Many-Body Physics in an Optical Lattice ClockbySarah L. Bromley
MPhys., University of St. Andrews, 2012
M.S., University of Colorado, 2017

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2018This thesis entitled:
Many-Body Physics in an Optical Lattice Clockwritten by Sarah L. Bromley
has been approved for the Department of Physics

Jun YeAna Maria Rey

Date

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

iii
Bromley, Sarah L. (Ph.D., Physics)
Many-Body Physics in an Optical Lattice Clock
Thesis directed by Prof. Jun Ye

In this work we study the effect of interactions in an optical lattice clock based on fermionic
Sr atoms. In current one-dimensional lattice clocks nuclear spin-polarized atoms are known to havecontact interactions of p-wave character and collective in nature. Here we focus on interactions
that will influence the design of future optical lattice clocks. We study the case where atoms are nolonger confined to a single nuclear spin state. By using samples of atoms with different distributions
among the ten nuclear spin states of Sr we show that these interactions are SU(N ) symmetric up to
a 3% uncertainty in our measurements. Through these measurements we are also able to determine
all the s-wave and p-wave scattering lengths.
We also study the case of nuclear spin-polarized interacting atoms that are allowed to tunnel
between different lattice sites where the electronic spin and the motion of these atoms become coupled. We observe spectroscopically the precession of the collective magnetization and evolution ofspin locking effects arising from the interplay between p-wave interactions and interactions inducedby the spin-orbit coupling. The many-body dynamics are captured by a spin model that describes
a broad class of condensed matter systems ranging from superconductors to quantum magnets.
By loading a dense sample of atoms into a magneto-optical trap we are able to observe longrange dipole-dipole interactions between our Sr atoms. These interactions will be important foratomic clocks based on a three dimensional lattice, such as the one recently demonstrated in ourlab. In these clocks it is possible to remove the contact interactions between the atoms by loading
only one atom per lattice site. In this case the dominant interactions will be from the long-range
dipole-dipole interactions that will take place between the atoms.

Dedication

To my parents for supporting me when I moved across the world and to Corielyn who agreed
to move back across the world with me.

v

Acknowledgements

When I first moved to Colorado I was surrounded by new things and new people. It took
me a while to get my bearings, but I did so with the help of a few awesome people: Clarissa, Carl,
Eric, Susanna, Jimmy, Effie, & Adam. For the first time I felt able to be completely myself aroundfriends and for your kindness and friendship I thank you.
Once I had my bearings my next task was to find myself a research group. I was persistentin wanting to join Jun’s group and was excited when he gave me the chance to join the Strontium
Team. I thank him for always pushing me to get the best out of myself. It can be annoying how
he always seem to be right about everything, but it has meant he was able to teach me a lot about
physics for which I am eternally grateful.
I have had the opportunity to work with an amazing Strontium Team over the last 5ish years
and I thank you all for being so great! When I first joined the group I was shown the ropes of
the Sr1 experiment by graduate student Mike Bishof and postdoc Xibo Zhang. Mike, Xibo, andI spent many late nights together in the lab for both the initial clock comparison between our
two Sr experiments and our SU(N ) work. They both taught me a lot about the running of theexperiments and the need to really know the everyday running of the experiment well.
On the Sr1 experiment we were quickly joined by Toby Bothwell and I have had the pleasure
to work with him for the majority of my PhD. He has helped tremendously to make improvements
to the control of the experiment and in leading the design of the next generation of the experiment.
I was also lucky to have a large overlap with postdoc Shimon Kolkowitz. Together we made a very
productive team and Shimon is one of the nicest and friendliest people I have met, as well as being

vi
a great physicist. Recently we have been joined on the experiment by Dhruv Kedar and postdoc
Colin Kennedy. They are both extremely friendly and hard-working people and I am sad that I
don’t have the oportunity to work with them both longer. Together with Toby they are alreadymaking an awesome team and I can’t wait to see what they do next.
At the same time we have always worked closely with the Sr2 experiment. When I firstjoined the experiment, Travis Nicholson, Ben Bloom, and Sara Campbell were all working hard tomeasure and improve their clock systematics. Travis always had such a great passion for the work
he was doing. Sara was always the bubbly, curious, and enthusiastic member of the team. She was
the “organizer” of the group and would take it upon herself to organize the yearly ski trip and evensome Ye lab t-shirts.
More recent additions to the Sr2 team include Ross Hutson and postdocs Ed Marti and Aki
Goban. Ross joined the Sr team a few months after I did and he was instrumental in implementing
LabRad into the new SrQ experiment. He is also investigating brilliant new ways for improving ourclocks. Ed is an incredibly clever individual and always willing to sit down and explain something
to you when you ask him questions. Aki is an incredibly hard worker and such a friendly person
to be around in lab.
I would also like to thank the stable lasers team as they help make our experiments better by
maintaining and improving our clock laser system. Wei Zhang has so much knowledge about thesestable lasers and he can still be found in the evenings at JILA. It is still unclear to me if he actually
still works at JILA or he just drops by to help out. Lindsay Sonderhouse somehow managed to
repair and keep the old frequency comb working for such a long time and now she is rewarded witha shiny new comb that can free up her time to work with the Sr atoms. John Robinson briefly
worked on the Sr1 experiment before joining the stable lasers teams and he is such a thoughtfulscientist who is making awesome next generation laser systems. Erik Oelker recently joined theteam as a posdoc and is already doing great work in the lab and leading the search for dark matterwith our clocks.
Ana Maria Rey has worked with our experimental team on almost all of the work describedvii
in this thesis and has been a great collaborator. She has helped us understand our observations andtaught us so much about what we are observing. As well as working closely with Ana Maria, wehave also worked closely with her group. In particular, I would like to thank Bihui Zhu, Johannes
Schachenmayer, Michael Wall, and Arghavan Safavi-Naini for answering my many questions.
I would also like to thank everyone else who works at JILA that I have not already mentioned.
JILA has such a great community of scientists, administrative staff, custodians, and technicians
which all contribute to the success of JILA. The different shops provide invaluable technical support
and guidance, and the administrative staff and custodians deal with all the different things goingon in the backgound and keep things running smoothly.
And lastly, I thank my family. To my parents I thank them for supporting my decision tomove so far away from home and their continued love and support even from a distance. I would
not have had the strength to move so far from home without their strong, loving upbringing. I
would also like to thank my brother and sister who have always been there for me whenever I
needed them. To the newest member of my family, Corielyn - I can’t believe I found you. Your
love, strength, and support inspires me to work hard and be the best person I can be. I know thatwhatever the future holds as long as you are by my side life will be good.

viiiContents

Chapter
1 Introduction to Optical Lattice Clocks

1

1.1

A History of Time, in Brief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Atomic Clocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

Clock Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.4

Quantum Projection Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.5

Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62 Strontium Cooling and Trapping
2.1

8

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.1.1

Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2

The Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92.3

Strontium Level Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.1

Hyperfine Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4

Blue MOT cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5

Red MOT cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6

The Optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Strontium Clock Evaluations
3.1

27

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.1

Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

ix
3.2

3.3

3.4

Measuring Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.1

Density Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.2

Lattice ac Stark Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.3

dc Stark Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.4

Zeeman Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.5

Blackbody Radiation Shift

3.2.6

Other Frequency Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Uncertainty Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.1

2013 Sr-Sr comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.2

2017/18 JILA-NIST comparison . . . . . . . . . . . . . . . . . . . . . . . . . 45

Comparing Clocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 Collective atomic scattering in a dense sample of 88 Sr
4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.1

4.2

4.3

4.4

49

Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.1

Strontium 88 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.2

Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2.3

System Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Experimental Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3.1

Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3.2

Linewidth Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3.3

Frequency Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Theoretical Model - The Coupled Dipole Model . . . . . . . . . . . . . . . . . . . . . 70
4.4.1

Coherences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4.2

Scattered Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.4.3

Motional Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

x

4.5

4.4.4

Multiple Scattering Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4.5

Frequency Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5 Many-body interactions in an optical lattice clock
5.1

82

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.1.1

Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.2

Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3

The Spin Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3.1

5.4

Collective Spin Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Interaction Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.4.1

χ+ - Promoting one atom from |gi to |ei . . . . . . . . . . . . . . . . . . . . . 89

5.4.2

ξ + - The exchange interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.5

Temperature Dependence of Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.6

Extensions to the Spin Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6 Spectroscopic Observation of SU(N ) physics
6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.1.1

6.2

92

Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

SU(N ) spin-orbital Hamiltonian in an energy space lattice . . . . . . . . . . . . . . . 94
6.2.1

Expansion of the Spin Hamiltonian - Spin Mixtures . . . . . . . . . . . . . . 96

6.3

Ramsey Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.4

Density-dependent frequency shift for nuclear spin mixtures . . . . . . . . . . . . . . 99

6.5

Temperature-dependent SU(N )-symmetric interactions . . . . . . . . . . . . . . . . . 100

6.6

Experimental-theory agreement and determination of the scattering parameters . . . 103

6.7

Coherent dynamic spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.7.1

Two-orbital dynamics in spin-polarized atoms . . . . . . . . . . . . . . . . . . 107

6.7.2

Spin-orbital SU(N ) dynamics in spin mixtures . . . . . . . . . . . . . . . . . 109

xi
6.8

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7 Spin-Orbit Coupling with Interactions
7.1

112

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.1.1

Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.2

Atoms trapped in a periodic potential . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.3

Lattice Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7.4

Coupling the spin and the orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.5

Connections to synthetic gauge fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.6

Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.7

Sideband Structure and Band Selection . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.8

Band Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.9

Carrier Transition and Band Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.10 Quasimomentum selection and Bloch Oscillations . . . . . . . . . . . . . . . . . . . . 136
7.11 Chiral Bloch Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.12 Ramsey Spectroscopy with SOC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.13 Ramsey Spectroscopy with SOC and Interactions . . . . . . . . . . . . . . . . . . . . 149
7.14 Frequency Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.15 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8 Conclusions and Outlook

160

Bibliography

162

Appendix
A Atom Number Calibration

173

A.1 Quantum Projection Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

xii
A.2 Transfer Efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

xiii

TablesTable
3.1

2013 Clock Systematic Evaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2

Comparison of the Sr1 Systematic Evaluations . . . . . . . . . . . . . . . . . . . . . 45

6.1

s- and p-wave scattering lengths in units of the Bohr radius . . . . . . . . . . . . . . 104

xiv

FiguresFigure1.1

Accuracy, Precision, and Systematic Uncertainty . . . . . . . . . . . . . . . . . . . .

3

2.1

Vaccum Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2

Strontium Energy Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3

Strontium Cooling Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4

1 P Hyperfine Structure
1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5

3 P Hyperfine Structure
1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6

461 nm Blue Master Laser Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.7

461 nm Locking Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.8

461 nm Blue Slave Laser Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.9

Repump Laser Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.10 689 nm Master Laser Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.11 689 nm Red Slave Laser Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.12 Unpolarized Clock Carrier Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.13 Axial Sideband Scans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.14 Radial Sideband Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.15 Magic Wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1

Locking to the Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2

Example Frequency Shift Measurement. . . . . . . . . . . . . . . . . . . . . . . . . 29

xv
3.3

Magic Wavelength Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4

dc Stark Shift Measurement Cartoon. . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5

dc Stark Shift Drift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.6

dc Stark Shift Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.7

Zeeman Frequency Shifts

3.8

Vacuum Viewport Temperature Stabilization . . . . . . . . . . . . . . . . . . . . . . 40

3.9

Temperature Stability During Clock Measurements . . . . . . . . . . . . . . . . . . . 41

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.10 Vacuum Lifetime Measurment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.11 Sr Frequency Chain for 2017/18 NIST Clock Comparison . . . . . . . . . . . . . . . 47
3.12 2013 Sr-Sr Frequency Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1

88 Sr Four Level Structure

4.2

Simplified Setup for measuring the Collective Scattering . . . . . . . . . . . . . . . . 53

4.3

Time of Flight Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4

Pixel Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.5

Temperature Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.6

Setup for Measuring the Forward Intensity . . . . . . . . . . . . . . . . . . . . . . . . 59

4.7

Forward Intensity Enhancement Measurement . . . . . . . . . . . . . . . . . . . . . . 59

4.8

Focused Probe Beam Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.9

Forward Enhancement Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.10 Transverse Blue Intensity Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.11 Transverse Red Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.12 Forward Linewidth Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.13 Transverse Linewidth Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.14 Blue Transition Lineshape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.15 Red Linewidth Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.16 Red Lineshape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

xvi
4.17 Frequency Shift from Collective Dipole-Dipole Scattering. . . . . . . . . . . . . . . 69

4.18 Fluorescence Intensity Theory Prediction . . . . . . . . . . . . . . . . . . . . . . . . 77
4.19 Forward Fluorescence Intensity Enhancement with Motion . . . . . . . . . . . . . . . 78
6.1

SU(N ) Interactions in an Energy Space Lattice. . . . . . . . . . . . . . . . . . . . . . 95

6.2

Ramsey Spectroscopy on the Bloch Sphere . . . . . . . . . . . . . . . . . . . . . . . . 98

6.3

Density Shift Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.4

SU(N ) Symmetry Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.5

Polarized Contrast Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.6

Spin Mixture Contrast Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.1

Axial Trapping Frequency Versus Retro-Reflected Power . . . . . . . . . . . . . . . . 116

7.2

Lattice Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.3

Spin-Orbit Coupled Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.4

Synthetic Gauge Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.5

813nm Breadboard Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.6

813 nm Horizontal Ingoing Lattice Mezzanine . . . . . . . . . . . . . . . . . . . . . . 124

7.7

Horizontal Lattice Retro Path Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.8

Sideband Scan from the Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.9

Sideband Scan from the Excited State, Ground Band . . . . . . . . . . . . . . . . . . 127

7.10 Sideband Scan from the Excited State, 1st Excited Band . . . . . . . . . . . . . . . . 128
7.11 Radial Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.12 Sideband Scans at Different Lattice Depths . . . . . . . . . . . . . . . . . . . . . . . 129
7.13 Band Relaxation from s-wave Collisions . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.14 Carrier Transition for Different Tunneling Rates. . . . . . . . . . . . . . . . . . . . 132

7.15 Linesplitting Theory-Data Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.16 Carrier Transition for Different π-Pulse Times . . . . . . . . . . . . . . . . . . . . . . 133
7.17 Carrier Transition for Different Pulse Areas . . . . . . . . . . . . . . . . . . . . . . . 134

xvii
7.18 Carrier Splitting for Different Tunneling Rates . . . . . . . . . . . . . . . . . . . . . 135
7.19 Bloch Oscillation Frequency Measurement . . . . . . . . . . . . . . . . . . . . . . . . 136
7.20 Bloch Oscillation Lineshape Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.21 Chiral Angle and q-Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.22 Rabi Flopping of Different Quasimomenta . . . . . . . . . . . . . . . . . . . . . . . . 140
7.23 Measurement of the Chiral Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.24 Ramsey Spectroscopy with SOC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.25 Ramsey Contrast for 2-Atom Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.26 Ramsey Spectroscopy with SOC - Data . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.27 Ramsey Spectroscopy with a Spin-Echo Pulse . . . . . . . . . . . . . . . . . . . . . . 146
7.28 Ramsey Spectroscopy with Spin-Echo Pulse for Different J. . . . . . . . . . . . . . 147

7.29 Diffusive Dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.30 Ramsey Spectroscopy with q-Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.31 Ramsey Spectroscopy with Interactions - θ1 = π/4 . . . . . . . . . . . . . . . . . . . 150
7.32 Ramsey Spectroscopy with Interactions - θ1 = π/2 . . . . . . . . . . . . . . . . . . . 151
7.33 Dependence of Interactions on Temperature . . . . . . . . . . . . . . . . . . . . . . . 153
7.34 Frequency Shift Without SOC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.35 Ramsey Spectroscopy with q-Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8.1

Future Experimental Chamber Design . . . . . . . . . . . . . . . . . . . . . . . . . . 161

A.1 QPN Measurement of the Atom Number . . . . . . . . . . . . . . . . . . . . . . . . . 174
A.2 QPN Measurement of the Atom Number 2 . . . . . . . . . . . . . . . . . . . . . . . . 174

Chapter 1

Introduction to Optical Lattice Clocks

1.1

A History of Time, in Brief
From the beginning of civilization time has been important for people. Time can bring withit changes in nature such as the seasons which can be important for reasons such as harvesting food.
With astronomical events, such as the seeming motion of the stars around the earth, occuring once
per year astronomical events become natural measures of time. The motion of the earth aroundthe sun or of the moon around the earth therefore give a natural system of time. For example, theancient egyptians used the rising of the star Sirius, that was absent from the sky for approximately
70 days per year, to mark the start of the year which they then split into 12 months of 30 days
each with an extra 5 days per year to make up a 365 day year [1].
The splitting of the day into 24 hours was first initiated by the Egyptians. Some of theoldest clocks ever found are from Egypt dating back to 2000 BC [2]. These early clocks werewater clocks and sundials and these types of clocks are still used around the world today. As time
progressed, the hour was split into 60 units of time (minutes) by the Babylonians. This splittingof time into smaller and smaller units is a natural progression as the need to measure smaller andsmaller time intervals is needed. Clocks have progressed from early clocks described above to clocks
such as pendulum clocks, and quartz crystal oscillator based clocks. However, such clocks are notgood frequency standards as their resonance frequencies depend on the physical parameters of theoscillator such as the size and shape of the quartz crystal. The frequency of such clocks can also
vary with time with aging.

2
Oscillators derived from frequency transitions in atoms do not have such problems. All atomsof the same element, and isotope, have identical transitions such that atoms are ideal frequency
standards. All transition frequencies have a central frequency ν0 with a linewidth ∆ν0 where wecan define a quality factor
Q=

ν0
∆ν0

(1.1)

Oscillators with higher Q can lead to more stable and accurate clocks meaning higher frequencies
and narrower transition linewidths are better for clocks. With some of the latest clocks based on
atoms the need for more accurate and precise clocks is still necessary for applications such as theglobal positioning system (GPS). In each of these satellites that we use every day to navigate there
are multiple atomic clocks which are required for location positioning.

1.2

Atomic Clocks
Atomic clocks are currently used to define time. The second is currently defined as 9,192,631,770

oscillations of the ground state hyperfine transition of cesium, and has been so since 1967 [3]. In thesame way that having more ticks on a ruler is a more precise way to measure distance, the higher
the frequency of the atomic transition the more precisely and accurately we can measure time. For
this reason, atomic clocks based on optical transitions with higher frequencies make better clocks
than microwave transitions which are lower frequencies that have a lower quality factor.
Atomic clocks are not only important for defining the unit of time but are also used in thedefinition of other units in the international system (SI). The SI units of meter, candela, and ampere
also depend on the definition of the second. For example, the meter is defined as the length of thepath travelled by light in vacuum during a time interval of 1/c of a second, where c is the speed oflight.
All time and frequency standards have several components in common. All are referenced toa periodic event that repeats at a constant rate. The clock then measures this periodicity. To makean atomic clock the atoms are laser cooled to reduce motional broadening of the transition. For the3case of optical lattice atomic clocks, the atoms are then confined in an optical lattice formed by astanding wave of light. The clock transition frequency is the periodic event, or oscillator, that theclock is referenced to and needs to be measured as accurately and precisely as possible. In order todo this, we need to understand all the different factors that change this transition frequency such as
electric and magnetic fields and even the interactions between the atoms themselves. Understanding
all these different frequency changes is vital for a clock and the environment for the clock needs to bewell controlled to limit the systematic uncertainties associated with the environmental changes. In
order to measure the frequency of the atomic transition a laser is stabilized to the atomic transition
and the frequency of the laser can then be measured. The ability to measure optical frequencies ismade possible due to the invention of the frequency comb, which earned Hall and Hänsch a shareof the 2005 Nobel prize for physics.
During the preceding paragraphs the terms “accuracy” and “precision” have been thrownaround without being specifically defined. It is important to understand these important concepts.
The accuracy of an atomic clock is how far away the measured value is from the true value. The

Figure 1.1: Accuracy, precision, and systematic uncertainty. A measurement will have a distributionof values (red line). The accuracy of a measurement is how far the center of the distribution isfrom the true value (dashed, line). The precision of the measurement is related to the width ofthe distribution where the narrower the width the more precise the measurement. The systematic
uncertainty of the measurement is related to how well the center of the distribution of measurements
can be determined.

4
precision of the measurement is the measure of the spread of values. The systematic uncertainty
is related to how well we can determine center of the distribution of measurements. Figure 1.1 isa cartoon showing these concepts of accuracy, precision, and systematic uncertainty. Technicallyspeaking accuracy should only be used when referring to a Cs atomic clock reference as that is
how time is defined for now, this will change when in the foreseeable future the definition of timechanges to be based upon on optical transitions. With other atomic frequencies the term systematic
uncertainty is employed instead.

1.3

Clock Stability
The precision of a clock is also known as the clock stability. In practice it is important to beable to measure the systematic uncertainties in a reasonable amount of time and it is importantthat a clock is stable in order to do this. The stability measures the noise of a clock where thenoisier the clock the longer it will take to average away this noise. The stability is most commonlydescribed in terms of an Allan deviation which gives the fractional frequency instability as a functionof the averaging time, τav . The Allan deviation involves a large number of measurements of theclock frequency and compares these frequencies. The Allan deviation, σy (τav ) can be written
mathematically by
"

j−1

X
1
σy (τav ) =(yi+1 − yi )22(j − 1)

#1/2
(1.2)

i=1

where j is the total number of measured values yi . If the cycle time of the experiment is Tcycle
then every 4Tcycle we get an independent measurement of the transition frequency by measuring
on either side of the mF = ±9/2 states as discussed in more detail in chapter 3. Overlapping
measurements of the transition frequency are routinely used which can give a new measurementof the transition frequency every 2Tcycle . Due to the overlapping nature of the measurements
any uncertainty measured in this way needs a correction factor as the measurements will not be
statistically independent [4].
For the overlapping Allan deviation where a measurement of yi made every 2Tcycle , we get5
an Allan deviation of σy (2Tcycle ). By taking an average of every two measurements a new Allan
deviation can be calculated, σy (4Tcycle ), where the time between these measurements is now 4Tcycle .
This can be repeated in the same way to get the Allan deviation at times of multiples of 2n 2Tcycle
for n an integer. For white noise sources this Allan deviation will keep decreasing linearly withtime. Due to the discrete nature of the measurements that are being made here, the high frequency
noise of the local oscillator is aliased by the slow sampling rate to frequencies close to the signal,
resulting in false frequency shifts in the clock. This effect is known as the Dick effect and limits
clock stability. The fundamental limit to the stability of the clock is quantum projection noise
which is discussed in the next section.

1.4

Quantum Projection Noise
The most fundamental limit to the atomic frequency measurements in optical lattice clocks

is the quantum projection noise (QPN) which is also referred to as atom shot noise. For a clock
operating at an average excitation fraction pe , and using uncorrelated atoms the number of atoms
found in the excited state follows a binomial, or coin toss, distribution. When tossing coins, theprobability of getting a head or tails is each pe = 0.5. When several coins are tossed at once,the proportion that land on heads will fluctuate about 0.5 each time another toss is made. If thenumber of coins is increased, the percentage that land on heads will fluctuate less. In a similarway, more atoms leads to less QPN.
From fluctuation in the excited state atom number follows Binomial statistics and it follows
that the variance of the measured excited state fraction isσ 2 [pe ] =pe (1 − pe )
N

(1.3)

where N is the atom number. This is the key reason that multi-atom atomic lattice clocks have an
advantage over single ion clocks: as the number of atoms increases the noise decreases. Typically,clocks are operated with pe = 0.5 as this is where the slope of the excitation probability with6
respect to frequency is maximized, giving a more sensitive measurement.
Since QPN is a white noise process, the noise amplitude averages down with the square rootof the number of points averaged. For an averaging time of τav and a cycle time of Tcycle , thenumber of points averaged is τav /2Tcycle , and the clock stability due to QPN is [5]χ0 1
σQP N =
πQ S/Ns

2Tcycle
χ0
=
τav
πQ

s

(1 − pe )N pes

Tcycle
χ0
=
τav
πQ

s

Tcycle
N τav

(1.4)

where Q = ν0 /δν is the transition line quality factor, pe = 1/2, and χ0 is a constant of order unitythat accounts for the details of spectroscopy. For Ramsey spectroscopy (see section 6.3) χ0 =1, but
experimentally we tend to use Rabi spectroscopy with only a single excitation from the clock laser.
This discussion of QPN has assumed that the states of the atom are uncorrelated and does not
apply for squeezed clock states for example.

1.5

Thesis Outline
The outline of this thesis is as follows. We will start by discussing the experimental setup,strontium structure, and cooling schemes employed in our one dimensional optical lattice clock inchapter 2. We will then move on to discuss the two systematic evaluations of our JILA Sr 1 clock,that have taken place during my time at JILA, in chapter 3. We will discuss how the frequency ofthe clock transition changes due to different systematic effects and the methods used to compare
the clocks. We will then move on to discuss some of the different experiments we have undertaken
to understand the interactions that can take place in our optical lattice clock that will be importantfor designing future generations of optical lattice clocks, as well as being interesting in their own
way. In chapter 4 we will discuss collective atomic scattering that takes place in a dense sample of
88 Sr atoms. For this chapter the atoms are not trapped within a lattice but we look at interactionsin the dense magneto-optical trap. The interactions that take place are dipole-dipole interactions
that are long range in nature and will be important for future generations of clocks such as in a
3D optical lattice clocks [6]. In such clocks the atoms can be prepared one atom per lattice site

7such that contact interactions can be prohibited. In this case the dominant interactions will thenbe from these long-range dipole-dipole interactions.
In chapter 5 we will briefly discuss the theory model developed by the group of Rey et al that
describes the interactions in our one dimensional optical lattice clock for atoms in the same nuclear
spin state and without tunneling taking place between the different lattice sites. In chapters 6 and
7 we will extend upon this model. In chapter 6 we will extend the model to include the case whereatoms can also be in different nuclear spin states and see how this changes how the atoms interact
with each other. In chapter 7 we will see how the interactions change when the atoms are allowed
to tunnel between the different lattice sites and the spin and motion become coupled and how thisaffects observables such as the contrast and the frequency shift in the clock spectroscopy signal.

Chapter 2

Strontium Cooling and Trapping

2.1

Introduction
The same vacuum chamber and basic setup has been used successfully in our lab for morethan ten years. In this chapter, we will give an overview of how we prepare strontium atoms forour experiments. In the majority of these experiments we cool the only stable fermionic isotope ofstrontium, 87 Sr, and load them into either a horizontal or a vertical optical lattice before manipulating them using the ultra-narrow clock laser transition at 698 nm. During one of our experiments
we use the bosonic 88 Sr isotope as this allows large dense samples of Strontium to be prepared.
The different laser systems that we use for cooling, trapping, and probing the atoms have all been
upgraded during the last few years and the new setups will be described in this chapter.

2.1.1

Chapter Outline

This chapter focusses on the preparation of the atoms ready for the more interesting physics
experiments to take place. In section 2.2 we will discuss the overall layout of the main experiment
including the location of the different lasers beams and which vacuum viewports are used for thedifferent beams. In section 2.3 the relevant energy levels of Sr are discussed including the particulartransitions needed for the cooling and probing of the strontium atoms. In section 2.3.1 we zoom inand look at the hyperfine structure of the 1 P1 and 3 P1 electronic levels that are used to cool theatoms.
We will then move on in section 2.4 to discuss the laser setup for cooling the atoms in the first9stage blue magneto-optical trap (MOT) based on the 1 S0 −1 P1 transition. As described in previous
theses [7, 8] the cooling light was derived from a 922 nm diode and tapered amplifier (TA) system.
This 922 nm light was then frequency doubled using two home-built setups using KNbO3 crystals.
This frequency doubled system was old and required long warm up times of several hours everyday. In 2015 we therefore upgraded to a diode based system that had been tested out previously
by the JILA Sr 2 experiment in our lab. The system uses one 461 nm ECDL laser to seed three
150 mW diode lasers.
In section 2.5 we discuss the laser setup for the second stage red MOT based on the 1 S0 −3 P1transition. This system was also upgraded from using a heatpipe to lock to the Sr resonance at 689
nm, to using a cavity based setup. The cavities used in this setup were from older laser setups thatwere used to create the clock laser and gives us ∼ Hz level linewidths and ∼ kHz level drifts perday. This laser is shared between the two Sr experiments in our lab and is used to seed the fourslave lasers used by the two experiments.
We will then discuss the loading of the atoms into the optical lattice at 813 nm that allows
us to trap the atoms in section 2.6. Commercial narrow linewidth Ti:Sapphire lasers have recently
become available and therefore during the last few years we have also upgraded from a TA systemof producing the lattice light that gave us ∼ 300 mW of light at the atoms to now have over 3 W
of light available at the atoms. This new Ti:Sapphire laser also has the advantage of not having
the noise pedestal associated with amplified spontaneous emission from a TA system. We will also
discuss in this section how we detect atoms and determine the fraction of atoms in each clock stateand how we can do further in lattice cooling that can be seen by scanning the clock laser

2.2

The Experimental Apparatus
A solidworks drawing of our experimental vacuum chamber is shown in Figure 2.1. The ovenwas last replaced in December 2010 with 15 grams of strontium being placed inside. Under typicaloperating conditions the oven is heated to 375◦ C and the nozzle of the oven is heated to 575◦ C tostop strontium buildup. The atomic beam leaving the oven is collimated using a stage of transverse

10

Figure 2.1: Vacuum chamber showing the position of the different laser beams. As the atomic
beam exits the oven it is collimated using a stage of transverse cooling before being slowed using
Zeeman slowing with a varying magnetic field and a counter-propagating laser beam. The atoms
are captured at the center of the spherical octagon in a magneto-optical trap (MOT) based on the
1 S −1 P transition before being transferred to a MOT based on the 1 S −3 P transition for cooling01
01
to µK temperatures. The blue cooling light is shown in blue and the overlapped red cooling lightand repumps are shown in red. The atoms are then usually transferred to a lattice, shown here
in green, that is either Horizontal (H1 axis) or close to vertical (V axis). The atoms can then beprobed with the clock laser, shown here as dashed lines, along the axis of the lattice. We also
alternatively probe along the H2 axis to measure the radial temperature

11
cooling that typically increases the atom number in the magneto-optical trap (MOT) by a factorof ∼ 2. The atoms then pass through two differential pumping apertures before passing through agate valve and entering the σ − Zeeman slower.
The atoms are slowed using a Zeeman slower beam detuned 1040 MHz below the 1 S0 −1 P1transition and kept in resonance as the atoms slow by the spatially varying magnetic field producedby the Zeeman slower. The atoms are then captured in a MOT based on the 1 S0 −1 P1 transition

2.3

Strontium Level Structure
Strontium is an alkaline-earth atom and therefore contains two valence electrons. This leads

to the formation of singlet and triplet states where transitions between them are weakly forbidden
due to the need for an electron spin to flip during such a transition. The lower lying energy levelsof strontium are shown in Fig. 2.2. The energy levels are shown to scale and are split to show the

Figure 2.2: Low lying enercy levels of strontium. The transitional energies are drawn to scale andsplit into singlet (left) and triplet (right) states

12
singlet states on the left and triplet states on the right.
Figure 2.3 shows the relevant cooling transitions that we use to cool and trap strontium.
The broad (Γ/(2π) = 32 MHz) 1 S0 −1 P1 transition is used to cool the atoms in a MOT to mK
temperatures. A small percentage of atoms (2 × 10−5 branching ratio) leak out of the blue MOT
cooling transition to the 1 D2 state [9]. The atoms in the 1 D2 state then decay to either the 3 P1
or 3 P2 state. Those that decay to the 3 P1 state can decay back quickly to the 1 S0 state but thosethat decay to the long-lived 3 P2 state need to be repumped to the ground state using the 707 nm
transition to the 3 S1 state where atoms can decay to any of the 3 Pj levels meaning a second repump
laser at 679 nm is also required to remove atoms that become trapped in the 3 P0 clock state. For
further cooling to µK temperatures the weaker Γ/(2π) = 7.5 kHz linewidth transition is used. For
87 Sr cooling the hyperfine structure of the atom needs to also be taken into account.

