal (imaginary) part of α and η00 (k) is the complex s-wave scattering phase shift describing
both elastic and inelastic components of the scattering process. Molecule formation is considered an inelastic process that removes the collision partners from the sample completely.
The collision energy E ≡ ~2 k 2 /(2µ) is related to the relative momentum2 (mass) ~k (µ) ofa collision pair and for our purposes, µ = mSr /2 with the atomic mass mSr . The scattering
length can be related to the s-wave scattering matrix S00 (k) via
S00 (k) = e2iη00 (k) ⇔ α(k) =1 1 − S00 (k).
ik 1 + S00 (k)(5.2)

For collisions between identical bosons, the elastic and inelastic collision cross sections are
related to the S matrix via [119, 121]πσel (k) ≡ 2 2 |1 − S00 (k)|2 ,kπσin (k) ≡ 2 2 (1 − |S00 (k)|)2 .
k

(5.3)

Note that the cross sections for nonzero k are not simply related to the squares of a and b,
but thatσel (k) = 8π|α(k)|2 f (k)2 ,σin (k) =2

8πb(k)f (k),
k

(5.4)

Note that here and in the literature, both k and ~k are referred to as the relative collision momentum.

100
with
1

f (k) ≡

1 + k 2 |α(k)|2 + 2kb(k)

.

(5.5)

The function f encapsulates the unitarity limit for both elastic and inelastic processes with
0 < f (k) < 1 and f (k) → 1 as k → 0.
These collision cross sections are related to the likelihood of a collision event with relative
velocity vrel ≡ ~k/µ occurring, in the sense that
Pc (|v i − v j |) = σ(|v i − v j |)|v i − v j |

∆t
,
Vc

(5.6)

is the probability of a collision event happening between particles i and j with velocities v
within an arbitrary collision volume Vc and time interval ∆t, as long as Pc ≪ 1. From theprobabilistic interpretation of the collision cross sections, we can derive differential equations
that describe the collision processes. Let V be the trap volume containing a sample of N
identical bosons. We subdivide V into collision volumes such that V = M Vc , and we choose
M such that the average number of atoms within each collision volume
M

1 X
Ni
Nocc =
M i=1

(5.7)

is small (Nocc ≪ 1). The number of particles influenced within a collision volume by collision
events during a sufficiently small ∆t is then given by a sum over distinguishable collisionpairs∆Ni = −2

Ni X
k−1
X
k=1 ℓ=1

Pc (|v k − v ℓ |)(5.8)

where the factor of 2 arises from the fact that two particles are influenced by each collision
event. For elastic collisions, this means that two particles change their velocity directions. For
the inelastic processes considered here, two particles are lost from the sample per collision
event. If we take the average of Eq. 5.8 over all collision volumes, we find the change inaverage occupation number∆Nocc ≡ h∆Ni ii = −2
∆t
≃ −2
Vc



* N k−1i X
X
k=1 ℓ=1

Ni (Ni − 1)2

i

Pc (|v k − v ℓ |)+hσ(vrel )vrel iT ,

i

(5.9)

101
where h·iT indicates a thermal average at temperature T over the relative velocity distribution
within the sample. The factor Ni (Ni −1)/2 stands for the number of distinguishable collisionpairs considered in the sum. For small Nocc , the collision volume particle number Ni is
distributed as a Poisson distribution and we use2
hNi2 i − hNi i2 = hNi i = Nocc ⇒ hNi (Ni − 1)i = Nocc

(5.10)

to find the average occupation number change due to inelastic collisions as
2
∆Nocc
Nocc
=−
hσin (vrel )vrel iT .
∆t
Vc

(5.11)We define the mean density n̄ ≡ Nocc /Vc , let ∆t → 0 and obtain the mean density evolution
due to inelastic collisionsn̄˙ = −hσin (vrel )vrel iT n̄2 ≡ −hKin iT n̄2 .

(5.12)

The above derivation lets us define inelastic and elastic collision rate constants~k
Kin (k) ≡σin (k),µ(5.13)~k
Kel (k) ≡σel (k),µwhere the rate Kin is defined as the factor multiplying the density squared in the density loss
equation and Kel is defined analogously. Note that the ratio of elastic to inelastic collisions
at a given collision energy
σel (k)|1 − S00 (k)|2k|α(k)|2
Kel (k)==[1 + k 2 |α(k)|2 + 2kb(k)]=Kin (k)σin (k)1 − |S00 (k)|2b(k)(5.14)

tends to zero as k → 0.
Note that the discussion in this Section has been a completely general discussion of elastic
and inelastic processes due to an energy-dependent complex scattering length and that no
approximations have been made.

5.2. OFR as an isolated decaying Feshbach resonanceIn the previous Section, we introduced a treatment of s-wave scattering under a complex
scattering length. Complex scattering lengths have been successfully used to describe scat-

102
tering resonances with loss, i.e. problems where free particles are coupled to a decaying
bound state. The Optical Feshbach Resonance is just such a case, where the bound stateis a metastable molecular level that decays spontaneously into free particles in the groundelectronic state.
For an isolated OFR, Bohn and Julienne [130] derive an S-matrix
S00 (k) =E/~ + ∆ + i[γ − Γs (k)]/2 2iη00 (k)e,E/~ + ∆ + i[γ + Γs (k)]/2(5.15)

where ∆ is the laser detuning from molecular resonance3 , γ is the intrinsic molecular decayrate, andΓs (k) ≡ 2kℓopt (k)γm ,(5.16)

is the laser-induced stimulated molecular linewidth in terms of the natural molecular linewidthγm = 2γa , where γa is the natural linewidth of the atomic transition. The strength parameterℓopt (k) has dimensions of length and thus is called the optical length. It can be expressed interms of molecular parameters and the driving laser intensity I as [136, 144]
λ3a |hn|Ei|2ℓopt (k) =
I,
16πc
k

(5.17)

where λa is the atomic transition wavelength, and c is the speed of light. The molecularFranck-Condon factor |hn|Ei|2 describes the wave-function overlap between the free particle
state with collision energy E and the bound molecular state wave function |ni. For small
collision energies, the Wigner-threshold law predicts that |hn|Ei|2 ∝ k and thus ℓopt should
be a very weak function of k. We have verified numerically that the Wigner-threshold law
holds for Sr collisions beneath E/kB = 10 µK, by comparing the isolated resonance formulas
against a full coupled-channels calculation [146]. Typical temperatures in our experimentare ∼3 µK, and we will neglect the dependence of ℓopt on k in the following. The opticallength is also independent of the oscillator strength of the underlying atomic transition and
3

The molecular detuning definition here includes the I-dependent AC Stark shift of the molecular line tosimplify the present discussion. We will derive a more complete description of the molecular detuning ∆later.

103
thus becomes a general parameter to compare the strength of molecular resonances acrossspecies.
In Eq. 5.15, we have allowed for extra molecular loss beyond the radiative decay, such thatγ > γm . We combine Eqs. 5.13, 5.15, and 5.16, neglecting the small background phase shiftη00 , and find expressions for the elastic and inelastic collision rates for an isolated resonance:2hℓeﬀ
Kin (k) =22µ (∆ + E/~) /γ + (1 + 2kℓeﬀ )2 /4(5.18)Kel (k) ≃ 2kℓeﬀ Kin (k),where we have defined an effective optical length
ℓeﬀ =

γm
ℓopt ,
γ(5.19)

that describes the OFR strength in the presence of extra molecular decay. The factor 2kℓeﬀ
in the denominator accounts for power broadening of the molecular response. The simple
relationship Kel /Kin = 2kℓeﬀ between elastic and inelastic collision rates only holds when we
can neglect the background scattering length abg . This relationship simplifies the discussion
drastically and we can define a small intensity regime, where
r
π
~
1
,
=
ℓeﬀ ≪
2hkiT
2 8µkB T(5.20)

for a three-dimensional Maxwell-Boltzmann distribution at temperature T . In this limit,
elastic collisions and power broadening can be neglected. The system dynamics can then be
described completely by two-body inelastic loss processes with collision rate
Kin →

ℓeﬀ
2h
,
µ (∆ + E/~)2 /γ 2 + 1/4(5.21)

which scales linearly with ℓeﬀ . A typical scale for inelastic collision rates in this regime canbe obtained by setting the detuning term to zero:
8hℓeﬀ≃ (3.8 × 10−12 cm3 /s) ×
Kin ∼
µ



ℓeﬀ
a0



,

(5.22)

at µ = mSr /2 = 44 amu. The inelastic collision rate is still dependent on the relative
momentum through the relative energy denominator. Measuring the inelastic loss rate in a
PA experiment with small intensities over long timescales thus allows a clean measurementof the line strength factor ℓeﬀ /I for a given molecular resonance.

104

5.3. Photoassociative Spectroscopy
We studied the photoassociative loss experimentally for three PA resonances in the excited1

S0 − 3 P1 0u potential of Sr2 . The ground state molecular potential (1 S0 − 1 S0 0g ) and theexcited molecular potential are shown in Fig. 5.2 with respect to the internuclear distance.
The PA laser couples two colliding ground state atoms to a high-lying vibrational state in
the metastable excited potential. The vibrational levels are labeled as negative numbers ncounting from the free particle threshold.
15
1

14

PA laser @ 689 nm

12
11
10

Difference potential (GHz)013Energy (103 cm−1 )S0 + 3 P1 0u

n = −2n = −3
Bound state
E/kB = 1µK
Condon point RC−1

n = −4

9
50

≈

1

0
10

20

S0 + 1 S0 0g
30

100

150

200
≈

Atomic distance (a0 )

40
50
60
70
Atomic distance (a0 )

80

90

100

Figure 5.2.: Molecular potentials coupled by a PA laser. The inset shows the difference
potential after subtracting the optical frequency. A free particle wave function is indicatedin solid blue and bound vibrational wave functions for the vibrational levels of interest are
shown in red. Note that the Condon points are at large interatomic distance.
The inset of the figure shows the difference between ground and excited molecular poten-

105
tials after subtracting the optical frequency. We investigated the n = −2, −3, and −4 levels,
close to the free particle threshold and thus at relatively small red detuning from the freeparticle atomic transition. The metastable molecules in these vibrational states are looselybound and the Condon points for photoassociation are situated at large interatomic distancesof (40 − 150) × a0 . For this reason, the Franck-Condon factor determining the optical lengthis not very dependent on the details of the short range molecular potential and a relatively
accurate calculation is based on knowledge of the long-range dispersion coefficients [144]. A
representative free particle wave function at a collision energy of E/kB = 1 µK is shownin solid blue and bound state wave functions are overlaid in solid red. The Franck-Condon
factor and thus ℓopt is determined by the overlap integral of these wave functions [144].

For small ℓopt , the effect of tuning the PA laser across a vibrational resonance is relativemomentum-dependent particle loss (see previous Section). Representative PA loss spectra
for small ℓopt are shown in Fig. 5.3. Here, the fractional atom loss is shown as a function of
PA laser detuning from the atomic transition after the PA laser interacted with the trappedsample for τPA = 200 ms. Note that the PA intensity is adjusted to compensate for theindividual line strength such that the overall loss is comparable. Atomic densities for all
spectra were similar ∼5 × 1011 cm−3 and the number of photons scattered per atom off theatomic transition τPA Γsc ≪ 1. The spectra look very similar, and we will find in the following
Sections that similar values of the optical length ℓopt describe all three spectra shown.

We will develop a model to extract the optical length and the center frequency of the PA
lineshapes for a thermal gas of 88 Sr. To obtain this data, we need a model of the density
distribution in the optical trap that does not assume thermal equilibrium. We will discuss
the trapping geometry and then derive a more general description for the density evolution
than Eq. 5.12. Integrating the density over the trap volume then provides us with a modelfor the atom number evolution with time which we can use to fit the PA spectra in Fig. 5.3.

Fraction of atoms remaining

106

PA detuning (MHz)PA detuning (MHz)−1084.5−222.5

−1084

−222

PA detuning (MHz)

−24.5

−24

1
0.8
0.6
0.4
0.2
0u n = −4524 mW/cm2ℓopt = 14(1) a0
Γsc = 0.05 s−1

0u n = −3
310 mW/cm2ℓopt = 14(1) a0
Γsc = 0.69 s−1

0u n = −2
4 mW/cm2ℓopt = 28(2) a0
Γsc = 0.77 s−1

Figure 5.3.: PA loss spectra in the small optical length regime for three vibrational levels.
The PA laser interacted with the atoms for 200 ms at the indicated PA laser intensity,
which has been adjusted to obtain similar optical length ℓopt . The scattering rate fromthe atomic transition Γsc is kept small such that atomic scattering does not influence theatomic density significantly.

107

5.4. Experimental Setup
This section gives a brief overview of the third generation Sr apparatus (Sr3) and summarizes
some of our experiences with the various components used.

5.4.1. Vacuum system & magnetic coils
An overview drawing of the apparatus is shown in Fig. 5.4 and in the following, capitalletters in parentheses refer to the labeled parts of the figure. The atomic beam source
is a conventional effusive oven design. The oven is heated from outside the vacuum withceramic clamshell heaters and the whole assembly is wrapped in high-temperature ceramicheat insulation. The oven temperature is controlled by commercial process controllers withfeedback from high-temperature thermocouples. To prevent vacuum part corrosion due toheat cycling, the front end of the oven is machined from a single piece of 316 stainless steel
with thick walls (A). A microtube oven nozzle [147, 148] drops in from the front and is heldin place by two ventilated screws. We thank F. Schreck of the University of Innsbruck forsending us the microtube material. The tubes are made of 304 stainless steel tubes with
200 µm inner diameter (ID). The tubes are acid-etched to 8 mm length and stacked in apressure-fit U-shaped holder. The oven front end is welded to a 45◦ elbow (B) that lets theback end of the oven hang down and prevents solid Sr from migrating towards the nozzle.
The elbow and the back end of the oven are connected by a Conflat flange with a Ni gasket(C). The back end is a simple cup filled with natural abundance Sr metal (D) that is watercooled with a cooling block in the very back (E). The water-cooling allows relatively rapid
cooling of the oven from operating temperature to room temperature with 1/e time constant∼2.5 h. The back (front) end is typically heated to 575 ◦ C (625 ◦ C). The 50 ◦ C temperature
differential combined with the 45◦ bend seems to have been enough to prevent nozzle clogging
for the last 3 yr of operation.
This design produces an atomic beam that separates into a collimated part (collimation
given by the opening angle of the individual tubes) and a diffuse part (given by the opening

108

Figure 5.4.: Sr3 vacuum system including magnetic coils without optics and electronics.
The parts labeled are described in the main text.

angle of the full nozzle diameter) [147, 149–151]. The major advantage of the microchannel
nozzle is that it saves a lot of source material by not transmitting as many atoms with largetransverse velocities that cannot be collimated easily even with transverse laser cooling.
The beam passes through a vacuum cube (F) containing a UHV compatible magneticallycoupled rotary feedthrough with a U-shaped attachment, driven by a stepper motor. This
setup acts as an atomic beam shutter by rotating the arms of the U into the atomic beampath. In our experiments so far, however, we have not found it necessary to shutter thebeam. The rotary shutter was implemented as a replacement for the previously-used invacuum Uniblitz shutter. These shutters have been used in earlier Sr experiments because
they allow full shuttering of both the atomic beam and oven BBR radiation on time scales< 1 ms. These shutters are notorious for failing catastrophically and are not compatible
with vacuum levels below 10−9 Torr, since they require in-vacuum electrical connections toa resin-enclosed driving coil. A typical failure mode is that the shutter blades crash into
one another and a part of the blade or driving mechanism breaks off. If that happens, thevacuum system will be contaminated with carbohydrates (it smells of burned plastic) and has
to be cleaned and rebaked from scratch. Use of these shutter systems is not recommended.

109
The beam passes through another six-way cross (G) which is used for transverse cooling.
The beam is collimated (and steered) by two retroreflected horizontal and vertical beams
at 461 nm and about −10 MHz detuning from 1 S0 -1 P1 . The collimated beam then passes
through a 5 inch long differential pumping tube with 1/4 inch ID between the oven and
the Zeeman slower regions (H). Both oven and Zeeman slower regions are pumped with
Varian StarCell 40 l/s ion pumps (I,J) and typical Zeeman slower region pressures can reach
10−10 Torr. The system contains ion gauges in the oven, Zeeman slower, and main chamber
regions. These are typically turned off and the pressure is estimated from the ion pumpcurrents continuously monitored with a Keithley precision voltmeter. The beam then passes
through a pneumatic all-metal gate valve (K) that separates the Zeeman slower tube andthe main chamber from the oven region.
The Zeeman slower (L) is 25 cm long and separated into two coils with opposite current
directions (not shown in figure). This design reduces the amount of wire that has to be used
and results in a much reduced resistive heat load. The design also allows using laser detunings
much closer to the atomic resonance. The zero-field detuning is ∼−640 MHz, which can beobtained with high efficiency by using an AOM around 300 MHz before doubling. Choosing
a relatively low AOM frequency and shifting the frequency in the infrared decreases the loss
drastically compared to using a low-efficiency and hard-to-align 1 GHz AOM at 461 nm.
Even though the resistive heat load is small, the Zeeman slower is wound on a water cooled
tube that encloses the actual vacuum tube. The coil design was optimized by an adaptivealgorithm that varied the number of transverse and longitudinal layers of wire for each coilto obtain the ideal field shape. Another water-cooled magnetic coil (M) is positioned at theend of the Zeeman slower which allows compensating for the residual longitudinal magnetic
field due to the Zeeman slower at the center of the vacuum chamber. All coils are driven
by constant-current power supplies stable to 10−3 that are always on [typical currents are(1 − 3) A].
The slowed atomic beam (design longitudinal velocity 40 m/s) then enters the main chamber (N) which is pumped with another 40 l/s ion pump (O). Main chamber pressures are in

110
the low 10−10 Torr to high 10−11 Torr regime. Unfortunately, the initial design did not include
additional Ti:sublimation pumps and the main chamber pressure would rise over the courseof the day to (5 − 8) × 10−10 . The oven load was reduced to negligible levels by encapsulating
the differential pumping tube region in a shaped aluminum block (not shown in figure) that iscooled by a Fisher Scientific ethylene-glycol circulating chiller to −20 ◦ C. Water-ice buildup
is manageable with good thermal insulation and the low humidity in Boulder. Typical lifetimes in the magnetically trapped 3 P2 state are 3 − 5 s. This lifetime is limited by outgassingof the surfaces in the main chamber. An improved main chamber vacuum design using two
125 l/s ion pumps and two Ti:sublimation pumps is being implemented. Accounting for theincreased main chamber surface area, the increase in ion pump speed alone should increase
the pumping speed at the center of the main chamber by a factor of 6.
The main chamber (N) is a Kimball spherical octagon made of 316 stainless steel andthe large 6 1/2 inch viewports are oriented vertically. To prevent losing access to the eight
2 3/4 inch viewports, the Zeeman slower was tilted upwards and is connected to one of
the sixteen 1 1/3 inch viewports. The viewport directly across from the atomic beam (P)is made of UV grade sapphire and heated to 150 ◦ C to prevent formation of a metal filmon the viewport. The heater is switched off during red MOT and dipole trap operation toprevent stray magnetic fields from influencing the atoms. The Zeeman slower laser beam(with a typical power of 30 mW) enters through the sapphire viewport and its collimation is
optimized to maximize loading of the atoms into the blue MOT. The blue MOT, red MOT
and crossed dipole traps are slightly offset from the Zeeman slower path and we have notobserved any effect of shuttering the atomic beam on our experiments so far.
The blue MOT is formed by three retroreflected laser beams at 461 nm, with typical beam
diameters of 11 mm, powers of ∼2 mW per beam, and detuning of −40 MHz from the1

S0 -3 P1 transition. The magnetic field gradient is 50 Gauss/cm and is produced by a pairof anti-Helmholtz coils (Q) that are driven by a feedback-stabilized constant voltage power
supply. The current is measured with a Hall probe and feedback is applied to two parallelwater-cooled MOSFETS. The large reverse breakdown voltage (Zener voltage 500 V) allows

111
shutting off the ∼10 mH coil pair within 250 µs with a PI based servo. The fast shutoff timeis necessary to switch the magnetic field gradient to 3 Gauss/cm for loading into the second
stage MOT. Residual magnetic fields at the center of the chamber are compensated by threefeedback-stabilized pairs of magnetic coils (X, Y, Z). The horizontal coil pairs (X, Y) areremoved by 65 cm from the chamber center to prevent limiting the optical access and they
allow compensation of magnetic fields up to several Gauss.

5.4.2. Blue laser & repumps
A schematic view of the laser beams and magnetic coils relevant to the blue MOT from the
top of the vacuum chamber is shown in the central part of Fig. 5.5. The blue light for Zeeman
slower and blue MOT is generated by second-harmonic generation (SHG) from infrared lightat 921.723 nm, as depicted in the top left inset of the figure.
Diode laser & Tapered Amplifiers. The infrared laser is an AR-coated diode byEagleyard4 in a Littman external cavity geometry. We use a JILA laser controller (JanHall’s highly stable current controller, JILA standard temperature controllers for base anddiode mount) to drive the laser diode. Typically, diodes are mounted in Thorlabs collimation
tubes and are collimated with aspheres. The C230TME-B aspheric lens works well with mostIR laser diodes. Gratings are typically Edmund ruled gold gratings (1200 lines/mm) for IR
wavelengths and Optometrics blazed holographic gratings (1200 or 1800 lines/mm) for red
wavelengths. The feedback mirror is held in a JILA mirror mount and we use a small
Thorlabs AE0203D04F piezoelectric transducer (PZT) to tune the horizontal mirror tilt.
We use a reasonably mechanically stable but very versatile mechanical design that holds
the diode, grating, and feedback mirror, originally designed by T. Ido. The diode head istemperature-stabilized with a thermoelectric cooler (TEC) and typical diode temperatures
are in the range of 16 − 30 ◦ C. The lower limit is set by the yearly humidity cycle in Boulder,
where the dew point can reach ∼16 ◦ C in August. The diode mount baseplate is separately
temperature-stabilized with a resistive heater to ∼5 ◦ C higher than both room temperature
4

Eagleyard AR-coated ridge waveguide laser model number EYP-RWE-0940-08000-0750-SOT01-0000.

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Figure 5.5.: Schematic overview of the 461 nm blue laser system for cooling and trapping.
See the main text for details. The central part of the figure shows a schematic top view ofthe vacuum system, magnetic coils, and laser beams for the blue MOT. The top left partof the figure shows the blue light generation, distribution, and spectroscopic lock using
saturated absorption. The bottom right inset shows the repump lasers necessary for blue
MOT operation.

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and diode head. The laser assembly sits in a simple enclosure and the beam exits through an
AR-coated microscope slide or commercial laser window. We use cylindrical lenses mounted
in cage systems to optimize the transverse beam shape. We typically use 60 dB of opticalisolation on our diode laser setups to prevent optical feedback issues. The Conoptics isolators
we use do not have µ-metal shielding, which makes them very magnetic, but also lot cheaper.
Zero-order waveplates for any wavelength can be obtained from Tower Optical with shortlead times.
The laser injects a 1.5 W Eagleyard tapered amplifier (TPA) chip in a custom mechanicalflexure mount (the original version was designed by K.-K. Ni). A TPA is a semiconductor
gain medium that widens adiabatically along the horizontal axis from input to output side.
The reason for this design is that it can generate large output power while circumventing
optical damage to the gain medium by keeping the laser intensity constant. The downsideof this approach is that the output beam shape of a typical TPA chip is square and shows
vertical stripes in addition to a diffuse background due to amplified spontaneous emission(ASE). The ASE background can be attenuated by coupling the light into a single-mode
optical fiber.5 To make the large optical power useful, it needs to be shaped and collimated.
We find that the Thorlabs C230TME-B aspheric lenses also work well for focusing a circularbeam into the TPA and collimating the output along the vertical axis. The horizontal axis
needs to be carefully collimated with an additional cylindrical lens. Although we have not
managed to break a TPA chip with backreflections, we typically tilt the collimating optics
and use 30 − 60 dB of optical isolation as well. To optimize coupling of TPA light into an
optical fiber, it helps to mount the collimating cylindrical lens on a linear translation stage.
The TPA output power can fluctuate quite drastically, but this seems to be mostly determined by the chip itself, and only to a lesser degree by the current stability of the power
supply. The intensity fluctuations can be servoed out by fast feedback to the TPA current
5

For the optical clock experiment, the broad ASE frequency spectrum in the TPA-amplified optical latticelight can become important, especially in a non-retroreflected lattice. High-efficiency transmissive gratings or a narrow bandwidth optical filter placed before the optical fibers can reduce the ASE problem
drastically.