Figure 2.3: Relevant transitions for cooling and probing Sr. The broad 32 MHz linewidth transition(1 S0 −1 P1 ) is used for slowing and cooling the atoms to mK temperatures. This transition is notcompletely closed and atoms decay via the 1 D2 to the 3 P1 state and the long-lived 3 P2 state. The
repump laser at 707 nm removes atoms from this long-lived state to the 3 S1 state where they decay
to all 3 Pj levels. An additional repump laser is needed at 679 nm to remove atoms from the 3 P0
clock state. Atoms are further cooled using the narrow 7.5 kHz natural linewidth transition down
to µK temperatures. The ultra-narrow clock transition at 698 nm that is used for probing theatoms is also indicated above.

13
2.3.1

Hyperfine Structure

The 1 P1 energy level structure for 88 Sr and 87 Sr are shown in Fig. 2.4. The isotope shift forthis transition is 49.2 MHz [10]. For the 87 Sr isotope there are three hyperfine levels. The shift
away from the central value is calculated using the equation∆EHF =

BQuad 34 K(K + 1) − I(I + 1)J(J + 1)
AHF S
K+
2
2
I(2I − 1)J(2J − 1)

(2.1)

K = F (F + 1) − J(J + 1) − I(I + 1)

(2.2)

where

with AHF S the magnetic dipole hyperfine constant and BQuad the electric quadrupole coupling
constant, F , J, and I are the quantum numbers associated with the total angular momentum, thetotal electronic angular momentum and the nuclear angular momentum respectively.
For the 1 P1 state of 87 Sr we have the quantum numbers J = 1, I = 9/2, F = {11/2, 9/2, 7/2}
and coupling constants AHF S = −3.4 and BQuad = 39 [11] to give the energy shifts shown in Fig.

Figure 2.4: Hyperfine structure of the 5p 1 P1 energy levels of strontium drawn to scale. For bosonic
88 Sr (left) the nuclear spin I =0 leads to only a single hyperfine energy level. For fermionic 87 Sr(right) the nuclear spin I = 9/2 leads to three hyperfine levels with splittings given above. The
natural linewidth is indicated by the width of the lines and is Γ = 2π × 32 MHz for this state.

14

Figure 2.5: Hyperfine structure of the 5p 3 P1 energy levels of strontium drawn to scale. For bosonic
88 Sr (left) the nuclear spin I =0 leads to only a single hyperfine energy level. For fermionic 87 Sr(right) the nuclear spin I =9/2 leads to three hyperfine levels with splittings given above. The
Γ = 2π × 7.5 kHz natural linewidth is narrower than the width of the lines.

2.4. The 3 P1 energy level structure for 88 Sr and 87 Sr are shown in Fig. 2.5. The isotope shift forthis transition is 62.2 MHz [12]. For the 87 Sr isotope there are again three hyperfine levels. For the
3P

1 state of

87 Sr we have the quantum numbers J = 1, I = 9/2, F = {11/2, 9/2, 7/2} and coupling

constants AHF S = −260.084 and BQuad = −35.658 [13] to give the energy shifts shown in Fig. 2.5.

2.4

Blue MOT cooling
The optical laser setups for the 1 S0 −1 P1 cooling transition are shown in Figs. 2.6, 2.7, and2.8. The master laser setup is shown in Fig. 2.6. A single mode master laser (Newport, Vortex
Plus TLB 6800) is split between the locking setup shown in Fig. 2.7 and between the slave diodelasers in Fig. 2.8. A small amount of light is also sent to the wavemeter for coarsely setting thefrequency of the master laser via piezo and current tuning of the master laser.
For the spectrometer setup ∼ 6.5 mW of light from the blue master laser is output from a fiber
launch with the power being monitored by a pickoff shortly after the output. The rest of the lightpasses through an acousto-optic modulator (AOM) giving a frequency shift of ωAOM 1 = 2π × 80
MHz, which provides an overall offset of the blue master laser from the Sr transition. The light is15

Figure 2.6: The optical setup of the 461nm master laser. The laser (Newport, Vortex Plus TLB
6800) pass through a double isolator setup (60 dB attenuation) before being shaped and split
between two different paths. The first path goes to the spectrometer to lock the laser to thestrontium transition and the second path goes to the slave laser setup for injection locking thethree diodes.

then split between the pump and probe beams. The probe beam passes through a ωEOM = 2π × 19
MHz EOM which adds sidebands to the laser for locking [14]. The pump beam passes througha species dependent AOM, with frequency ωSr87/88 , where magnetic mirrors are used to switch
between cooling 87 Sr and 88 Sr. The spectroscopy cell used is a hollow cathode lamp (Hamamatsu
L2783-38NE-SR) and uses Doppler-free spectroscopy for locking.
The main principle behind Doppler-free spectroscopy is velocity selection. There will be onlya small subset of atoms with velocities that can interact with both the pump and probe beams. If
both pump and probe beams are at the same frequency then only atoms with near zero velocity,
along the beam propagation direction, can interact with both beams. Given that the absorption
profile in the rest frame of the atoms has a natural linewidth Γ0 we should expect the frequency
range of the laser that can interact with the atoms to be approximately ω0 ± Γ0 /2. This means

16

Figure 2.7: Some of the light from the blue master laser is fiber coupled to the setup shown above
for locking to the strontium transition. An AOM is used to give an overall offset of +82.2 MHz fromthe 88 Sr transition. The light is then split between two paths. The probe path passes through the
EOM before passing through the strontium cell and onto the photodiode. The second path passes
through a species dependent AOM to add an additional offset for the pump beam. Magnetic mirror
mounts are used to switch between the two strontium isotopes with the 88 Sr AOM at a frequencyof +80 MHz and the 87 Sr AOM at a frequency of +175.7 MHz.

17
that we can ignore the increased linewidth due to Doppler motion.
In our experimental setup, the pump laser is sent through an isotope-dependent AOM before
going through the vapor cell. The pump laser frequency thus experiences a shift of +ωSr87/88 suchthat ωpump = ωprobe + ωSr87/88 . To find the central velocity at which both the pump and the probe
beams are resonant with the atoms we need to consider how the frequency of each beam is shifted
00
due to the Doppler effect. Let ωprobeand ωpump
be the frequencies of the two lasers in the rest

frame of the atom. In terms of the frequencies in the lab frame they are given byvz 0ωprobe= ωprobe 1 −
c
0
ωpump
=

ωprobe + ωSr87/88

(2.3)


1+

vz 
c

(2.4)

0
0
In the rest frame of the atom, on resonance, ωprobe= ωpump= ω0 . Solving these 2 equations forvelocity gives
vz = −cωSr87/88
ωSr87/88 + 2ωprobe

(2.5)

0
0
Putting this expression for vz back into the equation for ωpump
and setting ωpump= ω0 gives

us the frequency we are locking from resonance asq2
ω02 + ωSr87/88
ω0 − ωSr87/88
ωSr87/88
ωprobe =+∼ ω0 −
2
2
2
ωSr87/88⇒ ωpump = ω0 +
2

(2.6)
(2.7)

making the approximation at the end because ω0 ∼ 2π × 400 THz and ωSr87/88 ∼ 2π × 100 MHz

We therefore know that we are locking our Blue Master laser −νAOM 1 − νSr87/88 below the
88 Sr resonance. In both cases this is -120 MHz below the resonance of the species we wish to probewith 87 Sr ∼48 MHz below 88 Sr which agrees with the detuning shown in 2.4.
The rest of the master light, that is not sent to the spectrometer, is used to injection lockthree different 150 mW laser diodes (Nichia, NDB4216E) with different offset frequencies. Oneof these injection locked diodes is used for the MOT beams with an overall offset from atomic
resonance of −40 MHz. This light is sent via a polarization maintaining fiber splitter (1 : 3 fiber
splitter, Evanescent Optics) to the experiment. The second of these diodes is used for Zeeman

18

Figure 2.8: The slave diode setup at 461 nm. Light from the 461 nm master laser (cyan) is used to
injection lock three diodes (Nichia NDB4216E). The diode light is shown in blue with the light being
sent to different parts of the experiment. One diode is used for the blue MOT beams and is offset by−40 MHz from atomic resonance. Another diode is used for the Zeeman slower slowing beam andis offset from atomic resonance by -1040 MHz. The third diode is used for both transverse coolingand probing the atoms. The probe beam has an offset of −10 MHz and the transverse coolingbeam passes through another +110 MHz AOM after the fiber to give the same offset frequency.

19

Figure 2.9: The 707 nm and 679 repump lasers remove atoms that become stuck in the long lived
3 P and 3 P states to allow further blue MOT cooling to take place. The two lasers are combined
20
on a polarizing beam splitter before passing through an AOM and enter the fiber to the experimentwith orthogonal polarizations. An alternative path takes the laser light to the wavemeter.

slowing the atoms that exit our strontium oven. The overall offset from resonance is −1040 MHz
from resonance. The third diode is used for both transverse cooling of the atoms (with a AOM
after the fiber) and probing the atoms both with a detuning of −10 MHz at the atoms. Each diodealso sends a small amount of light to the wavemeter to check the mode of the laser is single mode
after locking.
The optical setup of the repump lasers required for cooling on the 1 S0 −1 P1 transition are
shown in Fig. 2.9. Both of the 707 nm and 679 nm lasers required for repumping are combined

20
on a polarizing beam splitter (PBS) before passing through an AOM in order to switch the beam
quickly on and off. This also means that the two beams are orthogonally polarized going into thefiber with one aligned to the fast-axis of the fiber and the other to the slow-axis. The repump
lasers are not locked but are set to the correct wavelength using the wavemeter and then optimized
using the atomic signal from the MOT that is measured using a photodiode. The laser currents
and piezos are swept for both ECDLs which allows the lasers to address all hyperfine structure andremain on resonance for many hours at a time.

2.5

Red MOT cooling
After blue MOT cooling to ∼mK temperatures the atoms are transferred to a broadband red

MOT based on the 1 S0 −3 P1 transition. The blue lasers and repump lasers are switched off and themagnetic field gradient is quickly reduced from 50 G/cm to 4 G/cm and the lasers are modulated
to increase their bandwidth and are broadened by ∼ 1 MHz. The gradient is held constant for∼ 100 ms and then the gradient is ramped over ∼ 100 ms to 11 G/cm to the single frequency red
MOT and at the same time the modulation is turned off.
The red master laser setup is shown in Fig. 2.10. The laser is stabilized with a prestabilization
cavity and a master cavity (not shown) to give a linewidth of ∼1 Hz. This is of course unnecessary,
but this cavity provides an excellent long-term stability so that our red MOT is reproducible everyday. A 300 MHz AOM is used to set the frequency of the master laser to a detuning of +40 MHz
compared to the 88 Sr 1 S0 −3 P1 transition. This red master laser is used to phase lock the lasers
needed for the red MOT transition, shown in Fig. 2.11.
The J = 0 1 S0 state has a small Landé g-factor compared to the excited state. The splittingof the different states by the applied magnetic field is large enough that not all of the magnetic
sublevels are trapped. In order to get around this problem we use an additional laser to mix thestates of the atoms so they can be continually cooled [7]. We refer to the MOT trapping laser as
laser B which works on the F = 9/2 → F = 11/2 transition and Laser A is used to interrogate
the F = 9/2 → F = 9/2 transition which mixes the states. DDSs are used to lock the lasers with

21

Figure 2.10: Setup of the 689 nm red master laser. The 689 nm light from the extended cavity
diode laser (ECDL) is split after the isolator with ∼ 10 % of the light being used to lock to aprestabilization cavity. The rest of the light then double passes through an 80 MHz AOM before
being split into two paths using a 300 MHz AOM. The −1 order is sent to the experiment and the0th order is sent to another high finesse master cavity for further stabilization.

22

Figure 2.11: Two 689 nm lasers are needed to operate the 1 S0 −3 P1 MOT. The F = 11/2 transition
is probed using Laser B which is the trapping laser of the MOT. The F = 9/2 state is probedwith Laser A which is used as a stirring laser with a further AOM that is used for nuclear-spin
polarizing the atoms. Laser B also has an AOM that can be used for on resonance probing of theatoms. The AOMs are all ∼ 80 MHz with those for laser B using the −1 order and those for laser
A using the +1 order. The lasers are locked using phase locks with different offset frequencies fromthe red Master laser (red lines).

23
appropriate offsets. For cooling 88 Sr the lack of hyperfine structure means that it is possible tooperate the red MOT using just one trapping laser and no additional stirring lasers.

2.6

The Optical lattice
After cooling the atoms in the blue and red MOTs to µK temperatures, the atoms are thentransferred to a one-dimensional optical lattice. In our case the lattice is on during the whole
cooling sequence, and when the red lasers are switched off the atoms are transferred to the lattice.
In most of the experiments discussed in this thesis the lattice is a vertical lattice (offset by 18◦ ) tosuppress tunneling between the different lattice sites. For our spin-orbit coupling work discussed
in chapter 7 a horizontal lattice is used to study the tunneling effects.
When a magnetic field is applied to the atoms the 10 nuclear spin states of Sr can be resolved
spectroscopically as shown in Fig. 2.12 using π-polarized clock light with the propagation axis of
the clock beam aligned with the direction of the lattice. The different heights of the unpolarized

Figure 2.12: Unpolarized clock scan with a magnetic field using π-polarized clock light The ten
nuclear spin states can be seen, along with Rabi sidepeaks, and are equally populated after MOT
cooling. The different heights of the excited peaks are due to different Clebsch-Gordon coefficientsfor the different mF states.

24
carrier clock transitions are caused by the different Clebsch-Gordon coefficients even though the
population, after the MOT cooling, is an incoherent mixture with the same number of atoms ineach state.
To determine the normalized fraction of atoms in each of the clock states we measure both thenumber of atoms in the ground clock state and the excited clock state. Once atoms are prepared in asuperposition or mixture of clock states the number of atoms in the ground state can be determinedby slashing on a 1 S0 −1 P1 blue probe laser beam which causes the atoms to fluoresce and heat up
and they are removed from the trap by this heating. The fluorescence signal is measured using aphotomultiplier tube (PMT). The number of atoms in the excited clock state can be measured byusing the repump lasers to move the atoms back down to the ground clock state where they canbe measured again using a blue probe beam. By also making a measurement of the background
fluorescence the fraction of atoms in the excited state can then be found.
While trapped in the lattice the atoms can be spin-polarized using a magnetic field along the

Figure 2.13: Axial sideband scans under different conditions. The green scan shows the axial
sideband scan without any in lattice cooling. With only sideband cooling the sideband scan isshown in blue. With only radial cooling the sideband scan is shown in yellow. With sideband andradial cooling the sideband scan is shown in red.

25
H2 axis (see Fig. 2.1) and a polarizing beam using the 1 S0 −3 P1 F = 9/2 → F = 9/2 transition
along the same axis. The atoms can be cooled in the lattice both axially and radially. For theaxial sideband cooling a component of the axial beam needs to be along the lattice axis and thefrequency can be set roughly using the known axial trapping frequency. The fine tuning of thefrequency is set by making the red detuned lattice sideband as low as possible. For the radialcooling the MOT trapping beams are used as molasses cooling and the frequencies are set by either
reducing the width of the radial profile of the trapped atoms or equivalently by reducing the widthof the axial sidebands. Figure 2.13 shows the effect of cooling on the axial sideband scan. Without
any in lattice cooling the sideband scan is shown by the green line. When radial cooling is applied(yellow line) the width of the axial sidebands are reduced due to axial radial coupling. If only
axial cooling is applied then the height of the red detuned sideband is reduced (blue line). With
both axial and radial cooling (red line) both the width of the sidebands and the height of the red
sideband are reduced. The clock laser that we use to probe the clock transition has been used since
before I joined the strontium team in 2013. The cavity and setup is described in detail in [7] andhas been an extremely reliable tool.

Figure 2.14: A radial sideband scan with νR = 190 Hz. The axial trapping frequency in this caseis νZ = 43 kHz.

26
As well as the axial sidebands the radial sidebands can be measured to determine the radialtrapping frequency. To measure the radial sidebands the clock beam needs to be misaligned fromthe lattice axis and an example of a radial sideband scan is shown in Fig. 2.14.
The lattice wavelength is set to the “magic wavelength” at 813 nm. This magic wavelength
occurs where the polarizability of the two clock states is the same meaning the two states are shiftedin frequency by the same amount. Figure 2.15 shows the scalar polarizability of the two clock statesin the wavelength range 700 − 900 nm. The vector polarizability is cancelled when mF = +9/2 andmF = −9/2 states are interrogated (see section 3.2.2).

Figure 2.15: The scalar polarizability of the two different clock state 1 S0 (red) and 3 P0 (blue). The
magic wavelength at 813 nm occurs when the polarizability of the two clock states are the same.

Chapter 3

Strontium Clock Evaluations

3.1

Introduction
In order to evaluate the frequency of our 1 S0 −3 P0 strontium clock transition we need tounderstand and measure all effects that shift the frequency of this transition. These effects are all
due to the environment in which the atom is located such as from electromagnetic fields, interactions
between different atoms, and the motion of atoms.
The basic principle around measuring these shifts is the same for most of these effects - wemust vary one parameter with all other conditions constant and measure how the frequency of
the transition varies. The way we do this is by preparing atoms in our optical lattice, applying
a magnetic field, and alternate preparing atoms in the mF = ±9/2 nuclear spin states as well aschanging the parameter of interest. For some shifts, for example the density dependent frequency
shift, it is possible to just probe one of the stretched nuclear spin states to measure the shift whichmeans the measurement can be made faster.
For the case of alternating between probing each of the stretched nuclear spin states, todetermine the resonance frequency the center of both stretched states are measured individually.
To do this one measurement of the excitation fraction is made on either side of each lineshape.
The difference between the two measured excitation fractions can then be used to predict where
the center frequency of the lineshape is. By doing this measurement for both mF = ±9/2 states
the center frequency can be determined (see Fig. 3.1). Once the center frequency is determinedfrom this measurement, digital feedback is used to change the clock laser probing frequencies to28stabilize them at 50% excitation fraction frequencies. This can be repeated for different conditions
such as different lattice intensities. By alternating high and low lattice intensities and measuring aresonance frequency for both lattice depths a shift in the resonance frequency can be measured andbe used to correct the transition frequency. To measure the center frequency using the stretchedstates four measurements are needed for each condition so we refer to this method of locking thelaser as an “8 point lock”.

Figure 3.1: Determining the strontium resonance frequency. Both the ±9/2 nuclear spin states areprobed on each side of the lineshape (red circles). Using the difference δ between the measured
excitation fractions the center of each lineshape is determined, and half way in frequency between
the two is the resonance frequency of Sr ν0 . A digital servo uses the measured splitting and center
frequency to feedback for the frequencies used to probe the next measurement of ν0 .

After many measurements of a frequency shift the distribution of measured shifts is Gaussianin nature. An example measurement is shown in Fig. 3.2. We can see the distribution is Gaussianand is peaked at −0.118 ∗ 10−15 , where we use fractional frequency units ∆ν/ν0 , where ν0 is
the resonance frequency of the clock transition. As more measurements are taken the systematic
uncertainty in how well we can determine the center frequency is reduced.

3.1.1

Chapter Outline

In this chapter we will discuss in detail some of the major causes of frequency shifts in
optical lattice clocks along with the comparison of different clocks. In section 3.2.1 we will discuss
the frequency shift that arises from contact interactions between the atoms that are trapped within

29

Figure 3.2: An example measurement of a density dependent frequency shift. After many measurements the distribution is Gaussian and the peak indicates the frequency shift. The systematic
uncertainty in the frequency shift is related to how well we can determine this center frequency.

the optical lattice. In section 3.2.2 we will discuss the effect the lattice light used to hold the atoms
has on the clock transition frequency. In section 3.2.3 we will discuss the dc Stark Shift which
can be caused by patch charges located on vacuum viewports, cavity mirrors, or other dielectricmaterials that are in close proximity to the atoms. In section 3.2.4 we will discuss the effectthat magnetic fields, that we need to apply to split the different nuclear spin states, have on thetransition frequency. In section 3.2.5 we will discuss the frequency shift caused by the surrounding
temperature environment of the atoms. This frequency shift can be large and it is important tomeasure the temperature accurately and precisely in order to make a strontium optical lattice clockwith low 10−18 fractional frequency uncertainty. In section 3.2.6 we will give a brief summary ofother known frequency shifts that effect the clock transition frequency for optical lattice clocks.
We will then move on in section 3.4 to discuss the two different clock comparisons that wehave undertaken during the last six years. The first of these clock comparisons (see section 3.3.1)
took place in 2013 where we compared both of the strontium clocks within our own lab. In section
3.3.2 we discuss the second of these clock comparisons which is ongoing and involves our colleagues

30
at the National Institute of Standards and Technology (NIST). In this comparison we are comparing
our Sr optical lattice clock to both a Yb optical lattice clock and an Al+ ion clock.

3.2

Measuring Systematic Uncertainties

3.2.1

Density Shift

The more atoms that we load into our optical lattice the more atoms we have to measure thetransition frequency. During a measurement each atom will collapse from a superposition state intoeither the excited or the ground clock states and the more atoms we have the better understandingof the initial superposition we will have. However, by having more atoms we are also loading more
atoms per lattice site. The atoms can then interact with each other via contact interactions that
we investigate further in chapter 6 for the case of atoms loaded into different nuclear spin states,
and in chapter 7 for the case of nuclear spin polarized atoms when tunneling is allowed. In chapter
4 we also look at interactions that are not caused by contact interactions but instead caused bydipole-dipole interactions between atoms.
While these studies will be important for the design of future clocks, in our current setup
the interactions that take place have been studied previously in detail [15, 16, 17]. Experimentallythe atom number can be calibrated in multiple ways as discussed in Appendix A. The density per
particle of the atoms in the lattice is given by
2n(r) =

22− x2 − y 2 − z 2
Nsite
σx
σy
σz
e
(2π)3/2 σx σy σz(3.1)

where Nsite is the number of atoms per lattice site, and σα are the 1/e widths of the atomic cloud.
For a thermal distribution of atoms, the widths in the lattice are given by [18]
r
σα =

p
~
× 2 hnα i + 1
2πM να

(3.2)

where
1

hnα i =e~ωkB T α

(3.3)
−1

31
is the average vibrational quantum number along the α direction. The average density of a latticesite is given by the density-weighted densityZ1

n(r)2 dr
Nsite
Nsite
=
3/2
(2π) σx σy σzn̄ =For the axial direction the atoms are cooled such that hnz i ≈ 0 giving σz =

(3.4)
(3.5)
q

~
2πM νz . In theradial direction, under standard clock operating conditions, we can approximate the exponentialq
2kB T
term in hnα i and write σx = σy = σR = (2πν.
2R) M
During the 2013 clock comparison our clock operating conditions were νz ≈ 46 kHz andνR = 240 Hz and the total number of atoms N = 700 atoms, where N = Nsite × L for L thenumber of lattice sites. For the 2017/18 comparison the lattice beam was expanded to reduce thedensity shift and we had the operating conditions of νz ≈ 46 kHz and νR = 90 Hz and N = 1000.
Comparing these two different lattice configurations we expect a ratio of the density shifts to be
N2017/18∆νρ (2017/18)=∆νρ (2013)
N2013

√

νz,2017/18 × νR,2017/18= 0.20√νz,2013 × νR,2013(3.6)

The actual measurments gave a ratio of 0.17 ± 0.03 consistent with the expected ratio (see Table
3.3.2)

3.2.2

Lattice ac Stark Shift

During clock spectroscopy, the transition frequency is perturbed by electromagnetic fields.
This means that even the light used to trap the atoms in the optical lattice causes perturbations to
the transition frequency. To avoid this, as mentioned in chapter 2, a “magic wavelength” opticallattice is used that moves the two clock states in the same way. The differential ac Stark shift
between the two clock states, ∆νac , is given by∆νac = U0 {∆κs + ∆κv mF ξ k̂ · B̂ + [3m2F − F (F + 1)(3|ˆ · B̂|2 − 1)∆κt ]}(3.7)

where U0 is the lattice trap depth, ˆ and k̂ are the polarization and wavevector of the lattice, ξ is

32

Figure 3.3: Measurement of the magic wavelength using mF = +9/2 (black) and mF = −9/2 (red).
The splitting of the two mF states indicates the presence of a residual vector Stark shift.

the ellipticity of the lattice polarization, B̂ is the direction of the applied bias field, and ∆κs , ∆κv ,and ∆κt are the coefficients of the differential scalar, vector, and tensor shifts, respectively. The
equation also uses the quantum numbers F and mF which are the total angular momentum andthe total angular momentum projection along the quantization axis given by the magnetic field.
For our lattice we use linearly polarized light, ξ = 0, with k̂ · B̂ = 0 ˆ · B̂ = 1. Doing this doubly
suppresses the vector Stark shift and by probing both the mF = ±9/2 states we can remove allcontributions from the vector Stark shift. The “magic wavelength” we use is therefore where thescalar and tensor components of the differential Stark shift cancel for mF = ±9/2.
To find this magic wavelength experimentally we vary the lattice intensity to change thedepth of the lattice and find the frequency shift for both mF = ±9/2. Such a measurement isshown in Fig. 3.3. As can be seen there is a residual vector shift which makes the shifts for thetwo states different. The magic wavelength is then the frequency of the lattice half way between
where the frequency shift is zero for the +9/2 state and for the −9/2 state. This measurement
gives roughly the magic wavelength and by performing an 8point lock (see section 3.1) with ±9/2
and by varying the lattice depth again between two values. Any residual lattice ac Stark shift can

33
be measured and if needed the frequency can be further tuned.
It is important to note that during the ac Stark shift measurement the density shift will bedifferent between the two lattice depths. The density shifts at each lattice depth therefore need
to be measured and taken into account in order to determine the true lattice ac Stark shift. Also,
during each of our clock evaluations the lattice ac Stark shift was found to drift with time. In
2013, a tapered amplifier (TA) system was used to create the optical lattice. These TA systems are
known to have a spectral noise pedestal due to amplified spontaneous emission (ASE). Even withthe use of a spectral filter the shift was found to drift linearly with time by ∼ 12 × 10−17 in 35 days.
During our second clock comparison it was thought that upgrading our lattice light source from aTA system to a lattice based on a Ti:Sapphire laser would reduce this problem and a filter would
not be required. Without any filtering a drift in the lattice ac Stark shift was still measured andthought to be due to a poorly performing pump laser of the Ti:Sapphire laser. By adding in twovolume bragg gratings the frequency shift was found to be consistent over time, and also consistent
with the first ac Stark measurement before the power in the laser started to deteriorate.
The lattice light may become a limiting factor for our optical lattice clock in the near future.
The recently built 3D optical lattice clock in our lab has measured an excited state lifetime of < 20s where the predicted lifetime should be ∼ 160 s. The lattice light is believed to be causing Raman
transitions from 3 P0 to 3 P1 allowing the atoms to decay to the ground state quickly. Other schemes
involving lower lattice intensities and larger lattice spacings may be required for future clocks totake advantage of the full excited state lifetime [19] .

3.2.3

dc Stark Shift

The dc Stark shift in an optical lattice clock was first seen by Lodewyck et al [20] and wasfound to be caused by a charge on an in-vacuum cavity mirror that was used to form the opticallattice. Due to the close proximity of the mirrors the fractional frequency shift was found to be at
the 10−13 fractional frequency shift level. This shift was removed by illuminating the mirrors with
UV radiation for two days. Even without an in vacuum cavity the vacuum chamber windows can

34
be close enough to measure a large frequency shift, especially as atomic clocks are pushed to betterlevels of accuracy.
In 2014, Bloom et al [21] reported a 1.3 × 10−16 shift caused by a charge on one of thevacuum viewports that was removed by filling the vacuum chamber with nitrogen gas. During the2013 evaluation of the systematic uncertainties on our experiment no dc Stark shift was observed.
However, during the 2017/18 systematic evaluation a large frequency shift was measured, and thenremoved, that will be discussed in more detail later in this section.
The dc Stark effect induces a frequency shift given by1∆νdc = − ∆α0 E 2
2

(3.8)

where E is the dc electric field and ∆α0 is the difference in the dc polarizability between thetwo clock states. The quadratic nature of the shift means that by reversing the direction of theelectric field will have no effect on the magnitude of the shift. However, when a static patch charge,
producing an electric field Ep , is present then the frequency shift for two opposite applied electricfields, Ea are given by1∆ν± = − ∆α0 (Ea ± Ep )2
2

(3.9)

Figure 3.4: Cartoon of the dc Stark measurement. On the left is a cartoon of the electric field when
one electrode is charged and the other is grounded. Due to the grounded chamber it can be difficult
to measure perpendicular axes. When equal and opposite voltages are applied to the electrodes,
the electric field can be more easily measured for perpendicular axes, due to less fringe effects.

35

Figure 3.5: Drift of the dc Stark shift with time. A Large dc Stark shift was measured and is seen
to discharge with time to below the 1018 fractional frequency level.

The difference between these two, ∆ν+ − ∆ν− , is proportional to 4Ea Ep . We can then get
the frequency shift by using∆νdc =(ν+ − ν− )2
16Va(3.10)

where Va is the applied voltage on the electrodes, where one has voltage Va and the other −Va . It
is important to use equal and opposite voltages on the two electrodes used in order to easily apply
orthogonal electric fields. The reason for this is shown in the cartoon in Fig. 3.4.
On the left of the figure shows the case when one electrode has a voltage and the other isgrounded. The electric field can be reversed by changing the voltage on the top plate from +V
to −V but overall the electric field does not go straight from one plate to the other due to thegrounded vacuum chamber. When the electrodes have equal and opposite voltages applied (right)
then the electric field is straighter and it is easier to apply orthogonal electric fields at the atoms.
When measuring the dc Stark shift it is convenient to use quadrant electrodes similar to
those discussed in [22]. Due to the asymmetry of our chamber the top and bottom viewports aremuch closer than the other viewports. This means that it is easier to place the electrodes close
to the atoms and their large size enables us to not block the laser beam access. These quadrant

36electrodes allow us to apply perpendicular electric fields to the atoms to measure the dc Stark shift
in all directions.
During our 2013 systematic evaluation no dc Stark shift was measured at the 10−18 level for
the Sr1 experiment. For the 2017/18 evaluation a large shift was measured that varied in time asshown in Fig. 3.5. The shift was measured over 17 days and was seen to exponentially decay to
below the 10−18 level over 15 days. The shift is believed to have occured due to charging a dielectricmaterial used between the electrodes and the thermistors on the viewports when the voltage wasapplied for a long time. By alternating the measurements between ±V and ground every cycle ofthe experiment we were able to stop charging them up.
To cancel the electric field we can apply a voltage to the electrodes that will cancel the field.
We can check that the field is cancelled by adding in an additional ±V to the electrodes. An
example of a cancelled dc Stark shift is shown in Fig. 3.6. It is seen to be parabolic as expectedfrom Equation 3.8 We note that even though voltages required to cancel the field are small, they
are reversed during atom preparation to avoid any build-up of charge.