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using e.g. a shunt transistor. For the blue laser, we generally intensity-servo the SHG light
directly because the SHG process introduces additional kHz-level noise. Although the chiptemperature matters, temperature stability is not nearly as critical as for a typical laser
diode. For these reasons, we use commercial Thorlabs current and temperature controllers
with our TPAs. The temperature controller drives a large-area TEC beneath the TPA mountto keep it stable slightly below room temperature.
Doubling Cavities. The infrared light is frequency-doubled in two linear build-up cavities using periodically-poled potassium titanyl phosphate (PPKTP). The Sr2 system uses asimilar cavity design but based on potassium niobate (KNbO3 ). For a linear cavity design,
one crystal surface is coated with a high-reflective coating and forms one of the mirror surfaces of the cavity [152]. The only other optical element necessary is a concave input couplerthat is highly reflective at the fundamental and transmissive at the second harmonic whichmakes such a setup much simpler than the typical bow-tie geometry. Because the crystal itself functions as a cavity mirror, the SHG process has to be temperature-tuned. For KNbO3 ,the phase-matching temperature is around 150 ◦ C, close to the depoling temperature. In
addition, there seem to be no TEC elements that can work at these temperatures and weuse a (unipolar) resistive heater instead. One expects better temperature stability and servo
bandwidth with a (bipolar) TEC at lower phase-matching temperatures. PPKTP has aphase-matching temperature around 30 ◦ C and high-efficiency SHG in a bow-tie doubler has
been demonstrated by the Paris group using a PPKTP crystal by Raicol [52, 153]. This success inspired us (and other groups) to try PPKTP in their doubling systems. Unfortunately,
many groups have found that KTP-based SHG is not particularly reliable, mostly becauseof an effect called gray-tracking [154–158]. This process is widely observed but poorly understood in cw operation, and similar effects in other types of crystals go by the names of
GRIIRA (green-induced IR absorption), BLIIRA (blue-induced IR absorption), formationof color centers, etc. The effect is very drastic: initially, the SHG process is very efficientand the SHG mode is a perfect TEM00 with high power. Depending on the amount of fundamental power used, the efficiency drops drastically within a few minutes to hours. The

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SHG mode starts to show dark striations and eventually, one can observe those dark regions
drifting around, which points to fluctuations on thermal timescales. Even with continual
readjustments to the crystal temperature, the efficiency keeps dropping until the output
becomes useless within a few hours.
Our observations are consistent with findings that gray-tracking is much reduced when
operating KTP at higher temperatures [159]. One of the hypotheses for the origin of graytracking is that color centers are formed in high-fundamental-intensity regions of the crystal.
The color center formation is also temperature dependent and the color centers preferentiallyabsorb the second harmonic. Since the linear buildup cavity contains a standing wave withalternating high- and low-intensity regions, the color center formation should be much more
severe than in the bow-tie configuration. Even with a fast temperature servo, these local
temperature inhomogeneities cannot be removed which is consistent with the dark striationsin the second harmonic mode.
The doubling efficiency can be restored by realigning the cavity such that a different region
in the crystal is used in the SHG process. In addition, heating the crystal up to ∼100 ◦ C overa full weekend seems to return it to its prior state. Unfortunately, the depoling temperature
for PPKTP is around 150 ◦ C, and there seems to be no particular advantage in using a hightemperature phase-matched PPKTP crystal over KNbO3 . To work with the system at hand,
we have added a small translation stage underneath the crystal mount that allows simple
adjustments to the cavity alignment. In combination with the overnight heating procedure,
the current crystals seem to work. Nevertheless, gray-tracking limits the output power of
each doubling cavity to 100 − 120 mW.
More recently, commercial SHG doubling systems have become available. Many groups
have bought the (fairly expensive) Toptica SHG system that seems to produce several
hundred mW of 461 nm reliably. Even more recently, NEL has started to sell a fibercoupled periodically-poled lithium niobate (PP LiNbO3 or PPLN) waveguide doubler for
about $10k [160]. With input powers of ∼350 mW (IR output power measured by tuning
the temperature off resonance), the doubler reliably produces ∼50 mW of second harmonic

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light. We have been using one of these devices for a few months and it seems to be very
reliable so far. Unfortunately, only long term use will show whether the doubling efficiency
degrades over time. Nevertheless, both the price and the operational simplicity of single-passdoubling over cavity-enhanced SHG makes this a very attractive alternative.
MOT beams. The SHG cavity length is stabilized to the IR laser by modulating thelaser current at ∼30 MHz. The cavity linewidth is larger than the sideband spacing and
we demodulate the IR transmission at the modulation frequency to obtain a Pound-DreverHall-like error signal. The cavity input coupler is mounted on a tubular PZT of matcheddiameter and we apply feedback to the PZT voltage. Once the cavity length is locked, cw
blue light is available and is split off from the IR light with a 45◦ dichroic mirror in front of
the input coupler.
The wavelength of the blue light is stabilized to a saturated absorption spectrometer (seetop right of Fig. 5.5) based on a Hamamatsu Sr hollow cathode lamp driven by a highvoltage power supply. The probe arm is modulated with an electro-optic modulator (EOM)at ∼20 MHz. The pump arm contains an acousto-optic modulator (AOM) that shifts thepump by +80 MHz with respect to the probe. The AOM RF signal is also chopped at ∼1 kHz
for lock-in detection of the spectroscopic signal. The error signal derived from the lock-in
detection allows stabilization of the blue laser wavelength by slow feedback to the IR laser
PZT (or current). Once feedback is applied, the blue light at the entry of the spectrometer
is now at −40 MHz with respect to the 1 S0 -1 P1 transition in the Sr cell. Tracing back, theblue light at the exit of the MOT cavity becomes detuned by −120 MHz.
The +80 MHz AOM in front of the optical fiber to the blue fiber splitter makes the blue
MOT beams −40 MHz detuned from 1 S0 -1 P1 in the Sr cell. All optical beams are transferredto the vacuum chamber via single-mode polarization-maintaining (SM/PM) angle-polished(APC/APC) optical fibers. The beams can be switched on fast timescales with AOMs andare shuttered mechanically in addition. The mechanical shutters are necessary to prevent
leakthrough into the diffracted AOM order. The human eye is very sensitive to 461 nm, andleakthrough can easily be seen by looking directly into the output of the optical fiber with

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the AOM turned off. Even a few photons of 461 nm will affect optical dipole trap lifetimes
drastically. The optical cavity length locks are fairly sensitive to acoustical vibrations. For
this reason, the blue laser setup was built on a honeycomb breadboard isolated from theoptical table by rubber stoppers. This setup provides enough mechanical isolation to stepon the optical table without unlocking the cavities. However, placing the fast and very loudUniblitz mechanical shutters next to the cavities makes the locks oscillate for hundreds of
ms, even when not mounting them directly to the blue laser breadboard. The new SRS
SR474 shutters do not have a coil-driven blade design, but rely on a torque-balanced barthat swings in front of the shutter opening. The motion of the bar is PID controlled and the
shutter speed can be set. We find that fast speeds still introduce vibrations, but that we canmount these shutters on the blue laser breadboard next to the doubling cavities when using
one of the slower settings (∼40 ms shutter time).
The beam for the blue MOT is split into three arms using a Schäfter+Kirchhoff cagemounted free space splitter based on waveplates and polarizing beam cubes. The fiber
splitter is a custom design that uses set screws in the fiber holders to point the beams. The
splitter is mounted to the optical table in good thermal contact and it is contained in aplastic enclosure. The main advantage of the design is that it (in principle) does not have to
be touched after initial alignment. Even though there are no thumb screws in the design, we
find that slow mechanical drifts deteriorate the fiber coupling between input and output onlong timescales. The transfer efficiency has degraded over the last three years and we havenot been able to recover it completely even with careful walks of the set screws. We conclude
that a full alignment of these systems is necessary every one or two years. To do that, one
needs CCD cameras that can be mounted to the output ports so that the different arms canbe aligned simultaneously by turning the set screws systematically. It remains to be seen if
this procedure can be done repeatably and without too much time investment. Nevertheless,with the fiber splitters we use for both the blue and red MOT beams, we typically do nothave to align the MOT beams at all.
Repumps. As mentioned in Sec. 1.1, we use repumping lasers on the 3 P2 -3 S1 and 3 P0 -3 S1

118transitions at 707 and 679 nm, respectively. The light is produced by two laser diodes (shownin the bottom right part of Fig. 5.5), similar to the 922 nm laser described above.
Until recently, the only 707 nm diode available was an AR-coated Toptica model. Unfortunately, 707 nm is a window of low efficiency for InGaAs-based laser diodes and even with
continual improvements over the last few years this model had low power and was muchless frequency-stable than all other diodes we use. Since then, we have switched to the new
Hitachi/Opnext HL7001MG diode with higher power and nominal wavelength of 705 nm.
This diode seems to be very reliable and is also used in the 698 nm spectroscopy lasers for the
optical clock experiments. The 679 nm light is produced by an AR-coated Hitachi/Opnext
HL6750MG.
Each day, the wavelength of the repump lasers is set to a particular value on a wavemeter.
The lasers are then tuned until we observe the largest enhancement in blue MOT fluorescence.
Without further stabilization, the laser wavelength is stable enough for several hours ofoperation. Both lasers are combined and launched into the same optical fiber. Since enough
optical power is available, we form a 1 cm diameter beam from a telescope mounted directly
to the vacuum chamber and no alignment is necessary.
Probe beam and absorption imaging. The zeroth order of the MOT AOM is coupled
into another fiber and goes to a probe setup closer to the optical chamber (see mid left ofFig. 5.5). The probe beam is used for absorption imaging of the atomic clouds in either the
red MOT or the dipole trap. A full overview of this commonly used technique is beyond
the scope of this Section, but good technical references to learn about the relevant details
are [161–163]. Appendix A summarizes some other relevant considerations.
The probe beam setup uses a double-passed AOM and a fast Uniblitz shutter and enters afinal fiber going to the “Probe in” port of the vacuum chamber. The imaging optics are a customized version of the Infinity Photo-Optical K2/SC long working distance microscope. The
front lens of the microscope is at ∼16 cm from the atomic cloud. We tested the microscopewith transmissive test targets and blue interference-filtered white light (10 nm bandwidth)
and recovered imaging resolutions of 4 µm even when including a vacuum viewport (AR119coated as the ones on the vacuum chamber). Due to spatial constraints, the microscope isseparated into the imaging lenses and a custom eyepiece mounted to the CCD camera, witha mirror in between.
We use an Andor iKon-M CCD camera with 1024 × 1024 pixels of 13 µm width. The
imaging system magnification corresponds to 2 µm/pixel in the image plane. All our images
so far are consistent with the 4 µm test-target resolution. The CCD chip on these cameras
can be TEC-cooled to −70 ◦ C without additional water cooling. Electronic noise at thesetemperatures is drastically reduced, but the cold temperatures require the chip to be invacuum. One of the main technical issues in absorption imaging with coherent laser light isfringing induced by interference between optical surfaces in the imaging path. The vacuum
window covering the CCD chip is especially susceptible to fringing. The window in ourcamera is custom AR coated for both 461 nm and 689 nm and the front surface is wedged at1◦ . With these precautions, fringing was very significantly reduced versus a camera coatedfor Rb absorption imaging that we had been using previously. The fringes change very slowly
with thermal drifts in the optical surfaces responsible. In principle, the fringe patterns canbe recognized and removed using image recognition techniques. We found that we could
reduce the residual fringe contrast by a factor of 3-4, but that the technique requires about
30 images taken under the same conditions to work reliably. With the AR coated wedged
window, fringing has been much less of an issue.
Absorption images are taken in the camera’s “Fast Kinetics” mode that allows splitting
the CCD into several regions. Images can be taken in the first region and then shiftedto the adjacent region with high speed. The next image can then be taken and the CCD
chip is shifted again. In this way, we take three images in quick succession. To prevent
contamination of the earlier images in the sequence, two thirds of the CCD chip are masked
off with a razor blade on a linear translation stage in a custom part mounted to the front of
the camera. A 1 inch diameter Uniblitz shutter is mounted to the front of the razor blade
mount and the K2/SC eyepiece mounts to the shutter. The eyepiece also contains a CVI
10 nm bandwidth interference filter centered at 461 nm to reduce stray light contamination.

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The filter does not seem to influence the shape of the absorption images we measure.
To initiate taking an absorption image, a dark-field image is taken first and the image isshifted once while the camera shutter is closed. The actual absorption image of the atomic
cloud is taken next. The camera shutter opens first (∼8 ms opening time), and the probe
shutter opens 3 ms before the probe AOM is gated for 50 µs to obtain the shadow imageof the atomic cloud. The probe shutter closes but the camera shutter remains open as theatomic cloud drops away under the influence of gravity and the image is shifted. The probe
shutter opens again and the AOM is gated to take the light-field image. Finally, the probe
shutter closes, the camera shutter closes, and the full CCD is downloaded from the camera
to the imaging program. We use a custom imaging program written in C++ that processes,
displays and stores the images on a file server. Using multithreading techniques, the program
can handle the full 1024 × 1024 pixel images (∼2.5 Mb 16-bit TIFF) at ∼1 s experimental
cycles.
The double-pass AOM, the fast shutter, and the final fiber in the blue probe setup are used
to minimize the effect of leakthrough of blue photons towards the atoms and the imaging
system. Leakthrough becomes a problem because of the finite probe shutter opening timeand charge builds up for each photon impinging on the CCD. The camera provides a cleaningmode that continuously shifts the CCD chip to remove these charges. This method is notparticularly useful in Fast Kinetics mode, however, since it cannot handle charge buildup in
between taking partial images. We tested for leakthrough streaking (blue photons hitting
the camera while the image is shifted) in between taking the shadow and light-field images
by running the imaging sequence without triggering the AOM. With the fast shutter and the
double-pass AOM, this effect was reduced to negligible levels.
Another important systematic effect in absorption imaging is forward scattering of blue
photons anywhere in the imaging path after the light has interacted with the atomic cloud [162].
We found that this effect can be drastically reduced by irising down the probe beam until
only a small region around the atomic cloud is illuminated on the CCD chip.

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5.4.3. Red lasers
The laser light for interacting with the 1 S0 -3 P1 transition is produced by diode lasers at
689 nm. The laser diodes used are Hitachi/Opnext models HL6738MG, AR coated in house.
The frequency stability necessary to interact with the 7.5 kHz wide intercombination line isderived from the Sr2 master laser described in Refs. [11, 12]. The top left part of Fig. 5.6
shows a schematic overview of this setup. The red master laser (dotted inset) is another
diode in Littman geometry that is stabilized to a high-finesse reference cavity and a saturated
absorption spectrometer (similar to the blue spectrometer above) based on a heated Sr cellfilled with Ar buffer gas [11]. For our purposes, we extract a laser beam from the master laserthat is at −40 MHz detuning with respect to 1 S0 -3 P1 in the Sr cell. In the Figure, α denotes
the laser used for the 1 S0 -3 P1 MOT, and β is used for the photoassociation experiment. The
MOT light is delivered to the vacuum chamber by another Schäfter+Kirchhoff fiber splitter(see middle right of the Figure) that also supports a second arm for a stirring laser in 87 Sr.
For both α and β, we form an optical beat note with the master light and phase-lock eachlaser by fast feedback to the laser diode current. For long-term stability, slow feedback to
the laser PZT actuator is also applied.
The local oscillator frequency for each phase-lock can then be tuned to set the detunings ofeach laser. The red MOT in 88 Sr requires only a single laser, but its detuning needs to change
from a broadband modulation to a single frequency [11, 18]. We use the following scheme totransfer atoms from the blue MOT to the red MOT. As the light from Zeeman slower, blue
MOT beams, and repump lasers is turned off with AOMs and mechanical shutters, the MOT
field gradient is switched from 50 Gauss/cm to 3 Gauss/cm. The gradient is kept constantfor 80 ms and then is ramped linearly to 10 Gauss/cm within 125 ms and kept at this value
for another 56 ms. At the end of this sequence, the magnetic field gradient and the red lightare switched off and atoms are loaded into the optical dipole trap. Residual magnetic fields
are zeroed by three pairs of compensation coils (see Figure).
As the magnetic field gradient switches from 50 Gauss/cm to 3 Gauss/cm, α light is applied
to the atoms and the α local oscillator is modulated to cover the Doppler-broadened velocity

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Figure 5.6.: Schematic overview of the 689 nm red laser system for cooling and trapping.
See the main text for details. The top part of the figure shows a schematic of the phaselocked diode lasers. The bottom part shows the magnetic coils and laser beams in aschematic top view of the vacuum chamber.

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profile in the ∼1 mK atoms released from the blue MOT. The modulation is produced by
the setup shown in the top right part of Fig. 5.6. An SRS DS-345 oscillator produces asinusoidal voltage at 36 kHz and its amplitude is controlled by an analog voltage from thecontrol computer. The sinusoidal voltage is summed into the control voltage of a VCO at∼120 MHz. The mean frequency of the VCO is stabilized by slow PI feedback to the control
voltage with a bandwidth that is smaller than the modulation frequency from the DS-345.
The error signal for the feedback loop is generated by a digital phase detector that compares
the output of the VCO to a DDS oscillator. The frequency of the DDS oscillator is controlled
by the control computer and it is referenced to the lab frequency reference (a Wenzel crystal
oscillator). A window comparator checks whether the modulation amplitude is small andswitches between the DDS oscillator and the stabilized VCO automatically. In this way, the
α local oscillator is a frequency-stabilized signal at 120.7 MHz with a kHz-level sinusoidal
frequency modulation. If the modulation is small, the signal becomes a very clean digitally
synthesized sine wave automatically.

As the magnetic field ramps between 3 and 10 Gauss/cm, the modulation peak-to-peakamplitude is decreased from 2.7 MHz and the center frequency is tuned closer to the atomic
line so that the blue edge of the modulation spectrum is fixed at −700 kHz with respect to1

S0 -3 P1 . At the end of the magnetic field ramp, the modulation amplitude is zero and α

remains at a single detuning of −700 kHz.

The second laser β is used for the photoassociation experiment described in this Chapter,but will function as a stirring laser in future experiments with 87 Sr [11, 20]. The local oscillator is derived from a referenced analog signal generator controlled by the control computer.
The phase-lock easily supports local oscillator jumps on the 10 MHz level and we can tune
β by about 1.5 GHz by slight adjustments to the PID gain. We find that tuning the local
oscillator frequency is much more reliable than actuating on the AOM before the opticalfiber.

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5.4.4. Dipole trap & photoassociation lasers
The optical dipole trap is formed by two crossed laser beams at 1064 nm. The light is
produced by a highly frequency-stable (kHz bandwidth) Innolight Mephisto. The setup is
shown schematically in the top left part of Fig. 5.7. The Mephisto produces a linearly polarized beam and we use 30 dB of free-space optical isolation (OFR/Thorlabs) before coupling
the beam into the input fiber of a Nufern fiber amplifier. When seeded with ∼100 mW oflight, the amplifier produces up to 40 W cw light at 1064 nm and maintains a frequency
bandwidth of 10 kHz. The output fiber is a large-mode-area armored optical fiber spliced
to a water-cooled 30 dB fiber isolator. The setup produces a fairly good TEM00 with good
linear polarization. The amplifier itself is contained in a water-cooled rack-mounted box.
Due to space constraints and safety considerations, the laser system was built on an optical
table enclosed in blackened aluminum sheeting removed from the main experiment.
The setup shown in Fig. 5.7 splits the amplifier’s output into two optical beams that arecontrolled by two AOMs and coupled into two SM/PM APC/APC optical fibers that deliver
the light to the vacuum chamber. The fiber coupling makes it necessary to add another 30 dB
of free space isolation to prevent the amplifier from shutting down due to backreflections.
The high optical power makes this system qualitatively different from all other laser systems
in this experiment. The mirrors used are high-power Nd:YAG mirrors from Thorlabs that
support both 0◦ and 45◦ incidence angles. All lenses and beam cubes are AR coated for1064 nm and lenses made of fused silica are used when possible to prevent beam pointing
drifts at high powers in BK7. Thermal power meters need to be used. Unfortunately, theamplifier mode shape depends on the amplifier current and we run the amplifier close tosaturation at 40 A. After ∼8 months of everyday usage at this current, we found that the
total output power has degraded to ∼32 W. Working with such high powers is impossible
without having some way to adjust the power down to manageable levels. For this purpose,
we added combinations of λ/2 waveplates, polarizing beam cubes and high power beam
dumps to harmlessly divert power without changing the mode profile.
We use high efficiency IntraAction AOM-302AF3 glass AOMs at 30 MHz with opposite

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Figure 5.7.: Schematic overview of the 1064 nm infrared laser system for the optical dipole
trap (ODT). See the main text for details. The left part of the figure describes the1064 nm system and delivery system. The top right inset describes an optional injectionlocked 689 nm laser to enhance the PA power available. The bottom right part shows aschematic top view of the vacuum chamber relevant for the OFR experiment.

126
orders to separate the optical frequency of both beams by 60 MHz. The AOMs are also
used to switch off the dipole trap beams and stabilize the power transmission through theoptical fibers. We have used these modulators very successfully for dipole traps at 914 nm
and optical lattices at 813 nm and obtained diffraction efficiencies above 90%. However,
we found that for high intensities, the diffraction efficiency tends to degrade on thermal
timescales. We added water-cooled baseplates to the AOMs to counteract this effect. Evenso, the AOM glass gets permanently damaged on timescales of several hours to days by opticalpowers above 15 W (even with 2 − 3 mm beam diameters). Specialized YAG modulatorsthat handle much higher intensities are available, but we found that other processes limit
the usable power and continued to use the IntraAction AOMs.
Since free-space propagation to the experiment was not an option due to spatial constraints, we had to bridge the distance with optical fibers. We required single-mode andpolarization-maintaining transmission to guarantee a well-defined and reproducible opticaltrap at the center of our chamber. This requirement limits the transmissible power severely.
Power transmission through single-mode optical fibers is limited mostly by stimulated Brillouin scattering (SBS), where light is scattered off phonons in the optical fiber [164]. Abovea certain threshold input intensity the transmitted power saturates and the rest is scattered
back. The SBS threshold scales linearly with fiber length length and is inversely proportional to the mode area. Typical Panda SM/PM fiber for 1064 nm (e.g. Nufern PM980-XP)
has an effective mode field diameter (MFD) of about 6 µm. We find that one can transmit
about 5 − 6 W through 5 m of this fiber. This is consistent with our finding that we could
transmit about 1 W through 25 m of fiber when trying to transfer a beam from an adjacentlab. Forcing more power into the fiber can result in slow degradation of the core, making
the fiber unusable. Specialized large-mode-area (LMA) fibers such as the one spliced to the
output of the fiber amplifier support higher SBS thresholds. The mode area is limited by
the desired spatial mode quality of the transmitted light. With larger mode area and longer
fiber length, more light leaks into higher transverse modes. We have tried working withseveral LMA fibers from Nufern, and have transmitted up to 12 W through 5 m of several

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fiber types with reasonable beam quality. At these powers, however, stable beam alignment
into the fiber tip becomes extremely critical. Even with the fiber launch in the enclosureand careful alignment, thermal fluctuations cause the beam pointing to vary slightly. At
these powers, any deviation from optimal alignment causes runaway heating in the fiber tip.
Typical FC connectors are constructed by epoxying the fiber core into the connector whichis then polished at the desired angle. With large heat load, the epoxy melts up to several
cm along the fiber and flows over the fiber core at the tip. The fiber loses transmission andthe epoxy burns the tip. We recleaved the fiber and machined a custom metal fiber holder
without epoxy. However, thermal contact between fiber tip and holder is still importantand in the subsequent test, the fiber tip itself melted. We decided to stay with the current
setup for the present experiment, but would recommend setting up such high power laser
sources with free-space coupling to the experiment if space allows. The great advantage offiber delivery are the well-defined beam profile and the spatially defined delivery point withrespect to the target at the center of the chamber. By rigidly mounting the fiber tip, pointing
fluctuations can be very small.
The fiber delivery system is sketched in the bottom left part of Fig. 5.7. We find that
adding both a λ/4 and λ/2 waveplate is necessary to compensate for slow changes in thefiber birefringence over time scales of several hours, even when the polarization was carefullyaligned with the fiber axis. The polarization at the output of the fiber is defined by the mount,but we find it expedient to include another λ/2 waveplate to optimize transmission after thefiber was unplugged. A polarizing beam cube defines the polarization completely and allows
servoing on the transmitted intensity independent of polarization drifts. An AR-coatedwedge picks off a small fraction of the intensity which is detected on a photodetector (PD).
The whole fiber launcher, waveplates, cube, photodetector, and beam diameter adjusting
lenses (not shown) are built in a cage-mounted system for mechanical stability. On the local
end, the photodetector voltage is compared against a stable reference and a PI servo feeds
back on the amplitude of the RF signal that drives the AOM. We also include a high-isolation
RF TTL switch for switching times limited only by the acoustic wave transfer time through128
the AOM crystal. Most optical fibers shown in Figs. 5.5 and 5.6 have similar setups to theone shown here.
Finally, the delivered ODT beam has to be shaped to provide the desired focus at a given
distance from the fiber tip. The beam shape can be measured and optimized easily with a
USB CCD camera on a linear translation stage in combination with a program that displays
and fits the mode in real time. To be useful for optimizing beams in a real experimental
setup, it is important that the CCD chip is very small and lightweight; DataRay sells sucha system. For almost the same price as the Thorlabs singlets, good 1 inch diameter singlets
or achromats are available from Optosigma or Newport. Lens quality matters especially forgood 461 nm beam quality over long distances. Lens quality becomes less crucial with larger
diameter and longer wavelength, and we found that the 2 inch diameter IR achromats by
Thorlabs produce reasonably aberration-free foci down to waists of 15 µm at the distances
required to focus on the atomic cloud from outside the vacuum chamber.
We also built an additional 689 nm diode laser that can be injection-locked by the lightdelivered from β for higher PA intensities. The setup is shown schematically in the top
right part of Fig. 5.7. The PA intensity is adjusted by placing neutral density filters before
the final fiber transmission to the chamber. This setup allows changing the PA intensity
by well-defined amounts over many orders of magnitude without changing the delivered PA
beam pointing and transverse mode.

5.4.5. Dipole trap geometry
As discussed in section 5.4.3, we prepare 88 Sr atoms in a MOT operating on the 1 S0 -3 P1
intercombination transition and obtain atomic clouds at typical temperatures of a few µK.
While the MOT is operating, the atomic cloud is overlapped with the dipole trapping beams
at λXODT = 1064 nm. After the MOT is switched off, ∼5 × 104 atoms remain in the crossed
dipole trap (XODT). The trapping geometry is shown in Fig. 5.8, where the XODT is formed
by a horizontal (H) and a vertical (V) beam which intercept in the x̂ − ẑ plane. The trapping
beams have 1/e2 waists wH = 63 µm and wV = 53 µm, respectively.

Potential (µK)

129

20
10
0
−100−50 0

50 100Potential (µK)

V beam coord. (µm)

20100−100−50 0

50 100

H beam coord. (µm)

Figure 5.8.: Geometry of the experiment in the absorption image plane. The directions
x̂, ŷ, and ẑ define the lab frame, where both gravity ĝ and bias magnetic field B̂ areparallel to ẑ. Symbols k̂ are beam directions, and ǫ̂ are beam polarization vectors, wheresubscripts H, V , and PA indicate horizontal, vertical, and PA beams. H and V Gaussian
beam profiles are shown in blue outline, PA Gaussian beam profile in red outline.

130
The drawing is shown in the focal plane of the absorption imaging system discussed inSec. 5.4.2 and the H-V trap is always adjusted to the same spot with respect to the MOT
before optimizing the loading from MOT to dipole trap for the day. Guaranteeing that the
XODT forms at the foci of the H and V beams is not critical because of their large Rayleighranges zRi ≡ πwi2 /λXODT > 8 mm, and the Gaussian beam isosurfaces at the beam waist arealmost cylindrical (blue outlines).
Gravity points along −ẑ and sets the trap depths of ∼7 µK and ∼15 µK along V and
H. The two graphs on the right hand side show cuts through the model potential, which
has been adjusted to match the trap frequencies measured via parametric resonance. An
isosurface (dark blue) of the model potential at 7 µK is shown in the zoomed-out portion.
The model potential is given by
U (x) = UH (x) + UV (x) + mSr gz,(5.23)

where Ui are the Gaussian beam potentials in the lab frame, given by
Ui (x) ≡ −UTi exp




2 
2
2
− 2 [ǫ̂i · x] + [(k̂i × ǫ̂i ) · x].
wi

(5.24)The trapping beams point in the directionsk̂H = − cos θH x̂ − sin θH ẑ,(5.25)k̂V = − cos θV ẑ + sin θV x̂,with θH = 16.0◦ and θV = 14.4◦ . Both beams are linearly polarized along ǫ̂H = ǫ̂V = ŷ,and their optical frequency differs by 60 MHz so that we can neglect any optical interference patterns distorting U . A bias magnetic field of Bz ≃ 100 mGauss defines the atomic
quantization axis.
The absorption image plane is spanned by cos π8 x̂ − sin π8 ŷ and ẑ. The PA beam (redoutline) propagates in the x̂− ŷ plane the along the horizontal axis of the absorption imaging
system,k̂PA = − cos θPA x̂ + sin θPA ŷ,(5.26)

131
with θPA = π/8. The PA beam waist is wPA = 41 µm and it is linearly polarized along theatomic quantum axis ǫ̂PA = ẑ. Typical kinetic energies are 2-4 µK, and in-trap cloud FWHM√
are 45-55 µm. Although the PA beam FWHM 2 2 ln 2 × wPA ≃ 97 µm is larger than theRRtypical cloud FWHM, we use a density-averaged intensity Iav = d3 xρ(x)I(x)/ d3 xρ(x)to characterize the PA intensity interacting with the atoms. Typical values are Iav ≃ (0.6 −0.7) × Ipk , where Ipk = πw2P2 is the Gaussian beam peak intensity for total beam power P .
PA

The PA beam also adds to the optical trap slightly because of its relatively small detuningfrom the atomic transition. For large intensities and small wPA and especially in a standing
wave configuration, this effect can become important. The additional trap depth introducedby the PA beam here is typically < 0.1 µK and we neglect it.

5.5. Trap potential calibration
The 88 Sr sample is loaded from a thermalized distribution in the MOT into a conservative
XODT potential. Although the sample is thermalized well by the MOT photons initially,
the 88 Sr background scattering length is so small that the sample behaves as an ideal gas on
experimental timescales. For this reason, inhomogeneities in kinetic energy persist and theergodic gas dynamics are completely determined by the XODT potential. For the temperatures and typical trap depths considered here, the sample is trapped in the potential welldefined by U . Nevertheless, the trap is not deep enough such that we can easily approximate it as harmonic. We calibrate the model potential by comparing its eigenfrequencies formeasured beam powers against parametric resonance measurements for each axis.
The trap eigenaxes are determined by diagonalizing the symmetric curvature matrix (theHessian) of the model potential around its minimum:(Hess U |min )ij =∂ 2U|min .
∂xi ∂xj(5.27)The eigenvalues of (2πmSr )−1 Hess U |min are the trap eigenfrequencies. The waists wH and wVused in the model potential are adjusted until the eigenfrequencies match the values obtained
from parametric resonance measured along each beam axis. The waist values obtained

132
thus are consistent with the values measured outside the vacuum chamber when accounting
for slight aberrations introduced by passing through the vacuum viewports. The vacuum
viewports are anti-reflection coated for 461 nm, 690 nm, and 813 nm, since the vacuum
chamber was designed for optical lattice clock experiments. The horizontal beam passes
through a thin vacuum viewport at normal incidence and its intensity transmission coefficient
at 1064 nm was measured as TH = 0.69. The vertical viewport is made of much thicker fusedsilica than the horizontal viewport due to its large diameter. In addition, the vertical beam
passes the viewport at θV and we measure TV = 0.58. The beam powers entering the vacuum
chamber are stabilized via feedback from photodetectors to acousto-optical modulators. The
modulators are located before the optical fibers that transfer the 1064 nm light from thelaser source to the vacuum chamber and are also used to switch off the trap to expand theatomic cloud for time-of-flight measurements. Typical optical powers before the vacuum
chamber are PV = 1.15 W and PH = 1.79 W. The real part of the 1 S0 atomic polarizability
when neglecting all but the 1 P1 contribution is Re α1 S0 (1064 nm) ≃ 239 a.u. [11], where theatomic unit of polarizability is 1 a.u. = 4πǫ0 a30 . The individual beam trap depth is thengiven by [165]
UTi =

Pi
Re α1 S0 (1064 nm)
πcǫ0 wi2(5.28)Taking the viewport transmission coefficients into account and using the typical beam powers
and calibrated waists, we obtainx̂1 ≃ 0.984 x̂ + 0.175 ẑ,x̂2 = ŷ,(5.29)x̂3 ≃ −0.175 x̂ + 0.984 ẑ,
where we ordered the eigenaxes x̂i according to the lab frame axes they are closest to. The
eigenfrequencies are{ν1 , ν2 , ν3 } ≃ {219, 290, 187} Hz.

(5.30)The potential along the 2-axis is the most harmonic since the transverse potential of bothtrapping beams adds to produce a very deep trap. The potential along the horizontal axis is133
the next deepest, but becomes lopsided because of gravity. The potential along the verticaltrap axis is very anharmonic and gravity introduces a single saddle point where atoms canescape (see Fig. 5.8). This asymmetry in trap depths along the trap axes produces a persistentinhomogeneity along the horizontal and vertical axes, even when loading the trap from athermal sample in the MOT. Unless care is taken to align the XODT with respect to the
MOT every day, the loading process does not prepare a comparable phase space distribution
each day. The difference between potential minima in the MOT and the XODT determines
the persistent kinetic energy inhomogeneities in the cloud. We developed a careful alignment
procedure that set the relative position of MOT and XODT in the absorption image plane andthen optimized the relative magnitudes of horizontal and vertical kinetic energies determinedfrom time-of-flight measurements.

5.6. Modeling the phase-space distribution
To understand and model the initial atomic phase-space distribution, we performed MonteCarlo simulations of loading the conservative model potential U with N classical particles
from a thermal sample at a given temperature. Letting each atom in the sample evolve inthe conservative trap for several trap oscillation cycles consistently produced inhomogeneousdensity distributions. For the relevant temperature range the resulting inhomogeneous density and momentum distributions consistently separated into symmetric distributions alongeach trap eigenaxis, since no thermalizing collisions were allowed to couple the different
axes. We also find that each axial potential energy follows the corresponding axial kinetic
energy faithfully and conclude that the trap is ergodic on timescales longer than a few trap
oscillation cycles. The resulting phase space distributions fit well to independent Gaussian
distributions along each trap eigenaxis. Nevertheless, it is not adequate to approximate
harmonic confinement where the RMS axial cloud extent is proportional to the axial RMS
velocity via the axial eigenfrequency. Even for typical temperatures of 3 µK, the cloud islarge enough to explore the anharmonic parts of the trapping potential.

134
We model the density distribution aswith atom number N =Rn(x) ≡ N f (x),(5.31)

d3 x n(x), position distribution


Y
1
x2i
p
f (x) =
exp − 2 ,2σxi
2πσx2ii(5.32)and position variances σx2i ≡ hx2i i − hxi i2 , where the angled brackets indicate an ensemble
average. The in-trap momentum distribution is also modeled as Gaussian


Y
1
p2i
p
g(p) =
exp − 2 ,2σpi
2πσp2ii(5.33)

with velocity variances σp2i ≡ hp2i i − hpi i2 . Although we do not assume that there is a global
temperature, the relationship between σx2i and σp2i is given by the (ergodic) trap and can be
determined from the Monte Carlo simulation when assuming the correctness of our model
potential. In this model, the expression for the phase space density becomesρ(x, p) = N h3 f (x)g(p),(5.34)

with Planck’s constant h = 2π~ and peak phase space density
ρ0 ≡ ρ(0, 0) = N

Y
i

~
.
σxi σp i(5.35)

Typically, we refer to the position and momentum variances as the widths wxi and per-axis
temperatures Ti , defined as
w xi ≡ σxi ,(5.36)

kB Ti /2 ≡ σp2i /(2m),with Boltzmann’s constant kB and atomic mass m. In these terms, the peak density n0 andpeak phase space density ρ0 becomen0 ≡ n(0) =NQ(2π)3/2

Y  ~2 m
ρ0 = N
wx2i kB Ti
ii w xi1/2.

,(5.37)

135
For comparison: in a harmonic trap with trap frequencies ωi and interaxis thermal equilibrium at temperature Ti ≡ T , these expressions assume the forms
kB T /2 = mωi σx2i /2 = σp2i /(2m),Y  mω 2 1/2Yin0 = N=Nmωi λT /h,
2πkB Tii
Y ~ωi,ρ0 = NkB T
i(5.38)√
with thermal wavelength λT = h/ 2πmkB T .

In summary, we find that both density and velocity distributions can be modeled as independent Gaussians along each trap eigenaxis. Because the trap is anharmonic, the connection
between cloud extent and RMS velocity is not simply given by the axial eigenfrequencies. We
need to measure both the spatial and the momentum distributions to obtain a full descriptionof the atomic sample.

5.7. Measuring the phase-space distribution
We use an interleaved experimental sequence, where two runs of the same experiment are
performed consecutively. At the end of the first experiment, the atomic cloud is imaged
via absorption while the cloud is confined in the XODT potential. This image provides us
with information about the density distribution at the end of the experiment. Appendix A
summarizes some important effects that need to be taken into account to make in-situ imaging and its analysis reliable. At the end of the second experiment, the XODT potential isswitched off and the atomic cloud is allowed to expand for typically 1.5 ms before we takethe absorption image. The second image provides information about the kinetic energy distribution. The images determine the projection of the density distribution and the kinetic
energy distribution into the absorption image plane. The atomic cloud is typically exposedfor 50 µs of light on the 1 S0 -1 P1 transition with intensities of 10% of the saturation intensity
Isat ≃ 40 mW/cm2 .

136

5.8. Inelastic loss as relative momentum knife
In the regime of small optical length, we can neglect elastic scattering and view the inelastic
loss as a “relative momentum knife”. At a particular detuning, the inelastic loss only removes
particle pairs with certain relative momenta. To investigate the effect on the single-particle
momentum distribution, we need to find out which parts of the single-particle distribution
contribute to a certain relative momentum class.

5.8.1. Single-particle and relative momentum distributions
The relative momentum pr of particles 1 and 2 having individual momenta p1 and p2 ,respectively, is given bypr ≡ p1 − p2 .

(5.39)

We assume that both particles are distributed according to the same Gaussian probability
distributionpi ∼ N (µ, C).

(5.40)

We will also assume zero mean µ = 0 and that the distribution is anisotropic but that theaxes are decoupled, such that the covariance matrix C is diagonal and given by
hp21 i 0
0




2
C =  0 hp2 i 0  .
2
0
0 hp3 i(5.41)

Then the differential probability for finding a single particle of momentum pi can be written
as

1 i
⊤ −1
i
dP (p ) = f (p )d p =exp − (p − µ) C (p − µ) d3 pi
3/2
2
(2π)
det C
"
#
!
−1/2
3
3
i 2
Y
X
(p
)
1
j
exp −
d 3 pi .
2πhp2j i=2
2 j=1 hpj i
j=1
i

i

3 i

1
√



(5.42)

Since the relative momentum is a sum of Gaussian-distributed momenta, the differentialconditional probability dP (pr |p1 ) of finding a relative momentum pr when the momentum

137
of particle 1 is known is also Gaussian. Its mean as a function of p1 is
hpr i = hp1 − p2 i = p1 − hp2 i = p1 ,(5.43)and its covariances are twice the single-particle covariances:
hpri prj i − hpri ihprj i = 2δij hp2i i.

(5.44)

Thus the conditional probability is
"

#
X 4
1
dP (pr |p1 ) = f (pr |p1 )d3 pr ∝ exp −(pr − p1j /2)2 d3 pr .
2 j hp2j i j(5.45)

By integrating over the marginal variable p1 we verify that the probability distributionfunction for the relative momentum pr is"#Z
X (prj )2
f (pr ) = d3 p1 f (pr |p1 ) ∝ exp −
.
2
i
hp
j
j

(5.46)

To simplify the problem, we would like to find an expression for the conditional proba-

bility that only involves the magnitude of the relative and single-particle momenta. In thefollowing, we will assume isotropy, such that
hp2j i ≡ σ 2 .

(5.47)

We note that conditional and joint probabilities of variables A and B are generally relatedby [166]
P (A, B) = P (A|B)P (B).

(5.48)

Thus the joint probability for pr and p1 is
dP (pr , p1 ) = dP (pr |p1 ) × dP (p1 )(5.49)

and we integrate over momentum shells S and S ′ of radii pr and p1 , respectively, to find thejoint probability for the magnitudes:
Z
Z
r 1
3 1
dP (p , p ) =dp
d3 pr f (pr |p1 )f (p1 )
′


ZS
ZS
(p1 − pr )2
1 r 1 2 −(pr )2 /σ 2r
dp1 dprdΩ (p p ) e
exp −
∝ dΩ
σ2
2p1 pr1 r 2 −2(pr )2 /σ 2 −(p1 )2 /σ 2 sinh σ 2∝ (p p ) ee
dp1 dpr ,2p1 prσ2(5.50)

138
where dΩr and dΩ1 indicate the angular differentials corresponding to integration over S and
S ′ . Since the single-particle probability is given by11dP (p )dp =Z(p1 )2S

d3 p1 f (p1 ) ∝ (p1 )2 e− 2σ2 dp1 ,(5.51)

we obtain the conditional probability of finding a particle pair with relative momentum ofmagnitude pr for a given single collision partner momentum of magnitude p1 :
1 r2(pr )2 (p1 )2 sinh 2pσ2p
dP (p |p ) ∝ (p ) exp − 2 −
dpr .
1 pr
2
2p
σ
2σ
2
r

1

r 2



(5.52)σFigure 5.9 shows the relative momentum magnitude pr distribution conditional to a known
value of the single particle momentum magnitude p1 in a thermal sample. For p1 → 0,
the relative momentum distribution is Maxwell-Boltzmann with single particle temperature√
reduced by 2. For increasing p1 , the distribution approaches a Gaussian distribution aroundp1 . From this picture, we conclude that relative momenta for small single-particle momentum
are determined by the second particle. For large single-particle momentum, the relative
momentum increases. We can now use the conditional relative momentum distribution topredict the effect of relative-momentum dependent loss on the single-particle momentum
distribution.

5.8.2. Effect of relative-momentum dependent loss
We are interested in a scenario where collision pairs with a given relative momentum pr are
removed from the sample and would like to predict the effect of such a removal on the single
particle momentum distribution. In particular, we are interested in determining which singleparticle momenta out of the initial distribution can contribute to a given relative momentum.
We use Bayes’ theorem [166] to invert the conditional probability:dP (p1 |pr )dP (pr ) = dP (pr |p1 )dP (p1 ) = dP (pr , p1 ).

(5.53)

This form can then be used to determine the conditional probability dP (p1 |loss) of finding aparticle of momentum p1 after removal of particles with momentum pr . If we define the loss

139

1.2

dPHpr Èp1 L

1.0
0.8
0.6
0.4
0.2
0.0
0

1

2

3

4

5

pr Σ
Figure 5.9.: Distribution of relative momentum magnitude pr , conditional on value of singleparticle momentum magnitude p1 for an isotropic Maxwell-Boltzmann single-particle mo√mentum distribution with σ = mkB T . The single particle momentum p1 /σ = 0 for theleftmost trace and increases in steps of 1 towards 5 (rightmost trace).
probability floss (pr ), the survival probability of a particle with p1 after relative momentum
loss according to floss (pr ) is
Z
Z
1
r 1
1
r
dP (p |loss) ∝ dP (p |p )dP (p )[1 − floss (p )] = dP (pr , p1 )[1 − floss (pr )],(5.54)

where we integrate over the relative momentum pr . If we normalize the left hand side of
Eq. 5.54 with respect to p1 , we findR∞22xe−x dx 0 dy [1 − floss (y)]e−2y y sinh 2xy
R∞dP (x|loss) = R ∞,dx xe−x2 0 dy [1 − floss (y)]e−2y2 y sinh 2xy
0(5.55)

with dimensionless momenta x ≡ p1 /σ and y ≡ pr /σ.

Figure 5.10(a) shows the effect of relative-momentum dependent loss on the single-particle
momentum distribution. Here, we have assumed a Gaussian floss centered at different valuesof pr /σ with a FWHM corresponding to 15 kHz in frequency units at a temperature of 3 µK.
The single-particle distributions are renormalized after the loss process according to Eq. 5.55.
The green trace shows the effect of loss when setting the center of the Gaussian loss at pr /σ =

140
0. The initial single-particle distribution is almost unchanged, since particles are removed
uniformly. The purple trace shows the effect of loss when pr /σ = 3. The initial distributionis almost unchanged, but in this case, this is because there are not many single-particle
momenta contributing to relative momenta of this magnitude. The red trace corresponds topr /σ = 1. Here, many particles are removed close to the most likely momentum, and thefinal distribution is shifted to higher momenta, corresponding to a higher average kinetic
energy. The blue trace corresponds to pr /σ = 2. Here, single-particle momenta larger thanthe most likely momentum are removed and the sample becomes colder.
The fractional change in the norm of the distribution in Eq. 5.55 with respect to having
no loss process is equal to the fractional atom number change
Z
Z ∞
N
8 ∞
2
−x2
=1−
dx xe
dy floss (y)e−2y y sinh 2xy,N0π 00and moments of the distribution can be calculated as
Z ∞n
xn dP (x|loss).
hx i ≡(5.56)(5.57)

0

However, it is more convenient to also define fractional changes of such moments with respect
to having no loss process present:  −1
√
Z ∞
Z ∞
π
N
hxn i
n+1 −x2−2y 2
dx x e= 1 − 1+n/2
dy floss (y)ey sinh 2xy ×
n
hx i0
2
Γ(3 + n/2) 0N00(5.58)For instance, the fractional change in temperature due to the relative momentum dependent
loss process is
T
hx2 i
= 2 .
T0
hx i0

(5.59)

Figure 5.10(b) shows the fractional atom number and temperature change under the same
conditions as for panel (a), when scanning the mean target relative momentum pr /σ. Most
atoms are removed when pr /σ targets the most likely single-particle momentum and thesample gets heated. For larger relative momentum, the sample becomes colder.
In conclusion: The relative momentum knife picture derived here describes removal ofatoms with a given relative momentum magnitude. Depending on which class of atoms are

141

0.6

dPHp1 ÈlossL

0.5
0.4
0.3
0.2
0.1
0.0
0

1

2

3

4

5

p1 Σ
(a) Renormalized single-particle distribution after loss according to a Gaussian loss probability floss with FWHM corresponding to a frequency widthof 15 kHz at 3 µK. The loss is centered at pr /σ = 0, 1, 2, and 3 for thegreen, red, blue, and purple traces, respectively.

(b) Fractional atom number (temperature) change N/N0 (T /T0 ) shown as
the red (blue) trace while scanning the loss center for the same conditions
as in panel (a).

Figure 5.10.: Effect of relative-momentum dependent loss on the single-particle momentum
distribution in an isotropic thermal sample.

142
removed, the sample gets heated or cooled. This simple static probabilistic model gives
insight into the effect of instantaneous particle removal. However, it cannot describe thesample dynamics for time scales that are longer than the typical trap oscillation periods
since the trap acts as a phase-space randomizer. To model the sample dynamics underrelative-momentum dependent particle removal, we need a model that can account for thetrapping potential.