Figure 3.6: A cancelled dc Stark shift measurement.

37
3.2.4

Zeeman Shifts

A magnetic field is applied in order to spin polarize the atoms into a single nuclear spinstate. Having the atoms in a single nuclear spin state has the advantages of suppressing s-wave
collisions and also gives a better signal to noise ratio during clock spectroscopy as all atoms can beinterrogated. The Zeeman shift has two components the first order Zeeman shift and the second
order Zeeman shift. The resulting Zeeman shift is shown in Fig. 3.7 for the 10 nuclear spin states.
The first order Zeeman shift is caused by a differential g-factor δg between the two clockstates. This first order shift for π-transitions is given byδν (1) = −mF Bδgµ0 /h(3.11)

where B is the magnetic field, µ0 is the Bohr Magneton, and h is Planck’s constant, with δgµ0 /h =−108.4 Hz/Gauss. This differential g-factor arises from state mixing of the 3 P0 clock state withother states due to hyperfine interaction mixing of states [8]. This shift is cancelled out whenboth spin states are interrogated. However a residual first-order shift can remain due to any smallchanges in the magnetic field strength which changes the measured center frequency from the true
center frequency.
For the second-order Zeeman shift, the frequency shift is proportional to B 2 and is dominatedby the interaction between the 3 P0 and 3 P1 states [8]. The shift is given byδν (2) = −AB2 B 2(3.12)

where AB2 = 0.233 Hz/G2 . Equivalently the shift can also be written in terms of the splitting
between the mF = +9/2 and mF = −9/2 states S asδν (2) = −BB2 S 2
where the coefficient is measured to be BB2 = 0.248(2) × 10−6 Hz−1 [21].

(3.13)

38

Figure 3.7: First- and second-order Zeeman frequency shifts for π-transitions of the different nuclear
spin states. The nuclear spin states are in order from mF = −9/2 (red line) to mF = +9/2 (black).

39
3.2.5

Blackbody Radiation Shift

The blackbody radiation (BBR) shift arises from the environment in which the atoms arelocated. The effect is similar to the lattice ac Stark shift but arises from the interaction of the atoms
with thermal BBR radiation instead of the optical lattice light. The BBR shift is the largest of theclock shifts under our usual operating conditions and strontium has the biggest BBR shift for the
optical atomic clocks currently being studied and therefore this systematic is particularly importantfor us. Whereas other systematic uncertainties are measured using the locking method discussed
in the introduction to this chapter, with the BBR shift we rely on measuring the temperature andthe homogeneity of the environment. It should be pointed out that some other groups have beenable to systematically vary the temperature environment to check the measurements are agreeing
with the theoretical model [23].
The BBR Shift, δνBBR , can be written in terms of two components asδνBBR = ∆νstat

T
300K

4


+ ∆νdynT
300K

6(3.14)

where the first term is a static shift and the second is a dynamic part. The static shift is froman effective dc electric field experienced by the atoms and the dynamic part is a correction to thisfirst term. The static coefficient was measured by Middleman et al to be ∆νstat = −2.13023(6)Hz. For the dynamic coefficient, before 2015 there were two calculated values from Middleman
et al [24] ∆νdyn = −147.6(23) mHz and from Safronova et al [25] ∆νdyn = −149.2(16) mHz.
For our 2013 clock evaluation we therefore took a weighted mean of the two values and used∆νdyn = (−148.3 ± 1.6) mHz where the error is the difference between the two calculated values.
In 2015 Nicholson et al were able to improve upon the uncertainty of the calculated value of∆νdyn . The reason for this relates to a proposal in by Safronova et al [25] who proposed a solution to
improving upon the uncertainty in the calculated value of this coefficient by measuring the lifetimeof the 3 D1 state of Sr. The reason for this is that the oscillator strength of the 2.6 µm 3 D1 −3 P0transition is the dominant source of uncertainty in the calculation of ∆νdyn due to its significant
overlap with the room temperature BBR spectrum [26]. From this new measurement the value of40

Figure 3.8: Temperature stabilization of the vacuum viewport. At least two thermistors are used
for each viewport. One of the thermistors is used in-loop to control the viewport temperature andthe others are used as witnesses. Peltier coolers are used to control the temperature to ∼ 22 ◦ C
and water lines are attached at the top to circulate water for removing any excess heat.

∆νdyn used during our 2017/18 clock evaluation was changed to (−148.7±0.7) mHz. The remaining
uncertainty in the blackbody radiation therefore comes from the measurement of the temperature.
The uncertainties of δνBBR will also be temperature dependent (∝ T 3 for the static contributionand ∝ T 5 for the dynamic contribution) meaning that cryogenic clocks have the prospect of leading
to more accurate clocks. For the 2013 evaluation we used 8 thermistors to record an emissivity
weighted average of the temperature around the vacuum chamber with the uncertainty related tothe drift of this average with time. For this same clock evaluation the second Sr experiment Sr2
took the approach of using in-vacuum temperature sensors combined with a BBR shielding box
around the vacuum chamber. One of these sensors was attached to a bellows and moveable in41
order to measure how the temperature distribution changed and to check the agreement with thetemperature model [4]. With the use of these in vacuum temperature sensors Sr2 were able toshow for the first time a total systematic uncertainty at the low 10−18 level. The temperaturestabilization of the vacuum viewports also allows the whole vacuum chamber to thermalize moreeasily.
During our 2017/18 systematic evaluation we wanted to test the limits of a room temperatureclock without in vacuum temperature sensors. For this approach we also built a BBR shielding
box around the whole chamber with the majority of the laser light being brought through opticalfibers into the box. In order to keep the BBR environment as uniform as possible we chose totemperature stabilize each of the vacuum viewports. The reason that we care in particular about
the vacuum viewports is that they have a much higher emissivity than the rest of the vacuum
chamber. The emissivity gives the ratio of heat emitted compared to a perfect blackbody suchthat a low emissivity surface does not emit as much BBR radiation and thus we care less about its
temperature.

Figure 3.9: Temperature measurements of the top (green and yellow) and bottom (red and blue)
vacuum viewports. These viewports have the highest emissivity weighted solid angle and hence
contribute most to the BBR shift and uncertainty. They are stable to within ±50 mK with thelarge spike during hour 12 during a break in the experimental data.

42
The temperature stabilization uses both Peltier coolers and water cooling to remove any
excess heat from the Peltier cooler. An example of the cooling setup is shown in Fig. 3.8 andfits over the vacuum viewport. Great effort was taken to control the temperature of the vacuum
chamber1 and an example of the temperature during clock operation is shown in Fig. 3.9. As
can be seen the temperature is stable to within ±50 mK and combined with the temperature
uncertainty of the thermistors the total uncertainty from BBR radiation was reduced by more thanan order of magnitude compared to our Sr1 2013 evaluation to ∼ 6 × 10−18 . The work on reducing
the BBR uncertainty is ongoing and is continuing to be improved upon during the writing of thisthesis.

3.2.6

Other Frequency Shifts

In this section we will briefly summarize some of the other frequency shifts that affect opticallattice atomic clocks. In the same way that the light forming the optical lattice can cause frequency
shifts, the clock light used to probe the atoms also causes a frequency shift. This can be measured
by using π-pulses of varying length and comparing how the clock transition frequency changes. It is
important when measuring this shift to take into account shifts associated with the AOM used forfiber noise cancellation. This AOM is switched on and off to create the probe pulse which causes aphase chirp due to the heating of the AOM crystal which also leads to the amplitude modulationof the clock light used for fiber noise cancellation [27].
As well as collisions between the different Sr atoms within the lattice, the Sr atoms can also
collide with any other atoms or molecules that are within the vacuum chamber. This shift can becalculated based on the methods discussed in Gibble et al [28]. For this measurement the lifetimeof the atoms in the vacuum needs to be known, and the measurement is shown in Fig. 3.10 andshows a lifetime in the optical lattice of 6.6 s. This will be a lower limit on vacuum lifetime of
the atoms and gives a shift and uncertainty below the 10−18 level. The movement of the atoms in
the lattice can also lead to Doppler shifts of the clock transition frequency. The first order doppler
1

The majority of this work was undertaken by Toby Bothwell and Dhruv Kedar.

43

Figure 3.10: Measurement of the vacuum lifetime measured in the optical lattice.

shifts are removed by referencing the probing clock laser to the mirror that retroreflects the lattice.
Second order Doppler shifts are calculated to not be important at the current level of systematic
uncertainty but could become important as the accuracy of clocks increases. Another important
effect that needs to be taken into account is the gravitational redshift. The frequency difference
between two clocks with a height difference of ∆h can be written as a fractional frequency difference
as
δν
g∆h
= 2
ν0
c

(3.15)

where g is the local acceleration due to gravity and c is the speed of light. The equation assumes
only small changes in height close to the earth’s surface. For our 2013 systematic evaluation theclocks were located on adjacent optical tables and a relative height difference between the two clocks
can be measured easily to the cm level. For the 2017/18 evaluation the two clocks are located ∼ 1
mile apart and so the height difference is measured by a geodetic survey conducted by the National
Geodetic Survey team.

44

3.3

Uncertainty Tables

3.3.1

2013 Sr-Sr comparison

The full uncertainty table for the 2013 systematic evaluation is shown in Table 3.3.1. On
the left are the nominal shifts and their uncertainties for the Sr1 experiment and on the rightshows the nominal shifts and their uncertainties for the Sr2 experiment. As can be seen the main
sources of uncertainty that are different are the BBR radiation shifts, the AOM phase chirp and thedensity shift. The main differences between the two Sr clocks are the use of in vacuum temperature
sensors which allow for the temperature uncertainty to be reduced. The Sr2 experiment also uses
Source of shift

∆νSr1

σSr1

∆νSr2

σSr2

BBR static
BBR dynamic
Density shift
Lattice Stark
Probe beam a.c. StarkFirst-order Zeeman
Second-order Zeeman
Residual lattice vector shift
Line pulling and tunnelingd.c. Stark
Background gas collisions
AOM phase chirp
Second-order Doppler
Servo error-4,832
-332
-84
-279
8
0
-175
0
0
-4
0
-7
0
1

45
6
12
11
4
<0.1
1
<0.1
<0.1
4
0.07
20
<0.1
4

-4,962.9-345.7-4.7-461.50.8-0.2-144.5
0
0
-3.5
0
0.6
0
0.4

1.8
3.7
0.6
3.7
1.3
1.1
1.2
<0.1
<0.1
2.1
0.6
0.4
<0.1
0.6

Totals-5,70453-5,921.2

6.4

Table 3.1: The uncertainties for the two experiments are quoted as 1σ standard errors and all
shifts and uncertainties are in fractional frequency units multiplied by 10−18 .

a lattice based on Ti:sapphire laser and a build-up cavity to create the optical lattice comparedto the TA system used by our Sr1 experiment for this evaluation. The high powered and more
spectrally pure Ti:sapphire laser allows for a more stable lattice ac Stark shift as well as the ability
to create a bigger lattice waist which reduces the density shift for a constant atom number. For thiscomparison the density shift for our Sr1 experiment is quoted for our working condition of ∼ 700

45
atoms and the Sr2 experiment used a few thousand atoms.
The BBR shift is calculated at 298 K for Sr1 and the box around the Sr2 experiments heats
up the box to be at a higher temperature of 300 K. For Sr1 the applied bias field leads to a splittingof 550 Hz whereas the Sr2 experiment worked at a lower magnetic field and also suffered from adrifting magnetic field that needed to be servoed and caused a residual first order Zeeman shift.
For both experiments a 160 ms pulse was used.

3.3.2

2017/18 JILA-NIST comparison
Source of shift∆ν2013σ2013∆ν2017/18

σ2017/18

BBR static
BBR dynamic
Density shift
Lattice Stark
Probe beam a.c. StarkFirst-order Zeeman
Second-order Zeeman
Residual lattice vector shift
Line pulling and tunnelingd.c. Stark
Background gas collisions
AOM phase chirp
Second-order Doppler
Servo error-4,832
-332
-84
-279
8
0
-175
0
0
-4
0
-7
0
1

45
6
12
11
4
<0.1
1
<0.1
<0.1
4
0.07
20
<0.1
4

-4,640.3-313.2
-14.5
-16.1
0.9
0
-178
0
0
0
0
-3
0
6.4

4.5
1.5
1
3.0
0.7
0.5
1.4
<0.1
<0.1
0.6
0.1
1
<0.1
0.9

Totals-5,70553-5169.7∼6

Table 3.2: Systematic evaluation table for Sr1 comparing the 2013 evaluation to the 2017/18
systematic evaluation. The uncertainties for the two experiments are quoted as 1σ standard errors
and all shifts and uncertainties are in fractional frequency units multiplied by 10−18 . It should
be noted that these numbers are from 2017 and are preliminary. Further improvement of thesesystematic uncertainties is already reaching the low 10−18 level.

During our 2017/18 systematic evaluation our Sr1 experiment was able to make an improvement of almost an order of magnitude upon our 2013 systematic evaluation. Table 3.3.2 compares
the two systematic evaluations for our Sr1 experiment. It should be noted that the measured systematics are from 2017 and they are constantly being improved upon. The majority of this comesfrom the improvement of the BBR shifts by more than an order of magnitude. During the second

46
evaluation the temperature control of the chamber was improved as discussed in section 3.2.5 withthe temperature of the vacuum viewports servoed to 22◦ C.
The volume of the lattice was also increased with the waist increasing from 32 µm to ∼ 85 µm
allowing for a lower density shift measurement along with longer pulses of 500 ms, and a carefullymeasured AOM phase chirp.

3.4

Comparing Clocks
It is important to compare different clocks as this allows us to check the agreement between

two clocks and can allow us to see if we are not accounting properly for any systematic offsets.
Comparing the frequencies of clocks depends on whether they are the same atom or not and canalso depend on where they are located relative to each other.
For our 2013 clock comparison [21] we compared two Sr optical lattice clocks that were locatedwithin the same lab. The same clock laser is used to probe both sets of atoms and there is only one
AOM that is different between the two experiments. In this case to compare frequencies all thatis needed is to measure the final frequency of each AOM and, after all corrections are considered,these two frequencies are all that need to be considered.
For our 2017/18 clock comparison our Sr optical lattice clock is being compared with twoother clocks at NIST ∼ 1 mile away. The two clocks that we are comparing to are the NIST
Yb optical lattice clock and the Al+ single ion clock. For this clock comparison the frequency
differences between the two clocks are more complicated and are shown in Fig. 3.11. This onlyshows the relevant frequencies at JILA and there are more frequency offsets at NIST where theatom-stabilized light for all clocks are measured using two independent frequency combs.
An example of some comparison data is shown in Fig. 3.12. The Allan deviation (see section
1.3) showing the stability of the two clocks is shown in (a) where the averaging of a single clock
is shown to be 3.4 × 10−16 at one second. Figure 3.12 (b) shows the frequency difference of thetwo clocks over the course of the month long measurement. The data shows agreement of the two
frequencies at the −2.8 × 10−17 level which is within the systematic uncertainty of the two clocks,

47
which is dominated by the systematic uncertainty of the Sr1 experiment. As of the writing of thisthesis the comparisons between our strontium optical lattice clock at JILA and the clocks at NIST
are still ongoing, but statistics indicate the Sr-Al clock’s frequency ratio is stable at the 1 × 10−18
level.

Sr	
  frequency	
  chainfbran(t)Si	
  cavity	  (1.5µm)

AOM
AOM

to	
  NIST

fSi3,FNC(t)1550nmfrep f0
Er:fiber frequency	
   comb

VCO

Sr	
  clock	
  laser	
  (698nm)

LF

Sr

atom	
  corrections

fstatic

AOM

AOM

AOM

fb,698 @698nm

fxfer

fS.L.

fSr,FNC

AOM
Distribution	
   center

Figure 3.11: The strontium frequency chain for sending atom stabilized light to NIST via the
BRAN fiber link. Highlighted in yellow are the relevant frequencies that need to be measured to
calculate any frequency offsets that are needed for the absolute frequency measurement by the
NIST frequency combs. Schematic courtesy of John Robinson.

48
10

-15

a

Allan Deviation

3.4× 10 − 16
√τ
10

-16

10

-17

10

-18

10

0

10

1

10

2

10

3

10

4

Averaging Time (s)b

3×10 -16

ν -ν (Fractional Frequency)
Sr2
Sr1

0
-3×10 -16
1×10 -16±5.4 x10 -17
0-1×10 -16
l
-Ju

17

0

:0

02

17

l
-Ju

0

:0

06

-J

18

00

1:

2
ul

-J

19

1:

0
ul

00
19

00

0
0
0
:0
:0
:0
07
05 g 03
g
g
u
u
u
-A
-A
-A
20
20
17
00

1:

5:

l0
-Ju

0
ug

-A

17

Coordinated Universal Time (UTC)√
Figure 3.12: (a) Allan deviation of the Sr1 and Sr2 comparison divided by 2 to reflect theperformance of a single clock. The red solid line is the calculated quantum projection noise forthis comparison. The green dashed line is a fit to the data, showing the worst case scenario forthe averaging of a single clock of 3.4 × 10−16 at one second. b) The absolute agreement between
Sr1 and Sr2 recorded at the indicated Coordinated Universal Time. The light-green region denotesthe 1σ combined systematic uncertainty for the two clocks under the running conditions at thattime. The top panel shows the frequency record binned at 60 s; in the bottom panel each solid
circle represents 30 min of averaged data. The green dashedlines represent the 1σ standard errorpinflated by the square root of the reduced chi-squared, χ2ν , for the weighted mean of these binned−17
comparison
data. The final comparison over 52,000 s of data showedpp agreement at −2.7(5) × 10
−1722
( χν =10.5) for the 30-min averaging time and −2.8(2)×10( χν =3.5) for the 60-s averaging
time. Reproduced from [21].

Chapter 4
Collective atomic scattering in a dense sample of 88 Sr

4.1

Introduction
Understanding interactions between light and matter in a dense atomic medium is a long-standing problem in physical science [29, 30] since the seminal work of Dicke [31]. In addition totheir fundamental importance in optical physics, such interactions play a central role in enabling a
range of new quantum technologies including optical lattice atomic clocks[21] and quantum networks[32].
The key ingredient in a dense sample is dipole-dipole interactions that arise from the exchangeof virtual photons with dispersive and radiative contributions, with their relative magnitude varying between the near-field and far-field regimes. The dispersive (real) part of these dipole-dipole
interactions is responsible for collective level shifts and the radiative (imaginary) part gives rise toline broadening and collective superradiant emission [33, 34, 35]. Intense theoretical efforts havebeen undertaken over many years to treat the complex interplay between the dispersive and radiative dynamics [36, 37, 38, 39, 40, 41, 42, 43, 44, 45]. However, experimental demonstrations thatprovide a complete picture to clarify these interactions have been elusive.
Collective level shifts and line broadening arising from the real and imaginary parts of dipoledipole interactions have recently been observed in both atomic [46, 47, 48, 49, 50] and condensed
matter [51] systems. The modification of radiative decay dynamics at low excitation levels has
also been observed using short probe pulses [52, 53, 54, 55], and interaction effects were manifested in coherent backscattering [56, 57]. While simple models of incoherent radiation transport50
have often been used to describe light propagation through opaque media [58, 59] and radiation
trapping in laser cooling of dense atomic samples [60], coherent effects arising from atom-atominteractions, which are necessary to capture correlated many-body quantum behavior induced bydipolar exchange, are beginning to play a central role. For example, the dipole-dipole interaction is responsible for the observed dipolar blockade and collective excitations in Rydberg atoms[61, 62, 63, 64, 65, 66, 67, 68]; it may also place a limit to the accuracy of an optical lattice clock andwill require non-trivial lattice geometries to overcome the resulting frequency shift [69]. Previous
theoretical efforts have already shown that physical conditions such as finite sample size, sample
geometry, and the simultaneous presence of dispersive and radiative parts can play crucial roles inatomic emission [36, 37, 39, 40, 70, 71, 72] .
In this work we use millions of Sr atoms in optically thick ensembles, taking advantage ofthe unique level structure of Sr to address motional effects, to study these radiative and dispersive
parts simultaneously. We demonstrate that a single, self-consistent, microscopic theory model
can provide a unifying picture for the majority of our observations. These understandings can help
underpin emerging applications based on many-body quantum science, such as lattice-based opticalatomic clocks [21, 73, 74], quantum nonlinear optics [66], quantum simulations [75], and atomic
ensemble-based quantum memories [76].

4.1.1

Chapter Outline

The outline of this chapter is as follows. In section 4.2 we will present the experimental detailsof the system used to measure the collective effects in 88 Sr. We will then discuss the experimental
results in section 4.3 which we shall separate into three different sections. Section 4.3.1 will discuss
the intensity measurements that we have made for both the blue and red transitions. Section
4.3.2 will discuss the linewidth broadening observations, again for both transitions. Section 4.3.3
will compare the measured density dependent frequency shifts of the blue transition to a priormeasurement of the frequency shift of the red transition [77]. In section 4.4 we will discuss in detailthe coupled dipole model which can be used to model the majority of our experimental observations.

51

4.2

Experimental Details

4.2.1

Strontium 88

We choose to use bosonic 88 Sr, whereas in the rest of the work described in this thesis we use
fermionic 87 Sr and there are a number of reasons we choose to do this. First of all 88 Sr contains
zero nuclear spin giving a relatively simple energy level structure compared to both 87 Sr and toalkali metal atoms. This simple structure makes bosonic 88 Sr, and other zero nuclear spin alkalineearth atoms, easier to model and to interpret the experimental observations. This simple structure
alongside the large natural abundance of the 88 Sr isotope makes cooling large dense samples ofatom to µK temperatures possible.

Figure 4.1: A schematic diagram of the four-level atomic structure shared by both the 1 S0 – 1 P1blue transition and the 1 S0 – 3 P1 red transition in bosonic 88 Sr.

Bosonic 88 Sr also has the advantage that we can study the effects of motion on the collective
emission from a dense sample of atoms. It contains both a strong 1 S0 – 1 P1 blue transition (λ = 461nm) and a spin-forbidden 1 S0 – 3 P1 red transition (λ = 689 nm) with a strict four-level geometry as
shown in Fig. 4.1. Throughout this chapter we will refer to these two transitions as the “blue” and“red” transitions respectively. The reason we can measure the effects of motion on the collective
emission can be seen by considering the Doppler broadening for both of these transitions. The full
width at half maximum of the Doppler broadening is given byr√
1 8kB T ln 2
∆νD == ∆D 8 ln 2
λ
M

(4.1)

where kB is the Boltzmann constant, M is the mass of an atom, T is the atomic temperature, λ52
is the wavelength of the transition, and ∆D is the Gaussian standard deviation of the frequency
broadening.
When the atoms are cooled to a typical temperature of T = 1.3 µK then the Doppler broadening of the blue transition is ∆νD = 57 kHz. This is almost three orders of magnitude smaller
than the natural linewidth of the blue transition, Γ0 = 32 MHz. This means that to an excellent
approximation, atomic motion is negligible and the atoms do not move between when they absorb
and when they emit a blue photon. On the contrary, the Doppler broadening of the red transition is∆νD = 38 kHz which is ∼ 5 times larger than the natural linewidth, Γ0 = 7.5 kHz. This means thatthe red transition is strongly affected by atomic motion. By comparing the behaviors of the sameatomic ensemble when probing the red and blue transition, we can thus collect clear signatures of
motional effects on collective scattering.

4.2.2

Experimental Setup

In the experiment up to 2 × 107 atoms are cooled to ∼ 1µK temperatures in our two-stage
magneto-optical trap (MOT), the first based on the blue transition and the second based on the redtransition. The atomic cloud is then released from the MOT and allowed to expand for a variable
time of flight (TOF), as discussed in section 4.2.3.2. This allows us to control the optical depth(OD) and density of the atomic ensemble while keeping the atom number, N , constant. The atoms
are then illuminated for 50(100) µs using a weak probe beam that is large compared to the size ofthe atomic cloud and resonant with the blue (red) transition.
A simplified schematic of the experimental setup is shown in Fig. 4.2. Two detectors measure
the resulting light scattered by the atoms. One detector is along the forward direction (detector
DF ) and the other along the transverse direction (detector DT , offset by ∼ 10◦ ). For a short TOF,
the atomic cloud is anisotropic and has an approximately Gaussian distribution with an aspectratio σx : σy : σz = 2 : 2 : 1, where σα are the root-mean-squared (r.m.s) radii.
We can use this same basic setup to perform a comprehensive set of measurements of fluorescence intensity emitted by a dense sample of 88 Sr atoms. We vary the atomic density, cloud

53

Figure 4.2: The atomic ensemble is weakly excited using a laser beam larger than the size ofthe atomic cloud and propagating along the x̂ with polarization along either ŷ or ẑ. The atomic
fluorescence is then collected simultaneously with two detectors: one in the forward direction, x̂,and the other offset by ∼ 10◦ from the perpendicular direction, ẑ.

54geometry, observation direction, and polarization of the probe laser and characterize the three key
parameters: the peak intensity, the linewidth broadening, and the linecenter shift.

4.2.3

System Parameters

A number of different system calibrations need to take place in order to know the initialconditions of our experiment and to obtain universal scaling parameters such as density and opticaldepth that can be compared between different experiments both in our lab and by others. In thissection we will describe how certain experimental parameters such as atom number and temperatureare calibrated and how they can be related to these universal parameters.

4.2.3.1

Atom Number

The atom number is calibrated using a method of transfer efficiencies [78] (see Appendix A).
The atom number in the blue MOT is found by measuring the lifetime of the blue MOT when
the repump lasers are blocked. Using the known branching ratios this measurement tells us thefraction of time each atom spends in the 1 P1 state and thus how many photons are emitted per
atom. Once the atom number in the blue MOT is known the transfer efficiency to the broad-band
red MOT can be calculated by comparing the PMT signal for the blue MOT and broadband red
MOT when the MOT beams are flashed on. Typically, this transfer efficiency is 48%.
The blue probe beam can then be used to calculate the transfer efficiency from broad-band
red MOT to single-frequency Red MOT in the same way. We note that for dense ensembles of atoms
the intensity can depend on the TOF due to the collective effects. In this case the transfer efficiency
is calculated for longer time of flights where the intensity tends to peak. For these measurements
this was typically 15 ms and gives a transfer efficiency for this stage of 65%.

4.2.3.2

Temperature

Unlike in most of our other experiments where we measure the temperature of the atoms using
the ratio of the areas of the different axial sidebands of the lattice, here we do not have a lattice

55
and thus measure the temperature using the free expansion of the atomic cloud. This temperature

Figure 4.3: Time of flight fluorescence images showing the expansion of the atomic cloud after
releasing the atoms from the magneto-optical trap.

calibration is performed at low density and in order to calibrate the temperature of the atoms the
pixel size of the camera that we are using (forward detector DF ) needs to be calibrated too. In
fact, the pixel size can be calibrated in the same measurement and then this allows us to determine
the size of the atomic cloud needed to obtain the temperature.
To perform this measurement, for different TOF, pictures of the atomic cloud are taken as
in Fig. 4.3. The vertical centers of each of these images is used to get the pixel size by knowing
the displacement of the atoms with time whilst falling under the influence of gravity
1z − z0 = v0 t + at2
2

(4.2)

with z0 the initial vertical position of the cloud, z the final position of the cloud after a TOF t, v0
the initial velocity of the cloud, and a the acceleration of the cloud. We let this fitting function
vary v0 , z, and a and from the extracted value of a we know our pixel size to be a/g where g = 9.81
ms−2 is the acceleration due to gravity. The pixel size of our camera is found to be (20.6 ± 0.2) µm
which is consistent with the expected size of ∼ 20 µm from the actual 10 µm pixels demagnified bya factor of two.
The Gaussian root-mean-squared radius of the velocity distribution, σvα , in the α = {x, y, z}
direction is associated with the temperature Tα of the cloud by
Tα =

M 2
σ
kB vα

(4.3)

56

Figure 4.4: The vertical center position of the atomic cloud is measured (purple, squares) versus
time-of-flight in order to calculate the pixel size which is found by fitting the data to Eq. 4.2(purple, line) to be (20.6 ± 0.2) µm.

Figure 4.5: An example measurement of the cloud temperature in the ŷ (purple, circles) andẑ (orange, squares) directions. The plotted Gaussian fitted r.m.s radii versus time-of-flight are
fitted with Eq. 4.4 (solid lines) to extract the atomic cloud temperatures to be Tz = 1.4µK and
Ty = 1.3µK.

57
where the Gaussian radius of the freely expanding cloud is related by the equationqσα (t) = σα,0 + σv2α t2(4.4)

where σα,0 is the initial Gaussian radius at time t = 0. Fitting Eq. 4.4 to the Gaussian radii as afunction of time can then be used to extract the temperature of the atomic cloud as shown in Fig.
4.5 giving typical temperatures of 1.3 µK.

4.2.3.3

Optical Depth and Density

In order to convert the measured atom numbers and cloud sizes into a more experimentallycomparable quantity such as density and optical depth we need to know how they scale with eachquantity. The distribution of atoms in our atomic cloud is approximately Gaussian and given by
the distribution22

n (x, y, z) = n0 e2y
− x 2 − 2σ2 − z 22σxewhere n0 is the peak density which satisfies the equation

(4.5)

y

e

R

dxdydzn (x, y, z) = N to give the peak

2σzdensity,
n0 =

N
3/2

(2π)

.

(4.6)

σx σy σz

For the case of optical depth the scaling with atomic cloud size is different. Throughout thischapter we will define the OD as the on resonance optical depth of the cloud. The optical depth
can be written as

OD = nσscatt L(4.7)

where n is the atomic density found by integrating the density distribution given in Eq. 4.5,2σscatt = 3λ2π is the resonant scattering cross section for a Lorentzian lineshape and L is the lengthof the medium through which the probe light travels. The addition of motion will reduce thepeak cross section and hence optical depth. If we consider the optical depth along the x̂ direction

58(direction of the probe beam, see Fig. 4.2) we can rewrite the OD as
OD =

N 3λ2
L
V 2π
Z

N

=

2

RRR

2
2
y
− x 2 − 2σ2 − z 2dxdydze

2σx

e

y

e

2σz

dxe

2
2
− x 2 3λ
2σx

2π
(4.8)

3N
= 2
2k σy σz=

3N
2 (kσ⊥ )2

where k is the wave vector of the light and we have defined σ⊥ as the radius perpendicular tothe direction of observation. This radius depends on the direction of observation and is given byσ⊥,T = σx = σy and σ⊥,F = (σy σz )1/2 , for the transverse and forward direction respectively.
As we can see the density and the OD scale differently with the cloud radii. The different
experimental data that we will show in the next three sections will be scaled to either OD or
density where the choice is dependent on the appropriate scaling of the quantity as determined by
the coupled dipole model (see section 4.4).

4.3

Experimental Observations

4.3.1

Intensity

The coherent effects manifest themselves most clearly in the fluorescence intensity in theforward direction, which is the direction along which the probe beam is travelling. The probe beam
synchronizes the dipoles so that in the forward direction the radiation from the different dipolesinterfere constructively to give a strong forward intensity peak. However, the forward direction
also has the experimental difficulty that we need to separate the probe beam from the fluorescenceemitted by the atoms. When measuring the intensity in the forward direction we use the additional
setup shown in Fig. 4.6 for this separation.
The probe beam is focused with a lens (L1 , focal length f1 = 15 cm) after passing throughthe atomic cloud and exiting the vacuum chamber. The probe beam is then blocked using a razor
blade which can be translated perpendicular to the probe beam. The same lens (L1 ) also collimates

59

L1

θ

L2

Figure 4.6: Setup for measuring the forward intensity. A lens (L1 , focal length f1 = 15 cm) justoutside the vacuum chamber focusses the probe beam onto a razor blade that can be translated
perpendicular to the direction of the probe beam to change the angle θ. This same lens collimates
the fluorescence emitted by the atomic ensemble and a second lens (L2 , focal length f2 = 15 cm)
images the fluorescence onto a detector

Figure 4.7: The enhancement of the intensity in the forward direction. The intensity in the forward
direction is measured for the blue transition (blue, squares) and red transition (red, squares) byusing the setup shown in 4.6 with the intensity normalized to the intensity at ∼ 7.5 mRad. The
enhancement of the intensity for the red transition is reduced due to motional effects. Solid lines
are theory fits using the coupled dipole model. The black arrow indicates the angle that the data
in Fig. 4.9 is taken.

60
the atomic fluorescence so that it can be imaged using a further lens (L2 , focal length f2 = 15 cm)
onto the forward detector, DF . The position of the razor blade can therefore be used to vary the
angular range of the detected fluorescence whilst at the same time blocking the probe beam.
We should note that in reality the setup is slightly more complicated than that shown in Fig.
4.6. The reason for this is that the forward CCD detector used for collecting the blue fluorescenceis actually insensitive to the red fluorescence. After the light is focused, using L2 , the light isspatially filtered using an iris and then the blue and red fluorescence split using a dichroic. The
blue fluorescence is still collected using the CCD camera used for the rest of the blue forward
measurements and the red fluorescence is collected using a photon counter.

4.3.1.1

Forward Intensity Enhancement

The angular distribution of the fluorescence in the forward direction can be characterized by∆x
the angle θ = arctan 15cm
which is the angle between x̂ and the edge of the razor blade. This angle

is changed by translating the razor blade edge from the center of the probe beam by ∆x. Figure
4.7 shows the intensity of the forward emission for the blue probe (blue, squares) and red probe(red, triangles) at different angles of theta which are normalized to that collected at θmax ≈ 7.5
mRad. The measurements at each position of the razor blade are averaged over multiple cycles ofthe experiment where we interleave both the blue and red probe beam and in each case compare
the fluorescence with and without atoms to remove any background noise.
For the case of the blue probe beam in the forward direction we see over a factor of 1000
enhancement in the normalized intensity for θ < 0.5 mRad. The enhancement is also present for the
red transition, but is reduced by almost two orders of magnitude at small θ due to motional effects.
However, the longer wavelength of the red transition leads to a wider angular area of enhancement.
A close look at Fig. 4.7 reveals that the angles θ for the two transition are actually offset
from each other, even though they are measured at the same razor blade positions. The reason forthis is that the two probe beams are offset from each other by 33 µm (∼ 0.2 mRad) as shown in
Fig. 4.8, giving different x̂ directions for the two probe beams and hence different angle θ.

61

Figure 4.8: Focused Probe Beam Calibration. The normalized intensity of the probe beams areplotted as the razor blade is scanned across the focus for both the blue (blue, squares) and red (red,triangles) probe beams. The fits (solid, lines) are Gaussian cumulative distribution functions given
by Eq. 4.9. From these measurements we can see that the probe beams are offset by ∼ 33 µm. The
focused r.m.s radius of the Gaussian probe beams is extracted from the fit to be 12.2 µm (16.4 µm)
for the blue (red) probe beam.

Figure 4.9: Intensity in the forward (x̂) and transverse (ẑ) directions for the blue (blue, squares) andred (red, triangles) probe beams as the atom number is changed taken at θ = 2 mRad as indicatedby the arrow in Fig. 4.7. The transverse intensity scales approximately with atom number N ,meaning that the forward intensity scales ∝ N 2 .

62
Figure 4.8 shows the intensity of the probe beam measured as the razor blade is translated
across the focus. The intensity is then fit with a Gaussian cumulative distribution function"#!
1x0 − x
GCDF =
1 + erf √
(4.9)
2
2σp
where x0 is the center position of the beam, and σp is the root-mean-square radius of the Gaussian
probe beam. The difference between the two values of x0 for the fits to the different probe beam
intensities gives the offset of the two beams. For the blue probe beam the extracted value of
σp = 12.2 µm and for the red probe beam σp = 16.4 µm.

4.3.1.2

Forward N 2 Intensity dependence

The observed forward intensity depends strongly on the atom number due to the constructive
interference of the dipole emission in that direction. In Fig. 4.9 we present measurements of theforward intensity Ix versus the transverse intensity Iz at a fixed angle of θ = 2 mRad (indicated by
the arrow in Fig. 4.7).
The intensities in each direction are normalized to the peak atom number used in Fig. 4.7. To
the first-order approximation, the transverse intensity scales linearly with atom number. Hence, theforward intensity of both the blue and red transitions scale approximately with the atom number
squared. The exact scaling for both transitions is reduced from N 2 due to multiple scattering
events (see section 4.4.4).

4.3.1.3

Transverse Polarization Dependent Intensity

Whereas the forward emission has no dependence on the probe beam polarization due tosymmetry arguments, the transverse fluorescence (along ẑ) should be highly sensitive to the probe
polarization and it is even classically forbidden if the probe beam is ẑ-polarized. However, multiple
scattering processes with dipolar interactions can completely modify this picture by redistributing
the atomic population in the three excited |mi states and thus scrambling the polarization of theemitted fluorescence.

63

Figure 4.10: Blue transition intensity ratio for ŷ- and ẑ-polarization in the transverse direction.
The classically forbidden intensity Izpol increases with OD due to multiple scattering events. Solidline is a theory prediction using the coupled dipole model, where the shaded area takes into accountthe uncertainty in the atom number.

Figure 4.11: On resonance intensity for the red transition for ŷ-polarization (red, squares) andẑ-polarization (pink, triangles) in the transverse direction. The classically forbidden intensity Izpol
increases with OD due to multiple scattering events. For OD > 22.5 the intensity decreases forboth polarizations.

64
Measuring the fluorescence directly along ẑ is also technically challenging in our setup. First,
as mentioned earlier our detector is offset by ∼ 10◦ from the vertical direction and this is due tothe fact we cannot block the vertical MOT beam path. The second is that it is only exactly along ẑ
that the fluorescence is classically forbidden but we need to take into account that the detector lensalso has a finite size. However, we still expect to see differences between the forward and transverse
directions due to the non-symmetrical intensity distribution of each radiating dipole.
The prediction of increasing transverse intensity with OD for ẑ-polarized light is shown in
Fig. 4.10 for the blue transition and Fig. 4.11 for the red transition. Figure 4.10 shows the onresonance intensity ratio of the two orthogonal intensities Iypol /Izpol of the blue transition and it
can be seen that this ratio decreases significantly with an increasing OD indicating the rapidly
rising fluorescence with a ẑ-polarized probe as OD increases. For the red transition, Fig. 4.11
shows how the intensity of the classically forbidden polarization (pink, triangles) increases with
OD. For OD > 22.5, the intensity of both polarizations decreases. It is possible that this is aneffect related to the atomic cloud moving away from the detector.

4.3.2
4.3.2.1

Linewidth Broadening
Blue Transition

For the blue transition, we have also investigated the linewidth broadening in both the forward
and transverse direction. The frequency of the blue probe beam is scanned across resonance toextract the fluorescence linewidth of the atoms simultaneously using both forward and transverse
detectors. In this case the razor blade is not present and instead a half-inch mirror at 45◦ is used
to pick off the probe beam and monitor the power.

For this case of the blue probe beam, whichis unaffected by motion, the expected lineshape is a Lorentzian
L (ν) =Γ
1
2
π (ν − ν0 )2 +


Γ 2
2

(4.10)

where Γ is the linewidth and ν0 is the resonance frequency. The fluorescence data at different OD
is then fitted using Eq. 4.10 with a varying amplitude. From this fit we can extract both the center

65

Figure 4.12: Linewidth broadening in the forward direction x̂ under two different atom numbers N(blue, squares) and N/4 (cyan, triangles).

Figure 4.13: Linewidth broadening in the transverse direction ẑ using two orthogonal polarizationsŷ (blue, squares) and ẑ (cyan, triangles) which is classically forbidden.

66
frequency ν0 and the linewidth Γ.
The fluorescence linewidth in the forward direction, which is determined primarily by the
OD of the atomic cloud, is plotted in Fig. 4.12. We show data for two different atom numbers
N = 1.7 (2) × 107 (blue, squares) and N/4 (cyan, triangles). To an excellent approximation the
linewidth data are observed to collapse to the same curve when plotted as a function of OD. At thesame time that we measure the linewidth broadening in the forward direction we can also measure
the linewidth broadening in the transverse direction using a separate detector. As mentioned earlier
the fluorescence scattering in the transverse direction is highly dependent on polarization. In thetransverse direction, ẑ, we can measure the linewidth broadening for both ŷ and ẑ polarization under
the same experimental conditions by interleaving scans with each polarization and the results forthe blue probe are shown in Fig. 4.13. The classically allowed polarization ŷ (blue, squares) showsa similar trend to the forward linewidth broadening, whereas the classically forbidden polarizationẑ gives rise to a much lower broadening for the same OD. For the classically allowed polarization,even though the data are taken under the same conditions as in Fig. 4.12 due to the anisotropic

Figure 4.14: Measured lineshapes of the blue transition for different OD. At low OD (cyan, circles)
the lineshape is Lorentzian. As the OD increases (blue, squares) the lineshape flattens out aroundthe center.

67
size of the atomic cloud the OD in the transverse direction is smaller than in the forward direction(see Section 4.2.3.3).
For the forward direction where we are able to reach higher OD, for the range of 0 < OD < 15
the lineshape is Lorentzian as shown in Fig. 4.14 (cyan, circles) for OD = 1.8. However, for larger
OD = 16 (blue, squares) the lineshape starts to flatten out near the center.

4.3.2.2

Red Transition

For the red transition, the importance of Doppler broadening requires the lineshape data to
be fitted with a Voigt profile which is a convolution between the Lorentzian natural lineshape and
the Gaussian Doppler broadened lineshape. It can be written in terms of the complementary error
function, erfc [x], asV (ν) ∝ e[−i(ν−ν0 )+Γ/2]2
2∆2
D




[−i (ν − ν0 ) + Γ/2]√
erfc
+ C.C.
2∆D(4.11)

where C.C. is the complex conjugate.
√
To fit using the Voigt function we fix the Gaussian linewidth, ∆νD = ∆D 8 ln 2 = 38 kHz
from our measured temperature and the resulting Lorentzian linewidth as a function of OD isshown in Fig. 4.15 for ŷ-polarized probe light and shows a strong increase of the linewidth with
OD. When the red data is taken a large magnetic field is turned on to separate the different |mistates. The reason for this is because the shift of the line resonance for the red transition is larger
than that for the blue transition when compared to the natural linewidth of the transition. By
only probing one state we are therefore able to remove any additional broadening that may occur
due to stray magnetic fields.

4.3.3

Frequency Shift

Whereas the transverse linewidth broadening is similar for the red and blue transition, thisbehavior is in stark contrast to the shift of the transition center frequency. Figure 4.17 shows thelinecenter frequency shift observed for the blue transition (blue, squares) and the red transition

68

Figure 4.15: Red Linewidth Broadening in the transverse (ẑ) direction for ŷ-polarization (red,squares) and ẑ-polarization (pink, triangles).

Figure 4.16: Red transition resonance fluorescence lineshape scans.

69(red, triangles). The blue frequency shift data is taken by extracting the center frequency from thesame Lorentzian lineshapes used to extract the linewidth data.
However, the red data is taken from [77] and measured under similar conditions to thosepresented here. Two counter-propagating probe beams are used and the frequency of the red probe
laser is measured using a frequency comb which was not the case for the red line broadening data
shown in Fig. 4.15. The frequency of this red laser can easily wander around on the 1 kHz level1
which is ∼ 0.1Γ0,red . On the contrary the frequency stability of the blue lasers is < 1 MHz whichis ∼ 0.03Γ0,blue .

Figure 4.17: Comparison of frequency shifts normalized to the corresponding natural linewidth forthe blue (blue, squares) and red (red, triangles) transitions. The blue frequency shift is consistent
with 0-0.004 of Γ0 at an atomic density of 1012 cm−3 . The red data shows more than 0.1Γ0 shift
at densities up to 0.7 × 1012 cm−3

For densities up to 1012 cm−3 the frequency shift of the blue transition is below the frequency
stability of our probe laser and hence consistent with a zero shift. For the red transition the shift
relative to the natural linewidth is more than 0.1Γ0,red at an atom density of 0.7 × 1012 cm−3 . The
density-related frequency shift significantly exceeds the predicted value based on general S-matrix
1This data was taken using an older red master laser which is locked to the Strontium transition using a heatpipe
and not the currently in use version based on a cavity resonance.

70
calculations of s-wave collisions [77]. These frequency shifts are relevant for future 3D optical latticeclocks [6]. Even though the 3D optical lattice clocks can work with one atom per lattice site which
will prevent collisional shifts, the long-range dipolar interactions will then be the most relevant
density dependent frequency shift.

4.4

Theoretical Model - The Coupled Dipole Model
Before we turn to a microscopic model to obtain a full and consistent understanding of

all these related experimental observations, we note that semiclassical models [79] treating theatomic cloud as a continuous medium of an appropriate refractive index can give an intuitive
explanation of the linewidth broadening in the forward direction. Classically, an incoming electricfield is attenuated as it propagates through the medium according to the Beer–Lambert law andthe forward fluorescence intensity is determined by the same mechanism. This simple semiclassicalmodel recovers the linear dependence of the forward width for small OD and predicts a nonlinear
dependence of the linewidth for large OD and a flattening of the line centre. However, we findthat this semiclassical approach cannot provide explanations for most aspects of the experimental
observations.
The Coupled Dipole model2 is a microscopic model that describes a system of N atoms thatare driven by a uniform electromagnetic field. Here, we consider the particular case of four-levelatoms that are each treated as discrete radiating dipoles located at frozen spatial positions, anddriven by a weak incident laser beam, where the atoms are coupled by retarded dipole radiation.
We will consider the simple four state model consisting of three degenerate upper levels and a single
ground state as shown in Fig. 4.1.
The Hamiltonian describing the system can be written in terms of an atomic part Ĥa , a part2

Also referred to as the coherent dipole model

71
that describes the field Ĥf , and one that describes the interaction between the two Ĥaf
ĤCD = Ĥa + Ĥf + Ĥaf
ĤCD = ~ω0Xb̂†j · b̂j +jwhere b̂†j b̂jX~ωλ â†λ âλ + i~

XX
j

λ

 (4.12)gλ · â†λ e−ikλ ·rj − âλ eikλ ·rj b̂j + b̂†j

λare the vector transition operators, or coherences, for the jth atom with transition

frequency ω0 and at position rj , â†λ (âλ ) are the photon creation (annihilation) operators for a
photon with wavevector kλ and frequency ωλ . The vector gλ is given by
 2
1/2
p ωλ
λ
g =
λ
20 V ~

(4.13)

where p is the electric dipole matrix element, λ is the unit polarization vector of the light, 0 is
the vacuum permittivity and V is the quantization volume of the electromagnetic field.

4.4.1

Coherences

In order to derive the equation of motion describing the evolution of the coherences we canfirst consider the equation of motion for some general atomic variable described by the operator Â.
The Heisenberg equation of motion for this operator is then given by
i
i ih
i ih
i
dÂ
ih
ih
=
ĤCD , Â =Ĥa , Â +Ĥf , Â +Ĥaf , Â .
dt
~
~
~
~

(4.14)

The first two terms are relatively simple and give, for the atomic Hamiltonian
i
i
Xh †
ih
b̂j · b̂j , ÂĤa , Â = iω0~(4.15)

jand for the electromagnetic field Hamiltonian we get
i X h
i
ih
Ĥf , Â =
ωλ â†λ âλ , Â = 0~(4.16)

λwhere we have used the fact that an atomic operator Â commutes with the electromagnetic field operators. The equation of motion for the Hamiltonian that couples the atom and the electromagnetic
field is given by
i

h
i
h
i 
XX
ih
Ĥaf , Â = −gλ · eikλ ·rj Â, ŝj âλ − e−ikλ ·rj Â, ŝj â†λ .
~j

λ(4.17)

72
where we define ŝj ≡ b̂j + b̂†j .
In order to simplify Eq. 4.17 further we can follow the method of Lehmberg [44] and James[45] and first consider the equation of motion for âλ . By again using the Heisenberg equation of
motion we end up with the resultXdâλ= −iωλ âλ +gλ · e−ikλ ·rj ŝj .
dt(4.18)

j

The solution to Eq. 4.18 can be shown to be given by−iωλ tâλ (t) = âλ (0) e

+

X

λ

−ikλ ·rj

Z t

g ·e


0
dt0 ŝj t0 e−iωλ (t−t ) .

(4.19)

0j

where to further simplify this solution, we need to make the assumption [80] that we can decompose
the coherence operators intob̂ (t) = B̂e−iωt(4.20)

where B̂ is an unknown operator whose time variation compared with e−iω0 t is slow enough that
we can remove it from the integral in Eq. 4.19. This leads to the solution of Eq 4.19 becomingâλ (t) = âλ (0) e−iωλ t − iXhigλ · e−ikλ ·rj b̂j ζ ∗ (ωλ − ω0 ) + b̂†j ζ ∗ (ωλ + ω0 ) .

(4.21)jwhere ζ ∗ (ω) is the Heitler zeta function [81] defined by−iζ ∗ (ω) ≡ −iP+ πδ (ω)ω(4.22)

with P the Cauchy principal part and δ (ω) the Dirac delta function.
Now we can consider the âλ term in equation 4.17 (ignoring the â†λ term), which gives us
−

XX
j

λ

h
i
h
i
XX
gλ · eikλ ·rj Â, ŝj âλ = −gλ · eikλ ·rj Â, ŝj âλ (0) e−iωλ t
j

+i

λ

XX
j,m

h
ih
i
gλ · e+ikλ ·rjm Â, ŝj b̂m ζ ∗ (ωλ − ω0 ) + b̂†m ζ ∗ (ωλ + ω0 )λ(4.23)

73
with rjm = rj − rm . We will consider the two terms in Eq. 4.23 separately. The first term can be
simplified using the rotating wave approximation to give
−

XX
j

h
i
i +
ip X hgλ · eikλ ·rj Â, ŝj âλ (0) e−iωλ t = +Â, b̂†j Ê 0 (rj,t )~(4.24)

jλ+where Ê 0 (rj , t) represents the positive frequency part of the vacuum field and is defined asX+Ê 0 (rj , t) =
i
λ

~ωλ
20 V

1/2

λ âλ ei(kλ ·rj −ωλ t)(4.25)For the second part of Eq. 4.23 we allow the volume V → ∞ which converts the summation
over λ into the integralXλV→(2πc)3Z∞

ω 2 dω

I
dΩk̂

2
X

(4.26)

ˆ=1

0

The seond integral can be written asV(2πc)3

I
dΩk̂

2
X

 2
eikλ ·rjm gλ =ˆ=1

1 ωp2 4ωf (k0 rjm ) = − 3 f (krjm )3
4π0 πc ~ 3Γπω0(4.27)where3Γ
f (krjm ) = −4h2

1 − (p̂ · r̂jm )i sin (kr

jm )

krjm

h
i
+ 1 − 3 (p̂ · r̂jm )2cos (krjm )(krjm )2−sin (krjm )!#(krjm )3
(4.28)

We can now write the second term in Eq. 4.23 as
+i

XX
j,m

h
ih
i
gλ · e+ikλ ·rjm Â, ŝj b̂m ζ ∗ (ωλ − ω0 ) + b̂†m ζ ∗ (ωλ + ω0 )

λ

=

X 

(4.29)hihi†b̂†j , Â b̂m − iΩ+
b̂
,
Â
b̂
j
m
jm

−iΩ−
jm − f (k0 rjm )

j,m

where
Ω±
jm =

Γ
P
ω03 π

Z∞
dω
0

ω3
f (k0 rjm )ω ± ω0(4.30)

74
A similar method can be used to rewrite the â†λ term in Eq. 4.17 to giveiihi X 
h
ih
†
+
†
b̂
b̂
−
iΩ
b̂
,
Â
,
Â
Ĥaf , Â =
−iΩ−
b̂
−
f
(k
r
)
m
0
jm
j
m
jm
jm
j
~
j,m

+

X 

h
i
h
i

−
†
†
+iΩ+
b̂
Â,
b̂
+
iΩ
b̂
Â,
b̂
−
f
(k
r
)
j
j
0 jm
m
jm
jm m

j,m

= −i

X

(4.31)


i
i h
i X
 h
h
f (k0 rjm ) b̂†j Â, b̂m + b̂†j , Â b̂†m
gjm Â, b̂†j b̂m −
j,m

jm
j6=m

where we have defined
3Γ
gjm =
4

h

− 1 − (p̂ · rjmˆ )2

i cos (k r

0 jm )

k0 rjmh+ 1 − 3 (p̂ · rjmˆ )2icos (k0 rjm )(k0 rjm )2−sin (k0 rjm )!!

(k0 rjm )3(4.32)The Ωjj terms have been removed from the equation and absorbed into the resonance frequency ofthe single atom. This self-interaction term is known as the Lamb shift and cannot be calculated
non-relativistically. The term gjm is the frequency shift due to the interaction between different
dipoles j and m.
This gives us a simplified version of the first term in Eq. 4.17. The second term can be foundin a similar way to givei ip X h
i +
h
i
Xh †
dÂ
−
= iω0
b̂j · b̂ , Â +Â, b̂†j Ê 0 (rj , t) + Ê 0 (rj , t) Â, b̂j
j
dt
~
j

−i

j

X
jm
j6=m

h

i

gjm Â, b̂†j b̂m −

X

 h
i h
i

f (k0 rjm ) b̂†j Â, b̂m + b̂†j , Â b̂†m

(4.33)

j,m

For the four level configuration exhibited by 88 Sr we now label the J = 0 ground state |gi =√|0, 0i and the excited J = 1 states using a Cartesian basis |ez i = |1, 0i, |ex i = (|1, −1i − |1, +1i) / 2√
and |ey i = i (|1, −1i + |1, +1i) / 2 with states labelled |J, mi in the standard angular momentum
basis. In the Cartesian basis we can now write the vector transition operators as b̂j = x̂b̂xj + ŷ b̂yj + ẑ b̂zj

75with b̂αj = |gij heα |. In this basis Eq. 4.37 becomes
dbαj
=
dt




X  αβΓ α i
αβαbj − Ωj δα,κ − i
gjm + ifjm
bβm
i∆ −
2
2
m6=j
β



X αβ
Γ α i
α
= i∆ −
bj − Ωj δα,κ − i
Gjm bβm .
22(4.34)m6=j
βThese two expressions can also be related as they are the Real and Imaginary parts of the dipoledipole interaction term
Gαβ
jm =

i
3Γ eik0 r hδα,β C (rjm ) + r̂αjm r̂βjm D (rjm )4 k0 r

(4.35)

with1
1+
k0 r (k0 r)2(4.36)

3i
3−
k0 r (k0 r)2(4.37)

C (r) = −1 − i

D (r) = 1 +We can solve Eq. 4.34 to find the steady state solutions which have the form
αβ

bαj =

X
Gjm
Ωj δα,κ /2+bβαα∆ + iΓ/2∆ + iΓ/2 m(4.38)

m6=j
β

We can solve this equation perturbatively assuming the interactions are weak using as an expansion
P
α(0)
α(1)
α
parameter G ≡ j6=m,α,β |Gαβ+bj +jm |/(N Γ) << 1 [82, 83] giving for Eq. 4.38 the form bj = bj
α(2)

bj

+ ..., where the zeroth and first order terms are given byΩκ eik·rj δα,κ /2∆α + iΓ/2(4.39)Ωκ δα,κ /2
K j eik·rj
i(∆α + iΓ/2)2 α,β(4.40)

α(0)

bj

α(1)

bj
j
where Kα,κ=αβ k·rjm.
m6=j Gjm e

P

=

=

76
4.4.2

Scattered Fluorescence

The electic field scattered by a dense sample of atoms in the far field can be written as [44, 45]

Ê+ (r̂s , t) ≈

Γ0 X r̂s × [r̂s × b̂j (t0 )]2p|r̂j − r̂s |(4.41)jwhere t0 = t − |r̂j − r̂s | /c. By using the vector identity u × (v × w) = (u · w)v − (u · v)w theintensity is then given byED
I (r̂s , t) = Ê+ (r̂s , t) Ê− (r̂s , t)≈

Γ20 X
[bj · b∗m − (bj · r̂s )(b∗m · r̂s )] e−iks ·(rj −rm )4p2 r2j,m≈(4.42)i
Γ20 X −iks ·rjm X h
α β
α β∗
δ
−
r̂
r̂
e
α,β
s s bj bm
4p2 r2j,mα,βwhere r̂α is the component of the unit vector r̂ along the α = x, y or z direction. The intensity canbe found by splitting the summation over j and m into the two cases where j = m and j 6= m. By
using the zeroth order solution to Eq. 4.38, the scattered intensity is driven only by the external
field and this leads to the intensity
I∝



1
2 −|ks −k0 |2 σ⊥N+Ne∆2 + Γ20 /4(4.43)The overall intensity distribution is Lorentzian with an amplitude dependent on atom number N .
The j = m term of Eq. 4.42 corresponds to the incoherent sum of single atom intensities to givethe first term in Eq. 4.43. The j 6= m term is an interference term and in our case of coherently
driven dipole moments this term is responsible for the build of of intensity ∝ N 2 in Eq. 4.43. The
phase coherence is restricted to a narrow angular region δθ ∼ 1/(k0 σ) = λ/σ in the direction of
the probe beam and the constructive interference is quickly reduced due to the random positionsof the atoms. The wavelength dependence of the angular region is consistent with the data shownin Fig. 4.7 where the red transition shows a larger angular region of forward enhancement.
Including the first order corrections the intensity of the scattered light in the transverse
direction is given byI∝N Ω2(∆ − Γ0 Re[Ḡ])2 + (Γ0 + 2Γ0 Im[Ḡ])2 /4(4.44)

77

Figure 4.18: Three dimensional intensity prediction from the coupled dipole model. The intensity
is peaked in the forward direction due to the constructive interference of the synchronized dipoles.
The intensity in the direction of the polarization is classically forbidden but finite due to multiple
scattering events. The speckled pattern is due to randomly positioned atoms and can be removed
by averaging over multiple atom configurations.

where Ḡ =αα −ik0 ·rjm /(N Γ).
j6=m Gjm e

P

For the forward direction the factor of N is replaced by

N 2 . We can see that the lineshape is still Lorentzian with modifications to the center frequency
and linewidth which are caused by the dipole-dipole interactions. For a relatively dilute cloud withaverage interparticle distance r̄  1/k, the far field interactions dominate and all terms beyond
1/r can be neglected. In this limit analytic expressions for the frequency shift and line broadening
are found to beODΓ04√
3 2πn0Γ0 Re[Ḡ] = −
Γ0
16k03

2Γ0 Im[Ḡ] =(4.45)(4.46)

This first-order approximation provides an intuitive picture about the role of dipolar effects
on the lineshape. We can see that for lower densities the frequency shift scales with density andarises from the real part of the dipole-dipole interactions. We can also see that the linewidth scaleswith OD, whch is seen experimentally (see Section 4.3.2), and arises from the imaginary part ofthe dipole-dipole interactions.

78
4.4.3

Motional Effects

The motional effects in the red transition are taken into account by introducing a random
detuning δν for each atom which can be sampled according to a Gaussian thermal distribution.
Assuming non-interacting and two-level atoms the zeroth order coherence given in Eq. 4.39 is
modified to become
(0)

bj

=

Ωeik·rj(∆ + δνj ) + iΓ0 /2(4.47)

We can now use this to determine the incoherent scattering in the forward direction to be
1
Iincoh = √
2π∆D

Z

2

−

d(δνj )|bj | e

δνj2
2∆2
D

(4.48)and the coherent part to be
1
Icoh =
2π∆2D

Z Z

δν 2

δν 2

− j2 − m2d(δνj )d(δνm )bj b∗m e 2∆D e 2∆D(4.49)

evaluating the integrals in Eq. 4.48 and Eq. 4.49 for the on resonance case gives us
Icoh=
Iincohpπ1228∆ /Γ0
D
erfc
2e

h

√ 1
2 2∆D /Γ0i

2∆D /Γ0

Figure 4.19: Suppression of the forward enhancement peak with motion.

(4.50)

79
This ratio, that is plotted in Fig. 4.19 tells us how the motion will suppress the forward interference
and depends on ∆D /Γ0 . For the blue transition ∆D /Γ0 ∼ 0 and for the red transition ∆D /Γ0 ∼ 2which qualitatively explains the suppression of the forward enhancement for the red transition.

4.4.4

Multiple Scattering Processes

For an atomic cloud with an increasingly large OD, dipolar interactions are stronger andmultiple scattering processes become relevant. The first order perturbative analysis then breaks
down and the full solution of Eq. 4.38 becomes necessary to account for multiple scattering processes. The first signatures arise from the forward fluorescence intensity, where its naive N 2 scaling
is reduced with an increasing atom number as a consequence of multiple scattering processes. The
effect is observed in both the red and blue transitions, and is expected to be more pronounced on
the red transition due to its longer wavelength. However, atomic motion leads to a lower effective
OD, which tends to suppress multiple scattering processes and thus helps to partially recover thecollective enhancement. The theory lines in Fig. 4.7 and Fig. 4.9 represent such quantitative
theory calculations for both transitions, which agree with the experiment.
Meanwhile, for the linewidth broadening observed in the forward direction, it becomes evident
that the scaling of the linewidth versus OD turns nonlinear at large values of OD. The experimental
data falls within the shaded area in Fig. 4.12, which represents the full solution with a 20%
uncertainty in the experimental atom number. Multiple scattering processes are also key to the
explanation of the measured fluorescence along the transverse direction, especially for the classicallyforbidden polarization ẑ. Indeed, for both intensity and linewidth broadening observed in thetransverse direction, under either ŷ or ẑ probe polarization, the full model (shown as shaded areas
in both Fig. 4.10 and Fig. 4.13) reproduces well the experimental results on the blue transition.
Taking into account motional dephasing, the transverse broadening for the red transition is also
well reproduced as shown in Fig. 4.15.

80
4.4.5

Frequency Shifts

From above we have found that the frequency shift arising from the dipolar coupling is
expected to scale with atomic density, which includes both a collective Lamb shift and the LorentzLorenz shift [84, 85]. For our experimental density this frequency shift, normalized to Γ0 , is . 10−3 ,which is consistent with the observed frequency shift for the blue transition. In contrast for the redtransition the measured density shift (normalized to Γ) is significantly larger than what is predictedfrom the current treatment of interacting dipoles; it is also much bigger than the unitarity limit ofs-wave scattering. Qualitatively, we expect that as the atoms move and approach each other, thelong-lived ground-excited state coherence in the red transition can be significantly modified by thecollisional process and open higher partial wave channels. We can thus expect a larger collisional
phase shift. This process can be further complicated by atomic recoil, light forces and Doppler
dephasing [86].