5.9. Inelastic loss from a Gaussian distribution
The treatment here is an abbreviated version of unpublished notes by T. Nicholson; see also
the online material of Ref. [10]. As argued in Sec. 5.1, we can describe the density evolution
by a nonlinear differential equation involving the momentum-averaged inelastic collision rate
constantṅ = −K̄in n2 − n/τbg ,(5.60)

where we have included a single-atom loss via the 1/e trap lifetime τbg . This equation canbe integrated to obtain the density evolution with time t:n(x, t) =n0 (x),
1 + K̄in τeﬀ n0 (x)(5.61)

with effective time scale τeﬀ = τbg (e−t/τbg − 1) and initial density distribution
n0 (x) = N0

3
Y
i=1

1
p

2πhx2i ie−x2i /2hx2i i

≡ npk

3
Y

2

2

e−xi /2hxi i ,(5.62)

i=1

with peak density npk . The fractional atom number at time t is then
Z
Z ∞
Z
3 q
Y
N (t)
3
−1
2
dr r2 n(r, t)
hxi i dΩr
= d x n(x, t) = N0N00i=1(5.63)
rZ ∞2 −1
dr r2 n(r, t),n≡
π pk 0pwhere we have defined rescaled coordinates ξi ≡ xi / hx2i i and the rescaled radius r ≡pP
2i ξi . In terms of these coordinates, the fractional atom loss becomes
r Z ∞
√ −u
Z ∞
2
2
ue
2
r2 e−r /2
N (t)(5.64)√,=drdu2 /2 =
−r
N0
π 0
π 0
1 + K̄in τeﬀ npk e
1 + K̄in τeﬀ npk e−u143with u ≡ r2 /2, which makes the integral more convenient for numerical calculations.
If we allow inhomogeneous kinetic energy distributions, the momentum-averaged inelastic
rate can be quite complicated to calculate. To simplify the integrals to a point where fitting
PA spectra becomes feasible, we have to assume a mean kinetic energy for the sample andperform the momentum average as if there were a single temperature describing the sample.
Under this assumption, the thermally-averaged inelastic collision rate can be written asZ√γ η e−η4hℓopt γm ∞K̄in (∆, u, ℓopt γm , T ) = √,dη 2
D + Γ2 /4
πµ
0
kB T
D ≡∆+
η − νrec − νs e−u/u0 ,h√Γ ≡ γ + 2kth ℓopt γm η,(5.65)

2
with dimensionless relative momentum magnitude η ≡ k 2 /kth
, thermal momentum ~kth =√2µkB T , center-of-mass PA photon recoil energy hνrec , and trapping laser ac Stark shift at

the center of the trap νs . The PA laser with optical frequency νl is detuned from the PA
resonance by∆ ≡ νl − [ν(1 S0 − 3 P1 ) + νn + ∆ν(I)],(5.66)

where ν(1 S0 − 3 P1 ) is the atomic transition frequency, and hνn is the energy of state n withrespect to the free threshold. The detuning term ∆ν(I) accounts for the ac Stark shift ofthe molecular resonance with respect to I.
For the fitting routine, we approximated the integral in Eq. 5.64 (Eq. 5.65) as a 10-term(53-term) sum using Gauss-Laguerre quadrature [167, 168]. The quantities (ℓopt γm ), T , u0 ,and a term added to the detuning to represent the line center were allowed to vary. The
parameter T is used as a check against the experimentally measured temperatures and agrees
well with the average of the axial temperatures. The Stark shift term νs e−u/u0 was included
to account for the broadening of the atomic loss profile to the blue side of a molecularresonance due to the ac Stark shift induced by the trap.

144

5.10. Results for small optical length
A typical PA loss feature for small ℓopt is shown in Fig. 5.11(a), where the final atom number
after application of PA light is shown with respect to PA detuning from 1 S0 -3 P1 . The peraxis kinetic energies for this scan correspond to a horizontal (vertical) temperature TH (TV ) =2 µK (3 µK) resulting in the typical thermal tail towards the red side of the resonance [169].
The solid line is a fit with the inelastic collision model derived in the last Section, for a
vacuum-limited trap lifetime τbg ≃ 1.3 s. We extract ℓopt and the position of the line center.
Panel (b) shows the time dependence of the same loss process. In the presence of resonant
PA light, we clearly observe two-body loss. To fit the decay curve, however, it is necessary
to perform the same thermal averaging process as for the lineshape to obtain a good matchwith the experimental data.
The measurements were performed for a range of optical lengths by adjusting the density
profile averaged PA intensity Iav . Multiple molecular resonances were measured and results
for n = −2 are shown in panel (c). The optical length data is fit with a linear function to
extract ℓopt /Iav , and the results are summarized in the table at the bottom. The fit coefficientℓopt /Iav is given by the free-bound Franck-Condon factor and decreases drastically withdecreasing n. Panel (d) exemplifies similar measurements done to determine the line shift∆ν of the same PA resonances with Iav . To obtain good signal-to-noise on these frequency
shifts, we decreased the PA exposure time and increased Iav to maintain a constant pulse
energy. Linear shift coefficients ∆ν/Iav and zero intensity line positions νn with respect tothe atomic transition are also shown in the table. The sign and magnitude of ∆ν/Iav are
consistent with the predictions in Ref. [144].

5.11. Elastic contributions to the cross section
For larger optical lengths, the elastic contribution to the scattering cross section becomes
important. In this regime, the simple inelastic loss picture becomes insufficient to describe
OFR spectra. Elastic collisions introduce cross-dimensional thermalization and (on longer

105
Final atom number

6

4

2

a
0

−24.5
−24
−23.5
PA laser detuning (MHz)

c

150
100
50
0

0

0.004
Iav (W/cm2 )

0.008

No PA
PA, -23.9 MHz

104

b
0

∆ν (MHz)ℓopt (a0 )Final atom number (104 )

145

0.5
Hold time (s)

0

1

d

−0.1
−0.2

0

0.2

0.4
0.6
Iav (W/cm2 )

0.8

γm
n ℓcc
ℓopt /Iavνn∆ν/Iav
opt /Iγ ℓopt /Iav
a0
a0
a0
kHz
( W/cm( W/cm( W/cm(MHz)( W/cm2)
2)
2)
2)-2 8.3 × 103 15.8(8) × 103 5.9(3) × 103 -23.98(1) −(251 ± 67)
68(3)
26(1)
-222.18(5)+(29 ± 2)
-3 38
-4 32
40(2)
15(1)
-1084.12(5)+(4 ± 1)Figure 5.11.: Photoassociative spectroscopy for n = −2. a, Typical PA loss feature in the
low intensity regime at Iav = 7 mW/cm2 , with density-profile averaged PA intensity Iav .
b, Time evolution of the trapped sample without PA light (blue squares) determines τbg .
The dynamics with resonant PA light (red circles) show distinct two-body loss curves thatcan be fit with a thermally averaged model (solid red curve). c, Linear increase of the
optical length ℓopt with increasing Iav for sufficiently small ℓopt . d, Molecular line center
shift ∆ν for large intensities. Both graphs are obtained by fitting loss spectra such asin panel a. For each n, OFR parameters from the coupled-channels calculation and theexperiment are summarized in the table at the bottom. Here, ℓcc
opt indicates the theoreticalresult, νn is the zero-intensity molecular line center with respect to 1 S0 -3 P1 , and ∆ν/Iavcharacterizes the molecular ac Stark shift with respect to I [144].

146
timescales) evaporation into the sample dynamics.
We show the atom loss with respect to PA laser detuning for n = −2 in Fig. 5.12(a). Bothin-trap size and kinetic energy are measured by absorption imaging. Far detuned from theresonance, the gas is almost ideal, as shown by the persistent kinetic energy inhomogeneity
along H and V in Fig. 5.12(b). In the center of the resonance (vertical dashed line), inelastic
collisions dominate and cause heating. For red detuning from the molecular resonance, thetemperatures approach each other.
The measured cloud widths wH and wV confirm that the potential energy follows the
kinetic energy [see Fig. 5.12(c)] since particles oscillate in the trap many times between
inelastic collisions. Similar measurements were performed for n = −3 and n = −4, andwe find that the same dispersive behavior in temperatures and widths appears around2hkiℓopt γm /γ ∼ 30% at 200 ms PA exposure times.
The data can be understood by a simple picture of competing elastic and inelastic collision
rates. As discussed in Sec. 5.2, the elastic and inelastic rates average differently with k in athermal sample, causing them to peak at different detunings from the molecular resonance.
Elastic collisions cause cross-dimensional thermalization [170] and tend to equalize TH and
TV . Since inelastic collisions predominantly remove cold atoms from the most dense partof the cloud, the resulting loss increases the system energy – a process referred to as antievaporation.
This behavior is confirmed by a Monte-Carlo simulation, where 5.5 × 104 classical particles
are simulated and each particle undergoes elastic and inelastic collisions according to the crosssections in Sec. 5.2. The simulation is described in detail in Appendix B. The solid lines
overlaid on the experimental data in Fig. 5.12 are the results of such a simulation. An average
ratio of elastic to inelastic collisions per particle from the simulation is shown in Fig. 5.12(d).
The dispersive shapes are sensitive to γ. Combined with the low ℓopt data in Fig. 5.11, theentire simulation reproduces the experimental data only when γ = 2π × 40(5) kHz > 2γa =15 kHz without other free parameters. We have ruled out magnetic field or PA laser noise
as a source of broadening. Instead, we conclude that this extra broadening is related to aa

red det.

7

blue det.

Kinetic energy (µK)

6
5
4
3
2
1
0

d

1.2

0.8

0.4

6

TH
TV

5
4
3

−24.5
−24
−23.5
PA laser detuning (MHz)hKel /Kini

1.6

b

2
−24.5
−24
−23.5
PA laser detuning (MHz)In-trap FWHM (µm)

Final atom number (104 )

147

70

c

wH
wV

60

50

40
−24.5
−24
−23.5
PA laser detuning (MHz)

−24.5
−24
−23.5
PA laser detuning (MHz)Figure 5.12.: Elastic contribution to the scattering cross section. Experimental data forn = −2 at Iav = 22 mW/cm2 (open circles) and results of a Monte-Carlo simulation (solidlines) using the single-channel elastic and inelastic cross sections in a crossed Gaussian
beam potential including gravity for ℓopt γm /γ = 140a0 . (a) Atom loss as a functionof PA laser detuning from the atomic 1 S0 -3 P1 resonance. In panels (b) and (c), blue(red) data points and solid lines indicate the corresponding quantities for the vertical(horizontal) trap axis. (b) Change in kinetic energy derived from time-of-flight images,and (c), potential energy change corresponding to varying in-trap density full widths at
half maximum (FWHM). (d) Simulated ratio of elastic and inelastic collisions per particle,
averaged over the PA exposure time.

148
faster molecular decay rate, which is consistent with our earlier measurements [137] and Rb
results [133].
We conclude that the isolated resonance approximation universally describes OFR in thevicinity of each resonance. We have validated the linear line strength scaling and linear
resonance shift with I and have observed a clear modification of both the inelastic andelastic parts of the scattering cross section.

5.12. Coupled-channels calculations
Our experimental work on optical Feshbach resonances was paralleled by detailed coupledchannels calculations by P. S. Julienne [10, 171]. The calculations predict the changes in realand imaginary parts of the scattering length using a model potential that only includes thenatural radiative decay. Single-channel optical length expressions can be fit to the numericalresults in the vicinity of each resonance. The optical length values extracted from thesefits are shown as ℓcc
opt in the Table in Fig. 5.11. The agreement between our data andthe theoretical predictions is fair; the theoretical value lies in between the experimentally
determined values for ℓopt and ℓeﬀ = ℓopt γm /γ.
As shown in the previous Sections, the single-channel theory has been very successfully
used to describe both inelastic and elastic cross sections in the vicinity of three different
molecular lines. The initial motivation for a more accurate description of the OFR process
was to extend the theoretical description beyond the single-channel model into the fardetuned regime.
In addition to the vibrational levels investigated here, the n = −1 vibrational state exists
at -0.4 MHz detuning from the free threshold, which leads to a PA resonance with a verylarge line strength ℓopt /I [137]. The isolated resonance theory indicates that operating withlarge I at O(105 γa ) detuning from the n = −1 state should allow modifications to thescattering length of O(100a0 ) [136] with very low inelastic losses. However, with intensities
up to 1 kW/cm2 we did not observe changes to the thermodynamic properties of the gas,

149
in contrast to theory predictions. The theory proposal relied on extrapolating the large line
strength of the n = −1 state to large detunings across multiple intermediate PA resonances.
This assumption seemed valid since the n = −1 resonance is several orders of magnitude
stronger than the intermediate resonances [137].
The discrepancy between theory and experiment stimulated a coupled-channels treatmentof an atomic collision in a radiation field that properly switches between the short range
molecular states and two field-dressed separated atoms [172, 173]. The two coupled excitedpotentials (0u , 1u ) have the form of Ref. [137] and the decay of the excited state is treatedby introducing an imaginary term −i~γm /2 in the molecular potential. The ground state
potential uses the dispersion coefficients of Ref. [174], has a scattering length of −1.4 a0 , andreproduces the bound state data of Ref. [140] to better than 0.4%. The coupled-channels
calculation does not assume isolated resonances, and all 0u and 1u molecular eigenstates
emerge from the calculation as interfering, decaying scattering resonances [121]. The resultsof the coupled-channels calculation are shown in Fig. 5.13.
Figure 5.13(c) and (d) show that the coupled-channels model reproduces the isolated
resonance expressions [121, 130] for the complex scattering length and the rate constants, aslong as the detuning is small compared to the spacing between molecular levels. However,
the elastic collision rate constant returns to its background value in between resonances
regardless of their relative strengths, as indicated by the dotted line showing the backgroundKelbg at E/kB = 4 µK in Fig. 5.13(a). The same is true of the real part of the scattering
length, as indicated by the dotted line for zero-energy abg in Fig. 5.13(d). These calculations
show that the OFR effect does not extrapolate across an intermediate resonance, even amuch weaker one, but that each molecular line behaves as an isolated resonance near its line
center.
It is instructive to write the complex scattering length in the limit of large detuning,|∆| ≫ γ + Γs , in the standard form for magnetic Feshbach resonances


w
i wγ
lim α(k) = abg 1 − +,k→0∆ 2 ∆2(5.67)

where the resonance width w ≡ −ℓopt γm /abg . To obtain a meaningful change in scattering

150

3

K (cm /s)

10
10

2

-10

-12

a

E/kB = 4 µK, I=10 W/cm
0u(-4)

1u(-1) 0u(-3)

Kin
Kel

-14

0u(-2)

10 bg
Kel
-1

Ratio Kel/Kin

1.6

0u(-2)

-0.8

b

1.2
0.8
0.4

-40
0
40
Detuning ∆/γm

-0.6

-0.4

-0.2

PA laser detuning (GHz)
(a0)
(a0)
0u(-4)
3 0u(-4)
c
d
10
5
-a
a
2
10
0
10
b
-5
2
1
53 W/cm
0.1

0

-80
0
80 -2000
02000
Detuning ∆/γm
Detuning ∆/γmFigure 5.13.: Coupled channels calculations, Figure courtesy of P. S. Julienne. (a) Elastic
and inelastic rate constants at E/kB = 4 µK and I = 10 W/cm2 , versus PA laser detuningfrom atomic resonance. Each resonance peak is labeled by its electronic symmetry 0u
or 1u and its vibrational quantum number. Between resonances, the coupled-channels
Kin is only approximate. (b) Ratio of the thermally averaged rate constants at 2 µK
for detuning ∆/γm near the 0u (−2) feature. The intensity I = 44 mW/cm2 to give thesame ℓopt = 360a0 as for the conditions in Fig. 5.12(b). (c) Zero energy limit of realand imaginary parts of the scattering length for detuning near the 0u (−4) feature. The
intensity I = 53.5 W/cm2 results in one PA photon scattering event during the typical
exposure time of 200 ms. (d) Same calculation as in panel (c), except on a linear scaleand large detuning. The circles show the coupled channels calculation, and the solid lines
show the analytic isolated resonance formula. The analytic isolated resonance expressions
are indistinguishable from the coupled channels results here.

151
length, ℓopt γm /∆ needs to be sufficiently large, and the imaginary part b = 12 lopt γm γ/∆2 needs
to be sufficiently small. Since Kin ≃ (2 × 10−12 cm3 /s) (b/a0 ), for a density of ρ = 1012 cm−3and b = 0.1a0 , Kin ρ gives the same loss rate (200 ms)−1 as the atomic light scattering rate forthe power I = 53 W/cm2 assumed for Figs. 5.13(c) and (d). Thus, the calculations predictthat changes in the scattering length on the order of 10 a0 ≫ |abg | should be possible for
detunings ∼ 100 γm for time periods ∼ 200 ms.

Chapter6

Summary & Outlook
n this thesis, we have discussed several precision measurements related to an opticalI

frequency standard based on neutral atomic Sr. We have presented detailed models that

led to a deeper understanding of the underlying physical phenomena. A lot of the workon optical standards tries to realize a quantum system that is as clean and as simple aspossible. All that should be required to understand the spectroscopic process is a driven
two-level atomic system without spontaneous decay. Effects that mar this simple picture areconsidered systematic effects that need to be understood, measured, and controlled.
Because the aim of an optical frequency standard is to prepare the cleanest driven two-levelsystem possible, the typical method of doing so is to consider the currently largest systematiceffect and try to understand it. Once a particular effect is successfully understood, it can bemodeled in a simple manner, and can be controlled and removed by careful system design.
This process might take a while, especially as the effects become smaller and smaller, but
it has been very successful so far in decreasing the frequency uncertainty towards the nextorder of magnitude.
The current main systematic under investigation is the density shift. As we have seen in
Chapter 3, fairly good models have been put forward to understand its origins. These models
have also led to new proposals on how to improve the system to suppress the density shift.
By now, this suppression has also been shown experimentally [51, 67]. Although a full clock

154
evaluation exploiting the density shift suppression will need to be done, these results promise
to get rid of one of the two limiting effects in the Sr uncertainty budget (see Sec. 1.2).
The second large uncertainty – the BBR shift – is currently under investigation in all majorlabs that work on Sr optical frequency standards. A better understanding of the theoreticalmodel for the atomic polarizabilities will reduce the BBR uncertainty, but requires carefulmeasurements of the key contributing electronic states [34]. Even with an improved polarizability model and a careful measurement of the environmental temperature, one cannot
confidently quote a small BBR uncertainty because of the non-blackbody character of thevacuum chamber environment. For this reason, the most likely approach to measure the
BBR shift is to build a specialized environment for the spectroscopic process. The moststraightforward approach would be to simply heat and cool the environment and measure
the clock frequency shift with respect to this parameter. The temperature range exploredby this approach is too small, however, to obtain good uncertainties since typical vacuum
chambers and experimental apparatuses cannot support temperature changes of more than afew 10 ◦ C. For these reasons, the main approach for constructing a specialized environmenthas been to build a small in-vacuum enclosure that can be cooled using liquid nitrogen or
helium [12, 36]. Atoms would then be transferred into such an enclosure using a moving optical lattice and spectroscopy could take place in a carefully constructed BBR environment
at a controlled (and small) temperature. The main difficulties for building such a system aretechnical in nature. One has to carefully consider how the enclosure is built to avoid leakageof room temperature BBR into the enclosure. In addition, pumping speed limitations insmall enclosures might be a considerable concern, even at cold temperatures. Characterization of the BBR shift will be a major step forward for optical lattice clocks, but it mighttake some time to settle on a reliable system design that does not impede clock operation insome other way.
Meanwhile, the high precision obtained in optical lattice clock experiments has led tointeresting proposals for investigations of condensed-matter systems that could be realizedwith alkaline earth atoms [126–128]. In the last few years, all isotopes of Sr have been

155
used to produce quantum degenerate gases [142, 143, 175–178]. A long-standing goal for
optical lattice clock experiments has been to operate with a sample in a Mott insulator
state, trapped in a three-dimensional optical lattice. This idea is conceptually intriguing,
because many-body effects would automatically be suppressed as long as the lattice is deep
enough to prevent tunneling and all atoms would be prepared in the lowest motional state ofeach site. Such a sample would be a real equivalent of an array of identical ion traps, wherea single particle is interrogated in a deep potential well. Even the very first experimental
strontium lattice clock paper [1] contained a drawing that shows single atoms trapped in an
eggcrate-shaped potential that can only be realized in a three-dimensional optical lattice.
Because of their conceptual attractiveness, these pictures can be very misleading. The
main detractor for building a serious 3D optical lattice clock is the fact that optical latticepolarizations cannot be kept parallel to the bias magnetic field in three dimensions. Of
necessity, one beam will have a polarization that is not colinear with the bias magnetic
field. This configuration breaks the symmetry that allows clock frequency shifts related tothe tensor polarizability in fermionic isotopes to be suppressed [22]. Although a bosonic
optical lattice clock in three dimensions has been built [179, 180], its accuracy is not nearly
as good as the systems realized with one-dimensional lattices. Bosonic isotopes require alarge magnetic field to induce the clock transition which introduces a large uncertainty due
to long-term magnetic field stability.
In addition, the Wannier-Stark ladder gravitational symmetry breaking between latticesites (see Cha. 2) cannot be obtained in an optical lattice of dimension two or greater. The
transverse harmonic confinement in an optical lattice beam will absorb the linear potential
shift due to gravity, and the energy splitting between lattice sites becomes parabolic as aresult. Because of the transverse harmonic confinement, sites at the center of the trap becomemuch closer to being degenerate than would be the case in a vertical one-dimensional opticallattice [56]. Tunneling can become a concern even in two-dimensional optical lattices – see
the online material for Ref. [51]. Care will have to be taken in finding a compromise between
tunneling (for shallow lattices) and lattice-intensity related detrimental effects such as photon

156
scattering at the lattice wavelength (for deep lattices).
In addition, the individual sites in a 2D or 3D optical lattice are actually much more
inhomogeneous than in a 1D optical lattice. Because of the transverse confinement, sites
away from the center of the trap will have different trap depths and frequencies than the sites
close to the center. These effects exacerbate the Rabi frequency inhomogeneities discussed
in Cha. 2, and averages over the particle distribution across different lattice sites becomemuch more important [51].
All of these effects will become even more important as laser coherence times become longer
and narrower linewidths are obtained using the next generation of highly stable spectroscopy
lasers. Preliminary investigations show that for long coherence times, new effects such as
different collision channels and higher-order light scattering processes will also start to besignificant [68, 181]. Finally, spectroscopy-induced dipole-dipole coupling between atoms inneighboring lattice sites [182] will become a concern when spectroscopic precisions approachthe 10 mHz level with novel spectroscopy laser designs [183–185].
For these reasons, it currently seems unlikely that the accuracy achieved in 1D opticallattice clocks will be surpassed by 3D optical lattice clocks in the near future. In addition,
operation of a 3D Mott insulator clock requires the preparation of a Mott insulating state
on competitive time scales. Current 1D experimental cycles are on the order of 1 s, which
already limits the optical clock precision via the optical Dick effect [186, 187] by aliasing
spectroscopy laser noise at the duty cycle into the measurement bandwidth. This limitation has led to a lot of work on duty-cycle optimization and sample reuse by the Parisgroup [54, 187–189]. Evaporative cooling to quantum degeneracy currently takes at least
several seconds [142, 143, 175–178], which would reduce the measurement precision evenfurther and make laser stabilization very difficult. The cooling time scale is unlikely to improve much since evaporation is necessarily tied to the trap frequency of the optical potential
in which the particles are cooled. Other cooling methods such as narrow-line laser coolingbased on spontaneous Raman scattering might prove more useful, but significant experimental work will be necessary to decide whether they are feasible [190, 191]. The preparation

157
time required makes an optical lattice clock based on a quantum degenerate gas unfeasible
unless some interleaved preparation scheme is employed where a second sample is preparedwhile the current sample is interrogated.
Nevertheless, most groups are working on building multiple Sr clock systems in the samelab to simplify the evaluation of systematic effects. Probing a condensed matter analog basedon quantum degenerate Sr with another experiment set up as an optical lattice clock has
many promising advantages. Such a setup could be used to realize novel and interestingalkaline earth quantum simulation schemes [126–128]. Probing such a system with the highprecision afforded by the optical lattice clock will then enable investigations of the system
dynamics at the Hz scale. Conversely, the understanding gained from careful investigationof the quantum degenerate gas system will help in furthering the optical lattice clock design
and may enable construction of a degenerate gas optical lattice clock in the future.
One tool for studying and manipulating such quantum degenerate gases of alkaline earth
atoms is the OFR effect. We investigated this effect in detail in a thermal gas of 88 Sr,
but as we have seen in Chapter 5, the real application of the OFR would be as a tool
to manipulate phase shifts or mean field energies in the degenerate gas regime. Because
there are no magnetic Feshbach resonances in the (interesting) states of alkaline earths, the
OFR is the sole candidate for manipulation of the interaction strength so far. The main
result of our studies is that the elastic scattering length modification under the OFR effectis inseparable from inelastic losses and that the effect is reliable only in the vicinity of avibrational resonance. But, we have also shown that it can still be usefully applied undera careful combination of parameters, even in a thermal gas. Whether the OFR could also
be used to further the study and operation of optical lattice clocks remains to be seen. The
relevant collision channel for the density shift is 1 S0 -3 P0 . One could, for instance, imagine
that the interaction energy might be modified by coupling this channel to a vibrational statein 3 P1 -3 P0 .
Even though the optical lattice clock is the simplest spectroscopic system we can prepare,each gain in systematic uncertainty brings new effects to light. Studying and learning to con-

158
trol these effects is what keeps work on a seemingly simple spectroscopy system interesting.
The benefit of pushing the accuracy to higher and higher levels is that novel effects unrelated
to the atomic physics can be studied. The main candidate for these studies are the relativistic effects of gravity and their manifestation in the red shift. We have shown in Cha. 4that optical clock experiments can easily be tied to investigations of large-scale fundamental
physics. The red shift manifests as a frequency shift of 1 × 10−16 m−1 , and optical clocks
are fast approaching the accuracy regime where gravity can be investigated on small lengthscales [192]. Projects for large scale investigations of gravity by placing optical standards
on a space vehicle are also under development [113]. Any advancement in miniaturization
and automation of the currently large, power-intensive, and manpower-hungry optical clock
systems will help these projects. Replacing the microwave-standards used in satellite systems
such as GPS with optical standards will enable precision mapping of gravity and improve
position triangulation on Earth’s surface from meters to millimeters or below. Astronomic
applications include ranging at interplanetary distances [193] and building phase-coherent
links between remote antenna arrays for very-long-baseline interferometry.