4.5Conclusions
We have shown clear experimental observations of collective emission from a dense sampleof 88 Sr atoms. These observations can be understood by treating the atoms as coherently coupled
dipoles where the key ingredient to understanding our observations is the dipole-dipole interactions
between the atoms. The probing laser beam synchronizes the dipoles such that the constructive
interference of the emitted fluorescence in the direction of the probe beam shows strong constructive
interference. This is seen experimentally as both an enhancement of intensity in the forward
direction and an N 2 scaling of the intensity.
By probing the 1 S0 – 1 P1 blue transition and comparing to the 1 S0 – 3 P1 red transition we areable to clearly see the effects of motion that are present for the red transition. The atomic motion
reduces the observed constructive interference. The linewidth broadening, caused by the imaginary
part of the dipole-dipole interactions, for both transition is similar with no major differences brought
about by the atomic motion. This is in contrast to the large difference in the frequency shift,

81
caused by the real part of the dipole-dipole interactions. For the blue transition the frequency shift
is consistent with the predictions of the coupled dipole model. For the red transition the frequency
shift is significantly larger than predicted by the coupled dipole model although it should be notedthat this measurement was taken under different conditions than the rest of the measurements inthis chapter.

Chapter 5

Many-body interactions in an optical lattice clock

5.1

Introduction
In this chapter we will provide a theoretical framework for studying the interactions of Stron-

tium atoms in a one-dimensional optical lattice clock, based on the work of Martin et al [15] andRey et al [87]. We will consider the case of nuclear spin polarized fermions in a deep lattice which is
the most relevant case for current optical lattice clocks. This model is the basis for the interactions
discussed in Chapter 6 and Chapter 7 which may become important for the operation of future
optical lattice clocks. In each of these chapters the spin model derived here will be generalized tothe cases under study, namely the case of interactions between spin mixtures and the interactions
between atoms when tunneling is allowed.

5.1.1

Chapter Outline

In section 5.2 we will start out with the Hamiltonian that describes the nuclear spin polarized
fermions in a one-dimensional optical lattice clock. We will the re-write this Hamiltonian in termsof a spin model (section 5.3). We will then discuss the interaction parameters in more detail (section
5.4) before discussing the temperature dependence of these parameters (section 5.5).

5.2

Hamiltonian
The simplest case of interactions in our one-dimensional lattice is the case of atoms trapped

within a deep lattice (tunneling, J = 0) and where all atoms are in the same nuclear spin state.

83
We denote the ground clock state (1 S0 ) as g and the excited clock state (3 P0 ) e. The Hamiltonian
describing the system can be written as
Ĥ = Ĥ0 + Ĥint + ĤL
 2

XZ
~
3
†
2
Ĥ0 =
d RΨ̂α (R)∇ + Vext (R) Ψ̂α (R)
2M
α
Z
h
i
1
d3 R Ψ̂†e (R)Ψ̂e (R) − Ψ̂†g (R)Ψ̂g (R)
+
2
Z
h
i
~Ω0
ĤL = −
d3 R Ψ̂†e (R)e−i(ωL t−kL ·r) Ψ̂g (R) + h.c.
2

(5.1)
(5.2)

(5.3)

Where Ψ̂α (R) is a fermionic field operator at position R for atoms with mass M in electronic
state α = {g, e}. These fermionic field operators obey the fermionic anti-commutator relations{Ψ†j (x), Ψk (x0 )} = δ(x − x0 )δjk . The Hamiltonian, ĤL , takes into account the interrogation of theatoms by the clock laser with Rabi frequency Ω0 that has frequency ωL and wavevector kL and isdetuned from atomic resonance by δ = ωL − ω0 .
For the interaction Hamiltonian, Ĥint , we consider only two possible partial wave interactions,s-wave and p-wave, as these dominate at our µK temperatures. Since we are considering spinpolarized fermions, that are in a symmetric nuclear state, we have one s-wave scattering length a−eg
describing collisions between two atoms in the anti-symmetric electronic state √12 (|gei − |egi). The
p-wave interaction can have three different scattering lengths bgg , bee , and b+
eg associated with thesymmetric electronic states |ggi, |eei, and √12 (|gei + |egi) respectively.
For the interaction Hamiltonian, the general expression for the interactions is written as [88]Z
X
1Ĥint =
d3 r1 d3 r2Ψ̂†α (r1 )Ψ̂†β (r2 )v(r1 , r2 )Ψ̂β (r2 )Ψ̂α (r1 )
(5.4)
2
α,β

where v(x, y) can be written asv(x, y) = Vs-wave + Vp-wave(5.5)

where the s-wave and p-wave interaction terms are written as
4πaαβ ~2δ(r)mπ~b3αβ ←−−∂3 →Vp-wave =∇ r δ(r) 3 r3 ∇ rm∂r
Vs-wave =

(5.6)
(5.7)

84
where the first term is for s-wave interactions and the second term is for p-wave interactions, withr = r1 − r2 with rj the position of the jth atom. The arrows above the gradient operators indicate
the direction in which the operator acts and r = |r|. By integrating over the Dirac delta functions,δ(r), we can rewrite Ĥint asĤint =Z
X
1 4π~2 a−eg(1 − δαβ ) d3 RΨ̂†α (R)Ψ̂α (R)Ψ̂†β (R)Ψ̂β (R)
2 m
αβ
Z
h


i
2
1 6π~ X 3+
bαβ d3 R ∇Ψ̂†α (R) Ψ̂†β (R) − Ψ̂†α (R) ∇Ψ̂†β (R)
2 m
αβ

i
 

· Ψ̂β (R) ∇Ψ̂α (R) − ∇Ψ̂β (R) Ψ̂α (R)
h



(5.8)

where the delta function indicates we have explicitly added in that we only have one s-wave channel
for nuclear spin-polarized fermions.
In the case of a deep lattice we can expand the field operators in a non-interacting atomPharmonic oscillator basis Ψ̂(R) = φz0 (z) n ĉαn φnx (x)φny (y), where ĉ†α,n (ĉα,n ) create (annihilate)a fermion in mode n = (nx , ny ) and electronic state α. We assume all atoms are in the ground axial
band of the lattice, φz0 (z), and are thermally populated amongst the radial modes of the lattice φn .
We can write these harmonic oscillator modes explicitly asz2
1
φz0 (z) = p√e− 2azπaz

φn (x) = p

1

(5.9)
2

− x2

2a
R



x
ar



Hn
e
(5.10)√
2n n! πarp
where Hn [x] is the nth Hermite polynomial, aR = ~/(mωR ) is the radial harmonic oscillatorplength with ωR the radial trapping frequency and az = ~/(mωz ) the axial harmonic oscillator
length with ωz the axial trapping frequency.
The interaction terms in Ĥint can be re-written as integrals over hermite polynomials. For
the s-wave interaction term the the x integral can be written as
"Z
#
r
2
Z
d%e−2% Hnx1 [%]Hnx2 [%]Hnx3 [%]Hnx4 [%]mωRpdxφnx1 φnx2 φnx3 φnx4 =~π 2nx1 +nx2 +nx3 +nx4 nx1 !nx2 !nx3 !nx4 !
rmωR≡
snx1 ,nx2 ,nx3 ,nx4~

(5.11)

85
where we have used the substitution % = x/ar and defined the term snx1 ,nx2 ,nx3 ,nx4 . The y integral
gives the same as the x integral and the z integral can be written explicitly asZ

1dx [φz0 ]4 = √

2π

r

mωz
~

(5.12)

So we can now rewrite the s-wave interaction term asZX
1 4π~2 a−eg(1 − δαβ ) d3 RΨ̂†α (R)Ψ̂α (R)Ψ̂†β (R)Ψ̂β (R)
2 m
αβ

=

αβ
n1 ,n2n3 ,n4

=

√ a−
eg √
(1 − δαβ )4 2π
ωR ωz Sn1 ,n2 ,n3 ,n4 ĉ†αn1 ĉ†βn2 ĉβn3 ĉαn4
4
ar

X ~

X ~
αβ
n1 ,n2n3 ,n44(5.13)(1 − δαβ )uSn1 ,n2 ,n3 ,n4 ĉ†αn1 ĉ†βn2 ĉβn3 ĉαn4√ √where we have defined Sn1 ,n2 ,n3 ,n4 ≡ snx1 ,nx2 ,nx3 ,nx4 sny1 ,ny2 ,ny3 ,ny4 and d u ≡ 4 2π ωR ωz a−
eg /ar .
∂∂∂For the p-wave interaction term, the ∇ = x̂ ∂x+ ŷ ∂y+ ẑ ∂zis the sum of derivatives in the

different directions. The z derivative term evaluates to zero and the x derivative integral gives



Z
Z
Z
dφnx1
dφnx2
dφnx3
dφnx4
z 4
dz [φ0 ]dyφny1 φny2 φny3 φny4 dx
φnx2 − φnx1
φnx4 − φnx3
dx
dx
dx
dx
r
1
mωz 1
1
=√
sny1 ,ny2 ,ny3 ,ny4 3 ×
~ ar
ar
2π
i h dH [%]ih dH [%]
R
dHnx2 [%]
dHnx4 [%] 
2
nx 3
n
x1
H
[%]
−
H
[%]
H
[%]
−
H
[%]
d%e−2%
n
n
n
n
x2
x1
x4
x3
d%
d%
d%
d%


p
+n
+n
+n
n
x
x
x
x
2
3
4 nx !nx !nx !nx !
π 2 1
1
2
3
4
r
1
mωz 1
1
sny1 ,ny2 ,ny3 ,ny4 3 pnx1 ,nx2 ,nx3 ,nx4
≡√
~ ar
ar
2π
(5.14)
where we have defined the term pnx1 ,nx2 ,nx3 ,nx4 . The y integral evaluates to the same expression.

86
We can therefore write the p-wave term as
Z
h


i
1 6π~2 X 3
bαβ d3 R ∇Ψ̂†α (R) Ψ̂†β (R) − Ψ̂†α (R) ∇Ψ̂†β (R)
2 m
αβ

i

 

· Ψ̂β (R) ∇Ψ̂α (R) − ∇Ψ̂β (R) Ψ̂α (R)hX ~ √ √b3αβ=
6 2π ωz ωR 3 Pn1 ,n2 ,n3 ,n4 ĉ†αn1 ĉ†βn2 ĉβn3 ĉαn44ar(5.15)

αβn1 ,n2n3 ,n4=X ~αβn1 ,n2n3 ,n44vαβ Pn1 ,n2 ,n3 ,n4 ĉ†αn1 ĉ†βn2 ĉβn3 ĉαn4
where we have defined Pn1 ,n2 ,n3 ,n4 ≡ sny1 ,ny2 ,ny3 ,ny4 pnx1 ,nx2 ,nx3 ,nx4 + pny1 ,ny2 ,ny3 ,ny4 snx1 ,nx2 ,nx3 ,nx4
√ √and vαβ ≡ 6 2π ωz ωR b3αβ /a3r . The full interaction Hamiltonian, Ĥint , can then be written in therotating frame asĤint =X ~αβ
n1 ,n2n3 ,n45.3

4[(1 − δαβ )uSn1 ,n2 ,n3 ,n4 + vαβ Pn1 ,n2 ,n3 ,n4 ] ĉ†αn1 ĉ†βn2 ĉβn3 ĉαn4(5.16)The Spin ModelAt typical operating conditions of νR = 450 Hz, νz = 80 kHz, and temperatures in the µK

regime, the mean interaction energy per particle is much weaker than the energy splitting between
neighboring single-particle vibrational modes. Thus to first-order, only collisions between atoms
that conserve the total single particle energy need to be considered. Such processes conserve thetotal number of particles per mode and in this case the many-body dynamics are mainly governedby the internal degrees of freedom of the atoms, i.e. their electronic and nuclear spin degree of
freedom, and the motional degrees are assumed frozen.
However, in the case of a purely harmonic spectrum of energy levels mode changing collisions
are allowed even under weak interactions due to (i) the linearly spaced energy levels of the harmonic
oscillator and (ii) the separability of the harmonic oscillator potential along the x and y directions.
Condition (i) would allow two particles in modes n = (nx , ny ) and m = (mx , my ) to collide into

87
modes n0 = (nx + k, ny + k 0 ) and m0 = (mx − k, ny − k 0 ) without violating energy conservaton
constraints. Condition (ii) would allow the same two particles to collide and scatter into modes
n0 = (nx , my ) and m0 = (mx , ny ).
In practice, however, the trapping potential does not give a fully harmonic trap due to theGaussian lattice laser profile. This anharmonic, inseparable, spectrum of energy levels then prevents
the processes described in (i) and (ii) and we are left with only two types of interactions that takeplace in the lattice:• Direct: No modes are changed n0 = n = (nx , ny ) m0 = m = (mx , my )• Exchange: Both motional eigenstates are exchanged n0 = m = (mx , my ) m0 = n =(nx , ny )This simplifies our interaction Hamiltonian and allows us to re-write it asĤint =X ~4αβn1 ,n2+[(1 − δαβ )uSn1 ,n2 ,n2 ,n1 + vαβ Pn1 ,n2 ,n2 ,n1 ] ĉ†αn1 ĉ†βn2 ĉβn2 ĉαn1(5.17)X ~αβ
n1 ,n24[(1 − δαβ )uSn1 ,n2 ,n1 ,n2 + vαβ Pn1 ,n2 ,n1 ,n2 ] ĉ†αn1 ĉ†βn2 ĉβn1 ĉαn2This can be re-written using the anticommutation relations of the fermionic operators and using thesymmetrization of the s-wave and p-wave coefficients, Sn1 ,n2 ,n1 ,n2 = Sn1 ,n2 ,n2 ,n1 and Pn1 ,n2 ,n1 ,n2 =−Pn1 ,n2 ,n2 ,n1 to giveĤint =

X ~[(1 − δαβ )uSn1 ,n2 ,n2 ,n1 + vαβ Pn1 ,n2 ,n2 ,n1 ] ĉ†αn1 ĉαn1 ĉ†βn2 ĉβn24αβn1 6=n2(5.18)

X ~−[(1 − δαβ )uSn1 ,n2 ,n1 ,n2 − vαβ Pn1 ,n2 ,n2 ,n1 ] ĉ†αn1 ĉβn1 ĉ†βn2 ĉαn24αβn1 6=n2

In order to be able to write the Hamiltonian as a spin model we use the spin operator
Ŝ =

1X †
ĉα σ αβ ĉβ2(5.19)

αβwhere Ŝ =

Ŝ X , Ŝ Y , Ŝ Z

and σ = (σ X , σ Y , σ Z ), where σ X , σ Y , and σ Z are the Pauli matrices

88
when written in the |gi, |ei basis as0
σX = 10
σY = i1
σZ = 0
1
 = |ei hg| + |gi he|0−i = (|ei hg| − |gi he|)/i00 = |ei he| − |gi hg|−1(5.20)(5.21)(5.22)The different components of the spin-1/2 operator can therefore be written as
N
XŜ X =ŜnXx =jN
X

ŜnYy =

N
X

j

j=1
N
X

Ŝ Z =

jj(5.23)j=1

j=1

Ŝ Y =

NX(ĉ†e,nx ĉg,nxj + ĉ†g,nx ĉe,nxj )/2(ĉ†e,ny ĉg,nyj − ĉ†g,ny ĉe,nyj )/(2i)(5.24)(ĉ†e,nz ĉe,nzj − ĉ†g,nz ĉg,nzj )/2(5.25)

jj

j=1

ŜnZz =

N
X

j

j

j

j=1

j=1

The full spin Hamiltonian 5.1 can now be re-written in the rotating frame, using these spin operaotrs, asĤ = −δN
X

ŜnZj −

j=1

+

j=1

N
X
j6=k

where Inj =NXΩnj ŜnYj +NXξnj nk Ŝnj · Ŝnk + χnj nk ŜnZj ŜnZk

j6=k

 K
Cnj nknj nk
ŜnZj Ink + ŜnZk Inj +Inj Ink
2
4
(5.26)†
α,β ĉα,nj ĉβ,nk is the identity operator. The coefficients of the interaction terms canP

be written asξn+j nk =χ+
nj nk =
Cn+j nk =Kn+j nk =Vnegj nk − Unegj nk
2ee
Vnj nk + Vnggj nk − 2Vnegj nk
2

Vneej nk − Vnggj nk
2
Vneej nk + Vnggj nk + Vnegj nk + Unegj nk
2(5.27)(5.28)(5.29)(5.30)

89
where
Vnαβ
= v αβ Pn1 ,n2 ,n2 ,n1 ≡ v αβ Pn1 ,n2
j nk(5.31)

Unegj nk = uSn1 ,n2 ,n2 ,n1 ≡ uSn1 ,n2(5.32)

capture the temperature dependence of the interactions.

5.3.1

Collective Spin Model

The spin Hamiltonian in Eq. 5.26 can be simplified further as we can replace the interaction parameters by their mode-averaged values due to only a weak dependence of the interaction parameters on the mode. We can also write the when the atoms are initially prepared in
the totally symmetric Dicke manifold [87]. The collective spin operator can then be written asPτŜτ =X,Y,Z = N
j=1 Ŝnj . The atoms will remain in the symmetric Dicke manifold because due to theenergy gap ξnj which suppresses transitions between the Dicke manifolds1 . This is the only termthat has a contribution from s-wave interactions. In general, the s-wave terms will dominate dueto the fact that p-wave collisions are suppressed by the centrifugal barrier.
We can now write our collective spin Hamiltonian as 2Ĥ/~ = −δ Ŝ Z − ΩŜ Y + ξ + Ŝ · Ŝ + χ+ Ŝ Z + C + (N − 1)Ŝ Z

5.4(5.33)

Interaction Parameters
In order to gain more of a physical understanding of the spin model we will here consider inmore detail what some of the interaction coefficients represent.

5.4.1

χ+ - Promoting one atom from |gi to |ei

To understand physically the meaning of the χ+ term in the interaction Hamiltonian it isuseful to consider an example. Let us consider the case where initially there is a total atom number
1

In other work ξ + is denoted as J ⊥90
NT with N atoms in |ei and (NT − N ) atoms in |gi. We now want to consider what is the energychange when we excite one of the atoms from |gi to |ei to give (N +1) atoms in |ei and (NT −1−N )atoms in |gi. This change in energy when promoting one atom from |gi to |ei is given by++∆E = N Vee + (NT − 1 − N )Veg
− (NT − 1 − N )Vgg − N Veg++= N (Vee + Vgg − 2Veg
) + (NT − 1)(Veg− Vgg )

(5.34)+= N (Vee + Vgg − 2Veg
) + ConstantThis can be understood as the atom that was initially in |gi and interacted with the other (NT −1 − N ) atoms in |gi atoms via Vgg interactions. This interaction is now no longer present. The+ interactions and this interaction is also
atom also used to interact with the N atoms in |ei via Veg

no longer present. The extra atom in |ei now has Vee interactions with the N other atoms in |ei+ interactions with the (N − 1 − N ) atoms left in |gi. So χ+ therefore represents the change
and VegTin energy when one atom is excited from |gi to |ei.

5.4.2

ξ + - The exchange interaction

As shown above the ξ + term corresponds to an exchange interaction. It represents that
there is a difference in energy between the triplet and the singlet states. As mentioned above theatoms are initialized in the fully symmetric |J, −Ji state (all atoms in |gi). After a homogeneous
laser excitation the atoms will still remain in the triplet manifold. The coupling between thisS = N/2 amnifold and the S = N/2 − 1 manifold is prevented by an energy gap ξ + Ŝ · Ŝ. The
different manifolds are connected by 2-body ee losses but atoms still remain in the fully symmetricstate. The energy offset between the two manifolds is due to the difference in energy between the|S = 1, m = 1i and |S = 1, m = 0i states we can write the offset as+−ξ + ∝ Veg
− Ueg

(5.35)

91

5.5

Temperature Dependence of Interactions
The thermally averaged two-body interactions are given by
D
E
√ √
a−
2
eg
−
Ueg
= Unegj nk = 4 2π ωR ωz
snj nk
ar
D
E
√ √
b3αβ
Vαβ = Vnαβ=6pnj nk2πωωz R 3 2 snj nk
j nk
ar(5.36)(5.37)For our 2D pancake shaped traps where the trap volume is proportional to the radial temperatureand the temperature dependence of the s- and p- terms is found numerically to be [89]snj nk =0.281
Ter0.498

pnj nk = 0.564Ter0.499(5.38)(5.39)

where Ter is the radial temperature in the effective harmonic oscillator units. We can see that forthe 2D traps the p-wave interactions, Vαβ , are temperature independent and the s-wave interaction− depends on temperature as 1/T
er .
term, Ueg

5.6

Extensions to the Spin Model
This model can be extended for the case of particles in different spin states which is discussed

in section 6.2.1. For the case of the experiment discussed in chapter 7, the only difference between
the experimental conditions discussed in this chapter and chapter 7 is the lattice depth. The
derivation discussed in this chapter assumes that the atoms are trapped in a deep lattice whichmeans we can make the assumption that the axial wavefunction is the ground state of a harmonic
oscillator.

Chapter 6

Spectroscopic Observation of SU(N ) physics

6.1

Introduction
Symmetries play a fundamental role in the laws of nature. A prominent example is the SU(3)symmetry of quantum chromodynamics, which governs the behavior of quarks and gluons. Whengeneralized to large N , SU(N ) is expected to generate exotic many-body behaviors emerging fromthe increased degeneracy and strict conservation laws. Owing to the strong decoupling between
the electronic-orbital and nuclear-spin degrees of freedom [90, 91], alkaline-earth (-like) atoms,
prepared in the two lowest electronic states (clock states with zero electronic angular momenta),
are predicted to exhibit nuclear spin (I) independence for their interatomic collisional parameters.
This property directly leads to a SU(N ≤ 2I + 1) symmetry for the interaction physics [92, 93,
94, 95]. Thanks to this symmetry, in addition to their use as ideal time keepers [21] and quantum
information processors [96, 97, 98, 99], alkaline earth atoms are emerging as a unique platform
for the investigation of high-energy lattice gauge theories [100], for testing orbital models used todescribe transition metal oxides, heavy fermion compounds, and spin liquid phases [101], and forthe observation of exotic topological phases [95, 102]. Progress towards these goals includes the
production of quantum degenerate alkaline-earth gases [103, 104, 105], imaging of individual spincomponents [106], control of interactions [104, 107, 108], and study of many-body spin dynamics[15].
Before this work, only indirect evidence for SU(N ) symmetry exists1 , including inference1

After completion of this work we also became aware of two other works [109, 110]

93
from suppressed nuclear spin-relaxation rates [106], reduced temperatures in a Mott insulator forincreased number of spin states [111], and the changing character of a strongly-interacting onedimensional fermionic system as a function of N [112]. Furthermore, these observations are limited
to the electronic ground state. The corresponding ground-state s-wave scattering parameter, agg ,has been determined from photo-association [113] and rovibrational spectroscopy [114], but theexcited state-related scattering parameters remain unknown.
In this chapter, we report a spectroscopic observation of SU(N ) symmetry and two-orbital
SU(N ) magnetism in an ensemble of fermionic 87 Sr atoms at µK temperatures and confined in
an array of two-dimensional (2D) disc-shaped, state-insensitive optical traps [115]. The trapping
potentials are approximately harmonic, with the axial (z) trapping frequency νz ∼ 80 kHz and theradial (x − y) frequency νR of ∼ 600 Hz; the slight anharmonicity makes the spacings between theenergy levels uneven. Axial and radial degrees of freedom are decoupled during the initial latticeloading and cooling. Under typical temperatures (1 µK< TR < 7µK, Tz ∼ 2µ K), atoms are cooled
to the motional ground state along the z direction. In contrast, the radial modes are thermally
populated. The SU(N ) symmetric spin degree of freedom is encoded in the 10 nuclear spin stateswith quantum number mI (Fig. 6.1(a)), and the pseudo-spin 1/2 orbital degree of freedom inthe two lowest electronic (clock) states (1 S0 and 3 P0 , henceforth |gi and |ei, respectively). Under
typical atomic occupancies (≤ 20 atoms per disc), temperatures and trap volume (∝ TR ), the meaninteraction energy per particle is at least two orders of magnitude smaller than the single-particle
vibrational spacing along any direction. The spectral resolution available with a laser of 1 × 10−16
stability [116] enables us to accurately probe these interactions while addressing individual nuclear
spin levels.

6.1.1

Chapter Outline

In section 6.2 we will start by describing the different interactions that can take place between
either two atoms in the same nuclear spin state or two atoms in different nuclear spin states. We
will extend the theory model of Chapter 5 to include the case of spin mixtures. In section 6.4 we94will discuss the frequency shifts and how they manifest for both spin polarized atomic samples andspin mixtures where we will show the atoms to be SU(N ) symmetric to the 3% level in section 6.5.
From our measurements we will then determine the different scattering parameters 6.6. In section
6.7 we will then discuss the dynamics via measurements of the contrast for both a nuclear spinpolarized sample of atoms and a nuclear spin mixture.

6.2

SU(N ) spin-orbital Hamiltonian in an energy space lattice
Under our operating conditions the atomic interactions are insufficiently energetic to transfer

atoms between the initially populated, slightly anharmonic motional eigenmodes, as discussed in
Chapter 5. To the first order approximation atoms remain frozen in these quantized motional levels
and the quantum dynamics takes place only in the internal degrees of freedom (spin and orbital)[15, 87, 117], in a way analogous to localized atoms in real-space lattice trapping potentials. Thisapproximation greatly simplifies the modeling of our system. Here, the large energy gap between
the interaction energy and the single-particle vibrational spacing, along with the anharmonicity
and non-separability of the optical trapping potential provided by the Gaussian laser beam profile,lead to an energetic suppression of mode-changing collisions. Moreover, the s-wave and p-wave(Fig. 6.1(b)) interactions, which are responsible for the dynamics, provide nonlocal interactions
when viewed within the energy-space lattice as they couple atoms without being overly sensitive tothe thermally populated motional levels. The decoupling between motional and internal degrees of
freedom combined with the sub-Hertz spectral resolution of the stable laser allows us to probe spinlattice models with effective long-range couplings in a non-degenerate Fermi gas, as schematicallyillustrated in Fig. 6.1(c). This system thus paves the way for study of quantum orbital magnetism
beyond the ultra-cold regime.
Spin models with long-range interactions have been implemented in dipolar gases [118] ortrapped ionic systems [119], but our system is further enriched by SU(N ) symmetry and holds
potential for addressing important open questions on many-body dynamics in spin-orbital models
under the co-presence of large degeneracy [101, 120, 121] and long-range interactions. By per-

95

Figure 6.1: Diagram of the interacting spin lattice. (a) Energy levels for the two lowest electronic
states (1 S0 and 3 P0 ) of 87 Sr atoms in a magnetic field, each with ten nuclear spin states, depictedby colors. This color scheme is used throughout this chapter to denote the interrogated state.
(b) Interactions between two fermionic atoms characterized by four s-wave (“a”) and four p-wave(“b”) elastic scattering parameters. The interactions are governed by symmetries in motional
states (bottom labels), nuclear spins (left labels), and electronic orbitals (white arrows). (c) (Leftpanel) Schematics of the interacting electronic orbitals (spin-1/2 arrows) distributed over a latticespanned by motional eigenenergies. In our system, the energy levels are populated according to a
Boltzmann distribution and the energy spacings are slightly anharmonic; the latter is crucial forfreezing atoms in their initially populated motional modes. For simplicity neither the anharmonic
lattice spacing, nor the Boltzmann population of modes are reflected in the figure. The shaded
connections illustrate the long-range nature of the interactions in energy space. In our theoretical
approach, those interactions are calculated using the matrix element overlap of the corresponding
modes. To the first order approximation, p-wave interactions can be treated collectively and canbe assumed to be of all-to-all type. Thus, we can replace the values of the coupling constants with
their thermal averages (27). Colored circles show the possibility of preparing statistical mixtures of
N nuclear spin states. (Right panel) Illustration of a few occupied eigenmodes in our optical trap
formed by a Gaussian beam.

96
forming Ramsey spectroscopy with various nuclear spin mixtures, we determine the nuclear spin
independence of the s-wave and p-wave interactions. Furthermore, we probe the non-equilibrium
dynamics of the orbital coherence, and the results are well reproduced by a two-orbital SU(N ) spinlattice model in quantized motional eigenenergy space.
Interactions between two 87 Sr atoms are governed by Fermi statistics with an overall wavefunction antisymmetrization under exchange in the motional, electronic, and nuclear spin degrees
of freedom (Fig. 6.1(b)). Consider a pair of interacting atoms (j and k) occupying two of thequantized eigenmodes of the trapping potential, nj and nk . If the atoms are in a nuclear spinsymmetric state they experience s-wave interactions only if their electronic state is anti-symmetric:√(|egi − |gei)/ 2. We denote the elastic scattering length characterizing those collisions as a−eg .
They can collide via p-wave interactions in three possible electronic symmetric configurations√{|ggi , |eei , (|egi + |gei)/ 2}, corresponding to the p-wave elastic scattering lengths bgg , bee , b+eg ,respectively. In contrast, if the two atoms are in an anti-symmetric nuclear spin configuration
they experience s-wave collisions under these three electronic symmetric configurations, with thecorresponding scattering lengths agg , aee , a+eg , respectively. Similarly, p-wave interactions occur in√(|egi − |gei)/ 2, corresponding to the scattering length b−eg . These eight parameters characterize
elastic collisions at ultralow temperatures, and SU(N ) symmetry predicts them to be independentof the nuclear spin configuration.

6.2.1

Expansion of the Spin Hamiltonian - Spin Mixtures

To understand the interactions of spin mixtures we can expand upon the model discussed inchapter 5 to now include the nuclear spin. Similar to the spin operators in Eq. 5.19 we introduce
the spin-orbital operators
Tj =

1 X
ĉαmnj σ αβ ĉβmnj
2
α,βm

(6.1)

97
and the nuclear-spin permutation operators
Snm (j) =Xĉ†αnnj ĉαmnj

(6.2)

α=e,g

where ĉ†αmnj (ĉαmnj ) creates (annihilates) a fermion in the mode nj , in electronic state α ∈ e, g
and with a nuclear spin m = 1, 2..., N ≤ 2I + 1. The nuclear-spin permutation operators satisfy
the SU(N) algebra [Snm (j), Spq (k)] = δj,k (δm,p Snq (j) − δn,q Spm (j)) and generate SU(N) rotations ofP
nuclear spins. We can now expand the field operators Ψ̂†αm (R) = φz0 (z) n ĉ†αmnj φnX φnY .
The Hamiltonian that governs the interactions, between atoms j and k, can be written as:
Ĥj,k = P̂ + Ĥ + + P̂ − Ĥ −±±Z Z
Tj · Tk + χ±Ĥ ± = ξj,kj,k T̂j T̂k + Cj,k(6.3)
T̂jZ + T̂kZ
2!
±I
+ Kj,k

(6.4)

P
m
n
[I± N
n,m=1 Ŝn (j)Ŝm (k)]±Here, I is the identity matrix, P̂ =are nuclear spin projector operators into2

the symmetric triplets (+) and anti-symmetric singlet (−) nuclear spin states, respectively. The
spin triplet projection operator, P̂ + , has an eigenvalue 1 when applied to a nuclear spin triplet stateand 0 when applied to a nuclear spin singlet state. Similarly the nuclear spin singlet projection
operator, P̂ − , has an eigenvalue 1 when applied to a nuclear spin singlet state and 0 when applied
to a nuclear spin triplet one. Equation 6.3 therefore states that if the nuclear spin of the atoms
is in (+) or (−), then they interact according to Ĥ + or Ĥ − , respectively. The coupling constants±±+−
ξj±, k , χ±j,k , Cj,k , Kj,k depend on the scattering parameters, aη and bη , η ∈ {ee, gg, eg , eg }, and the

wavefunction overlap of the j and k-atoms radial vibrational modes in the 2D traps.
The total spin-orbital interaction Hamiltonian is the sum over all pairs of atoms, Ĥ SO =P
1/2 j6=k Ĥj,k , and commutes with all the SU(N ) generators, Snm (j). For the case of spin-polarized
fermions the Hamiltonian reduces to 5.26. The spin-orbital Hamiltonian, Ĥ SO , is thus invariant
under transformations from the SU(N ) group (i.e., SU(N ) symmetric). This implies that the
number of atoms in each of the nuclear spin sublevels is conserved. The N is chosen by the initialstate preparation of the nuclear spin distribution and can vary from 1 to 2I + 1 = 10 in 87 Sr(I = 9/2). In this experiment we control N via optical pumping through another electronic state.

98
In addition to elastic interactions, 87 Sr atoms exhibit inelastic collisions. Among those however, only the e − e ones have been observed to give rise to measureable losses [122]; we denotethese two inelastic scattering lengths as γee and βee for s-wave and p-wave, respectively. We set
other inelastic parameters to zero based on their negligible contributions in measurements [122].

6.3

Ramsey Spectroscopy
To understand the density shift and contrast measurements undertaken in this chapter wemust first look at Ramsey spectroscopy which is used to measure both of these observables. Ramsey
spectroscopy can be understood in a Bloch sphere picture (see Fig. 6.2) where we consider the twopoles of the Bloch sphere to be our two spin states |gi and |ei. We can first consider Ramsey
spectroscopy more generally for a single group of atoms, that are not interacting with each other.
The atoms are initially prepared in |gi and an initial clock pulse, with detuning δ, rotates the Blochvector around the Ŷ axis of the Bloch sphere and creates a superposition of the two clock states.
Here let us assume that an equal superposition of the two states is created which is to say the
initial pulse has an area of θ1 = π/2. In the rotating frame of the laser, the atom will then evolvearound the Bloch sphere with a rate proportional to the detuning between the atom and the laser.

Figure 6.2: Ramsey Spectroscopy on the Bloch Sphere. An initial π/2 pulse creates a superpositionof |gi and |ei. During the dark time τ the atom evolves around the Bloch sphere at a rate proportional to the detuning between the laser and the atom resonance. A second π/2 pulse projects theatoms into the population of |gi and |ei99
A second laser pulse after a time τ , with an area θ2 = π/2, also around Ŷ then rotates the
Bloch vector to have a projection along the Ẑ which can be measured as a population difference
between the two clock states. By repeating this experiment many times and varying the phasedifference between the two pulses the projection of the atoms will show an oscillatory behavior
where the amplitude of the oscillations is known as the fringe contrast and is the projection of thelength of the Bloch sphere in the X̂ − Ŷ plane. When interactions are introduced, during the darktime τ the atoms will accumulate a phase shift around the Bloch sphere with respect to the case ofa non-interacting sample. It is this phase shift that we refer to as a density dependent frequency
shift

6.4

Density-dependent frequency shift for nuclear spin mixtures
We first test SU(N ) symmetry in a two-orbital system by measuring the density-dependent

frequency shift of the clock transition under various nuclear spin population distributions. We use
a Ramsey sequence to measure interactions [15] under an external magnetic field that produces
Zeeman splittings much larger than the interaction energy. The sequence starts with all atoms in|gi. Only atoms in a particular nuclear spin state are coherently excited and interrogated, whereas
atoms in other states (spectators) remain in |gi. We denote the number of interrogated atoms by
Nitot , the number of spectator atoms by NStot , and define a population ratio f = NStot /Nitot and theinterrogated fraction xi = Nitot /(Nitot + NStot ). We control orbital excitation fraction, pe , by varying
the initial pulse area, θ1 , where 0 < θ1 < π. After a free evolution time, τ = 80 ms, the second
pulse of area θ2 = π/2 is applied for subsequent readout of the electronic orbital distributions.
The resonance frequency shift is recorded as a function of the atomic number in the trap, which
can be varied in a controlled manner. We operate with highly homogeneous atom-laser couplingsuch that the orbital excitation is the same for all interrogated atoms. Consequently, in a fully
spin-polarized sample, the s-wave interactions are suppressed and the p-wave interactions dominate
the free evolution dynamics [15]. The s-wave interactions are allowed only when spectator atoms
populate other nuclear spin states. The existence of SU(N ) symmetry in the scattering parameters100
results in an independence of the density shift with respect to both the interrogated nuclear spinstate and the distribution of spectator atoms among the various spin components. Moreover, inthe presence of a large magnetic field (which energetically suppresses excitation of the spectator
atoms), the density shift should be only a function of Nitot , pe , and NStot .
In Fig. 6.3(a), we compare the fully spin-polarized case (mI = +9/2) against three otherscenarios with different spin mixtures under TR = 6 − 7 µK. The observed density shifts as a linear
function of pe , when scaled to the same number of interrogated atoms (Nitot = 4000), show three
features: (I) the linear slope, l, depends only on Nitot , (II) the offset with respect to the polarizedcase increases linearly with f , and (III) both l and the offset are independent of how the atoms
are distributed in the nuclear spin levels. The last point is verified, for example, by measuring thesame shifts when interrogating 29% of the total population in either +9/2 or +7/2.

6.5

Temperature-dependent SU(N )-symmetric interactions
To determine the temperature dependence for the density shift and for additional confirmationof the observed nuclear spin independence, we interrogate other nuclear spin states, −9/2 or −3/2,
at a lower TR ∼ 2 µK, when the distribution across all spin states is nearly even (Fig. 6.3(b)). The
measured density shifts scaled to Nitot = 4000 are again similar to each other, providing further
direct experimental evidence for SU(N = 10) symmetry. At this lower TR , the slope still depends
only on Nitot , but there is a smaller offset of the density shift relative to the polarized case whenxi varies. To quantify the TR dependence, we plot together all measured ratios, l/l0 , where l0 is
the linear slope for the polarized case. We see that (IV) the ratios collapse into a single value
independently of f and TR for fixed Nitot , yielding l/l0 = 1.00 ± 0.03 (Fig. 6.4(a)). This result
agrees well with the SU(N )-predicted ratio of unity and verifies this symmetry to the 3% level.
We emphasize that the test of SU(N ) symmetry (at the 3% level) is based directly on themeasured interactions that are independent of nuclear spin configurations, and it does not require
accurate knowledge of some common-mode system calibrations. For quantum simulations of SU(N )physics, it is important to precisely test this symmetry to a level much below all relevant energy

101

Figure 6.3: (a) and (b) (Left) Measured density shifts (symbols) and for different nuclear spin
configurations at (a) TR = 6 − 7 µK and (b) ∼ 2 µK. (Right) Illustration of the interrogated states(black arrows) and population distributions among various nuclear spin states. For consistency, theshifts are scaled for Nitot = 4000. The solid and dotted lines show theory calculations for the corre−
sponding xi and TR as indicated in the plots, with b−eg and aeg . The gray band in (b) corresponds
to p∗e0 , the excitation fraction for zero density shift in a polarized sample. The spectator atoms
generate a temperature-dependent density shift, which is independent of pe of the interrogatedatoms and thus manifests as a net offset from the purely polarized density shift.

102

Figure 6.4: (a) Ratio of the slope of the frequency shift between the spin mixed and polarized
samples. The dotted lines represent the standard error. (b) The difference in the zero-shift excitation fraction between the spin-mixed and polarized samples. The solid and dashed lines are theory−
fits used to determine b−eg and consequently aeg by the analytic relations between s- and p-wave
scattering parameters (see Supplementary Materials). In (a) and (b), two values of TR are used:
2.3(2) µK (open symbols) and 6.5(4) µK (filled symbols). In addition to conditions used for Fig.
6.3, other spin configurations are studied: open up triangles (xi = 49%), open diamond (41%),
open right triangles (46%), open and filled hexagons (26%), open and filled left triangles (24%),
filled pentagons (29%), and filled stars (12%).

103
scales. Although our measurement uncertainty (3%) has not reached the ultimate theoreticalprediction (0.1%) resulting from a small admixture of the 3 P0 state with higher lying P stateswith finite electronic angular momenta [90, 91, 92], it is already sufficient for realizing a SU(N )symmetric, unity-filling spin lattice system. Further reduction in our experimental uncertainty canbe achieved by enhancing the measurement precision with improved laser stability. We observethat l decreases only by 10% when TR is raised from 2 µK to 6 µK, verifying its insensitivity to TR .
We also determine the excitation fraction where the shift is zero for a spin mixture, p∗e , andcompare it to that of a polarized sample, p∗e0 (gray band, Fig. 6.3(b)), for various interrogated spinstates (colors in Fig. 6.4(b)). The difference shows the following features: (V) it collapses onto asingle line (for a given TR of either 2.3 or 6.5 µK) as a function of f , which provides further evidence
for the spin-independence of the interactions; (VI) at NStot = 0 (fully polarized), the two lines crosseach other at the origin, as expected from the TR -insensitivity of the p-wave interactions. The
proportionality constant of (p∗e0 − p∗e ) to f is finite for 6.5 µK (lower line), and decreases to almost
zero for TR ∼ 2.3 µK (upper line). This near zero proportionality constant for TR ∼ 2.3 µK reflects
an accidental cancellation of the spectators s- and p-wave interaction effects at this temperature.

6.6

Experimental-theory agreement and determination of the scatteringparameters
In the presence of a large external magnetic field that produces differential Zeeman split-tings much larger than the interaction energy, those terms in the Hamiltonian that exchange thepopulation between the occupied spin-orbital levels are energetically suppressed and the populations of different spin-orbital levels are conserved. Hence, the Hamiltonian is dominated by
Ising-type interactions that preserve the spin-orbital population. In this regime the many-bodydynamics for a single trap with N atoms can be captured under a collective approximation that
replaces the coupling constants with their corresponding thermal averages, On±j ,nk → O± [87]. For
the experimentally relevant case where only Ni atoms in spin mI are interrogated and where
NS atoms in the other spin components remain in |gi, the effective many-body Hamiltonian

104
Table 6.1: s- and p-wave scattering lengths in units of the Bohr radius
Channels-wavep-wavegg

96.2(1)74.6(4)

Determination Methods-wavep-waveeg+

169(8)-169(23)eg−

68(22)-42+103−22ee (elastic)

176(11)-119(18)ee (inelastic)

γee = 46+19−32

βee = 121(13)s-wavep-waves-wavep-waves-wavep-waveBothTwophoton-associative
spectroscopy(23)androvibrational
spectroscopy(24)Analytic relation between s-wave andp-wave parameters (34)
Analytic relation (34)
Density shift in a polarized sample (20)Density shift in a spin mixture at different temperatures (this work)
Analytic relation
Analytic relation (34)
Density shift in a polarized sample (20)Two-body loss measurement (33) andanalytic relation (34)during τ simplifies substantially. It consists of two parts, Ĥ SO = Ĥi + ĤS . The first part,Ĥi = χ+ (T̂ Z )2 +C + T̂ Z Ni , describes the p-wave interactions between the interrogated atoms [15, 87],
P
α=X,Y,ZmI
where T̂ α=X,Y,Z = N
are collective orbital operators acting on the Ni interroj ŜmI (j)T̂jgated atoms. The density shift induced by these interactions, ∆ν i = Ni (C + − cos (θ1 )χ+ ), with
χ+ =3b3ee +b3gg −2(b+eg )hP iTR
2

and C + =b3ee −b3gg
hP iTR , depends linearly on the number of excited2

atoms Ni pe . Here hP iTR corresponds to the thermal average of the p-wave mode overlap coefficients. Assuming a Boltzmann distribution of initially populated radial motional modes, we havehP iTR ∝ (TR )0 (insensitive to TR ) [15]. For a spin polarized sample, the observed density shifts arewell reproduced by theory (solid black lines in Figs. 6.3(b) and (c)) based on the same p-wave parameters as determined in [15]. The second part, ĤS = NS ΛT̂ Z , describes the interactions between
the interrogated and spectator atoms with both p- and s-wave contributions. The related density
+

−

+

−

+

−

shift is ∆ν S = ΛNS , with Λ = C +C −ξ 2−ξ −χ −χ =+−−(b )3 +(beg )3 −2b3gg
a+
eg +aeg −2agg
hSiTR + eg
hP iTR .
4
4

The s-wave thermal average, hSiTR , decreases with TR as hSiTR ∝ (TR )−1 .
This model fully reproduces the experimental observations as summarized in points (I-VI)
stated in section 6.4 and 6.5, and shown in Fig. 6.3 anf Fig. 6.4. To quantitatively compare withthe experiment, we perform a Poissonian average of the atom number across the array of 2D traps

105
and use the average excitation fraction to account for the two-body e − e losses [15, 122] during
τ . The capability of the SU(N ) spin lattice model to reproduce the experimental observations alsoenables us to determine the remaining s- and p-wave scattering parameters. For each of the fourchannels, η ∈ {ee, gg, eg + , eg − }, the s-wave and p-wave parameters relate to each other throughthe characteristic length, āη , of the van der Waals potential [123] given by2πāη =Γ(1/4)2



2µC6
~2

1/4
.

(6.5)

The C6 coefficients for the η = gg, ee and eg channels were found to be 3107(30) a.u., 5360(200)a.u., and 3880(80) a. u., respectively, where a.u.= Eh a60 , with Eh the Hartree energy and a0 being
the Bohr radius. A multichannel defect theory predicts that, for a single van der Waals potential,
η, the complex scattering lengths for the s-wave interactions, Aη = aη − iγη , and the complex
scattering volumes for the p-wave interactions, Bη3 = b3η − iβη3 , are related by the van der Waals
length by the relation
Aη
=1+
āη



Bη
ā

3 "

Bη
ā

#−1

3

+ 2.128.

(6.6)

Thus, after we determine āη using the available van der Waals C6 coefficients, only four elastic
scattering parameters remain independent. Among those,agg , bee , b+eg (and thus their respective por s-wave counterparts) are known [15], leaving only one unknown parameter associated with theeg- channel. Using the theoretical predictions and a single parameter to fit the data in Fig. 6.4(b),−we extract a−eg and beg . Table 6.1 lists all the scattering parameters determined from the prior andcurrent measurements.
Recently [124] the s-wave scattering lengths eg + and eg − have been measured in a 3D opticallattice clock. In the eg + channel this was measured to be (161 ± 1.1)a0 and in the eg − channel thiswas measured to be (69.2 ± 1.0)a0 , where a0 is the Bohr radius. The excellent agreement with thevalues we report here (in Table 6.1) shows us the validity of the analytic relations that were used
to determine some of the scattering parameters in this work.

106

6.7

Coherent dynamic spectroscopy
We perform coherent dynamic spectroscopy to explore the development of many-body corre-

lations as a key effect of Ising orbital magnetism, in the combined orbital and nuclear spin degrees
of freedom, generated by the p-wave and s-wave interactions. This allows us to further validate theSU(N ) spin-orbital model, Eq. 6.3, as a description for our system. The many-body correlations
that build up during the free evolution manifest as a decay of the e − g orbital coherence in theqtot
X i2 + hT̂ Y i2 that we measure as a function ofform of Ramsey fringe contrast C(τ ) = 2/Ni
hT̂tottot
X,Yτ . Here T̂tot
is the sum of T̂ X,Y over the 2D traps. We extract a normalized C by comparing

the high-atom-number raw contrast against that of the low-atom-number. This normalization removes single-particle decoherence effects. The decay of C during the free-evolution period has beenshown to be a particularly suitable observable for characterizing the role of interactions during the
dynamics [125]. For example, contrast measurements in an array of polar molecules pinned in a
3D lattice provided clear signatures of dipolar interactions and their description in terms of a spin
exchange model [118, 126].
In the presence of a large magnetic field, the decay of C has two sources. The first arises
from within the interrogated atoms: p-wave elastic interactions, two-body e − e losses, higher-orderinteraction-induced mode-changing processes, as well as dephasing induced by the distribution of
atoms across traps. All these p-wave effects are accounted for in our work using the same p-wave
parameters that were determined first in Ref. [15] and then reconfirmed with our density shiftmeasurements in this work (Fig. 6.3). The second source comes from spectators, which act on theinterrogated atoms at a given site as an inhomogeneous and density-dependent effective magnetic
field along z, with both s- and p-wave contributions. The effective magnetic field is static if theatoms are frozen in their motional states, but can vary with time in the presence of higher-ordermode-changing processes. The p-wave interaction plays a dominant role at high TR = 5 − 6 µ K,
while the s-wave interaction, which has a stronger dependence on the mode distribution, becomes
important at lower TR . We focus first on the nuclear spin polarized case (Fig. 6.5) to benchmark

107
our model, and then use various population distributions among nuclear spin states to investigate
the interplay between orbital and spin degrees of freedom (Fig. 6.6).

6.7.1

Two-orbital dynamics in spin-polarized atoms

To understand in detail the orbital dynamics induced by p-wave interactions, we first study
a nuclear spin polarized sample (xi = 100%) for TR = 5 − 6 µK and θ1 = π/4,(Fig. 6.5(b)).
To separate the effects of dephasing and many-body correlation in the contrast decay, we apply
a π echo pulse in the middle of the Ramsey sequence (Fig. 6.5(a), lower panel). The π echopulse modifies the contrast decay in a θ1 -dependent way, because of the enhanced e − e loss after
the echo pulse (note the number normalization in C) for θ1 < π/2 as well as the fact that thep-wave contribution to contrast decay contains both θ1 -independent and -dependent terms. The
θ1 -independent contribution is generated by the term C + T̂ Z Ni . This term is responsible for thedensity-dependent dephasing between atoms distributed in different 2D traps. The echo pulse
removes it, as well as other dephasing effects of technical origin. The θ1 -dependent contribution is
generated by the term χ+ (T̂ Z )2 in the Hamiltonian, and can lead to many-body orbital correlations
that are not removable by echo.
For θ1 = π/4, the Ramsey contrast decays more slowly with an echo pulse (6.5(b)). This
positive echo effect can be attributed to the suppressed dephasing from inhomogeneous atomic
densities across different 2D traps (θ1 -independent contribution) and to the faster number loss
with echo.
In a polarized sample where p-wave interactions dominate, the contrast decay is expected to
be insensitive to TR . This is confirmed with measurements performed at TR = 2.6 µK (Fig. 6.5(c))
that show similar decay behaviors to those at 5.4 µK (Fig. 6.5(b)). In addition, we find an excellent
agreement between the data and the theoretical model that uses temperature-insensitive p-wave
parameters. Figure 6.5(d) plots the ratio of contrasts with and without echo for different pulse
areas and illustrates both the positive echo effect in suppressing contrast decay for θ1 = π/4, as
well as the negative effect for θ1 = 3π/4 when the echo enhances contrast decay. The negative echo

108

Figure 6.5: Evolution of orbital coherence in nuclear spin-polarized samples. (a) (Upper panel)
Ramsey sequence with varying θ1 and τ ; (Lower panel) sequence with an echo (π) pulse. The groupof circles illustrates the orbital configurations for interrogated atoms (black circles). ((b) and (c))
Normalized Ramsey contrasts for θ1 = π/4, xi = 100%, under two different radial temperatures,
TR = 5.4 µK and 2.6 µK, respectively. The contrast is normalized by comparing the high-atomnumber raw Ramsey fringe contrast, C (defined in the main text), against the low-atom-number
raw contrast. The high total atom numbers (measured at a very short free evolution time, τ = 20ms) are in the range of 2200 ∼ 3100 for Figs. 6.5 and 6.6 (day-to-day variation over four months).
However, for each specific case the data without and with echo were taken on the same day andtheir atom numbers are matched to within 3 ∼ 7%. These atom numbers are recorded and serve
as inputs to our theory calculation of Ramsey contrast decay. The filled symbols are for echomeasurements and the empty symbols without echo. The solid and dashed lines show theory
calculations with echo and without echo, respectively, using a two-orbital model with independently
determined parameters (based on measurements shown in Fig. 6.4 and previous studies, see Table
6.1). Under the conditions of ((b) and (c)), the dominant source for contrast decay arises fromp-wave interactions between the interrogated atoms. (d) Effects of echo, characterized by the ratioof contrast with echo to that without echo, for 1 = π/4 (anti-diagonal ellipse and solid line), π/2(horizontal ellipse and short dashed line), and 3π/4 (diagonal ellipse and short dotted line), underxi = 100%.qError bars represent 1σ standard error multiplied by the square root of the reduced
chi-squared

χ2reduced .

109effect can be attributed to both the development of many-body orbital correlations for θ1 = 3π/4[15] and the reduced e − e loss after the echo. All measurements are well reproduced by our spinlattice model.

6.7.2

Spin-orbital SU(N ) dynamics in spin mixtures

To investigate the interplay between orbital and spin degrees of freedom, we perform similar
spectroscopic measurements in spin-mixed samples (Fig. 6.6(a)). We study the spin-mixed cases
under TR = 5 − 6 µK, with θ1 = π/4 and the interrogated fraction xi = 14% and 56%,(Figs.
6.6(b) and (c), respectively). Here, the data show similar positive effect of an echo pulse in thepresence of spectator atoms. Because p-wave interactions between interrogated atoms are reduced
as the interrogated fraction decreases, the overall contrast decay becomes slower. Based on thedetermined scattering parameters, our model predicts that spectator atoms cause almost negligible
decoherence effects at this high TR = 5 − 6 µK.
When we decrease TR to ∼ 2 µK, the rise of the s-wave contribution causes significant decoherence effects coming from the spectator atoms. Figure 6.6(d) illustrates the influence of spectators
for the xi = 14% minority case where contrast decay is clearly faster than in Fig. 6.6(b). The inclusion of off-resonant mode-changing collisions as higher order corrections is required for temperatures
below 5 µK to accurately reproduce the experimental observations. These mode-changing collisions
can be visualized as relocating pairs of atoms in the energy-space lattice shown in Fig. 6.1(c),analogous to interaction-induced tunneling processes in a real space lattice. The echo pulse suppresses the part of contrast decay arising from mode-preserving collisions between spectators andinterrogated atoms, but it cannot reverse the decay due to mode-changing processes. In Fig. 6.6(d),the measured contrast decay with echo enables us to determine a single parameter characterizing
the mode-changing processes.
For a further and independent test of our model, we explore another case with xi = 56% andTR ∼ 2 µK, so that both the interrogated atoms and spectator atoms have important contributions
to the contrast decay. The data are well described by the same theory model (Fig. 6.6(e) and Fig.

110

Figure 6.6: Evolution of orbital coherence in nuclear spin-mixed samples. (a) (Left panel) Ramsey
sequence with varying θ1 and τ ; (Right panel) sequence with an echo (π) pulse. The group of
circles illustrates the orbital configurations for interrogated atoms (black circles) and spectator
atoms (colored circles). ((b) and (d)) Normalized Ramsey contrast for θ1 = π/4, xi = (14 ± 2)%,
under TR = 5.4 µK and 2.1 µK, respectively: measurements with echo (filled symbols) and withoutecho (empty symbols). ((c) and (e)) Normalized contrast for θ1 = π/4, xi = (56 ± 6)%, underTR = 5.0 µK and 1.6 µK, respectively. The solid and dashed lines show theory calculations withecho and without echo, respectively, using a two-orbital model with independently determinedparameters (based on measurements shown in Fig. 6.4 and previous studies, see Table 6.1). In thismodel, the spectators act as an effective inhomogeneous magnetic field causing dephasing to theinterrogated atoms. Without mode changing collisions the dephasing is static and removable byecho; with mode changing collisions it is assumed and confirmed experimentally that echo does notremove the time-dependent dephasing. The theory uses a single fitting parameter forq the modechanging processes for all plots. Error bars represent 1σ standard error multiplied byχ2reduced .

111S1) without any variation of the pre-determined parameters, demonstrating a firm understandingof the system dynamics.

6.8

Conclusions
The experimental exploration of exotic SU(N ) physics is just starting. The unique capabilityof precision laser spectroscopy has so far allowed us to explore Ising orbital magnetism at relatively
high temperatures. We expect to explore the full Hamiltonian including the exchange interactions
by controlling the atomic density, temperature and the magnetic field to engineer various spin-spinand spin-orbital dynamics. This will allow us to push the frontier of emergent many-body quantum
physics at increasingly high temperatures, as well as the study of time-resolved dynamics in the
SU(N ) Kondo lattice and Kugel-Khomskii models [92, 127, 128] in the quantum gas regime.

Chapter 7

Spin-Orbit Coupling with Interactions

7.1

Introduction
Spin-orbit coupling (SOC) links a particle’s velocity to its quantum mechanical spin andis essential for numerous complex condensed matter phenomena including topological insulatorsand Majorana fermions. Engineering SOC in cold atom systems therefore allows us to study suchmaterials within the clean environment associated with these systems.
In this work we experimentally explore SOC that occurs naturally within a one-dimensional
optical lattice clock when tunneling is allowed. Previous studies of SOC and topological states withalkali atoms has been highly successful for the case of non-interacting systems [129, 130, 131, 132,133]. However, the detrimental effects of spontaneous emission and heating [134, 135] have limited
the study of SOC with interactions in alkali atoms to initial studies in a bulk gas [136, 137], in a
lattice modulated BEC [138, 139], and with two particles in a lattice[140]. Encoding the effective
spin degree of freedom in the long-lived electronic clock states significantly reduces the detrimental
effects of dissipation. We can take advantage of this feature to study the interplay between SOC
and the strong collective interactions in an optical lattice clock (OLC). Some of the physics we
explore is described in a theory proposal [135], although experimental work naturally follows its
own lead.
In our one-dimensional OLC many-body effects arise from the cooperation and competition
between p-wave and s-wave interactions (as discussed in chapters 5 and 6), along with single-particle
SOC dynamics. The spin-motion coupling we engineer in the OLC primarily affects how spins

113
interact with each other, without any thermalization effects in the lattice. This unique condition
sets up an effective spin system that provides a simpler view of the complex interplay between SOC
and many-body interactions. Meanwhile, it grants us immediate access to quantum magnetism atµK motional temperatures.
The many-body dynamics are described by a collective XXZ spin model [15, 141], whichcontains both exchange (s- and p-wave) and Ising (p-wave) terms. The dynamics of collective
XXZ models have largely been studied theoretically in condensed matter physics, for example inthe context of superconductivity through the Anderson pseudospin mapping [142], which identifies
Cooper pairs and holes as the two components of an effective pseudospin. Only limited experimental
studies have been conducted so far, and they have been restricted mainly to weak quenches [143].
The ultra-narrow clock transition in our OLC enables the preparation, control, and spectroscopic
resolution of the dynamics in a broad parameter space, including quenches over a large dynamic
range.
SOC with strong interactions between a pair of atoms has been realized in a lattice [140].
Here, we instead use a large atom number, N , to tune the strength of the interactions to enter astrong, collective interacting regime well beyond single-particle SOC dynamics. We observe thatboth s-wave and p-wave interactions induce precession of the collective magnetization. Furthermore,
the exchange interactions compete with the SOC-induced dephasing and promote spin alignment
and locking. Similar interaction-induced spin locking effects have been observed in other trappedgas experiments [144, 145], and were recently shown to play a crucial role in the stabilization oftime crystal phases in trapped ions [146] and impurity centers in diamond [147]. In those cases,
however, dephasing arose from spatial inhomogeneities, in contrast to our system where dephasing
is a direct consequence of an intrinsic modification of the band structure by SOC.

7.1.1

Chapter Outline

The outline of this chapter is as follows. We will start by reviewing the characterization of
the trapping potential in section 7.2 for the case where the two beams forming the lattice have

114
different powers. We will then review the band structure of the atoms in a lattice in section 7.3,
which is important when considering the lineshape when tunneling is allowed in the lattice. We
will then discuss the spin-orbit coupling further in section 7.4 and the relation to synthetic gauge
fields in section 7.5. The experimental details will then be discussed in section 7.6 and we willshow the single particle effects of SOC in our optical lattice clock. We will look at the effects oftunneling on the axial sidebands in section 7.7 and the signatures of Band relaxation that we cansee from the sidebands in section 7.8. The effects of tunneling on the carrier transition are discussed
in section 7.9 and how the observations can be explained using the band structure of the atoms
in the lattice. We will show that these features lead to the ability to select atoms with specificquasimomentum and observe Bloch oscillations for a tilted lattice in section 7.10. By selecting
different quasimomentum groups we can Rabi flop the atoms in order to determine the chiral Blochvector angles in section 7.11, which are the eigenstates of the spin-orbit coupled Hamiltonian.
We will then move on to discuss further dynamics using Ramsey spectroscopy. We will
start with the single particle Ramsey contrast in section 7.12 for different tunneling rates before
increasing the atom number to introduce the collective interactions in section 7.13. We will thenlook at the frequency shifts in section 7.14 and compare them to those studied previously, in chapter
6, with zero tunneling rate.

7.2

Atoms trapped in a periodic potential
For the case of a vertical one-dimensional lattice the gravitational energy shift between neigh-boring lattice sites (∼ 1 kHz) lifts the degeneracy between adjacent potential wells which strongly
inhibits tunneling [148]. In this case here where we want to study the effects of tunneling we therefore move from a vertical lattice to a horizontal lattice configuration. We will consider the case ofan unbalanced lattice where the power of both beams that form the lattice can be different.
The trapping potential near the waist of a one-dimensional lattice that is travelling along the

115ẑ direction can be written as22
U (z, r) = − Uconst + Uz cos2 (kz) e−2r /ω0

(7.1)

with
√
4α P1 P2Uz =πc0 ω02√
P1 + P2 − 2 P1 P2
Uconst = απc0 ω02

(7.2)
(7.3)

where k = 2π/λ where λ is the lattice wavelength, c is the speed of light, and 0 is the permittivityof free space. P1 (P2 ) is the ingoing (reflected) lattice power, and α = α(λ) is the wavelength
dependent a.c. polarizability, r is the transverse distance from the trap center and ω0 is the latticebeam waist. To find solutions to this equation we can Taylor expand the cosine and exponential
terms keeping terms up to the fourth spatial power to give


 2

1
2r
2k 2 r2 z 2
U (z, r) ≈ Uz −1 + k 2 z 2 + k 4 z 4 + (Uz + Uconst )
−
.
3
ω2
ω02

(7.4)

In the axial direction the potential is approximated as a one-dimensional harmonic oscillator witha quartic distortion and in the radial direction we just have a harmonic oscillator. The r2 z 2 termin Eq. 7.4 is an axial radial coupling term.
The harmonic approximation to the trapping frequencies of the lattice can be found byequating the harmonic terms in Eq. 7.4 to the energies of a harmonic oscillator. For example in
the axial direction
1
k 2 Uz z 2 = M ωz2 z 2
2 r
1
2k 2 Uz⇒ νz =
2π
M

(7.5)
(7.6)

with νrec = h/(2M λ2L ) the recoil frequency. Similarly, the radial trapping frequency is found to bes
1
4(Uz + Uconst )
νR =
.
(7.7)
2π
mω02

116

Figure 7.1: Axial trapping frequency, νz , against retro-reflected power P2 , which is measured using
1/4
a photodiode. The data (purple, squares) is fitted with a P2 curve (purple line) as given in Eq.
7.8.