The atomic frequency standard of the future will not matter much for civil time keeping,but I believe that Sr based precision spectroscopy will remain interesting for years to come.

Appendix

A

Absorption imaging of trapped
particles
his appendix summarizes the relevant physics of a probe beam scattering off a sample ofT

trapped atoms. Absorption imaging of trapped particles is subject to many systematiceffects. Nevertheless, in-situ absorption imaging can be made to work reliably provided thatthese effects are understood, measured, and accounted for.

A.1. Beer’s law and classical scattering cross section
Assume that a probe beam of intensity Iin enters an absorbing medium and that the intensity
after transmission has been reduced to Iout . Then the medium’s optical depth (OD) is definedas
OD = − ln

Iout
Iin

(A.1)

If the medium absorbs with a constant probability per unit length, the medium has a linear
absorption coefficient α and the intensity along the beam coordinate z attenuates exponentially according to Beer’s law
Iout (z) = Iin e−αz .

(A.2)

160
Assuming that the medium consists of a dilute collection of identical scatterers, the absorption coefficient should be proportional to their density n and one can define a scatteringcross section
σ = α/n(A.3)

that summarizes the scattering properties of an individual scatterer. For a dilute monatomic
gas interacting with a resonant monochromatic field, the resonant scattering cross section canbe calculated from the linear response of a two-level system consisting of ground state |gi andexcited state |ei. If the field’s polarization is perfectly aligned with the two-level system’squantum axis, and the field is perfectly monochromatic, one finds from the equivalenceprinciple that [194]
3λ2eg
σeg =
,
2π

(A.4)

where λeg is the wavelength of the atomic transition.

A.2. Polarization, selection rules, and atomic response
function
We will argue that the quantum mechanical equivalent of Eq. A.4 essentially gives the sameanswer. The derivation follows the solid treatment in Ref. [195] and will be carried out insome detail to expose each assumption going into our final result. We would like to determine
the numerical value of the factor 3∗ ∈ [0, 3] as defined by Siegman [196] and check whether
λeg in Eq. A.4 should be λeg or λeg /(2π).
We start with a differential formulation of Fermi’s Golden Rule for the transition rate perphoton energy E from a combined light and atomic state |ii to another combined light andatomic state |f i under the perturbation H ′ :dWf i =

2π
|hf |H ′ |ii|2 S(E)P (E)dE,
~

(A.5)

where P (E) describes the atomic response to the application of a photon of energy E, i.e.
it is the probability of a successful absorption process at photon frequency ω ≡ E/~. The

161
energy distribution S(E) is the area-normalized power spectrum applied to the atom.

A.2.1. Atomic matrix element
In the dipole approximation, the Hamiltonian H ′ coupling a two-level atom of mass m to asingle-mode light field with vector potential A, polarization ǫ̂, wave vector k, and frequency
ω can be written as [195]
e
e
H =
p·A=
mc
m
′

r


2π~ 
a(p · ǫ̂)eik·r + a† (p · ǫ̂∗ )e−ik·r ,
Vω

(A.6)

where e is the electron charge, p (x) is the atomic momentum (position) operator, and a (a† )is the photon mode annihilation (creation) operator in the quantization volume V . The firstterm describes the absorption of a photon and the second term describes the emission of aphoton. Here, we will only consider first order perturbation theory for H ′ , where a photon
is absorbed and the two-level atom undergoes the corresponding transition of interest. We
consider an initial state |ii ≡ |gi ⊗ |ni ≡ |g, ni of the combined system, where the atom isin ground state |gi and the photon field is in the n-photon Fock state |ni. One photon isabsorbed and the atom undergoes a transition to excited state |ei such that the final state
is |f i ≡ |ei ⊗ |n − 1i ≡ |e, n − 1i. The transition matrix element becomes
r
r
e 2π~
e 2π~n′ik·r
hf |H |ii =he, n − 1|a(p · ǫ̂)e |g, ni =he|(p · ǫ̂)eik·r |gi.
m Vω
m
Vω

(A.7)

As long as the particle motion can be treated classically (beyond the Lamb-Dicke regime),
we can ignore the operator character of the phase factor and separate it from the electronic
degree of freedom. We findehf |H ′ |ii ≃mr

2π~n ik·re he|p · ǫ̂|gi,
Vω

(A.8)

and we can write down the Heisenberg equation for the position operator r and atomic
Hamiltonian Ha :[r, Ha ] = i~

dr
i~p
=
.
dt
m

(A.9)

We use that |ei and |gi are eigenstates of Ha and findhe|p · ǫ̂|gi = imωeg he|r · ǫ̂|gi ≡ −imωeg he|D · ǫ̂|gi,e(A.10)

162
for the bare transition frequency ωeg defined as the eigenenergy difference between the atomic
states |ei and |gi, and the dipole operator D ≡ −er. Thus Eq. (A.7) becomesr
2π~n′hf |H |ii = −iωeg he|D · ǫ̂|gi.
Vω(A.11)

Note that we have not assumed that the driving photon at frequency ω is exactly at the baretransition frequency ωeg .
We proceed by assuming that the atomic states are well described by a total angularmomentum J and magnetic quantum number m, so that|gi ≡ |Jg mg i

(A.12)|ei ≡ |Je me i.
In this case, the dipole matrix element Eq. (A.11) can be separated into spherical tensor
components. The scalar product between dipole operator and polarization vector separates
into [195]D · ǫ̂ =+1
X(−)q (ǫ̂ · ê−q )Dq1 ,(A.13)

q=−1

with polarization quantum number q corresponding to a spherical tensor basisê0 ≡ ẑ√(A.14)ê±1 ≡ ∓(x̂ ± iŷ)/ 2
with quantum axis along ẑ. In the spherical tensor basis, the dipole matrix elements can befurther simplified via the Wigner-Eckart theorem1
hJe me |Dq1 |Jg mg i = (2Je + 1)−1/2 hJg 1mg q|Je me ihJe kD1 kJg i.
1(A.15)The numerical and phase factors in front of the reduced matrix element in the Wigner-Eckart theorem havevarious definitions in the literature. It is important to use the definition of the Wigner-Eckart theorem,
the Clebsch-Gordan coefficient in terms of 3j-symbols and the reduced matrix element from the samereference to get consistent results. We follow Ref. [195] here which is consistent with Refs. [194, 197–200]
in both Wigner-Eckart theorem and definition of Clebsch-Gordan coefficients in terms of 3j-symbols.
Sakurai [71] uses a different phase prefactor but that might be a misprint. Other books use a different
factorization of the Wigner-Eckart theorem and are not consistent with the presentation here [57, 201–
204].

163
with radial integral hJe kD1 kJg i (called the reduced matrix element) and Clebsch-Gordan
coefficientpJg 1 Je.
hJg 1mg q|Je me i = (−)me 2Je + 1 
mg q −me(A.16)For a given combination of quantum numbers, the Clebsch-Gordan coefficient enforces the
dipole selection rules |Je − Jg | ≤ 1, mg + q = me , and Je = Jg ⇒ q 6= 0. In particular,
this means that for a given combination of quantum numbers, there is only one polarization
component q = me − mg with |q| ≤ 1 that can drive the dipole transition.
Combining Eqs. A.15-A.16, we findhJe me |Dq1 |Jg mg i = δq,me −mg (ê−q · ǫ̂)(−)me +q 

Jg

1

Je

mg q −me hJe kD1 kJg i.

(A.17)The reduced matrix element can be related to the Einstein A coefficient of the particulartransition [195]34ωeg|hJe kD1 kJg i|2
.
Aeg =
3~c3
2Je + 1

(A.18)For the transition rate Eq. (A.5), we only require the magnitude-squared of the atomic
dipole matrix element. We find2π
4π 2 n 2|hf |H ′ |ii|2 =ω × |he|D · ǫ̂|gi|2~
V ω egJg
4π n 2
ωeg × δq,me −mg |ê−q · ǫ̂|2 
Vω
m
2

=

×

g

3~c3
(2Je + 1)Aeg .
3
4ωeg

1

Je

q −me

2

 ×

(A.19)The Fermi’s Golden Rule differential transition rate for absorption of photons with energy
E = ~ω while the atom undergoes the transition |gi → |ei becomesdWeg = Φ × 3(π~c)2

Aeg
×
ωeg


× δq,me −mg |ê−q · ǫ̂|2 (2Je + 1) 

×

1
S(E)P (E)dE,
E

Jg

1

Je

mg q −me2

 ×(A.20)

164
where we have defined the photon flux Φ ≡ cn/V as the density of photons in the quantization
volume traveling at the speed of light. To obtain a useful expression for the scattering rate,
we have to assume a particular form of the atomic response P (E).

A.2.2. Atomic response and laser spectrum
In the case of a (homogeneously broadened) Lorentzian line centered at ωeg with linewidth
γe , we have an area-normalized response

P (E = ~ω) =~γe /(2π)
12=(~ω − ~ωeg )2 + (~γe /2)2π~γe 1 + 4(ω − ωeg )2 /γe2

(A.21)

Note that the homogeneous linewidth γe includes decay to other levels. If the lineshape isPdominated by the transition’s natural linewidth, γe = Ae = j Aej , with Einstein coefficient
Ae and branching ratios Aej /Ae to all other levels |ji.

In the limit of a monochromatic spectrum at the transition frequency the spectrum becomes S(E) → S0 (E) ≡ δ(E − ~ωeg ) and we retrieve a total transition rateJg
3λ2eg Aeg
Weg = Φδq,me −mg |ê−q · ǫ̂|2 (2Je + 1) 
2π γe
m

g

1

Je

q −me

2

 .

(A.22)

If the probe laser spectrum is broadened with respect to S0 (E), the transition rate Weg will
be diluted by a factorR∞
Z ∞dES(E)P (E)/E
ωeg /ω
0dωS(ω),=
ξ ≡ R∞
1 + 4(ω − ωeg )2 /γe2
dES0 (E)P (E)/E00where the laser spectrum is area-normalized toR∞0(A.23)dωS(ω) = 1. Figure A.1 shows thedilution factor ξ for several power spectral density shapes assuming that the laser spectrum
is centered around ωeg but broadened. Detuning from resonance is easily included in ξ byshifting S(ω).

165

1
0.8
ξ

0.6
0.4
Square : ξ = atan(r)/r
Lorentzian: ξ = 1/(1 + r)√√√
2
Gaussian: ξ = π/( 8r)e1/(8r ) erfc(1/( 8r))

0.2
0

0

1

2

3

r = γlaser /γe
Figure A.1.: Transition rate dilution factor ξ from laser spectrum for square spectrum(red), Lorentzian spectrum (blue), and Gaussian spectrum (green).

A.2.3. Scattering cross section
The scattering cross section for the atomic transition |gi → |ei is defined as the ratio of
scattering rate Weg to photon flux Φ. From Eq. (A.22), we findσeg ≡ Weg /Φ =3λ2eg Aeg
2π γeξδq,me −mg |ê−q · ǫ̂|2 (2Je + 1) 

Jg

1

Je

mg q −me2 .

(A.24)

In our case, the atomic states under investigation are |gi = 5s2 1 S0 and |ei = 5s5p 1 P1 in
88

Sr. The quantum axis is provided by a magnetic field of several hundred mGauss along thevertical axis in the lab frame. The ground state is spinless (J = 0) and the excited state has
three magnetic substates (J = 1), so that there are three different transitions to consider.
The transitions are not quite closed, since 5s5p 1 P1 decays to 5s4d 1 D2 which decays to themetastable 5s5p 3 P2 . However, the branching ratio from 3 P1 to 1 D2 is on the order of 10−5so that we will ignore losses to 1 D2 for the photon numbers per atom considered here. The
natural linewidth is Ae = 2π × 30 MHz which dominates the line profile at µK temperatures.
For these reasons, we only consider radiative decay and set γe = Ae = Aeg .

166
In addition, we have Jg = 0, mg = 0, and Je = 1, such that2Jg 1 Je 7→ 1 ∀ q.
(2Je + 1) 
mg q −me(A.25)

If we apply an arbitrary (but pure) polarization ǫ̂ with ǫ̂∗ · ǫ̂ = 1, then
+1
X

q=−1

|ê−q · ǫ̂|2 = 1,

(A.26)

3λ2eg
ξ,
2π

(A.27)and the scattering cross section becomesσeg =in agreement with the classical expression Eq. (A.4).

A.3. Atomic saturation
When the light intensity approaches the saturation intensity
Isat =

πhc
γ,
3λ3eg

(A.28)

the atomic system does not absorb linearly anymore. Instead, a full treatment based on
the Maxwell-Bloch equations becomes necessary to describe pulse propagation in a saturable
medium. In the present case, we operate at a few ten percent of Isat and would like tocalculate first order corrections to the linear response. The treatment in this section follows
Ref. [205, 206].
As in section A.1, we assume pulse propagation along z and find an equation for thesaturation parameters0 (z) ≡ I(z)/Isat = 2Ω2 /γ 2 ,(A.29)ds0
s0= −αBeer.
dz
1 + s0(A.30)

on resonance as [205]167This equation can be integrated by separation of variables [206]Z soutZ z
1 + s0ds0= −αBeer
dz
s0
sin
0
sout
+ (sout − sin ) = −αBeer z ≡ −ODBeer .
lnsin(A.31)The left hand side can be related to the measured optical depth ODexp ≡ − ln(sout /sin ),which allows us to correct for the effect of atomic saturation:
ODBeer = ODexp + (1 − e−ODexp )sin ,(A.32)

if we know the probe intensity interacting with the atoms. Detuning and laser spectrum is
accounted for by the factor ξ we derived in the previous Section.

A.4. Time evolution of sample with probe pulse
To obtain large signal-to-noise from the trapped sample, it is necessary to scatter many
photons from each individual atom. For each absorbed photon, an atom gets a momentum
kick in the direction of the probe beam with the photon recoil momentum prec = h/λeg ,corresponding to a recoil velocity vrec = h/(mλeg ). In addition the atom will receive a random
momentum kick when the photon gets reemitted. In this Section, we derive a compromisefor the number of photons scattered per atom that still gives good signal-to-noise, but does
not influence the sample too much.

A.4.1. Photon rescattering
The response of the atomic cloud to the rescattering process amounts to a random walk in
momentum space, overlaid with linear acceleration along the probe beam direction. The
random walk can be modeled using a multivariate Ornstein-Uhlenbeck process [207], where
the position and momentum of each particle is a random variable influenced by the trapping
potential and the white momentum noise. For a particle in a one-dimensional harmonic trapwith trap frequency ω, the Itô stochastic differential equation for position x and velocity v

168
is








0
=
−ω 2
dv

dx


 


0 0
x
1
dw
   dt +  √   x 
0
v
0
dwv
D

(A.33)

with velocity diffusion constant D, which is the mean-square velocity change per unit time,and zero-mean white noise increments dwx and dwv . The equation is of the form
dx = Ax dt + B dw,(A.34)

with constant system matrix A and constant noise covariance matrix B. The equation canbe solved for the mean hxt i and the covariance matrix Cov(xt ) [207]. We obtain
hxt i = e−At hx(0)iCov(xt ) = e−AtCov(x0 )e

−A⊤ t

Z t
′
⊤
′
+ dt′ e−A(t−t ) BB ⊤ e−A (t−t ) .

(A.35)

0This form allows us to find equations for the position and velocity variances under the22
influence of the probe pulse in a harmonic trap (assuming that σxv(0) = σvx(0) = 0),σv2 (0) 2
2ωt − sin 2ωtsin ωt + D
2
ω4ω 3
2ωt + sin 2ωt
σv2 (t) = σx2 (0)ω 2 sin2 ωt + σv2 (0) cos ωt + D.
4ω

σx2 (t) = σx2 (0) cos2 ωt +(A.36)For long times ωt ≫ 1, this result amounts to a linear increase in both position and velocity
uncertainty with superposed oscillations at the trap frequency.
For a three-dimensional anisotropic harmonic trap (without anharmonic corrections orinteraxis coupling) and trap frequencies ωi , the above results hold true for each pair of
position and velocity coordinates (xi , vi ) and per-axis diffusion constant Di .
The overall system energy E can then be calculated as
E(t) =

Xmi

2σv2i (t) += E(0) +mωi2 2
σ (t)
2 xi

m X
t
Di .
2 i

(A.37)The rate of energy gain is
ΓE ≡

mX
dE
=
Di ,
dt
2 i

(A.38)

169
reemphasizing the role of the diffusion constants Di as the per-axis mean square velocity
gain per unit time. If we define the total diffusion constant D, we can write it as the productof a scattering rate times the square of the recoil velocity
D≡

X
i

2
Di ≡ Γsc vrec
= Γsc

h2
.
m2 λ2eg(A.39)In this context, we are interested in the short term behavior of the atomic cloud under
the influence of photon scattering with rate Γsc . In the limit ωt ≪ 1, the above equations
become independent of the trapping frequency and we obtain the free space random walk
results
σx2i (t) ≃ σx2i (0) +Di t 3
3(A.40)

σv2i (t) ≃ σv2i (0) + Di t
We are interested in having a constant number of photons Np ≡ Γsc tp interact with theatoms during the probe time tp to obtain a certain signal size on the camera. However, we
would also like to minimize size changes of the cloud. Under these conditions, comparing
the per-axis cloud size change
Np (t′p )2δσx2i (t′p )==δσx2i (tp )Np t2p ′ 2tp,
tp(A.41)

tells us that we should work with low exposure times and high probe intensities, as long as
atomic transition saturation does not cause too many problems.
2
If we assume that photons are rescattered isotropically, we have Di = D/3 = Γsc vrec
/3 and22
vrec tptp222
σxi (t) = σxi (0) + Np
= σxi (0) + Np 3.28 nm ×.
(A.42)
3µs

A.4.2. Doppler shift
So far, we have only treated the probe photon reemission, but have ignored the linear acceleration of the atomic cloud along the probe direction due to the absorption of the probe
photon. The continual absorption of probe photons along the probe direction results in a
linear increase in each atom’s mean velocity along the probe direction
hv(t)i = vrec Γsc tp = vrec Np .

(A.43)

170
In the low saturation limit, the photon scattering rate is dependent on the probe detuning
δp viaΓsc = Γ0sc /[1 + (2δp /γ)2 ],(A.44)

for on-resonance scattering rate Γ0sc and atomic linewidth γ. The Doppler effect changes thedetuning during the course of the probe pulse via
2δp /γ =2πh
4πhv(t)i =Np ≡ βNp ,λeg γmλ2eg γ(A.45)

with β ≡ 1.422 × 10−3 . Assuming that the atoms are at rest initially (δ = 0), the corresponding fractional change in per-axis diffusion constant Di isδDi(βNp )2
Γsc (0) − Γsc (tp )=≃ β 2 Np2 ≃=Di
Γsc (0)1 + (βNp )2



Np
700

2

.

(A.46)

In summary: relying on the free-space diffusion formalism above necessitates scattering muchless than 700 photons from each atom.

A.4.3. Cloud displacement
Each atom’s mean position along the probe direction z is given by
t2p
tphz(t)i = Γsc vrec = Np vrec .
2
2
This means that we can neglect the cloud displacement as long as hzi ≪Np ≪σz (0)
2σz (0)≃ (203 µs/µm).
vrec tp
tp(A.47)pσz2 (0), or(A.48)

A.4.4. Radiation trapping
If a photon gets scattered in a region of high atomic density, the surrounding atoms are likely
to rescatter the scattered photon. In this way, radiation can be trapped within a sample ofhigh density and photon energy can be stored.
The reabsorption probability of the spherical wave emanating from an atomic scatterer
can be estimated by imagining a sphere of radius r around the scattering atom. This sphere171
contains Na = 4π
r3 n̄ atoms, where n̄ is the average atomic density within the sphere. Each3
atom within the sphere contributes a resonant rescattering cross section σ = 3λ2eg /(2π) to thetotal rescattering cross section Na σ. As soon as the total rescattering cross section becomescomparable to the surface of the sphere, the rescattering probability approaches unity. Using
this argument, we can define a mean-free radius
2
4πrfree≡ σNa = σ

4π 3
r n̄
3 free

−1

n̄
2π
.
≃ (29.6 µm) ×⇒ rfree =n̄λ2eg1012 cm−3(A.49)

If the mean free radius is much smaller than the cloud size, probe photons will be scatteredmultiple times within the cloud and enhance the photon scattering rate Γsc and thus the
detrimental effects of size changes.

Appendix

B

Monte-Carlo Collision Simulation
ith current processor speeds, it has become feasible to simulate the thermodynamics

Wof a dilute thermal gas of ultracold atoms directly. The main problem remains thesimultaneous evolution of N = 104 − 105 particles, since even the simulation of two-particle
interactions requires O(N 2 ) checks at each time step. A popular method for simulationsin the collisionless flow regime due to Bird [208] reduces this complexity by discretizing
space into small volumes and only handling local interactions. Among other applications,
it has been successfully used to model evaporative cooling [209, 210], particle evolution in
non-harmonic traps [211], and cross-dimensional thermalization [170, 212].
In the following, we will briefly describe the algorithm, analyze a few simple test cases, andshow results for the full OFR interaction model including both elastic and inelastic collision
processes.

B.1. Bird’s method
The application of Bird’s method to ultracold atom systems has been described in many
theses; good discussions can be found in [170, 211]. The program used here is implemented
in C++ and makes extensive use of the GNU Scientific Library (GSL).
The algorithm proceeds in time steps of length τ . Between each time step, the particles
of mass m are evolved in the trapping potential U (x) according to the classical equations of

174
motion:ẋ = v(B.1)

v̇ = −∇U (x)/m.
Convenient units for the simulation are ms, µm, and recoil energy, for time, position,and potential, respectively. The particles are propagated using an embedded Runge-Kutta
method [98, 213], using analytical derivatives of U . Implemented potentials include anisotropic
harmonic traps and arbitrary superpositions of Gaussian beams (excluding interference effects). Including gravity in the potential is optional.
After each time step, space is discretized into a grid of spacing d. The coordinate system’s
origin lies close to the center of the trap, and each site is indexed by a tuple (i, j, k) ∈ Z3 .
Each particle at position x is associated with the closest grid site(i, j, k) = [x/d],(B.2)

where the brackets indicate rounding to the closest integer. We define a spherical volume
 3
4π d
V =
(B.3)
3 2
around each grid point and the grid spacing d is automatically adjusted such that the sampleaverage of the occupation number (mean particle number per cell in the trap volume) is
hNocc i = 0.01,(B.4)

which is well below 10% [170].
To simplify the collision checks, a list of particle identifiers is sorted according to theparticles’ associated grid indices. After the sorting operation, the particles at the same grid
point are consecutive in the index list to enable quick access. Using quicksort [98], reusing
the sorted list from time-step to time-step to reduce the number of sort operations, andsorting only once per time step allows the algorithm to be fairly fast.
Within each sphere, a list of Nocc (Nocc − 1)/2 potential collision pairs is made. For eachdistinguishable pair of atoms with velocities v 1 and v 2 , the collision probability is calculated

175
asPcoll = |v 1 − v 2 |στ /V,(B.5)

where σ is the total collision cross section for the process under consideration. A random
number q is then drawn uniformly from [0, 1], and if Pcoll > q, the collision happens. For
successful elastic collisions, the particle velocities are rotated with a random Euler rotation
matrix. For successful inelastic collisions, both particles are removed from the simulation.
The corresponding collisional cross sections σel and σin can be velocity dependent or can bemade to depend on other external parameters. The OFR effect is simulated by inserting thecross sections from the Bohn and Julienne theory [121, 130].
Finally, the time step τ is servoed using a simple integrating feedback loop such that the
sample averaged collision probability during τ remains small, typically hPcoll i ≃ hNocc i ×
0.05 ≃ 5 × 10−4 . These numerical parameters have been found to make the method reliable
and stable [170]. When simulating velocity-dependent cross sections in a thermal distributionof atoms, it is hard to predict what the actual collision rate per time step is. With feedback,the simulation becomes a factor of 5-10 faster by checking for collisions only as often as
required.
The remaining important point is that the simulation obviously works with distinguishable
particles. To mimic results for indistinguishable thermal bosons in the same spin state(J = 0), we have to modify the corresponding cross sections. The simulation is basedon distinguishable particles identified by an id tag. To translate the simulation results tothermalization of identical bosons in the same spin state, we need to include a bosonicenhancement factor in the elastic scattering cross section. Simulating thermalization ofindistinguishable bosons requires using a scattering cross section σel = gα 4πa2 with gα = 2
for s-wave scattering length a. The same bosonic enhancement factor gα appears in the
inelastic cross section. A thorough and careful discussion of the origin of these factors canbe found in Ref. [119].
However, many authors make no clean distinction between the inelastic scattering eventcross section and the inelastic collision rate. Even such basic terminology as event rate and176collision rate is not uniquely defined. This confusion arises from different factors of twocanceling in the final equations for the density evolution under inelastic loss. The density
evolution equation generally used is of the form of Eq. 5.12:ṅ = −Kin n2 = −σin vrel n2 .

(B.6)

Unless this full definition is given, the meaning of Kin can be ambiguous. Cross sections
σ are always related to the probability of a collision event in the sense of Eqs. 5.6 or B.5.
Multiplication by the relative velocity thus gives a collision event rate for a collision pair at agiven relative velocity. Particle pairs are thus lost at rate Kin , but particles are lost at a rate
2Kin , since two particles are lost per collision event. But, since there are only N (N − 1)/2
possible distinguishable collision pairs (see Eq. 5.9), the final rate coefficient appearing in
the evolution equation for the single-particle density (Eq. B.6) is equal to the pair-loss rate.
For the same reason, we need to be careful in comparing simulation results for elastic
collisions to common statements like: “It takes about three elastic collisions to thermalizea particle.” [162, 170, 211, 212] The meaning of this statement is that each of N particles
participates in roughly κ = 3 collisions during the 1/e time Γ−1therm it takes for kinetic energy
inhomogeneities to equilibrate. If we fit an exponential decay to the disappearance of aelthermal inhomogeneity, we can count the number of elastic collision events Nevent
during an

exposure time of τPA and obtain a rough estimate for κ fromκ≃el
2Nevent
1
×.
NτPA Γtherm(B.7)

B.2. Inelastic collisions
The inelastic collisions were calibrated against the analytical model of Sec. 5.9 in a simplified
test. The optical trap was replaced by an isotropic harmonic oscillator with trap frequencies(νx , νy , νz ) = (200, 200, 200) Hz, to make the problem as simple as possible. Gravity wasremoved and the trap was cut off at a radial distance of 200 µm corresponding to a trap
depth >1 mK along each axis. An initial distribution containing 50,000 atoms was synthesized, assuming Maxwell-Boltzmann statistics with both velocity and position variances

177
corresponding to a temperature of 3 µK along each of the three trap axes. Both the velocity
and position variances are stationary if the simulation is run without allowing any collision
processes.
We then allow inelastic collisions with a constant (relative velocity independent) crosssection and turn off any elastic collisions in the simulation. To make comparisons between
elastic and inelastic collisions more obvious, we parametrize the inelastic cross section by
an artificial “inelastic length” b and set σin = 8πb2 . The simulation is run for 500 ms withb = 100 a0 , and the position and velocity variances along each trap axis are calculated ateach time step. Figure B.1(a) shows the increasing position variances along x, y, and z as
red, blue, and green traces, respectively. Similarly, panel (b) shows the increase in velocity
variances with time. Panel (c) shows the fractional atom loss as the solid black trace. A fit
with the thermally averaged model in Sec. 5.9 (dashed red trace) shows fair agreement andreturns a collision rate that is about 10% larger than what is used in the simulation. Thisoverestimation is consistent with the slight overestimation (underestimation) of the black
curve at times shorter (longer) than 70 ms (150 ms).
The analytical model does not describe the effect of inelastic loss on the momentum distribution, but only includes density-dependent loss from a Gaussian distribution. Similarlevels of agreement are obtained when fitting the simulated mean density as a function oftime with the solution of n̄˙ = −Kin n̄2 . In contrast, the Monte-Carlo simulation calculates
the effect of inelastic losses on the full phase-space distribution in the presence of a particular
trapping potential. We conclude that the agreement between model and simulation is goodenough to extract optical lengths from inelastic loss data at the 10-20% level. Typical experimental error bars are much larger (see Fig. 5.11) and we use many PA spectra to reduce
the statistical error on ℓopt /Iav . The same level of agreement between the analytical modeland the Monte-Carlo simulation is achieved when modeling the full model potential in thesimulation and fitting the resulting atom loss traces.
Panel (d) emphasizes the fact that potential and kinetic energies follow each other. The
slope of lines fitted to the velocity variances as a function of the position variances is pro-

Velocity variance Hum2 ms2 L

Position variance Hum2 L

178

240
220
200
180
0

100

200

300

400

400
380
360
340
320
300
280
0

500

Time HmsL(a) Sample potential energy increases with inelastic loss.

(c) Inelastic particle loss (black) and fit with thermally averaged model from Sec. 5.9.

100

200

300

400

500

Time HmsL(b) Sample kinetic energy increases with inelastic
loss.

(d) Potential and Kinetic energy are tied together
on average over time scales longer than a few
trap oscillation cycles.

Figure B.1.: Simulation of inelastic loss in a harmonic trap using an initial MaxwellBoltzmann distribution at 3 µK.

179
portional to the square of the axial trap frequencies, as expected in a harmonic trap. Comparisons against more realistic anisotropic harmonic traps and the full model potential werealso performed. Agreements between the analytical model and the simulation are similar.

B.3. Elastic collisions
To calibrate the Monte-Carlo simulation against known behavior under elastic collisions, thetrap was changed to an anisotropic harmonic oscillator with trap frequencies close to themodel potential eigenfrequencies at the bottom of the trap (νx , νy , νz ) = (240, 310, 180) Hz.
We introduce an anisotropy in both position and momentum space and let the X and Y axes
have a Boltzmann distribution with Tx0 = Ty0 = 3 µK. The Z axis is initially set to Tz0 = 5 µK.
By the initial axial temperatures Ti0 we mean that both the potential energy m(2πνi )2 hx2i i/2
and kinetic energy mhvi2 i/2 is set to kB Ti0 /2 along each axis initially. Elastic collisions tend
to equilibrate the temperature and from a treatment based on Enskog’s equation, we expectthat the axial temperatures Ti approach each other exponentially [162]. This thermalization
process can be modeled asṪi = −
with solution

X Γtherm
j6=i3(Ti − Tj ),Ti (t) = T̄ (0) + e−Γtherm t [Ti (0) − T̄ (0)],and mean temperature T̄ ≡

P

(B.8)

(B.9)

j Tj . The thermalization constant Γtherm is then related to thenumber of collision events required for thermalization κ via [162]Γtherm =2n̄σel hvrel iT̄,κwith mean density n̄ and thermally averaged relative velocitys
8kB T̄
.
hvrel iT̄ =πµ(B.10)(B.11)

Since these quantities are thermally averaged and the initial distribution is not thermalized,
we expect Γtherm to be time-dependent and also depend on the initial inhomogeneity. Following Ref. [212], we can define an initial inhomogeneity parameter as the ratio of axial energies

180
Ei (valid for an anisotropic harmonic trap):Ω=

Ex,yhvx2 iTx0
hv 2 i + (2πνx )2 hx2x i= 0.6.
=== x2
Ez t=0hvz i + (2πνz )2 hx2z i
hvz2 iTz0(B.12)For this value of Ω, we find a prediction of κ ≃ 2.5 from the analytical model in Ref. [212].
The simulation results for σel = 8π(100a0 )2 are shown in Fig. B.2. As seen in panel (a), no
atoms are lost due to evaporation or inelastic losses. However, the mean density decreases
by about 5% because the density along X and Y dilutes slightly as the Z distributionµK = 3.67 µK andconcentrates. The axial kinetic energies equilibrate towards T̄ = 3+3+5
3
we extract Γ−1therm ≃ 65 ms from fitting Eq. B.10 to the data in panel (c).
The axial potential energies in panel (d) follow the kinetic energies and equilibrate at thelevels given by the individual trap frequencies, as shown by panel (e). As argued above,the collision event rate changes slightly as the sample equilibrates [see panel (f)]. Initially,
the rate of collision events is about Γel (0) ≃ 860 ms−1 and then settles towards Γel (∞) ≃840 ms−1 as the sample thermalizes. From Eq. B.10, we find κ∞ ≃ 2.23. Rescaling towards
the value at short times [212] we find
κ0 = κ∞Γel (0)≃ 2.3,
Γel (∞)(B.13)

which is in fair agreement with the prediction κ ≃ 2.5 [212].
We conclude that the Monte-Carlo simulation agrees reasonably well with what we expectfrom the known results for both elastic and inelastic collision processes in harmonic traps forconstant cross sections. We then use it in the main text to model the collisional dynamics of
the OFR effect in the full model potential and with the full detuning- and velocity-dependent
cross section expressions.

181(a) The atom number stays constant throughout
the simulation.

(c) The kinetic energy equilibrates to the meanover all axes.

(e) The potential energy follows the kinetic energy
on timescales longer than a few trap oscillations.

(b) The mean density decreases because the X andY distributions are heated.

(d) The cloud widths settle according to the axial
trap frequency and the kinetic energy.

(f ) The collision event rate decreases since themean density goes down.

Figure B.2.: Simulation of cross-dimensional thermalization in an anisotropic harmonic
trap. The initial potential and kinetic energy along the X and Y axes are k2B × (3 µK),
the Z axis distributions are heated to 5 µK.

Appendix

C

Related Publications

By now, the first generation of graduate students in optical lattice clock experiments
has graduated and the interested reader is referred to the many excellent and pub-

licly available theses on high-resolution laser spectroscopy, absolute frequency measurements,
remote optical comparison of frequency standards, and the relevant atomic physics and technology.

Publicly available theses on optical lattice clocks:• JILA Sr clock: [11–14]• LNE SYRTE Sr clock: [52, 53, 147] (French), [54] (English)• LENS Sr clock: [214] (English)• NIST Yb clock: [215]• LNE SYRTE Hg lattice spectroscopy: [216] (English)

List of publications related to this work:• S. Blatt, T. L. Nicholson, B. J. Bloom, J. R. Williams, J. W. Thomsen, P. S. Julienne,
and J. Ye, Thermodynamics of the Optical Feshbach Resonance Effect, arXiv:1104.0210v1,
submitted to Physical Review Letters [10].
• M. D. Swallows, M. Bishof, Y. Lin, S. Blatt, M. J. Martin, A. M. Rey, and J. Ye, Suppressionof Collisional Shifts in a Strongly Interacting Lattice Clock, Science 331, 1043 (2011) [51].
• M. D. Swallows, G. K. Campbell, A. D. Ludlow, M. M. Boyd, J. W. Thomsen, M. J. Martin,
S. Blatt, T. L. Nicholson, and J. Ye, Precision measurement of fermionic collisions using

184
an 87 Sr optical lattice clock with 1 × 10−16 inaccuracy, IEEE Transactions on Ultrasonics,
Ferroelectrics and Frequency Control 57, 574 (2010) [186].
• S. Blatt, J. W. Thomsen, G. K. Campbell, A. D. Ludlow, M. D. Swallows, M. J. Martin, M.
M. Boyd, and J. Ye, Rabi spectroscopy and excitation inhomogeneity in a one-dimensional
optical lattice clock, Physical Review A 80, 052703 (2009) [9].
• G. K. Campbell, M. M. Boyd, J. W. Thomsen, M. J. Martin, S. Blatt, M. D. Swallows, T.
L. Nicholson, T. Fortier, C. W. Oates, S. A. Diddams, N. D. Lemke, P. Naidon, P. Julienne,
Jun Ye, and A. D. Ludlow, Probing Interactions Between Ultracold Fermions, Science 324,
360 (2009) [46].
• G. K. Campbell, A. D. Ludlow, S. Blatt, J. W. Thomsen, M. J. Martin, M. H. G. de Miranda,
T. Zelevinsky, M. M. Boyd, J. Ye, S. A. Diddams, T. P. Heavner, T. E. Parker, and S.
R. Jefferts, The absolute frequency of the 87 Sr optical clock transition, Metrologia 45, 539(2008) [7].
• A. D. Ludlow, S. Blatt, T. Zelevinsky, G. K. Campbell, M. J. Martin, J. W. Thomsen, M. M.
Boyd, and J. Ye, Ultracold strontium clock: Applications to the measurement of fundamental
constant variations, The European Physical Journal Special Topics 163, 9 (2008) [31].
• S. Blatt, A. D. Ludlow, G. K. Campbell, J. W. Thomsen, T. Zelevinsky, M. M. Boyd, J. Ye,
X. Baillard, M. Fouché, R. Le Targat, A. Brusch, P. Lemonde, M. Takamoto, F.-L. Hong, H.
Katori, and V. V. Flambaum, New Limits on Coupling of Fundamental Constants to Gravity
Using 87 Sr Optical Lattice Clocks, Physical Review Letters 100, 140801 (2008) [8].
• A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda,
M. J. Martin, J. W. Thomsen, S. M. Foreman, Jun Ye, T. M. Fortier, J. E. Stalnaker, S. A.
Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, Sr
Lattice Clock at 1 × 10−16 Fractional Uncertainty by Remote Optical Evaluation with a Ca
Clock, Science 319, 1805 (2008) [31].
• T. Zelevinsky, S. Blatt, M. M. Boyd, G. K. Campbell, A. D. Ludlow, and J. Ye, Highly
Coherent Spectroscopy of Ultracold Atoms and Molecules in Optical Lattices, ChemPhysChem
9, 375 (2008) [217].
• J. Ye, S. Blatt, M. M. Boyd, S. M. Foreman, E. R. Hudson, T. Ido, B. Lev, A. D. Ludlow,
B. C. Sawyer, B. Stuhl, and T. Zelevinsky, Precision measurement based on ultracold atoms
and cold molecules, International Journal of Modern Physics D 16, 2481 (2007) [218].
• M. M. Boyd, T. Zelevinsky, A. D. Ludlow, S. Blatt, T. Zanon-Willette, S. M. Foreman, andJ. Ye, Nuclear spin effects in optical lattice clocks, Physical Review A 76, 022510 (2007) [94].
• T. Zelevinsky, M. M. Boyd, A. D. Ludlow, S. M. Foreman, S. Blatt, T. Ido, and J. Ye, Opticalclock and ultracold collisions with trapped strontium atoms, Hyperfine Interactions 174, 55(2007) [219].
• A. D. Ludlow, X. Huang, M. Notcutt, T. Zanon-Willette, S. M. Foreman, M. M. Boyd, S.
Blatt, and J. Ye, Compact, thermal-noise-limited optical cavity for diode laser stabilization
at 1 × 10−15 , Optics Letters 32, 641 (2007) [27].

185• M. M. Boyd, A. D. Ludlow, S. Blatt, S. M. Foreman, T. Ido, T. Zelevinsky, and J. Ye, 87 Sr
Lattice Clock with Inaccuracy below 10−15 , Physical Review Letters 98, 083002 (2007) [94].
• T. Zanon-Willette, A. D. Ludlow, S. Blatt, M. M. Boyd, E. Arimondo, and J. Ye, Cancellationof Stark Shifts in Optical Lattice Clocks by Use of Pulsed Raman and Electromagnetically
Induced Transparency Techniques, Physical Review Letters 97, 233001 (2006) [220].
• M. M. Boyd, T. Zelevinsky, A. D. Ludlow, S. M. Foreman, S. Blatt, T. Ido, and J. Ye, Optical
Atomic Coherence at the 1-Second Time Scale, Science 314, 1430 (2006) [43].
• A. D. Ludlow, M. M. Boyd, T. Zelevinsky, S. M. Foreman, S. Blatt, M. Notcutt, T. Ido, andJ. Ye, Systematic Study of the 87 Sr Clock Transition in an Optical Lattice, Physical Review
Letters 96, 033003 (2006) [2].

Bibliography[1] Masao Takamoto, Feng-Lei Hong, Ryoichi Higashi, and Hidetoshi Katori.
An optical lattice clock.
Nature 435, 321, 2005.
doi:10.1038/nature03541.
[2] Andrew D. Ludlow, Martin M. Boyd, Tanya Zelevinsky, Seth M. Foreman, Sebastian Blatt, Mark Notcutt, Tetsuya Ido,
and Jun Ye.
Systematic study of the 87 Sr clock transition in an optical lattice.
Physical Review Letters 96, 033003, 2006.
doi:10.1103/PhysRevLett.96.033003.
[3] Rodolphe Le Targat, Xavier Baillard, Mathilde Fouché, Anders Brusch, Olivier Tcherbakoﬀ, Giovanni Rovera, and Pierre
Lemonde.
Accurate optical lattice clock with 87 Sr atoms.
Physical Review Letters 97, 130801, 2006.
doi:10.1103/PhysRevLett.97.130801.
[4] N. Lemke, A. Ludlow, Z. Barber, T. Fortier, S. Diddams, Y. Jiang, S. Jeﬀerts, T. Heavner, T. Parker, and C. Oates.
Spin-1/2 optical lattice clock.
Physical Review Letters 103, 063001, 2009.
doi:10.1103/PhysRevLett.103.063001.
[5] Xavier Baillard, Mathilde Fouché, Rodolphe Le Targat, Philip G. Westergaard, Arnaud Lecallier, Yann Le Coq, Giovanni D. Rovera, Sebastien Bize, and Pierre Lemonde.
Accuracy evaluation of an optical lattice clock with bosonic atoms.
Optics Letters 32, 1812, 2007.
doi:10.1364/OL.32.001812.
[6] Ch. Lisdat, J. S. R. Vellore Winfred, T. Middelmann, F. Riehle, and U. Sterr.
Collisional losses, decoherence, and frequency shifts in optical lattice clocks with bosons.
Physical Review Letters 103, 090801, 2009.
doi:10.1103/PhysRevLett.103.090801.
[7] Gretchen K. Campbell, Andrew D. Ludlow, Sebastian Blatt, Jan W. Thomsen, Michael J. Martin, Marcio H. G. de Miranda, Tanya Zelevinsky, Martin M. Boyd, Jun Ye, Scott A. Diddams, Thomas P. Heavner, Thomas E. Parker, andSteven R. Jeﬀerts.
The absolute frequency of the 87 Sr optical clock transition.
Metrologia 45, 5, 539, 2008.
doi:10.1088/0026-1394/45/5/008.
[8] S. Blatt, A. Ludlow, G. Campbell, J. Thomsen, T. Zelevinsky, M. Boyd, J. Ye, X. Baillard, M. Fouché, R. Le Targat,
A. Brusch, P. Lemonde, M. Takamoto, F.-L. Hong, H. Katori, and V. Flambaum.
New limits on coupling of fundamental constants to gravity using 87 Sr optical lattice clocks.
Physical Review Letters 100, 140801, 2008.
doi:10.1103/PhysRevLett.100.140801.
[9] S. Blatt, J. W. Thomsen, G. K. Campbell, A. D. Ludlow, M. D. Swallows, M. J. Martin, M. M. Boyd, and J. Ye.
Rabi spectroscopy and excitation inhomogeneity in a one-dimensional optical lattice clock.
Physical Review A 80, 052703, 2009.
doi:10.1103/PhysRevA.80.052703.

188[10] S. Blatt, T. L. Nicholson, B. J. Bloom, J. R. Williams, J. W. Thomsen, P. S. Julienne, and J. Ye.
Thermodynamics of the optical Feshbach resonance effect.
arXiv:1104.0210v1 2011.
[11] Martin M. Boyd.
High Precision Spectroscopy of Strontium in an Optical Lattice: Towards a New Standard for Frequency and Time.
PhD thesis, University of Colorado, 2007.
URL http://jilawww.colorado.edu/yelabs/pubs/theses.html.
[12] Andrew D. Ludlow.
The Strontium Optical Lattice Clock: Optical Spectroscopy with Sub-Hertz Accuracy.
PhD thesis, University of Colorado, 2008.
URL http://jilawww.colorado.edu/yelabs/pubs/theses.html.
[13] Seth M. Foreman.
Femtosecond Frequency Combs for Optical Clocks and Timing Transfer.
PhD thesis, University of Colorado, 2007.
URL http://jila.colorado.edu/thesis.
[14] Sebastian Blatt.
Precision Spectroscopy in 1D Optical Lattices.
Masters thesis, Leopold-Franzens-Universität Innsbruck, 2005.
[15] Xinye Xu, Thomas Loftus, Matthew Smith, John Hall, Alan Gallagher, and Jun Ye.
Dynamics in a two-level atom magneto-optical trap.
Physical Review A 66, 011401, 2002.
doi:10.1103/PhysRevA.66.011401.
[16] Xinye Xu, Thomas H. Loftus, John L. Hall, Alan Gallagher, and Jun Ye.
Cooling and trapping of atomic strontium.
Journal of the Optical Society of America B 20, 968, 2003.
doi:10.1364/JOSAB.20.000968.
[17] Xinye Xu, Thomas Loftus, Josh Dunn, Chris Greene, John Hall, Alan Gallagher, and Jun Ye.
Single-stage sub-Doppler cooling of alkaline earth atoms.
Physical Review Letters 90, 193002, 2003.
doi:10.1103/PhysRevLett.90.193002.
[18] Thomas H. Loftus, Tetsuya Ido, Martin M. Boyd, Andrew D. Ludlow, and Jun Ye.
Narrow line cooling and momentum-space crystals.
Physical Review A 70, 063413, 2004.
doi:10.1103/PhysRevA.70.063413.
[19] Thomas Loftus, Tetsuya Ido, Andrew Ludlow, Martin Boyd, and Jun Ye.
Narrow line cooling: Finite photon recoil dynamics.
Physical Review Letters 93, 073003, 2004.
doi:10.1103/PhysRevLett.93.073003.
[20] Takashi Mukaiyama, Hidetoshi Katori, Tetsuya Ido, Ying Li, and Makoto Kuwata-Gonokami.
Recoil-limited laser cooling of 87 Sr atoms near the Fermi temperature.
Physical Review Letters 90, 113002, 2003.
doi:10.1103/PhysRevLett.90.113002.
[21] J. Ye, H. J. Kimble, and H. Katori.
Quantum state engineering and precision metrology using state-insensitive light traps.
Science 320, 1734, 2008.
doi:10.1126/science.1148259.
[22] Martin Boyd, Tanya Zelevinsky, Andrew Ludlow, Sebastian Blatt, Thomas Zanon-Willette, Seth Foreman, and Jun Ye.
Nuclear spin effects in optical lattice clocks.
Physical Review A 76, 022510, 2007.
doi:10.1103/PhysRevA.76.022510.
[23] A. V. Taichenachev, V. I. Yudin, C. W. Oates, C. W. Hoyt, Z. W. Barber, and L. Hollberg.
Magnetic field-induced spectroscopy of forbidden optical transitions with application to lattice-based optical atomic clocks.
Physical Review Letters 96, 083001, 2006.
doi:10.1103/PhysRevLett.96.083001.