Figure 7.1 shows that experimentally by just lowering the retro-reflected power, P2 , the axial
1/4

trapping frequency decreases ∝ P2

as expected from Eq. 7.6. Inserting the values of Uconst andUz into Eq. 7.6 and Eq. 7.7 gives√
132πα P1 P2νz =2πω0 λLc0 Ms√
4α P1 + P2 + 2 P1 P2
1νR =πc0 M
2πω02
s

(7.8)

(7.9)

where we can see that for a balanced lattice these equations reduce to the more familiar equations
[7]
r
1
32παP
νz =
2πω0 λL
c0 M
r
1
16αP
νR =
2
2πω0 πc0 M(7.10)(7.11)

In our experiment we want to be able to keep the radial confinement approximately constantwhilst changing the tunneling rate. The axial confinement is due to the interference term between√
the the two counter propagating laser beams that are making the lattice (∝ P1 P2 ). On the other117
hand the radial confinement is present even in a dipole trap. We can therefore have tunneling andstrong radial confinement by changing the power of only one of the lattice beams. Experimentally
this can be done by double passing the retro-reflected lattice beam through an acousto opticalmodulator (AOM).

7.3

Lattice Band Structure
In order to derive the coupling matrix which describes the motion of particles in a lattice westart with the Schrödinger equation for a particle with a wavefunction ψn (z) in a periodic potential
given by V (z) = V0 sin2 (kz)−~2 d2
ψn (z) + V (z)ψn (z) = En ψn (z),2M dz 2

(7.12)

where En is the particle energy eigenvalue, with n the energy band index. Bloch’s theorem tells usthat the eigenfunction of a particle in a periodic potential can be written as ψnq (z) = eiqz Unq (z),where q is the quasimomentum and Unq (z) is a periodic function with the same periodicity as V (z).
Inserting this solution into the Schrödinger equation of Eq. 7.12 gives us
~2
−
2M




d2
d
2
+ 2iq
− q Unq (z) + V (z)ψn (z) = En ψn (z).
dz 2dz

(7.13)

We can write the periodic functions as a Fourier series
Unq (z) =∞X

al e2iklz ,(7.14)l=−∞where al is the amplitude of the traveling wave forming the lattice, and k is the wavevector of thelight forming the lattice. The equation now becomes
∞
∞
∞
X
X
~2 X
al (q + 2kl)2 e2iklz +V (z)al e2iklz = En
al e2iklz .
2M
l=−∞

l=−∞

(7.15)l=−∞Using the exponential definition of a sine function we can rewrite the lattice potential as
V (z) = V0 sin2 (kz) = −i
V0 h 2ikze+ e−2ikz − 2 ,4(7.16)

118

Figure 7.2: Lattice Band structure for increasing lattice depth. The four lowest bands are shown.
The energy width of each band is proportional to the tunneling rate which increase with decreasing
depth.

inserting this into Eq. 7.15 and considering all terms with e2iklz we get the following recurrence
relation~2
V0
V0
al (q + 2kl)2 −[al−1 + al+1 ] += En al ,2M
4
2(7.17)

which can be written as an infinite square matrix with quasimomentum dependent eigenvalues
En (q). To solve for the eigenenergies we need to truncate the matrix which can be written as


V0
V0
~2
2
0
...
0
 2m (q − 2kl) + 2 − 4



.


.
..
..


− V40
− V40
...




2
..
..


V
~
2
0
(7.18). 2m (q) + 2
.
0
0






..
..
..


.
.
− V40
− V40
.




2
~
0
...
0
− V40 2m
(q + 2kl)2 + V20
The lowest eigenenergy solutions to this matrix are shown in Fig. 7.2 for different lattice depths asa function of quasimomentum. These solutions are the band structure of the atoms in the latticewhere the width of each band is 4J, with J the tunneling rate.

7.4

Coupling the spin and the orbit
As mentioned in the introduction spin-orbit coupling occurs naturally in a one-dimensional

optical lattice clock during clock spectroscopy when tunneling is allowed. When an atom in the

119

Figure 7.3: Spin-orbit coupled band structure (solid, bichromatic) from the coupling of the bareclock state |g, qi0 (red, dashed) and the momentum-shifted clock state |e, q + φi0 (blue, dashed).
The band splitting is given by the Rabi frequency, Ω, and has a bandwidth 4J with J the tunnelingrate. The collective density of states (green, solid) shows divergences around q ∼ 0 and q ∼ π.

ground state, g, with quasimomentum, q, and ground band 0, which we denote |g, qi0 , absorbs aclock laser photon it receives a momentum kick and ends up in |e, q + φi0 . The momentum kick is
given by
φ=

2π~ a
λL
7π
=π
≈
λc ~
λc
6

(7.19)

in units of ~/a, with a = λL /2 the lattice spacing. For convenience we are now going to performa gauge transformation on the band structure of the lattice trapped atoms described in section
7.3. In the tight-binding limit these bands can be approximated as sinusiodal functions E0 (q) =−2~J cos(q). These bands are transformed such that |g, qi0 → |g, qi0 remains unchanged and|e, qi0 → |e, q + φi0 meaning that all transitions in quasimomentum space now seem to conservequasimomentum.
Figure 7.3 shows the gauge transformed clock states under the rotating wave approximation
with clock laser detuning δ = 0. The ground band of the ground clock state |g, qi0 (red, dashed)
and the ground band of the gauge transformed excited clock state |e, q + φi0 (blue, dashed) become
coupled when the clock laser is on, forming spin-orbit coupled bands (solid, bichromatic). The band120
splitting is given by the Rabi frequency Ω and the bandwidth is 4J.
The probability of a transition between the two bands at a specific q is determined by thejoint transition density of states. The joint density of states (green, solid) diverge around q ∼ 0
and q ∼ π which is the location of the saddle points in the energy difference between the two band
dispersion curves. These divergence points in the density of states of a crystalline lattice are called
Van Hove singularities (VHSs) and are well known from optical absorption spectra in solids andscanning tunneling microscopy [149, 150].
The resulting Hamiltonian can be diagonalized in quasimomentum space using this gaugetransformation. The transformed spin-orbit coupled Hamiltonian is given by
ĤSOC /~ = −N
X

BSOC (qi , nri , Ω, δ) · Ŝi ,(7.20)

i=1

where the components of S are the spin-1/2 angular momentum operators Ŝ X,Y,Z for the two clockstates. BSOC (qi , nri , Ω, δ) is an effective quasimomentum-dependent magnetic field given by
BSOC (qi , nri , Ω, δ) = 0, BnYz (Ω), BnZz (q, δ)(7.21)= [0, Ω, (Enz (q) − Enz (q + φ))/~ + δ]

7.5

Connections to synthetic gauge fields
To connect our system to related works on synthetic gauge field we can treat the clocktransition as a synthetic dimension as shown in Fig. 7.4. The clock laser with Rabi frequency Ω
excites atoms from |gi → |ei along the synthetic dimension. This causes a change in momentumof the atom absorbing the photon of φ (see Eq. 7.19). Each lattice site experiences the clock laser
with a different phase and this excitation in the synthetic dimension can be thought of as a complex
site dependent tunneling rate in a synthetic dimension. In the real dimension of the lattice, theatoms can tunnel between the different lattice sites with a tunneling rate J. In the case of an atom
following a closed trajectory about a single plaquette with the path |g, ni → |e, ni → |e, n − 1i →|g, n − 1i → |g, ni, will accumulate a phase φ which resembles the flux experienced by a charged
particle in the presence of an external magnetic field.

121

Figure 7.4: Connecting our experiment to work on synthetic gauge fields. (a) We can treat the |gi
and |ei clock states as a synthetic dimension. Together with the real space dimension of the lattice
site, these two dimensions form plaquettes and can be thought of as a two-dimensional synthetic
lattice which the atoms are confined to. (b) A single plaquette demonstrating the synthetic gauge
field. After traversing around the plaquette the atom picks up a phase φ. The atoms tunnel in thereal space dimension by tunneling with rate J. The clock laser, with Rabi frequency Ω, imprints asite dependent phase on the atoms which can be thought of as a complex site dependent tunnelingrate along the synthetic dimension.

122

7.6

Experimental Setup
In this experiment up to 1.5 × 104 nuclear spin-polarized atoms are cooled to ∼ 2 µK andloaded into a one-dimensional horizontal lattice aligned along the ẑ axis. The lattice is formed
using a high power (P1 ≈ 3 W) incoming beam and a variable retro-reflected beam. Although
the basic setup for these different experiments is the same with a strong incoming beam and aweak retroreflected beam the actual experimental setup was improved between the earlier set ofexperiments [151] and the experiments studying interactions [89].
First we will briefly describe the initial setup. The initial lattice setup involved free spacecoupling the 813 nm Ti:sapphire light to the chamber which was thought to be the best approach fro
maximum power at the atoms. The lattice is focused and gives both an astigmatic and ellipticallyshaped beam with waists of 40 µm and 50 µm in the two directions. After passing through thechamber, the lattice passes through a polarizing beam splitter (PBS) before being collimated andpasses through one or two non-polarizing beam splitters (NPBS) with reflection:transmission of
90 : 10 and 50 : 50 before passing through a λ/4 waveplate and then being retro reflected. The
NPBSs give an overall maximum power of up to ∼ 50 mW that is used for the retro beam and thewaveplate combined with the PBS gives us the fine control of the power.
For studying interactions with SOC it is necessary to be able to change the lattice depth
during each sequence. For this reason the Ti:sapphire breadboard layout is modified to the setup
shown in Fig. 7.5 which allows the power to be fiber-coupled to up to two fibers to the experiment.
The acousto-optical modulator (AOM) allows us to servo the intensity of the lattice. In thisexperiment we only use one fiber but the setup allows for flexibility in future experiments. The
simple fiber output near the experiment is shown in Fig. 7.6, where the beam is focused andcombined with the MOT beams using a dichroic. This fiber-coupled setup gives us the advantagethat any day-to-day pointing stability from the Ti:sapphire laser is not converted to a pointing
stability for a lattice and instead the fiber coupling just needs to be optimized. The fiber output
also has the advantage that it cleans up the beam shape and we are able to get a symmetric123

Figure 7.5: Lattice laser setup. The Ti:Sapphire laser (M-squared, SolsTis) is pumped using an 18
W pump laser (Lighthouse Photonics, Sprout) to give up to 6 W of light at 813 nm. The beam iscollimated using cylindrical lenses and all the power can be dumped into a power meter or sent to
the experiment. The beam to the experiment then passes through an optical isolator before being
split between two paths. One of these paths sends a small amount of light to the wavemeter and thefrequency comb (for locking). The other path sends light to the lattice with the option of sending
light through two different fibers to the vertical or horizontal lattice. The retro reflected latticelight passes through the rejection port of the isolator and is monitored/servoed using a photodiode.

124

Figure 7.6: The experimental layout of the ingoing horizontal lattice mezzanine. The 813 nm lightis outcoupled from a fiber before being focused by a lens onto the atoms. The beam enters thechamber along one of the MOT beams (H1 axis) and is combined using a 2” dichroic.

Gaussian beam profile with a waist of 40 µm.
The retro-reflection setup for the horizontal lattice is shown in Fig. 7.7. The power of theincoming lattice is monitored using a photodiode which is used to feedback the lattice power to the
AOM shown in Fig. 7.5 to servo the lattice. Two AOMs are used in the retro path to dynamicallycontrol the retro-reflected intensity. Two AOMs are needed in order to not cause a frequency change
in the retro-reflected beam which would mean that no lattice is formed. The first AOM in the setup
uses the +1 order and the second AOM uses the −1 order for this reason. A λ/4 waveplate is used
to set the maximum retro-reflected power of the lattice and the power of the retro-reflected light ismonitored using a separate photodiode. The retro-reflected light then passes through the chamberand back through the fiber launch shown in Fig. 7.6 and back to the breadboard shown in Fig.7.5
where the power can also be monitored using the rejection port of the optical isolator.
The clock laser beam is combined with the retro-reflected lattice beam using a dichroic. Both

125

Figure 7.7: The setup of the retro reflected path of the horizontal lattice. After the incoming latticebeam passes through the chamber the following setup is used to control the power of the retroreflected lattice beam. A pickoff monitors both the incoming lattice intensity and the retroreflectedlattice intensity. Two AOMs are used in order to keep the frequency of the retro reflected latticebeam the same at the atoms with each AOM using a different diffraction order. An iris and a mirror
are used to remove the zeroth orders of the AOMs. The clock laser travels through a periscope fromthe main experimental table up to the mezzanine shown here. The zeroth order of the clock lightis retroreflected for fiber phase noise cancellation and the −1 order of the clock beam is combinedwith the lattice using a dichroic.

126
beams are combined with the MOT beams using a small half inch mirror. A dichroic cannot be
used in this case because we want the red 689 nm MOT beams to be transmitted and the 698 nm
clock laser needs to be reflected into the chamber. To maximize the power a mirror is therefore
used and the lattice is offset horizontally from the red MOT beam with both overlapping at theatoms.

7.7

Sideband Structure and Band Selection
A typical coarse scan of the atoms across resonace by changing the frequency of the clock

laser is shown in Fig. 7.8 for Uz /Erec = 30.5. From this scan we can observe a strong narrow central
carrier peak and several sideband transitions. At typical operating temperatures the majority of
the atoms are prepared in the ground band (0) of the ground state, g, which we denote |gi0 . The
central peak is referred to as the carrier transition and represents atoms undergoing a transition
from |gi0 → |ei0 where the position of the peaks can be understood due to the band structure of
atoms in the lattice.
The sideband transitions are suppressed by the Lamb-Dicke parameter and are caused bytransitions between different bands. The blue detuned sideband transition corresponds to the transition |gi0 → |ei1 and a smaller transition is seen at even higher positive detunings corresponding
to |gi0 → |ei2 . The red detuned sideband corresponds to the transition |gi1 → |ei0 where thepopulation of |gi1 depends on the temperature of the atoms. The ratio between the red and blue
sidebands can therefore be used to determine the temperature of the atoms in the lattice [152].
With these carrier and sideband transitions corresponding to different band transitions we
can then use these to experimentally prepare atoms in different bands of the lattice. For example
we can excite atoms using a clock laser detuned to the blue sideband, remove all remaining groundstate atoms using the blue probe beam and we are left with only atoms in the first excited band|ei1 . Figure 7.9 shows a sideband scan for atoms in |ei0 and Fig. 7.10 shows a sideband scan foratoms in |ei1 (solid) and for reference a scan for atoms in |ei0 is also shown (dashed line). We
can see that the red sidebands for atoms prepared in different bands are offset from each other127

Figure 7.8: Sideband scan from the ground state. At our temperatures the majority of atoms are in
the ground band. The transitions from the |gi0 state are shown in the inset using the band structureof the |gi and |ei bands. The carrier transition is dominated by the |gi0 → |ei0 transition, thesideband from |gi0 → |ei1 (yellow, solid) is blue-detuned and the |gi0 → |ei2 sideband (green, solid)
can also be seen. Atoms that are not initially in the ground band are detected as a red-detuned
sideband with atoms transitioning from |gi1 → |ei0 .

Figure 7.9: Sideband Scan from the excited state ground band. A π-pulse initializes atoms in |ei0
with the transitions from |ei0 shown in the inset using the band structure of the |ei and |gi bands.
The carrier transition is dominated by the |ei0 → |gi0 transition, the sideband from |ei0 → |gi1
(yellow, solid) and the |ei0 → |gi2 sideband (green, solid) can be seen on the left. The sideband on
the right is decreased from Fig. 7.8 due to the initial π-pulse preparation.

128

Figure 7.10: Sideband scan from the excited state 1st excited band. A strong clock pulse detuned
to the blue sideband initializes atoms in |ei1 . The transitions from |ei1 are shown in the inset using
the band structure of the |ei and |gi bands. The carrier transition is dominated by the |ei1 → |gi1
transition, the sidebands from transitions from |ei1 → |gi2 (green, solid), |ei1 → |gi0 sideband(purple, solid), and |ei1 → |gi3 sideband (cyan, solid) can all be seen. For comparison the sideband
scan from Fig. 7.9 is also shown (black, dashed).

by a lattice recoil, Erec , and this is due to the anharmonicity of the lattice. This anharmonicityof the lattice axial bands makes transitions from |ei0 → |gi1 to be different from transitions from|ei1 → |gi2 . For the case of atoms in the 1st excited band shown here, the red detuned transition
corresponds to the |ei1 → |gi2 transition, the carrier to |ei1 → |gi1 transition and the blue detuned
transition to |ei1 → |gi0 .
We can see that the sidebands are smeared out towards the carrier transition and this isdue to the radial-axial coupling within the lattice. Experimentally we can see this coupling by
exciting atoms from the blue sideband using different clock laser detunings within the sideband.
By preparing atoms in |ei1 , and removing all other atoms, using different excitation detunings thedifference between the different selections can be seen most clearly by then measuring the radial
temperature of the atoms in |ei1 . An example of this is shown in Fig. 7.11. For atoms excitedat the peak of the blue detuned sideband we see a lower radial temperature compared to atoms
excited closer to the carrier frequency.

129

Figure 7.11: By exciting atoms on the blue detuned sideband we can select different radial groupsof atoms. The blue line is the case when atoms are excited at the peak of the blue detuned sideband
giving a temperature of 2.5 µK. The red line is when atoms are selected from the blue detuned
sideband, but at a frequency closer to the carrier frequency giving a temperature of 6.4 µK

Figure 7.12: Three different sideband scans at lattice depths Uz /Erec = 5.5 (green), Uz /Erec = 17.4
(red), and Uz /Erec = 43.9(blue). As the lattice depth decrease the sideband scans gets closer to anaxial Doppler scan.

By lowering the lattice depth the sidebands move towards the carrier. Figure 7.12 shows the

130effects of lowering the retro-reflected lattice power P2 on the sideband structure. For lower lattice
depths the sidebands merge together to form a gaussian distribution of atoms.

7.8Band Relaxation
During the course of these experiments it was discovered that sometimes when the atoms

were being prepared in the first excited band |ei1 band relaxation was observed. This was observed
both in the splitting of the carrier transition (see section 7.9) and the location of the sideband
transitions for different axial bands (see section 7.7). At first this was thought to be due to theramping of the lattice but the amount of relaxation was found to be independent of the ramping
time and also found to occur without any ramping taking place.
It was found that this band relaxation is due to s-wave collisions between atoms in |gi0 andatoms in |ei1 (Fig. 7.13). In both Fig. 7.13(a) and (b) a sideband scan for atoms in |ei0 (blue,
dashed) and |ei1 (red, dashed) are shown. If the atoms in |gi are removed straight after band
selection, as in Fig. 7.13 (b) to |ei1 then no band relaxation is observed. If however the atoms
in |gi0 are not removed straight away and are held in the lattice with the |ei1 atoms before being
removed then band relaxation is clearly seen in Fig. 7.13(a).

7.9Carrier Transition and Band Mapping
For J = 0 the excited state fraction, pe , for detuning, δ, is described by the function|Ωeg |2pe (δ) =sin2|Ωeg |2 + [δ − ωz (ne − ng )]2


 q
t
2
2
|Ωeg | + [δ − ωz (ne − ng )]2(7.22)For the case of J 6= 0 we need to take into account the changing density of states. To do this we
can convolve the lineshape given above with the joint transition density of states as a function of
δ, which is given byD(δ) =



 √

 0

1
[4J sin(φ/2)]2 −δ 2

− |4J sin(φ/2)| ≤ δ ≤ |4J sin(φ/2)|(7.23)
otherwise

131

Figure 7.13: Band relaxation caused by s-wave collisions. The red and blue traces, in both plots,show the red detuned sideband for the |ei1 and |ei0 bands respectively. (a) Atoms are excited
to |ei1 and are held for 200 ms before removing atoms in |gi0 and scanning the sideband (green
trace). Atoms can be seen to be present in both |ei1 and |ei0 due to band relaxation caused by
the presence of ground state atoms during the hold time. (b) Atoms are excited to |ei1 and theground state atoms are removed before holding for 200 ms and then scanning the sideband (green
trace). Atoms can be seen to be present in only |ei1 and no band relaxation is observed.

132
The convolution can be solved numerically and Fig. 7.14 shows the resulting convolutions fordifferent tunneling rates using a 160 ms π-pulse. For J/(2π) = 0 Hz the convolution gives thenormal Rabi lineshape given in Eq. 7.22. As J increases the lineshapes broadens and two peaks
appears. These two peaks arise due to the divergences in the density of states shown in Fig. 7.3,
also known as Van Hove Singularities.

Figure 7.14: The theoretical carrier transition for different tunneling rates and a 160 ms π-pulse.
The solutions are numerically integrated convolutions of the joint transitional density of states andthe sinc2 Rabi lineshape.

A comparison between data and a theory prediction is shown in Fig. 7.15 for a 5 ms π-pulse
and a tunneling rate J/(2π) = 87 Hz. The π-pulse is set using a high lattice (J = 0 Hz) before
lowering the lattice depth and scanning the detuning of the clock laser δ and the excitation fraction
is measured in the usual way. We can see that the theory agrees well with the experimental data.
Figure 7.16 shows the effect of changing the π-pulse time on the carrier transition. For longer(weaker) pulses the splitting of the Van Hove Singularities is still visible spectroscopically. As thestrength of the pulse is increased the splitting is no longer visible anymore. The reason for this isthat the Fourier limited linewidth is increasing and becomes larger than the Van Hove splitting.
The results of changing the pulse area are shown in Fig. 7.17 for the carrier transition in the

133

Figure 7.15: Carrier transition in the ground band |gi0 → |ei0 with a comparison of the measuredlineshape (red, squares) for a 5 ms π-pulse with J/(2π) = 87 Hz. The theory prediction (red, solidline) is from the convolution of the joint transitional density of states and the sinc2 Rabi lineshape.
The overall amplitude has been scaled by 0.9 for better theory-experiment agreement.

first excited band |ei1 → |gi1 . Two different pulse areas are shown including a π-pulse (red lines)

Figure 7.16: Carrier transition in the ground band, |gi0 → |ei0 , for J/(2π) = 30 Hz with different
Rabi frequencies. The data is shown as dashed lines and the theory is solid lines for π-pulse timesof 3 ms (red), 5 ms (orange), 10 ms (green), and 20 ms (blue). Each theory line has been scaledby an overall amplitude which is the same for each.

134
and a 3π-pulse (blue lines) where the experimental data (dashed) agrees well with the theory lines(solid). It should be noted that for other pulse areas the amplitude scaling does not agree [151]Figure 7.17: Carrier transition in the first excited band, |ei1 → |gi1 , for different pulse areas withdata shown as dashed lines and theory as solid lines and J/(2π) = 370 Hz. The π-pulse (red) and
3π-pulse (blue) agree well with theory lines.

135

Figure 7.18: Carrier transition splitting for different lattice depths for the |gi0 → |ei0 transition(purple, circles) and the |ei1 → |gi1 transition (orange, squares). The splitting is extracted using
fits from a convolution of a lorentzian lineshape and the density of states and the lattice depth isextracted from the sideband scan.

136

7.10

Quasimomentum selection and Bloch Oscillations
From previous sections we have found that the splitting of the lineshape corresponds todifferent quasimomenta at different detunings. Therefore, just as the spectroscopically resolved
sidebands enable band preparation, the quasimomentum dependence of the clock transition frequency enables the selective preparation and subsequent manipulation of atoms with particularquasimomenta. Following initialization in the |gi0 state, a clock pulse with Rabi frequency Ω < 2∆,
where ∆ = 4J| sin(φ/2)| is the largest momentum-induced detuning from the bare clock transition
frequency, is applied to the carrier transition with a clock laser detuning δ ∗ ≤ ∆. Only atoms
with quasimomenta in a window centered around q ∗ ≈ arccos(δ ∗ /∆) and with a width bounded by
2πΩ/∆ will be excited to |ei0 . Atoms outside this window will be left in |gi and a strong 5 ms pulseof 461 nm light removes them from the lattice. Following a variable hold time, a second π-pulse

Figure 7.19: The excited state fraction of the Van Hove Singularities (VHSs) versus holding timewith fits is shown for a tilted lattice. When only atoms from the left VHS are excited (q ∼ 0), theleft VHS peak height (red, circles) and the right VHS (blue, triangles) oscillate with time. Whenatoms are excited at the center of the lineshape the VHS peak heights (green, squares) oscillate twice
as fast as when q ∼ 0 atoms are initially prepared. In both case we see the oscillations amplitudedecrease with time due to diffusion of quasimomentum and after long times all quasimomentum
are occupied.

137

Figure 7.20: With a tilt in the lattice the quasimomentum selection of atoms undergoes Blochoscillations. A range of quasimomentum can be selected by applying a π-pulse to the line splitting.
Shown above are for atoms selected from the left Van Hove singularity (top) and in the center ofthe lineshape (bottom) and then removing any atoms that are not excited to the excited state.

138
can be used to measure the new lineshape. If the lattice is tilted with respect to gravity then theatoms will undergo Bloch oscillations during the hold time, with their quasimomenta evolving as
q(t) = q0 + νB t

(7.24)

where q0 is the initial quasimomentum of the atom and the value q(t) is restricted to the first
Brillouin zone. The Bloch oscillation frequency νB is given byνB =mgλL sin(θL )4π~(7.25)

with g the acceleration due to gravity and θL is the angle of the lattice tilt measured from thehorizontal.
Figures 7.19 shows how the excitation fraction of the VHSs change with time when undergoing
Bloch oscillations. When δ ∗ is detuned to the left VHS the excitation fraction of the left VHS (red,circles) and right VHS (blue, triangles) can be seen to oscillate with a period of 70 ms indicating a
Bloch oscillation frequency of 14 Hz and a lattice tilt θL = 16 mRad. When the atoms are excitedin the center of the lineshape, δ/(2π) = 0, the Bloch Oscillations appear to be twice as fast (green,squares). The reason for this is that when atoms are excited in the center of the lineshape thesolution of q ∗ from above has two solutions. This leads to two groups of atoms being excited andrunning through quasimomentum space giving the appearance of the oscillations being twice as
fast. The complete lineshape data from which this data is extracted is shown in Fig. 7.20 for thecase of atoms excited on the left VHS (blue, solid lines) and excitation at the center of the lineshape(red, solid lines).

7.11

Chiral Bloch Vector
The eigenstates of the Hamiltonian ĤSOC (Eq. 7.20) can be described by Bloch vectors in

the Ŷ − Ẑ plane of the Bloch sphere (see Fig. 7.21). They point along the direction of the quasimomentum dependent dependent magnetic field BSOC (qi , nri , Ω, δ) with their orientation specified

139

Figure 7.21: (a) Chiral Bloch vector on the Bloch spher points in the Ŷ -Ẑ plane at an angle ofθB with respect to the Ẑ axis in the direction of the quasimomentum dependent magnetic field,
BSOC (qi , nri , Ω, δ). This magnetic field causes a rotation of the Bloch vector around this direction.
(b) The tunneling rate is set to J/(2π) ∼ 16 Hz and a 50 ms π-pulse is used to select atoms withdifferent q by setting the clock laser detuning to those indicated by the arrows.

by the chiral Bloch vector angle, θB , with respect to the Ẑ axis and is given byΩθB = arctan(Enz (q) − Enz (q + φ))/~ + δ(7.26)

where the q dependence of θB is a manifestation of chiral spin-momentum locking [135, 153]. In
the description of the system as a synthetic gauge field the chiral Bloch vector angle is directly
connected to the topological nature of chiral edge modes of the two-dimensional Hofstadter model[154, 153]. Coupling multiple nuclear spin states with our synthetic gauge fields should enable therealization of topological bands and exotic phases in higher dimensions.
We can characterize the q dependence of the chiral Bloch vector angle using the same quasimomentum selection technique that was used to observe Bloch oscillations in Section 7.10. For thesemeasurements the lattice tilt was adjusted to minimize the Bloch oscillation frequency νB ≤ 3 Hzsuch that the motion is not dominated by Bloch oscillations. In order to extract a chiral angle we
therefore need to determine the effective detuning δef f = δ + (Enz (q) − Enz (q + φ))/~ which is

140

Figure 7.22: Rabi flopping of different quasimomenta. The line splitting is set to 125 Hz(J/(2π) ∼ 16 Hz) and quasimomenta are selected with five equally spaced detunings, red→purple
data δ/(2π) = {−62.5, −31.25, 0, 31.25, 62.5} Hz using a 50 ms π-pulse. Each q-selection then undergoes Rabi flopping with the clock laser detuned to the right VHS, δ/(2π) = +62.5 Hz and a 5
ms π-pulse. The shapes are data and the lines are theory fits.

Figure 7.23: (a)Spin-orbit coupled band structure during the Rabi flopping, showing the q selection
regions as dashed lines corresponding to the colors in Fig. 7.22. (b) The extracted Chiral angles
for the different q-selections by fitting the data in Fig. 7.22. The black solid line is a theory fit
using Eq. 7.26.

141
made up of the normal laser detuning δ and the additional detuning brought about by tunneling.
We can therefore perform a Rabi flopping measurement on our selected quasimomenta and extract
an effective detuning.
For this measurement the tunneling rate is set to J/(2π) ∼ 16 Hz and five separate experiments are performed. In each case atoms are prepared in |ei0 using a weak 50 ms π-pulse(Ω = 2π × 10 Hz) and all remaining |gi atoms are removed. For the five experiments, atoms are
selected with a clock laser detuning of δ/(2π) = {−62.5, −31.25, 0, 31.25, 62.5} Hz as indicated inFig. 7.21 and thus selecting different quasimomenta groups. Each of these quasimomenta thenundergo Rabi flopping as a second pulse with Ω = 2π × 100 Hz is applied with δ/(2π) = 62.5
Hz. The resulting Rabi flopping data are shown in Fig. 7.22 where the colors correspond to theq-selection detunings shown in Fig. 7.21.
From this data we can fit each individual curve with a cosine fitting function
f itChiral = A + B

hqiΩ2
2 + δ2 t
cos
Ω
ef f
2
Ω2 + δeff(7.27)

where A is a constant, B = B(t) is a time varying exponential envelope common to all curves, andwe set Ω = 2π × 100 Hz. From these fits we therefore extract the effective detuning δef f and use
this to calculate the chiral angle by




Ω

θB = arcsin  q22Ω + δeff(7.28)The extracted chiral angles are shown in Fig. 7.23 plotted aginst the theory prediction of Eq. 7.26
alongside the band structure during Rabi flopping. We should note that the theory line assumes
only a single value of q is initially selected but a range of q values around the central value areexperimentally excited, thus the theory fit is not ideal. The theory lines shown in Fig. 7.22 include
the different values of q.