189[24] N. Poli, Z. W. Barber, N. D. Lemke, C. W. Oates, L. S. Ma, J. E. Stalnaker, T. M. Fortier, S. A. Diddams, L. Hollberg,
J. C. Bergquist, A. Brusch, S. Jeﬀerts, T. Heavner, and T. Parker.
Frequency evaluation of the doubly forbidden 1 S0 →3 P0 transition in bosonic 174 Yb.
Physical Review A 77, 050501, 2008.
doi:10.1103/PhysRevA.77.050501.
[25] D. Wineland and Wayne Itano.
Laser cooling of atoms.
Physical Review A 20, 1521, 1979.
doi:10.1103/PhysRevA.20.1521.
[26] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland.
Quantum dynamics of single trapped ions.
Reviews of Modern Physics 75, 281, 2003.
doi:10.1103/RevModPhys.75.281.
[27] A. D. Ludlow, X. Huang, M. Notcutt, T. Zanon-Willette, S. M. Foreman, M. M. Boyd, S. Blatt, and J. Ye.
Compact, thermal-noise-limited optical cavity for diode laser stabilization at 1 × 10−15 .
Optics Letters 32, 641, 2007.
doi:10.1364/OL.32.000641.
[28] Seth M. Foreman, Kevin W. Holman, Darren D. Hudson, David J. Jones, and Jun Ye.
Remote transfer of ultrastable frequency references via fiber networks.
Review of Scientiﬁc Instruments 78, 021101, 2007.
doi:10.1063/1.2437069.
[29] Seth M. Foreman, Andrew D. Ludlow, Marcio H. G. de Miranda, Jason E. Stalnaker, Scott A. Diddams, and Jun Ye.
Coherent optical phase transfer over a 32-km fiber with 1 s instability at 10−17 .
Physical Review Letters 99, 153601, 2007.
doi:10.1103/PhysRevLett.99.153601.
[30] T. Fortier, N. Ashby, J. Bergquist, M. Delaney, S. Diddams, T. Heavner, L. Hollberg, W. Itano, S. Jeﬀerts, K. Kim,
F. Levi, L. Lorini, W. Oskay, T. Parker, J. Shirley, and J. Stalnaker.
Precision atomic spectroscopy for improved limits on variation of the fine structure constant and local position invariance.
Physical Review Letters 98, 070801, 2007.
doi:10.1103/PhysRevLett.98.070801.
[31] A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen,
S. M. Foreman, J. Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke,
K. M. Beck, and C. W. Oates.
Sr lattice clock at 1 × 10−16 fractional uncertainty by remote optical evaluation with a Ca clock.
Science 319, 1805, 2008.
doi:10.1126/science.1153341.
[32] Wayne M. Itano, L. L. Lewis, and D. J. Wineland.
Shift of 2 S1/2 hyperfine splittings due to blackbody radiation.
Physical Review A 25, 1233, 1982.
doi:10.1103/PhysRevA.25.1233.
[33] Sergey Porsev and Andrei Derevianko.
Multipolar theory of blackbody radiation shift of atomic energy levels and its implications for optical lattice clocks.
Physical Review A 74, 020502, 2006.
doi:10.1103/PhysRevA.74.020502.
[34] S. Porsev, Andrew Ludlow, Martin Boyd, and Jun Ye.
Determination of Sr properties for a high-accuracy optical clock.
Physical Review A 78, 032508, 2008.
doi:10.1103/PhysRevA.78.032508.
[35] M. S. Safronova, Dansha Jiang, M. G. Kozlov, and U. I. Safronova.
Blackbody radiation shifts and magic wavelengths for atomic clock research.
In 2010 IEEE International Frequency Control Symposium (FCS), p. 59. 2010.
doi:10.1109/FREQ.2010.5556374.
[36] Thomas Middelmann, Christian Lisdat, Stephan Falke, Joseph S. R. Vellore Winfred, Fritz Riehle, and Uwe Sterr.
Tackling the blackbody shift in a strontium optical lattice clock.
arXiv:1009.2017v1 2010.

190[37] H. Katori.
In P. Gill, editor, Proceedings of the 6th Symposium on Frequency Standards and Metrology, pp. 323–330 (World Scientiﬁc,
Singapore), 2002.
[38] Hidetoshi Katori, Masao Takamoto, V. Pal’chikov, and V. Ovsiannikov.
Ultrastable optical clock with neutral atoms in an engineered light shift trap.
Physical Review Letters 91, 173005, 2003.
doi:10.1103/PhysRevLett.91.173005.
[39] H. J. Kimble, C. J. Hood, T. W. Lynn, H. Mabuchi, D. W. Vernooy, and J. Ye.
The quantum internet.
In R. Blatt, J. Eschner, D. Leibfried, and F. Schmidt-Kaler, editors, Proceedings of the Fourteenth International Conference on Laser Spectroscopy (ICOLS99), Innsbruck, pp. 80–99 (World Scientiﬁc, Singapore), 1999.
[40] Irène Courtillot, Audrey Quessada, Richard Kovacich, Anders Brusch, Dmitri Kolker, Jean-Jacques Zondy, Giovanni
Rovera, and Pierre Lemonde.
Clock transition for a future optical frequency standard with trapped atoms.
Physical Review A 68, 030501, 2003.
doi:10.1103/PhysRevA.68.030501.
[41] Masao Takamoto and Hidetoshi Katori.
Spectroscopy of the 1 S0 -3 P0 clock transition of 87 Sr in an optical lattice.
Physical Review Letters 91, 223001, 2003.
doi:10.1103/PhysRevLett.91.223001.
[42] Masao Takamoto, Feng-Lei Hong, Ryoichi Higashi, Yasuhisa Fujii, Michito Imae, and Hidetoshi Katori.
Improved frequency measurement of a one-dimensional optical lattice clock with a spin-polarized fermionic 87 Sr isotope.
Journal of the Physics Society Japan 75, 104302, 2006.
doi:10.1143/JPSJ.75.104302.
[43] M. M. Boyd, T. Zelevinsky, A. D. Ludlow, S. M. Foreman, S. Blatt, T. Ido, and J. Ye.
Optical atomic coherence at the 1-second time scale.
Science 314, 1430, 2006.
doi:10.1126/science.1133732.
[44] Anders Brusch, Rodolphe Le Targat, Xavier Baillard, Mathilde Fouché, and Pierre Lemonde.
Hyperpolarizability effects in a Sr optical lattice clock.
Physical Review Letters 96, 103003, 2006.
doi:10.1103/PhysRevLett.96.103003.
[45] X. Baillard, M. Fouché, R. Le Targat, P. G. Westergaard, A. Lecallier, F. Chapelet, M. Abgrall, G. D. Rovera, P. Laurent,
P. Rosenbusch, S. Bize, G. Santarelli, A. Clairon, P. Lemonde, G. Grosche, B. Lipphardt, and H. Schnatz.
An optical lattice clock with spin-polarized 87 Sr atoms.
The European Physical Journal D 48, 11, 2007.
doi:10.1140/epjd/e2007-00330-3.
[46] G. K. Campbell, M. M. Boyd, J. W. Thomsen, M. J. Martin, S. Blatt, M. D. Swallows, T. L. Nicholson, T. Fortier, C. W.
Oates, S. A. Diddams, N. D. Lemke, P. Naidon, P. Julienne, J. Ye, and A. D. Ludlow.
Probing interactions between ultracold fermions.
Science 324, 360, 2009.
doi:10.1126/science.1169724.
[47] A. M. Rey, A. V. Gorshkov, and C. Rubbo.
Many-body treatment of the collisional frequency shift in fermionic atoms.
Physical Review Letters 103, 260402, 2009.
doi:10.1103/PhysRevLett.103.260402.
[48] Kurt Gibble.
Decoherence and collisional frequency shifts of trapped bosons and fermions.
Physical Review Letters 103, 113202, 2009.
doi:10.1103/PhysRevLett.103.113202.
[49] Zhenhua Yu and C. J. Pethick.
Clock shifts of optical transitions in ultracold atomic gases.
Physical Review Letters 104, 010801, 2010.
doi:10.1103/PhysRevLett.104.010801.

191[50] Y. B. Band and I. Osherov.
Collisionally induced atomic clock shifts and correlations.
arXiv:1011.3311v1 2010.
[51] M. D. Swallows, M. Bishof, Y. Lin, S. Blatt, M. J. Martin, A. M. Rey, and J. Ye.
Suppression of collisional shifts in a strongly interacting lattice clock.
Science 331, 1043, 2011.
doi:10.1126/science.1196442.
[52] Rodolphe Le Targat.
Horloge à réseau optique au Strontium: une 2ème génération d’horloges à atomes froids.
PhD thesis, LNE-SYRTE, 2007.
URL http://tel.archives-ouvertes.fr/tel-00170038.
[53] Xavier Baillard.
Horloge à réseau optique à atomes de Strontium.
PhD thesis, Université de Paris VI, 2008.
URL http://tel.archives-ouvertes.fr/docs/00/26/73/09/PDF/These\_X\_Baillard.pdf.
[54] Philip G. Westergaard.
Horloge à réseau optique au Strontium: en quête de la performance ultime.
PhD thesis, L’Edite de Paris, 2010.
URL http://tel.archives-ouvertes.fr/tel-00541420/en/.
[55] K. E. Cahill and R. J. Glauber.
Ordered expansions in boson amplitude operators.
Physical Review 177, 1857, 1969.
doi:10.1103/PhysRev.177.1857.
[56] Pierre Lemonde and Peter Wolf.
Optical lattice clock with atoms confined in a shallow trap.
Physical Review A 72, 033409, 2005.
doi:10.1103/PhysRevA.72.033409.
[57] R. Shankar.
Principles of Quantum Mechanics (Springer, New York), second edition, 1994.
ISBN 0-306-44790-8.
[58] Milton Abramowitz and Irene A. Stegun, editors.
Handbook of Mathematical Functions (Dover, Mineola, New York), 1972.
ISBN 0-486-61272-4.
[59] Pradip K. Ghosh.
Ion Traps (Clarendon Press, Oxford), 1995.
ISBN 0-19-853995-9.
[60] J. Slater.
A soluble problem in energy bands.
Physical Review 87, 807, 1952.
doi:10.1103/PhysRev.87.807.
[61] W. Kohn.
Analytic properties of Bloch waves and Wannier functions.
Physical Review 115, 809, 1959.
doi:10.1103/PhysRev.115.809.
[62] M. Glück, A. R. Kolovsky, H. J. Korsch, and N. Moiseyev.
Calculation of Wannier-Bloch and Wannier-Stark states.
The European Physical Journal D - Atomic, Molecular and Optical Physics 4, 239, 1998.
doi:10.1007/s100530050205.
[63] Markus Glück, Andrey R. Kolovsky, and Hans Jürgen Korsch.
Wannier-Stark resonances in optical and semiconductor superlattices.
Physics Reports 366, 103, 2002.
doi:10.1016/S0370-1573(02)00142-4.

192[64] Ana Maria Rey.
Ultracold bosonic atoms in optical lattices.
PhD thesis, University of Maryland, Department of Physics, 2004.
URL http://jila.colorado.edu/∼arey/papers/thesis.pdf.
[65] Quentin Beauﬁls, Gunnar Tackmann, Xiaolong Wang, Bruno Pelle, Sophie Pelisson, Peter Wolf, and Franck Pereira Dos
Santos.
Laser controlled tunneling in a vertical optical lattice.
arXiv:1102.5326v1 2011.
[66] Ana Maria Rey.
Collisional frequency shifts in 1D bosonic atoms.
unpublished 2010.
[67] Michael Bishof, Yige Lin, Matthew D. Swallows, Alexey V. Gorshkov, Jun Ye, and Ana Maria Rey.
Resolved atomic interaction sidebands in an optical clock transition.
arXiv:1102.1016v2 2011.
[68] N. D. Lemke, A. D. Ludlow, J. von Stecher, J. A. Sherman, A. M. Rey, and C. W. Oates.
p-Wave cold collisions in a Yb lattice clock.
arXiv:1105.2014v1 2011.
[69] T. Holstein and H. Primakoﬀ.
Field dependence of the intrinsic domain magnetization of a ferromagnet.
Physical Review 58, 1098, 1940.
doi:10.1103/PhysRev.58.1098.
[70] L. Childress, M. V. Gurudev Dutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J. Wrachtrup, P. R. Hemmer, and M. D. Lukin.
Coherent dynamics of coupled electron and nuclear spin qubits in diamond.
Science 314, 281, 2006.
doi:10.1126/science.1131871.
[71] J. J. Sakurai.
Modern Quantum Mechanics (Addison-Wesley, Reading, Massachusetts), second edition, 1994.
ISBN 0-201-53929-2.
[72] T. L. Gilbert.
Classics in magnetics A phenomenological theory of damping in ferromagnetic materials.
IEEE Transactions on Magnetics 40, 3443, 2004.
doi:10.1109/TMAG.2004.836740.
[73] Massimiliano d’Aquino.
Nonlinear Magnetization Dynamics in Thin-films and Nanoparticles.
PhD thesis, Università degli studi di Napoli “Federico II”, Facoltà di Ingegneria, 2004.
URL http://www.fedoa.unina.it/148/.
[74] Ivan Cimrák.
On the Landau-Lifshitz Equation of Ferromagnetism.
PhD thesis, Ghent University, Faculty of Applied Sciences, Faculty of Mathematical Analysis, 2005.
URL http://biblio.ugent.be/input/download?func=downloadFile&fileOId=494022.
[75] J. Webb, M. Murphy, V. Flambaum, V. Dzuba, J. Barrow, C. Churchill, J. Prochaska, and A. Wolfe.
Further evidence for cosmological evolution of the fine structure constant.
Physical Review Letters 87, 091301, 2001.
doi:10.1103/PhysRevLett.87.091301.
[76] R. Srianand, H. Chand, P. Petitjean, and B. Aracil.
Limits on the time variation of the electromagnetic fine-structure constant in the low energy limit from absorption lines
in the spectra of distant quasars.
Physical Review Letters 92, 121302, 2004.
doi:10.1103/PhysRevLett.92.121302.
[77] Thibault Damour and Freeman Dyson.
The Oklo bound on the time variation of the fine-structure constant revisited.
Nuclear Physics B 480, 37, 1996.
doi:10.1016/S0550-3213(96)00467-1.

193[78] S. G. Karshenboim.
Some possibilities for laboratory searches for variations of fundamental constants.
Canadian Journal of Physics 78, 2000.
doi:10.1139/cjp-78-7-639.
[79] S. G. Karshenboim, V. V. Flambaum, and E. Peik.
Atomic clocks and constraints on variation of fundamental constants.
In G. W. F. Drake, editor, Handbook of Atomic, Molecular and Optical Physics, pp. 455–463 (Springer), 2005.
ISBN 0-387-20802-X.
[80] S. N. Lea.
Limits to time variation of fundamental constants from comparisons of atomic frequency standards.
Reports on Progress in Physics 70, 1473, 2007.
doi:10.1088/0034-4885/70/9/R01.
[81] Savely G. Karshenboim and Ekkehard Peik, editors.
Lecture Notes in Physics: Astrophysics, Clocks and Fundamental Constants, volume 648 (Springer, Berlin), 2004.
ISBN 3-540-21967-6.
[82] E. Peik, B. Lipphardt, H. Schnatz, T. Schneider, Chr. Tamm, and S. Karshenboim.
Limit on the present temporal variation of the fine structure constant.
Physical Review Letters 93, 170801, 2004.
doi:10.1103/PhysRevLett.93.170801.
[83] M. Fischer, N. Kolachevsky, M. Zimmermann, R. Holzwarth, Th. Udem, T. W. Hänsch, M. Haas, U. D. Jentschura, andC. H. Keitel.
New limits on the drift of fundamental constants from laboratory measurements.
Physical Review Letters 92, 230802, 2004.
doi:10.1103/PhysRevLett.92.230802.
[84] M. Zimmermann, M. Fischer, N. Kolachevsky, R. Holzwarth, T. Udem, T. W. Hänsch, M. Abgrall, J. Grünert, I. Maksimovic, S. Bize, H. Marion, F. Pereira Dos Santos, P. Lemonde, G. Santarelli, P. Laurent, A. Clairon, and C. Salomon.
High-resolution laser spectroscopy and time variation of fundamental constants.
Laser Physics 15, 997, 2005.
[85] E. Peik, B. Lipphardt, H. Schnatz, C. Tamm, S. Weyers, and R. Wynands.
Laboratory limits on temporal variations of fundamental constants: An update.
In H. Kleinert, R. T. Jantzen, and R. Ruﬃni, editors, Proceedings of the 11th Marcel Grossmann Meeting on General
Relativity, pp. 941–951. 2008.
[86] John Turneaure, Cliﬀord Will, Brian Farrell, Edward Mattison, and Robert Vessot.
Test of the principle of equivalence by a null gravitational red-shift experiment.
Physical Review D 27, 1705, 1983.
doi:10.1103/PhysRevD.27.1705.
[87] John Prestage, Robert Tjoelker, and Lute Maleki.
Atomic clocks and variations of the fine structure constant.
Physical Review Letters 74, 3511, 1995.
doi:10.1103/PhysRevLett.74.3511.
[88] A. Bauch and S. Weyers.
New experimental limit on the validity of local position invariance.
Physical Review D 65, 081101, 2002.
doi:10.1103/PhysRevD.65.081101.
[89] N. Ashby, T. P. Heavner, S. R. Jeﬀerts, T. E. Parker, A. G. Radnaev, and Y. O. Dudin.
Testing local position invariance with four cesium-fountain primary frequency standards and four NIST hydrogen masers.
Physical Review Letters 98, 070802, 2007.
doi:10.1103/PhysRevLett.98.070802.
[90] Th. Schmidt.
Über die magnetischen Momente der Atomkerne.
Zeitschrift für Physik 106, 358, 1937.
doi:10.1007/BF01338744.
[91] H. Kopfermann.
Nuclear Moments (Academic Press, New York, New York), 1958.

194[92] V. V. Flambaum and A. F. Tedesco.
Dependence of nuclear magnetic moments on quark masses and limits on temporal variation of fundamental constantsfrom atomic clock experiments.
Physical Review C 73, 055501, 2006.
doi:10.1103/PhysRevC.73.055501.
[93] E. J. Angstmann, V. A. Dzuba, and V. V. Flambaum.
Atomic clocks and the search for variation of the fine structure constant.
arXiv:physics/0407141v1 2004.
[94] Martin M. Boyd, Andrew D. Ludlow, Sebastian Blatt, Seth M. Foreman, Tetsuya Ido, Tanya Zelevinsky, and Jun Ye.
87 Sr lattice clock with inaccuracy below 10−15 .
Physical Review Letters 98, 083002, 2007.
doi:10.1103/PhysRevLett.98.083002.
[95] T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M.
Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C.
Bergquist.
Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place.
Science 319, 1808, 2008.
doi:10.1126/science.1154622.
[96] Philip R. Bevington and D. Keith Robinson.
Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, New York), third edition, 2003.
ISBN 0-07-247227-8.
[97] Matthew Lybanon.
A better least-squares method when both variables have uncertainties.
American Journal of Physics 52, 22, 1984.
doi:10.1119/1.13822.
[98] William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery.
Numerical Recipes in C++ (Cambridge University Press, Cambridge), second edition, 2002.
ISBN 0-521-75033-4.
[99] R. J. Carroll, D. Ruppert, and L. A. Stefanski.
Measurement Error in Nonlinear Models (Chapman & Hall, London), 1995.
ISBN 0-412-04721-7.
[100] Norman A. Draper and Harry Smith.
Applied Regression Analysis (Wiley, New York, New York), third edition, 1998.
ISBN 9812530347.
[101] David R. Williams.
NASA planetary factsheet 2005.
URL http://nssdc.gsfc.nasa.gov/planetary/factsheet/.
[102] M. R. House.
Orbital forcing timescales: an introduction.
Geological Society, London, Special Publications 85, 1, 1995.
doi:10.1144/GSL.SP.1995.085.01.01.
[103] P. Bretagnon and G. Francou.
Planetary theories in rectangular and spherical variables. VSOP87 solutions.
Astronomy and Astrophysics 202, 309, 1988.
[104] W. M. Smart.
Celestial Mechanics (Wiley, New York), 1953.
[105] David W. Allan, Neil Ashby, and Cliﬀord C. Hodge.
The science of timekeeping.
Hewlett-Packard Application Notes 1289, 1997.
[106] T. M. Fortier, N. Ashby, J. C. Bergquist, M. J. Delaney, S. A. Diddams, T. P. Heavner, L. Hollberg, W. M. Itano, S. R.
Jeﬀerts, K. Kim, W. H. Oskay, T. E. Parker, J. Shirley, J. E. Stalnaker, F. Levi, and L. Lorini.
Improved limits on variation of the fine structure constant and violation of local position invariance.
In Frequency Control Symposium, 2007 Joint with the 21st European Frequency and Time Forum, p. 663 (IEEE International), 2007.

195
ISBN 9781-424-4064-7-0.
doi:10.1109/FREQ.2007.4319157.
[107] Thomas Dent.
Eötvös bounds on couplings of fundamental parameters to gravity.
Physical Review Letters 101, 041102, 2008.
doi:10.1103/PhysRevLett.101.041102.
[108] Thomas Dent, Steﬀen Stern, and Christof Wetterich.
Competing bounds on the present-day time variation of fundamental constants.
Physical Review D 79, 083533, 2009.
doi:10.1103/PhysRevD.79.083533.
[109] Holger Müller, Achim Peters, and Steven Chu.
A precision measurement of the gravitational redshift by the interference of matter waves.
Nature 463, 926, 2010.
doi:10.1038/nature08776.
[110] Michael A. Hohensee and Holger Müller.
Redshift anomalies with universal free-fall.
arXiv:1008.0611v2 2010.
[111] Peter Wolf, Luc Blanchet, Christian J. Bordé, Serge Reynaud, Christophe Salomon, and Claude Cohen-Tannoudji.
Atom gravimeters and gravitational redshift.
Nature 467, E1, 2010.
doi:10.1038/nature09340.
[112] Peter Wolf, Luc Blanchet, Christian J. Bordé, Serge Reynaud, Christophe Salomon, and Claude Cohen-Tannoudji.
Does an atom interferometer test the gravitational redshift at the Compton frequency?
arXiv:1012.1194v3 2010.
[113] P. Wolf, Ch. J. Bordé, A. Clairon, L. Duchayne, A. Landragin, P. Lemonde, G. Santarelli, W. Ertmer, E. Rasel, F. S.
Cataliotti, M. Inguscio, G. M. Tino, P. Gill, H. Klein, S. Reynaud, C. Salomon, E. Peik, O. Bertolami, P. Gil,
J. Páramos, C. Jentsch, U. Johann, A. Rathke, P. Bouyer, L. Cacciapuoti, D. Izzo, P. Natale, B. Christophe,
P. Touboul, S. G. Turyshev, J. Anderson, M. E. Tobar, F. Schmidt-Kaler, J. Vigué, A. A. Madej, L. Marmet, M.-C.
Angonin, P. Delva, P. Tourrenc, G. Metris, H. Müller, R. Walsworth, Z. H. Lu, L. J. Wang, K. Bongs, A. Toncelli,
M. Tonelli, H. Dittus, C. Lämmerzahl, G. Galzerano, P. Laporta, J. Laskar, A. Fienga, F. Roques, and K. Sengstock.
Quantum physics exploring gravity in the outer solar system: the SAGAS project.
Experimental Astronomy 23, 651, 2008.
doi:10.1007/s10686-008-9118-5.
[114] E. Tiesinga, B. J. Verhaar, and H. T. C. Stoof.
Threshold and resonance phenomena in ultracold ground-state collisions.
Physical Review A 47, 4114, 1993.
doi:10.1103/PhysRevA.47.4114.
[115] S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn, and W. Ketterle.
Observation of Feshbach resonances in a Bose-Einstein condensate.
Nature 392, 151, 1998.
doi:10.1038/32354.
[116] Ugo Fano.
Sullo spettro di assorbimento dei gas nobili presso il limite dello spettro d’arco.
Il Nuovo Cimento 12, 154, 1935.
doi:10.1007/BF02958288.
[117] Ugo Fano.
On the absorption spectrum of noble gases at the arc spectrum limit.
arXiv:cond-mat/0502210v1 2005.
[118] Herman Feshbach.
A unified theory of nuclear reactions. II.
Annals of Physics 19, 287, 1962.
doi:10.1016/0003-4916(62)90221-X.
[119] James Patrick Burke, Jr.
Theoretical Investigation of Cold Alkali Atom Collisions.
PhD thesis, University of Colorado, 1999.
URL http://jila.colorado.edu/thesis/.

196[120] C. J. Pethick and H. Smith.
Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge), second edition, 2008.
ISBN 9780-521-84651-6.
[121] C. Chin, R. Grimm, P. S. Julienne, and E. Tiesinga.
Feshbach resonances in ultracold gases.
Reviews of Modern Physics 82, 1225, 2010.
doi:10.1103/RevModPhys.82.1225.
[122] Chris H. Greene.
Universal insights from few-body land.
Physics Today 63, 40, 2010.
doi:10.1063/1.3366239.
[123] Francesca Ferlaino and Rudolf Grimm.
Forty years of Efimov physics: How a bizarre prediction turned into a hot topic.
Physics 3, 2010.
doi:10.1103/Physics.3.9.
[124] Wolfgang Ketterle and Martin W. Zwierlein.
Making, probing and understanding ultracold Fermi gases.
In M. Inguscio, W. Ketterle, and C. Salomon, editors, Ultracold Fermi Gases, Proceedings of the International School ofPhysics ”Enrico Fermi”, Course CLXIV, Varenna, 20 - 30 June 2006, p. 95. 2008.
[125] Immanuel Bloch, Jean Dalibard, and Wilhelm Zwerger.
Many-body physics with ultracold gases.
Reviews of Modern Physics 80, 885, 2008.
doi:10.1103/RevModPhys.80.885.
[126] Michael Hermele, Victor Gurarie, and Ana Maria Rey.
Mott insulators of ultracold fermionic alkaline earth atoms: Underconstrained magnetism and chiral spin liquid.
Physical Review Letters 103, 135301, 2009.
doi:10.1103/PhysRevLett.103.135301.
[127] A. V. Gorshkov, M. Hermele, V. Gurarie, C. Xu, P. S. Julienne, J. Ye, P. Zoller, E. Demler, M. D. Lukin, and A. M. Rey.
Two-orbital SU(N) magnetism with ultracold alkaline-earth atoms.
Nature Physics 6, 289, 2010.
doi:10.1038/nphys1535.
[128] Michael Foss-Feig, Michael Hermele, and Ana Maria Rey.
Probing the Kondo lattice model with alkaline-earth-metal atoms.
Physical Review A 81, 051603, 2010.
doi:10.1103/PhysRevA.81.051603.
[129] P. Fedichev, Yu. Kagan, G. Shlyapnikov, and J. Walraven.
Influence of nearly resonant light on the scattering length in low-temperature atomic gases.
Physical Review Letters 77, 2913, 1996.
doi:10.1103/PhysRevLett.77.2913.
[130] John Bohn and P. Julienne.
Semianalytic theory of laser-assisted resonant cold collisions.
Physical Review A 60, 414, 1999.
doi:10.1103/PhysRevA.60.414.
[131] Kevin Jones, Eite Tiesinga, Paul Lett, and Paul Julienne.
Ultracold photoassociation spectroscopy: Long-range molecules and atomic scattering.
Reviews of Modern Physics 78, 483, 2006.
doi:10.1103/RevModPhys.78.483.
[132] F. Fatemi, K. Jones, and P. Lett.
Observation of optically induced Feshbach resonances in collisions of cold atoms.
Physical Review Letters 85, 4462, 2000.
doi:10.1103/PhysRevLett.85.4462.
[133] M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruﬀ, R. Grimm, and J. Hecker Denschlag.
Tuning the scattering length with an optically induced Feshbach resonance.
Physical Review Letters 93, 123001, 2004.
doi:10.1103/PhysRevLett.93.123001.

197[134] Gregor Thalhammer, Matthias Theis, Klaus Winkler, Rudolf Grimm, and Johannes Hecker Denschlag.
Inducing an optical Feshbach resonance via stimulated Raman coupling.
Phys. Rev. A 71, 033403, 2005.
doi:10.1103/PhysRevA.71.033403.
[135] Matthias Theis.
Optical Feshbach Resonances in a Bose-Einstein Condensate.
PhD thesis, Fakultät für Mathematik, Informatik und Physik, Leopold-Franzens-Universität Innsbruck, 2005.
URL http://pc150-c704.uibk.ac.at/ultracold/doc/thesis_matthias_theis/thesis_matthias_theis.pdf.
[136] R. Ciurylo, E. Tiesinga, and P. S. Julienne.
Optical tuning of the scattering length of cold alkaline-earth-metal atoms.
Physical Review A 71, 030701, 2005.
doi:10.1103/PhysRevA.71.030701.
[137] T. Zelevinsky, M. M. Boyd, A. D. Ludlow, T. Ido, J. Ye, R. Ciurylo, P. Naidon, and P. S. Julienne.
Narrow line photoassociation in an optical lattice.
Physical Review Letters 96, 203201, 2006.
doi:10.1103/PhysRevLett.96.203201.
[138] K. Enomoto, K. Kasa, M. Kitagawa, and Y. Takahashi.
Optical Feshbach resonance using the intercombination transition.
Physical Review Letters 101, 203201, 2008.
doi:10.1103/PhysRevLett.101.203201.
[139] Rekishu Yamazaki, Shintaro Taie, Seiji Sugawa, and Yoshiro Takahashi.
Submicron spatial modulation of an interatomic interaction in a Bose-Einstein condensate.
Physical Review Letters 105, 050405, 2010.
doi:10.1103/PhysRevLett.105.050405.
[140] Y. Martinez de Escobar, P. Mickelson, P. Pellegrini, S. Nagel, A. Traverso, M. Yan, R. Côté, and T. Killian.
Two-photon photoassociative spectroscopy of ultracold 88 Sr.
Physical Review A 78, 062708, 2008.
doi:10.1103/PhysRevA.78.062708.
[141] A. Stein, H. Knöckel, and E. Tiemann.
The 1S+1S asymptote of Sr2 studied by Fourier-transform spectroscopy.
European Physical Journal D 57, 171, 2010.
doi:10.1140/epjd/e2010-00058-y.
[142] Simon Stellmer, Meng Tey, Bo Huang, Rudolf Grimm, and Florian Schreck.
Bose-Einstein condensation of strontium.
Physical Review Letters 103, 200401, 2009.
doi:10.1103/PhysRevLett.103.200401.
[143] Y. de Escobar, P. Mickelson, M. Yan, B. DeSalvo, S. Nagel, and T. Killian.
Bose-Einstein condensation of 84 Sr.
Physical Review Letters 103, 200402, 2009.
doi:10.1103/PhysRevLett.103.200402.
[144] R. Ciurylo, E. Tiesinga, and P. S. Julienne.
Stationary phase approximation for the strength of optical Feshbach resonances.
Physical Review A 74, 022710, 2006.
doi:10.1103/PhysRevA.74.022710.
[145] Jeremy M. Hutson.
Feshbach resonances in ultracold atomic and molecular collisions: threshold behaviour and suppression of poles in scattering lengths.
New Journal of Physics 9, 152, 2007.
doi:10.1088/1367-2630/9/5/152.
[146] Paul S. Julienne.
Notes on 88 Sr optical Feshbach resonances.
unpublished 2010.

198[147] Irène Courtillot.
Première observation de la transition fortement interdite 1 S0 -3 P0 du strontium, pour une horloge optique à atomespiégés.
PhD thesis, Université de Paris VI, 2003.
URL http://tel.archives-ouvertes.fr/tel-00004039/en/.
[148] Eric Wille.
Preparation of an optically trapped Fermi-Fermi mixture of 6 Li and 40 K atoms and characterization of the interspecies
interactions by Feshbach spectroscopy.
PhD thesis, Universität Innsbruck, Fakultät für Mathematik, Informatik und Physik, 2009.
URL http://ultracold.at.
[149] I. I. Rabi, J. M. B. Kellogg, and J. R. Zacharias.
The magnetic moment of the proton.
Physical Review 46, 157, 1934.
doi:10.1103/PhysRev.46.157.
[150] Norman F. Ramsey.
Molecular Beams (Oxford University Press, London), 1956.
[151] Jacques Vanier and Claude Audoin.
The Quantum Physics of Atomic Frequency Standards (Adam Hilger, Bristol), 1989.
ISBN 0-85274-434-X.
[152] Bruce G. Klappauf, Yannick Bidel, David Wilkowski, Thierry Chanelière, and Robin Kaiser.
Detailed study of an efficient blue laser source by second-harmonic generation in a semimonolithic cavity for the coolingof strontium atoms.
Applied Optics 43, 2510, 2004.
doi:10.1364/AO.43.002510.
[153] R. Le Targat, J.-J. Zondy, and P. Lemonde.
75%-efficiency blue generation from an intracavity PPKTP frequency doubler.
Optics Communications 247, 471, 2005.
doi:10.1016/j.optcom.2004.11.081.
[154] B. Boulanger, M. M. Fejer, R. Blachman, and P. F. Bordui.
Study of KTiOPO4 gray-tracking at 1064, 532, and 355 nm.
Applied Physics Letters 65, 2401, 1994.
doi:10.1063/1.112688.
[155] R. Blachman, P. F. Bordui, and M. M. Fejer.
Laser-induced photochromic damage in potassium titanyl phosphate.
Applied Physics Letters 64, 1318, 1994.
doi:10.1063/1.111920.
[156] Benoı̂t Boulanger, Jean-Philippe Fève, and Yannick Guillien.
Thermo-optical effect and saturation of nonlinear absorption induced by gray tracking in a 532-nm-pumped KTP opticalparametric oscillator.
Optics Letters 25, 484, 2000.
doi:10.1364/OL.25.000484.
[157] Zhi M. Liao, Stephen A. Payne, Jay Dawson, Alex Drobshoﬀ, Chris Ebbers, Dee Pennington, and Luke Taylor.
Thermally induced dephasing in periodically poled KTP frequency-doubling crystals.
Journal of the Optical Society of America B 21, 2191, 2004.
doi:10.1364/JOSAB.21.002191.
[158] S. Wang, V. Pasiskevicius, and F. Laurell.
Dynamics of green light-induced infrared absorption in KTiOPO4 and periodically poled KTiOPO4 .
Journal of Applied Physics 96, 2023, 2004.
doi:10.1063/1.1738528.
[159] A. Skliar (Raicol).
private communication 2008.
URL http://www.raicol.com.
[160] Daisuke Akamatsu, Masami Yasuda, Takuya Kohno, Atsushi Onae, and Feng-Lei Hong.
A compact light source at 461 nm using a periodically poled LiNbO3 waveguide for strontium magneto-optical trapping.
Optics Express 19, 2046, 2011.
doi:10.1364/OE.19.002046.

199[161] W. Ketterle, D. S. Durfee, and D. M. Stamper-Kurn.
Making, probing and understanding Bose-Einstein condensates.
arXiv:cond-mat/9904034v2 1999.
[162] Jacob Lyman Roberts.
Bose-Einstein Condensates with Tunable Atom-atom Interactions: The First Experiments with 85 Rb BECs.
PhD thesis, University of Colorado, Department of Physics, 2001.
URL http://jilawww.colorado.edu/yelabs/pubs/theses.html.
[163] Jeﬀ Conrad.
Depth of field in depth 2006.
URL http://www.largeformatphotography.info/articles/DoFinDepth.pdf.
[164] Andrey Kobyakov, Michael Sauer, and Dipak Chowdhury.
Stimulated Brillouin scattering in optical fibers.
Advances in Optics and Photonics 2, 1, 2009.
doi:10.1364/AOP.2.000001.
[165] Rudolf Grimm, Matthias Weidemüller, and Yurii B. Ovchinnikov.
Optical dipole traps for neutral atoms.
arXiv:physics/9902072v1 1999.
[166] Edwin T. Jaynes.
Probability Theory (Cambridge University Press, Cambridge), 2003.
ISBN 0-521-59271-2.
[167] Vladimir Ivanovich Krylov.
Approximate Calculation of Integrals (Dover Publications, Mineola, New York), 2005.
ISBN 0-486-44579-8.
[168] Philip J. Davis and Philip Rabinowitz.
Methods of Numerical Integration (Dover Publications, Mineola, New York), 2007.
ISBN 0-486-45339-1.
[169] R. Ciurylo, E. Tiesinga, S. Kotochigova, and P. Julienne.
Photoassociation spectroscopy of cold alkaline-earth-metal atoms near the intercombination line.
Physical Review A 70, 1, 2004.
doi:10.1103/PhysRevA.70.062710.
[170] Jonathan Michael Goldwin.
Quantum Degeneracy and Interactions in the 87 Rb-40 K Bose-Fermi Mixture.
PhD thesis, University of Colorado, 2005.
URL http://jila.colorado.edu/thesis/.
[171] P. S. Julienne.
to be published 2011.
[172] P. S. Julienne and F. H. Mies.
Nonadiabatic theory of atomic line broadening: Redistribution calculations for Sr(1 P →1 S)+Ar.
Physical Review A 34, 3792, 1986.
doi:10.1103/PhysRevA.34.3792.
[173] Reginaldo Napolitano, John Weiner, and Paul S. Julienne.
Theory of optical suppression of ultracold-collision rates by polarized light.
Physical Review A 55, 1191, 1997.
doi:10.1103/PhysRevA.55.1191.
[174] S. G. Porsev and A. Derevianko.
High-accuracy calculations of dipole, quadrupole, and octupole electric dynamic polarizabilities and van der Waals coefficients C6 , C8 , and C10 for alkaline-earth dimers.
Journal of Experimental and Theoretical Physics 102, 195, 2006.
doi:10.1134/S1063776106020014.
[175] Meng Khoon Tey, Simon Stellmer, Rudolf Grimm, and Florian Schreck.
Double-degenerate Bose-Fermi mixture of strontium.
Physical Review A 82, 011608, 2010.
doi:10.1103/PhysRevA.82.011608.

200[176] Simon Stellmer, Meng Khoon Tey, Rudolf Grimm, and Florian Schreck.
Bose-Einstein condensation of 86 Sr.
Physical Review A 82, 041602, 2010.
doi:10.1103/PhysRevA.82.041602.
[177] B. DeSalvo, M. Yan, P. Mickelson, Y. Martinez de Escobar, and T. Killian.
Degenerate Fermi gas of 87 Sr.
Phys. Rev. Lett. 105, 030402, 2010.
doi:10.1103/PhysRevLett.105.030402.
[178] P. G. Mickelson, Y. N. Martinez de Escobar, M. Yan, B. J. DeSalvo, and T. C. Killian.
Bose-Einstein condensation of 88 Sr through sympathetic cooling with 87 Sr.
Phys. Rev. A 81, 051601, 2010.
doi:10.1103/PhysRevA.81.051601.
[179] Tomoya Akatsuka, Masao Takamoto, and Hidetoshi Katori.
Optical lattice clocks with non-interacting bosons and fermions.
Nature Physics 4, 954, 2008.
doi:10.1038/nphys1108.
[180] Tomoya Akatsuka, Masao Takamoto, and Hidetoshi Katori.
Three-dimensional optical lattice clock with bosonic 88 Sr atoms.
Physical Review A 81, 023402, 2010.
doi:10.1103/PhysRevA.81.023402.
[181] M. Bishof et al.
in preparation 2011.
[182] D. Chang, Jun Ye, and M. Lukin.
Controlling dipole-dipole frequency shifts in a lattice-based optical atomic clock.
Physical Review A 69, 023810, 2004.
doi:10.1103/PhysRevA.69.023810.
[183] D. Meiser, Jun Ye, D. Carlson, and M. Holland.
Prospects for a millihertz-linewidth laser.
Physical Review Letters 102, 163601, 2009.
doi:10.1103/PhysRevLett.102.163601.
[184] D. Meiser and M. J. Holland.
Steady-state superradiance with alkaline-earth-metal atoms.
Physical Review A 81, 033847, 2010.
doi:10.1103/PhysRevA.81.033847.
[185] M. J. Martin, D. Meiser, J. W. Thomsen, Jun Ye, and M. J. Holland.
Extreme non-linear response of ultra-narrow optical transitions in cavity QED for laser stabilization.
arXiv:1105.2373v1 2011.
[186] Matthew Swallows, Gretchen Campbell, Andrew Ludlow, Martin Boyd, Jan Thomsen, Michael Martin, Sebastian Blatt,Travis Nicholson, and Jun Ye.
Precision measurement of fermionic collisions using an 87 Sr optical lattice clock with 1 × 10−16 inaccuracy.
IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 57, 574, 2010.
doi:10.1109/TUFFC.2010.1449.
[187] Philip Westergaard, Jerome Lodewyck, and Pierre Lemonde.
Minimizing the Dick effect in an optical lattice clock.
IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 57, 623, 2010.
doi:10.1109/TUFFC.2010.1457.
[188] Jérôme Lodewyck, Philip Westergaard, and Pierre Lemonde.
Nondestructive measurement of the transition probability in a Sr optical lattice clock.
Physical Review A 79, 061401, 2009.
doi:10.1103/PhysRevA.79.061401.
[189] Jérôme Lodewyck, Philip G Westergaard, Arnaud Lecallier, Luca Lorini, and Pierre Lemonde.
Frequency stability of optical lattice clocks.
New Journal of Physics 12, 065026, 2010.
doi:10.1088/1367-2630/12/6/065026.

201[190] J. I. Cirac, M. Lewenstein, and P. Zoller.
Collective laser cooling of trapped atoms.
Europhysics Letters 35, 647, 1996.
doi:10.1209/epl/i1996-00165-4.
[191] S. Blatt.
Feasibility of narrow line laser cooling of Sr in optical dipole traps by spontaneous Raman scattering.
unpublished 2009.
[192] C. W. Chou, D. B. Hume, T. Rosenband, and D. J. Wineland.
Optical clocks and relativity.
Science 329, 1630, 2010.
doi:10.1126/science.1192720.
[193] Jun Ye.
Absolute measurement of a long, arbitrary distance to less than an optical fringe.
Optics Letters 29, 1153, 2004.
doi:10.1364/OL.29.001153.
[194] B. H. Bransden and C. J. Joachain.
Physics of Atoms and Molecules (Pearson Education, Edinburgh), second edition, 2003.
ISBN 0-582-35692-X.
[195] D. Budker, D. F. Kimball, and D. P. DeMille.
Atomic Physics : an exploration through problems and solutions (Oxford University Press, Oxford), 2004.
ISBN 0-19-850950-2.
[196] A. E. Siegman.
Lasers (University Science Books), 1986.
ISBN 0-935702-11-3.
[197] A. R. Edmonds.
Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, New Jersey), second edition, 1960.
[198] A. Messiah.
Quantum Mechanics (Dover, Mineola, New York), 1999.
ISBN 0-486-40924-4.
[199] I. I. Sobel’man.
Introduction to the Theory of Atomic Spectra (Pergamon Press, Oxford), 1972.
[200] D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii.
Quantum Theory of Angular Momentum (World Scientiﬁc, Singapore), 1988.
ISBN 9971509962.
[201] D. M. Brink and G. R. Satchler.
Angular Momentum (Clarendon Press, Oxford), second edition, 1968.
[202] M. E. Rose.
Elementary Theory of Angular Momentum (Dover, New York), 1995.
ISBN 0-486-68480-6.
[203] H. Friedrich.
Theoretical Atomic Physics (Springer, Berlin), second edition, 1998.
ISBN 3-540-64124-6.
[204] E. Merzbacher.
Quantum Mechanics (Wiley, New York), third edition, 1998.
ISBN 0-471-88702-1.
[205] L. Allen and J. H. Eberly.
Optical Resonance and Two-Level Atoms (Dover, New York), 1987.
ISBN 0-486-65533-4.
[206] T. Cumby.
Notes on OD saturation 2010.

202[207] C. W. Gardiner.
Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Springer, Berlin), third edition, 2004.
ISBN 3-540-20882-8.
[208] G. A. Bird.
Molecular gas dynamics and the direct simulation of gas flows (Clarendon Press, Oxford), 1994.
ISBN 0-19-856195-4.
[209] Huang Wu and Christopher J. Foot.
Direct simulation of evaporative cooling.
Journal of Physics B: Atomic, Molecular and Optical Physics 29, L321, 1996.
doi:10.1088/0953-4075/29/8/003.
[210] Huang Wu, Ennio Arimondo, and Christopher Foot.
Dynamics of evaporative cooling for Bose-Einstein condensation.
Physical Review A 56, 560, 1997.
doi:10.1103/PhysRevA.56.560.
[211] Michael Eric Gehm.
Preparation of an optically-trapped degenerate Fermi gas of 6 Li: Finding the route to degeneracy.
PhD thesis, Duke University, 2003.
URL http://www.phy.duke.edu/research/photon/qoptics/theses/.
[212] J. Goldwin, S. Inouye, M. Olsen, and D. Jin.
Cross-dimensional relaxation in Bose-Fermi mixtures.
Physical Review A 71, 043408, 2005.
doi:10.1103/PhysRevA.71.043408.
[213] J. Stoer and R. Bulirsch.
Introduction to Numerical Analysis (Springer, New York), third edition, 2002.
ISBN 0-387-95452-X.
[214] Marco Giacinto Tarallo.
Development of a Strontium optical lattice clock.
PhD thesis, Univesità degli Studi di Pisa, Facoltà di Scienze Matematiche Fisiche e Naturali, 2009.
URL http://etd.adm.unipi.it/theses/available/etd-05182009-164852/.
[215] Zeb Barber.
Ytterbium Optical Lattice Clock.
PhD thesis, University of Colorado, 2007.
URL https://jilawww.colorado.edu/pubs/thesis/barber/.
[216] Michael Petersen.
Laser-cooling of Neutral Mercury and Laser-spectroscopy of the 1 S0 -3 P0 optical clock transition.
PhD thesis, Université Pierre et Marie Curie, Paris VI, 2009.
URL http://tel.archives-ouvertes.fr/tel-00405200/en/.
[217] Tanya Zelevinsky, Sebastian Blatt, Martin M. Boyd, Gretchen K. Campbell, Andrew D. Ludlow, and Jun Ye.
Highly coherent spectroscopy of ultracold atoms and molecules in optical lattices.
ChemPhysChem 9, 375, 2008.
doi:10.1002/cphc.200700713.
[218] J. Ye, S. Blatt, M. M. Boyd, S. M. Foreman, E. R. Hudson, T. Ido, B. Lev, A. D. Ludlow, B. C. Sawyer, B. Stuhl, and
T. Zelevinsky.
Precision measurement based on ultracold atoms and cold molecules.
International Journal of Modern Physics D 16, 2481, 2007.
doi:10.1142/S0218271807011826.
[219] T. Zelevinsky, M. M. Boyd, A. D. Ludlow, S. M. Foreman, S. Blatt, T. Ido, and J. Ye.
Optical clock and ultracold collisions with trapped strontium atoms.
Hyperﬁne Interactions 174, 55, 2007.
doi:10.1007/s10751-007-9564-x.
[220] Thomas Zanon-Willette, Andrew Ludlow, Sebastian Blatt, Martin Boyd, Ennio Arimondo, and Jun Ye.
Cancellation of Stark shifts in optical lattice clocks by use of pulsed Raman and electromagnetically induced transparency
techniques.
Physical Review Letters 97, 233001, 2006.
doi:10.1103/PhysRevLett.97.233001.