7.12

Ramsey Spectroscopy with SOC
To observe the dynamics of our spin-orbit coupled system we can perform Ramsey spec-

troscopy. Ramsey Spectroscopy for a single group of non-interacting atoms was discussed in section

142
6.3, now we can consider the case of Ramsey spectroscopy when SOC is included. An initial pulse
with a strong Rabi frequency, and hence short pulse time and large spectral bandwidth, of area θ1and δ = 0 excites all atoms into a coherent superposition of clock states that are then allowed tofreely evolve during τ . Although the clock laser is off during this period, the atoms accumulate aphase in the rotating frame of the laser and thereby retain the imprinted optical phase. As a result, the atoms continue to experience the SOC induced effective magnetic field BSOC (qi , nri , Ω=0,
δ=0) throughout the dark time τ . One observable we measure using this procedure is the Ramsey
rD E
D E2
2
fringe contrast, defined as C = 2
Ŝ X + Ŝ Y /N , which is the length of the projection ofhithe collective magnetization in the X̂ − Ŷ plane of the Bloch sphere. Here Ŝ = Ŝ X , Ŝ Y , Ŝ Z are
P{X,Y,Z}.
collective spin operators with Ŝ {X,Y,Z} = N
i=1 Ŝi
The concentration of atoms at the two Van Hove singularities allows us to qualitatively understand the ensuing dynamics as arising mainly from these two groups of atoms, with quasimomentaq ∼ 0 and q ∼ π, and corresponding detunings of δ± , respectively. Figure 7.24 depicts the Bloch
sphere visualization of Ramsey spectroscopy for the case when the two groups are non-interacting

Figure 7.24: Ramsey Spectroscopy with SOC for two groups of atoms with q ∼ 0 and q ∼ π. An
initial strong π/2 pulse creates a superposition of |gi and |ei. During the dark time τ the atom
evolves around the Bloch sphere at a rate proportional to the detuning between the laser and theatom resonance. A second π/2 pulse projects the atoms into the population of |gi and |ei

143

Figure 7.25: Normalized collective Bloch vector components and contrast for Ramsey spectroscopy
with 2-atom groups. S Y remains zero for all times as the Y components for the two groups of
atoms cancel. S X oscillates as the two Bloch vectors rotate around the Bloch sphere leading to acontrast that varies between zero and one.

and for θ1 = π/2. For a variable evolution time τ , the atoms with opposite detunings δ± evolvearound the equator of the Bloch sphere in opposite directions (dashed blue arrows). Consequently,
the length of the collective spin vector (solid blue arrow) changes, but the vector direction remains
parallel or anti-parallel to X̂.
Mathematically we know that the rate of precession of the Bloch vector is proportional to thedetuning. As we set the laser detuning δ = 0 we know that the extra detuning is coming from theband structure such that δef f = δ + ∆En (q) = ∆En (q). For the case of our model of two groups of
atoms at q ∼ 0 and q ∼ π the components of the Bloch vector are given by
1
X
S2-atom= sin(θ1 ) (cos [δef f τ ] + cos [δef f τ ])2
1
Y
S2-atom= sin(θ1 ) (sin [δef f τ ] + sin [δef f τ ]) = 0
2(7.29)(7.30)As the initial pulse is around S Y the atom groups with opposite detunings evolve symmetricallyfrom the X̂ axis in opposite directions and the S Y components from each group of atoms cancel toYXgive a collective S2-atom= 0 at all times. The S2-atomcomponent evolves in time and is shown in144
Fig. 7.25 for θ1 = π/2 (red, solid). The contrast (green, solid) starts off at 1 and tends towards zeroas Jτ /(2π) increases. The contrast then revives and goes back to full scale. We can understandthese zeroes of the contrast as the two groups of atoms accumulating a phase difference of π leading
to the length of the collective Bloch vector to be zero (C = 0).
Representative contrast curve measurements are shown in Fig. 7.26 for tunneling rates J1 /(2π) =3.2 Hz (green triangles) and J2 /(2π) = 17.6 Hz (blue circles) as a function of τ . This data wastaken in the non-interacting regime by using a small number of atoms (N < 500). The collapses
and revivals in the contrast can be readily understood from the simple model of the two atomgroups. The detuning, δ± ≈ ±4J determines the precession rate around the Bloch sphere and wethus expect the contrast to collapse and revive with a periodicity proportional to 1/J. Figure 7.26
also shows the contrast data with the x-axis scaled as a function of Jτ /(2π), illustrating that thecontrast curves for different J values then collapse onto a single curve.
An obvious feature of the observed contrast evolution is the long term decay, which is not
captured by the simple two-group approximation. While the joint density of states is the largest at
the Van Hove Singularities, all q values are in fact populated, with atoms contributing at detunings
in-between δ± . Integrating over the contributions from all q the components of the collective Blochvector and contrast are given by1XSAll
q = sin(θ1 )Zπcos [∆En (q)τ ] dq = sin(θ1 )J0 [4Jτ sin (φ/2)](7.31)

sin [∆En (q)τ ] dq = 0(7.32)qX )2 + (S Y
2
C = (SAllq
All q ) = sin(θ1 )|J0 [4Jτ sin (φ/2)] |(7.33)

2π

YSAll
q = sin(θ1 )

1
2π

−π
Zπ
−π

where J0 is a zeroth order Bessel function of the first kind, as shown in Fig. 7.26.
To validate that the collapses, revivals, and overall decay in contrast are due to BSOC (qi , nri , Ω, δ),
we can remove its effect by adding a spin echo pulse to the Ramsey sequence. Any dephasing fromthe static SOC-induced effective magnetic field during the first τ /2 period of free evolution will
re-phase during the second τ /2 free evolution period, due to the π echo pulse, which flips the sign of145

Figure 7.26: (a) Ramsey Spectroscopy with SOC for different tunneling rates as a function of darktime, τ . The data shows an oscillation with a magnitude given by a Bessel shaped oscillation. (b)The data for different, finite, tunneling rates scales to the same curve when scaled to Jτ . The solidlines are theory curves, and the dashed line is an exponential fit with a decay constant of 0.6 s, andthe error bars are 1σ confidence intervals.

146

Figure 7.27: Ramsey spectroscopy with a spin-echo pulse. A spin echo pulse (orange, diamonds)
removes the static dephasing caused by SOC for a Ramsey sequence for the same J (purple circles,3
theory is solid purple line). The spin-echo data is a fit ∝ e−(τ /τd ) .

phase accumulation. Figure 7.27 shows the effect of spin echo (orange diamonds) for J/(2π) = 4.2
Hz. The spin echo eliminates the collapses and revivals from the Ramsey fringe contrast (purple
circles), and prolongs the overall contrast decay. However, the observed decay in contrast at longtimes is still fast compared to the contrast decay time for J = 0 (∼ 1 s). Contrast decays underspin echo for different values of J are shown in Fig. 7.28. The spin echo decays do not collapse
to a single curve when the free evolution time is scaled to Jτ /(2π), indicating that the additional
dephasing does not scale linearly with J, and that quasimomentum is not conserved at large τ .
It is this additional dephasing that results in the suppression of the revivals in contrast at lower
tunneling rates as shown in Fig. 7.26 (green triangles).
For J > 1 Hz, the SOC-induced dephasing dominates over all other dephasing mechanisms in
our clock. While our Hamiltonian accounts for the static dephasing, a systematic investigation of
the relevant range of J reveals that the spin echo data has an additional decay with the functional
3

form of ∝ e−(τ /τd ) as shown in Fig. 7.27. This form of spin echo decay is well known from NMR
and solid-state spin defect experiments [155, 156], where the dephasing is the result of a slow,

147

Figure 7.28: (a) Spin echo decay of contrast for four different J values as a function of τ (b) Same3
data as in (a) (for non-zero J) as a function of Jτ . All J > 0 curves decay as ∝ e−(τ /τd ) , where τd
is a nonlinear function of J, implying an extra diffusive dephasing, and do not collapse to a single
curve when the free precession time is rescaled to Jτ . For J = 0 the echo data decays ∝ e−(τ /τd0 ) .
All error bars are from individual contrast fits, and all solid lines for the J > 0 (J = 0) echo data3
are fits ∝ e−(τ /τd ) (e−(τ /τd0 ) ).

148

Figure 7.29: A log-log plot of the decay rates (1/τd ) as a function of J for different lattice depths
show a J dependence consistent with a 1/τd ∝ J 1/2 . The error bars come from fits of τd and thesolid line is a best fit curve using 1/τd ∝ J 1/2 .

random diffusion of magnetic field with time. In our case this corresponds to a diffusion of the
SOC effective magnetic field BSOC (q), indicating that the quasimomentum is not conserved atlarge τ . The extracted decay rate (1/τd ) for different J is shown in Fig. 7.29. The scaling is
consistent with 1/τd ∝ J 1/2 . We observe the same scaling of decay rate with J for the Ramsey
sequences, and include this decay in our theory model as an empirical parameter. The most likely
mechanism for this empirically observed diffusion of BSOC is the coupling of axial motion to thethermally populated radial modes. The spatial inhomogeneity in J across the lattice due to thefinite Rayleigh range of the lattice beams may also contribute.
We can also use the quasimomentum selection techniques discussed above in section 7.10 toshow that the zeroes in the Ramsey contrast are due to the cancellation of the different VHSs. The
data in Fig. 7.30 (orange, squares) shows the Ramsey contrast in the case where atoms are selected
preferentially from one of the VHSs. For comparison the case without quasimomentum selection is
also shown (purple, circles). We can see that for the case of quasimomentum selection we no longer
get any zeroes in the contrast and the contrast remains finite.

149

Figure 7.30: Ramsey spectroscopy with q-selection. Atoms are preferentially selected from the right
VHS and the other atoms are removed (orange, squares). For comparison the contrast curve forthe same tunneling rate is shown without quasimomentum selection (purple, circles).

7.13

Ramsey Spectroscopy with SOC and Interactions
Having characterized the single particle Ramsey dynamics under SOC, we introduce inter-actions by increasing the atomic density. As shown in Figs. 7.31 and 7.32, signatures of strong
spin interactions start to emerge as N increases. The blue circles are the case of no interactions(N < 500) and red squares are the case where we introduce interactions by increasing the atom
number by more than an order of magnitude (N ∼ 1 × 104 ). We observe in Fig. 7.31 that for
an initial Ramsey pulse θ1 = π/4, an increase in atomic density qualitatively alters the dynamics,
suppressing the collapses in contrast observed for the low density case. For θ1 = π/2 we observethat interactions shift the zeroes of the contrast compared to the non-interacting case (Fig. 7.32),
and by further increasing the density and reducing the tunneling rate, we see that the first collapseof the contrast can be suppressed altogether. We note that for J = 0 the contrast decay has beenpreviously seen to be highly sensitive to the initial Ramsey pulse area and no contrast revival withinteractions has been observed [15, 141].
In order to quantitatively understand the complex interplay between interactions and SOC,

150

Figure 7.31: (a) Ramsey spectroscopy with interactions for an initial pulse of area θ1 = π/4.
Without interactions (blue, circles) the overall amplitude is reduced from the θ1 = π/2 case becauseof the sin(θ1 ) amplitude of the contrast. With interactions (red, squares) the contrast no longer
goes to zero due to the rotation caused by the exchange interactions. (b) On the Bloch sphere,
as the VHSs (dashed arrows) rotate around the Bloch sphere they become distinguishable andexchange interactions induce rotations shown as purple and green trajectories. For θ1 = π/4 thisrotation leads to the Bloch vectors for the two VHSs being of unequal length in the X̂ − Ŷ planeso that the collective spin vector (solid arrow) remains finite.

we consider the spin model that has previously been successfully used to understand many-bodyinteractions in optical lattice clocks [15, 141]. During these measurements all atoms are initiallyprepared in the |gi spin state, and each atom occupies a single motional mode in the lattice.
The initial mode distribution is preserved during clock interrogation as the collision energy isinsufficient to alter the motional eigenstates. Since the motional degrees of freedom are frozen,
we can treat the single-particle modes as corresponding lattice sites spanning an energy space.
Thus, s-wave and p-wave contact interactions are mapped into non-local, infinite-range collective
interactions between the electronic pseudospins in the energy-space lattice [15]. The Hamiltonian
for our system, including interactions, then becomes Ĥ = ĤSOC + Ĥint , with Ĥint given by,Ĥint /~ =

χ +  Z 2 C +ξ+Ŝ+(N ) Ŝ Z +Ŝ · Ŝ.
L
L
L(7.34)

− /2
The spin couplings χ+ = (Vgg + Vee − 2Veg ) /2, C + = (Vee − Vgg ) /2, and ξ + = Veg − Ueg151(a) 1.0J/(2π) = 3.3 Hz

Contrast

0.8
0.6

(c)

0.4

θ1 = π/2
^

ez

0.2
^

0.0

ey
^

0

(b) 1.0

0.05

0.1

0.15

Jτ/(2π)

0.2

ex

0.25

J/(2π) = 1.3 Hz

AboveContrast

0.8
0.6

^

ey
^

ex

0.4
0.2
0.0

0

0.05

0.1

0.15

Jτ/(2π)0.2

0.25

Figure 7.32: Contrast curves for θ1 = π/2. (a) J > |N ξ + /L| = 3.5 Hz and the interactions cause
the zero in the contrast to be pushed to larger Jτ . (b) J ∼ |N ξ + /L| = 5.6 Hz and interactions
prevent static dephasing and the contrast approaches zero only at long times. Solid lines are theory
including atom loss and diffusive dephasing (see Fig. 7.28). (c) On the Bloch sphere for θ1 = π/2
the exchange induced rotation is symmetric and the two Bloch vectors are the same length in theX̂ − Ŷ plane and are shown as green and purple trajectories on the Bloch sphere.

152−
depend on Vαβ and Uαβwhich are the p-wave and s-wave mean interaction parameters, respectively.

L is the number of lattice sites, and thus N/L represents the mean number of atoms per site. Dueto the temperature of the atoms in the lattice being > 1µK the s-wave and p-wave interactions are
similar in magnitude (see Fig. 7.33).
The term proportional to ξ + encapsulates the exchange interaction process mediated by boths-wave and p-wave collisions. For the nuclear spin-polarized identical fermions initially preparedin the lattice, and in the absence of SOC, this term becomes a constant of motion and is thus
irrelevant to the dynamics. However, when J 6= 0, the effective q-dependent SOC magnetic field
BSOC (qi , nri , 0, δ) causes the initially spin polarized atoms to dephase with respect to each other,
thereby introducing exchange interactions between them, which directly compete with the singleparticle SOC dynamics.
The p-wave interaction terms proportional to χ+ and C + generate a collective Ising Hamiltonian which commutes with Ŝ2 and have previously, in the absence of SOC, been shown to induce
many-body spin dynamics [15] for any superposition of e and g. These terms are unchanged in thepresence of SOC, and have a negligible effect on the spin contrast for the experimental conditions
and timescales we study here.
Throughout this work, we find that the explored experimental timescales are in a regime
where the mean field approximation is valid. In this approximation the interaction terms can betreated as an additional time-dependent magnetic field generated by the collective spin vector, Bint .
This allows us to factor out a collective spin operator from Ĥint as given in Eq. 7.34 in order toarrive at the mean field Hamiltonian including both interactions and SOC:
ĤMF/~ = −

NX
i=1

where Bint =

h

2ξ +
L

D

BSOC (qi , nri , Ω, δ) · Ŝi +N
XBint · Ŝi ,(7.35)

i=1

E + D E  + +D E
i
Z + N C + . The X̂ and Ŷ components can
Ŝ X , 2ξL Ŝ Y , 2 ξ +χ
Ŝ
L
L
+

be written together as a collective, evolving, transverse magnetic field, 2ξL hŜ⊥ (t)i around which
individual atom Bloch vectors rotate. This term competes with the single particle dephasing term,
BSOC (qi , nri , Ω, δ), and forces the pseudo-spins to remain aligned, causing interaction dependent

153

Figure 7.33: Dependence of interactions on temperature and lattice depth. (a) Under current
experimental temperatures the exchange interactions (ξ + ) are of the same order as the Ising interactions (χ+ ). (b) Interaction strengths at T = 1µK (solid lines) and T = 10µK (dashed lines) for
the interaction parameters χ+ (purple), ξ + (green), and C + (Blue) at different lattice depths. The
strong dependence of the exchange interactions on temperature is due to the s-wave contribution.
The p-wave interactions which enter C and χ+ are independent of temperature.

154changes to the contrast. The Ẑ component of the interaction magnetic field is a constant of motion ++ +ξ +and gives rise to a collective precession of the Bloch vectors at a rate N CL − χ Lcos θ1 , wherehŜ Z i = −N/2 cos θ1 . When the tunneling rate J is zero, all the terms proportional to ξ + in Ĥ M F
will not affect the contrast or frequency shift.
The competition between the interaction-induced transverse magnetic field and the static
SOC dephasing is shown schematically in Fig. 7.31 and Fig. 7.32 under the simple two-group
approximation for θ1 = π/4 and θ1 = π/2, respectively. For θ1 = π/4, the collective rotationdifferentially changes the projected length of the individual Bloch vectors on the transverse plane,
generating a net |hŜ Y (t)i| > 0. As a result, when the vectors are π out of phase, they no longer
completely cancel, leaving a finite contrast at all times, as opposed to the complete collapse observed
for the non-interacting case where |hŜ Y (t)i| = 0. This is apparent in the data shown in Fig. 7.31,
where the contrast remains finite for the interacting case (red circles, N ξ + /L = −2.0 Hz, N χ+ /L =1.2 Hz).
For θ1 = π/2, due to symmetry, the rotation of the Bloch vectors on the Bloch sphere (red,dashed arrows in Fig. 7.32) around the collective spin vector (red, solid arrow) does not change therelative transverse length of the vectors – which imposes |hŜ Y (t)i| = 0. The effects of interactions
are shown in Fig. 7.32 for varying strengths of interactions (N ξ + /L) compared to J. When theinteractions are still small compared to the tunneling (J > |N ξ + /L|, with N ξ + /L = −3.5 Hz andN χ+ /L = 1.3 Hz) (Fig. 7.32, (a)), they cause no qualitative change to C compared to the noninteracting case, except for a weak rephasing of the spins that slightly delays the contrast collapse
and decreases the revival amplitude. This is manifested as an interaction-induced shift of the timeof the first contrast zero, ∝ N 2 (ξ + )2 /L2 J 2 .
If J is decreased such that J ∼ |N ξ + /L|, then the exchange interactions produce a qualitatively different behavior, as shown in Fig. 7.32(b). For J/ (2π) = 1.3 Hz, the non-interacting case(blue circles) shows the characteristic collapse and revival. In contrast, the interacting case (withN ξ + /L = −5.6 Hz, N χ+ /L = 3.4 Hz), shows no collapse whatsoever, instead exhibiting only amonotonic decay with Jτ . The suppression of the collapse and revivals is a result of the exchange-

155
induced rephasing of the spins. Ideally, this type of spin locking would preserve the coherence
indefinitely, as can been seen directly from the interacting Hamiltonian (7.34), where for large ξ +the initial state is an eigenstate. Indeed, long-term synchronization has been previously observedin other cold atom experiments [144] with dominant s-wave interactions. In our OLC we also needto account for competing mechanisms.
One important decoherence mechanism is atom loss due to inelastic two-body e − e p-wave
collisions [87, 122], which becomes particularly relevant for a large N . The effect of the losses on
the contrast, however, is largely compensated when the contrast is normalized by the total atom
number, as we do throughout this work. The most relevant contribution to decoherence for thecurrent experiment is the single-particle diffusive dephasing observed in Fig. 7.28. Its effect on
the contrast can already be seen in the non-interacting case (blue circles) and is exacerbated when
operating at the low tunneling rates required to enter the J ∼ |N ξ + /L| regime. We anticipate thatquasimomentum conservation, and signatures of spin-locking at longer times, will be achievable ina 3D optical lattice, where coupling to the thermally populated radial modes would be eliminated.

7.14

Frequency Shift
To complete our full characterization of the spin system and to disentangle the interaction

dynamics from decoherence, we also study the effects of interactions on the phase accumulated
D E D Eby the collective spin vector during the free precession time τ , tan(∆ν2πτ ) = Ŝ Y / Ŝ X . In
optical lattice clocks this is traditionally described by a density-dependent frequency shift [15, 16](∆ν).
For J = 0 (no SOC), ξ + is a constant of motion, and the density shift arises entirely fromthe Ising p-wave interactions. In Fig. 7.34 a density shift measurement without SOC shows a clear
linear dependence on the fraction of the atoms in the excited clock state (also shown on the Bloch
sphere), fully consistent with previous studies [15, 141]. These works were all performed in theregime J = 0, where ∆ν has been well characterized and found to be independent of the dark time
between the Ramsey pulses. In this work the measured shift in the absence of SOC agrees with the

156

Figure 7.34: (a) With J = 0 (no SOC) atoms interact via p-wave collisions only, leading to afrequency shift linearly dependent on excitation fraction, Pe . (b) On the Bloch sphere for θ1 < π/2
the collective vector for atoms with interactions (red) rotates to give a negative phase shift withrespect to the non-interacting (blue) vector. For θ1 > π/2 the interactions lead to a positive shift.

prediction from the mean-field Hamiltonian (Eq. 7.35) ∆νJ=0 = N



χ+
C+
L − L cos θ1, where theshift depends linearly on the fraction of atoms in the excited clock state (Pe = (1 − cos θ1 ) /2).
In contrast, interactions in the presence of SOC give rise to a frequency shift that is dependent
on the dark time between the Ramsey pulses (Fig. 7.35(a)). From this data the frequency shift
extrapolated to zero excitation fraction is seen to diverge when the single-particle contrast decays
to zero (see Fig. 7.35(b)).
For J > 0 (with SOC), the situation becomes more complicated. To develop an intuitive
understanding, we return again to the two atom-group model, where a simple analytic expression
can be derived to first order in interactionsN ξ+
tan(4Jτ )∆ν = ∆νJ=0 −cos(θ1 ) 1 −.
L
4Jτ

(7.36)The same exchange term that produces the time-dependent collective transverse field responsible
for modifying the contrast dynamics also results in a frequency shift. This term diverges whencos(4Jτ ) = 0, which physically corresponds to the case when the two non-interacting atom group

157

Figure 7.35: (a) Density shifts for J/(2π) = 2.2 Hz (with SOC) for different Jτ measured byvarying τ , as indicated in panel (b). The magnitude and sign of the density shift can be seen tovary with time. (b) The non-interacting contrast curve, including additional grey contrast data,
and the ∆νdensity shift for 4000 atoms corresponding to the data in (a) including theory
Pe =0

curves for J > 0 (solid, black) and J = 0 (dashed, black). (c) The divergence can be understoodby considering the Bloch vectors for the different VHSs (dashed arrows) with τ . As the contrast
goes through zero for no interactions (blue), the SOC induced exchange interactions (red) preventthe collective spin vector (red, solid arrow) from going to zero. As the non-interacting collective
spin vector (blue, solid) goes through zero it changes sign, causing a change in sign of the slope ofthe density shift. All error bars are 1σ confidence intervals.

158
vectors are π radians out of phase on the Bloch sphere, as illustrated in Fig. 7.35(c). When thespins rephase and the contrast becomes finite again, the exchange-induced shift diminishes. It
completely turns off in the two atom-group approximation when the spins re-align. However, forthe experimentally relevant case of a thermally populated band with all q values participating, thedensity shift will change in magnitude with time but will not disappear completely, since the spins
do not completely rephase.
The experimentally measured dependence of the SOC density shift on τ at a finite tunnelingrate of J/ (2π) = 2.2 Hz is shown in Fig. 7.35(a)-(b) for N ξ + /L = −2.7 Hz and N χ+ /L = 1.6 Hz.
The observed shift is not entirely linear in excitation fraction, indicating that the interactions canno longer be described by first order perturbation theory, and higher order corrections are required.
Figure 7.35(b) compares the contrast to the extrapolated density shift for zero excitation fraction∆ν
for the same data as in Fig. 7.35(c). The extracted quantity ∆νshows a divergence
Pe =0

Pe =0

around the zero of the contrast, consistent with Eq. 7.36. The highly non-trivial functional form of
the density shift indicates that SOC induced exchange interactions will be a major factor in opticallattice clocks if the effects of tunneling are not suppressed. However, the experimentally observed
density shift and contrast, which encapsulate the magnetization dynamics, can be well describedby theory. This agreement highlights the fact that for the experimentally relevant timescales,
the complex interplay between SOC and many-body dynamics can be understood and explored
precisely.

7.15

Conclusions
In conclusion, we have explored the emergence of complex dynamics with interacting fermions

under engineered spin-orbit coupling in a Sr optical lattice clock. The many-body dynamics are
fully characterized by a collective XXZ Hamiltonian aside from extra dephasing arising from nonconserved quasimomenta. In the future we plan to suppress this dephasing by using more sophisticated pulse sequences[155] or by employing a 3D optical lattice [6], where the p- wave interactions
would also be suppressed. The lower temperatures associated with loading a Fermi-degenerate gas

159
in a 3D lattice geometry will also enable the study of SOC in higher dimensions, precise control of
the SOC phase [157, 140] φ, and exploration of a new strongly interacting regime where the collective XXZ model is no longer applicable, and where richer exotic behaviors including topologicalsuperfluids[158] and Kondo correlated metallic phases can emerge[159].

Chapter 8

Conclusions and Outlook

The work that is presented in this thesis will be useful for future generations of clocks basedon 3D lattice geometries. These clocks are cooled to degeneracy by allowing multiple nuclear
spin states to be present, making the SU(N ) interactions discussed in chapter 6 relevant. Initialstudies of these SU(N ) symmetric interactions have taken place in a 3D lattice[124] with thesemeasurements of few body interactions improving upon the precision of the measured scatteringparameters measured in chapter 6. The close agreement between the values measured in these twostudies confirms the validity of the numerical expressions that were used to determine some of thesescattering parameters.
As these 3D optical lattice clocks move towards Mott and band insulator states [19] the longrange interactions between atoms studied in chapter 4 between atoms on different lattice sites will
become the leading density dependent interaction. The spin-orbit coupling discussed in chapter 7
will also become important as the excited state atomic lifetime in these 3D lattice clocks is reduced
due to Raman scattering of atoms from 3 P0 to 3 P1 . Experimental schemes using lower lattice
depths may therefore be necessary to take full advantage of the full ∼ 160 s lifetime of the excitedclock state.
As mentioned in the introduction, our Sr1 clock experiment has been working successfully
for more than ten years. The future plans of the experiment mean that it is necessary to upgrade
the experimental vacuum chamber to allow for more flexibility with future experiments as well as abetter vacuum lifetime. The new chamber design is shown in Fig. 8.1. It consists of a two-chamber

161
design where one chamber will be used for the MOTs and the other will be the science chamber.
This two-chamber design along with the commercial Zeeman slower from AOSense allow us to avoid
the use of a heated window, which avoids the uncertainty associated with the BBR shift from thisheated window. The atoms will be optically transported between the two chambers and the atoms
can then be cooled to degeneracy before being transported to within an optical cavity where theatoms will be trapped within either a 2D or 3D optical lattice.
The multi chamber design allows the preparation of atoms whilst measurements of the atoms
are taking place. The only deadtime associated with the measurement will be due to the time
it takes to move the atoms from the cooling region in the center of the science chamber to thecavity. It is predicted the deadtime will be as short as 25 ms. The cavity, with high finesse at 689
nm, will be used to create one of the lattice dimensions of the lattice and will be used to create aspin-squeezed clock.

Figure 8.1: The future experimental vacuum chamber design will allow for semi-continuous measurements of degenerately cooled atoms inside a cavity. The cavity will be used to study spin-squeezingof atomic clocks.

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Appendix A

Atom Number Calibration

A.1

Quantum Projection NoiseAs discussed in section 1.4, binomial statistics tells us that the standard deviation of themeasured excitation fraction is given byr
σQP N =pe (1 − pe )
N

(A.1)

where N is the number of atoms measured and p is the mean excitation fraction. We can therefore
use this as a calibration of our atom number. The atom number that we measure will be proportional
to the voltage, VP M T , that we measure using a photomultiplier tube (PMT) and we can therefore
rewrite our standard deviation of the measured excitation fraction assrpe (1 − pe )pe (1 − pe )σQP N =≡NaQP N VP M T(A.2)

where aQP N is a conversion factor between measured voltage and atom number that we wish to
determine for our atom number calibration.
However, during our measurements we are also susceptible to other noise such as electronicnoise that we need to take into account. When we measure the excitation fraction we do this bymeasuring three independent voltages to give the excitation fractionpe =VExc − VBgnd
VExc − VBgnd=
VGnd + VExc − 2VBgnd
VP M T(A.3)

where VExc is the PMT voltage from the excited state atom number, VGnd is the PMT voltage fromthe ground state atom number, and VBgnd is the background voltage of the PMT. If we assume

174

Figure A.1: Measurement of the standard deviation of the excitation fraction, σp , versus
VP M T /(pe (1 − pe )) where VP M T is the voltage measured using a PMT from atomic fluorescence
proportional to the atoms number N . Both the atom number and excitation fraction are variedsystematically to obtain the data (black, squares) and the data is fit (red, circles) with Eq. A.7 toextract the conversion factor between measured voltage and atom number aQP N = 16, 800

Figure A.2: When the data in Fig. A.1 is scaled using the calibrated atom number, it can be seenthat the data agrees well with the predicted pe (1 − pe ) theoretical curve.

175that each of these measurements contains electronic noise each with standard deviation δC whichis shared by VExc , VGnd , and VBgnd then the standard deviation from electronic noise is given by
s
2
2
2


∂p
∂p
∂p
2
2
σelectronic =(δC) +(δC) +
(δC)2
(A.4)
∂VExc
∂VGnd
∂VBgnd
s

2 


1−p 2
−p
−1 + 2p 2= |δC|++(A.5)
VP M T
VP M T
VP M T|δC| p2 − 6p + 6p2(A.6)=
VP M T
Combining this result with the standard deviation from QPN we get the total standard deviation
to bes
σQP N +electronic =p(1 − p)+aQP N VP M T|δC|VP M T2(2 − 6p + 6p2 )(A.7)

Experimentally these measurements can be made by changing both the atom number and theexcitation fraction in a systematic way. An example measurement is shown in Fig. A.1 and A.2.
In this case a π pulse of 1 ms was calibrated and then both the length of the blue MOT loading
time was changed, to change the atom number, for five different pulse times ranging from 0 ms to
1 ms were used to measure the standard deviations of the excitation fraction and gives an atom
number calibration of (16, 800 ± 1, 100) atoms/VP M T

A.2

Transfer Efficiencies
This method of calculating the atom number depends upon knowing the number of atoms

loaded into the blue MOT and then calculating the transfer efficiencies for the different coolingstages. To estimate the number of atoms in the blue MOT we can observe the fluorescence emitted
during the MOT. We know that the 1 S0 −1 P1 cooling transition is almost closed with a small leakage
rate to the 1 D2 where some atoms end up in the metastable 3 P2 state. The branching ratios arewell known such that measurements of the blue MOT lifetime, in the absence of repumps, cantherefore be used to extract an effective saturation parameter for the blue MOT beams. This gives
us a measurement of how long the atoms spend in the 1 P1 state and thus how many photons are176
emitted per atom. The saturation parameter is given in terms of the measured MOT lifetime, τ ,by [78]
2
1+4 ∆
Γ
S0 =
τ /τ0 − 1

(A.8)

where τ0 = 0.00156, and S0 = 0.37 in our most recent measurement.
The excited state fraction can be written as
1
pe =
2



s
1 + s + 4∆2 /Γ2


(A.9)

When we are measuring the fluorescence from the blue MOT we are using a detector that only
measures a fraction of the solid angle fSA . The number of photons emitted per atom is Aρexc whereA = 32 MHz/2π where the number of atoms in the blue MOT is given by
NblueM OT =number of photons detectedfSA Aρexc(A.10)

where we also use the fact that our detector, along with our SRS preamp, has a calibrated signalof 104 nW per Volt at 461 nm. The latest measurement was taken under the same configuration
as discussed in section A.1 giving an atom number of (15, 500 ± 1000) atoms/VP M T where both
independent measurements agree within error bars.
This measurement gives us the number of atoms in the blue MOT. To calculate the numberof atoms in the red MOT we need to calculate the number of atoms transferred to the broadband
red MOT. The probe beam that we use is too small compared to the blue MOT so in order tomeasure the transfer efficiency from the blue MOT we need to use the blue MOT beams instead. It
is very easy to saturate the PMT in this case so the voltage needs to be turned down considerablycompared to our usual measurement of atoms in the lattice or red MOT. Once the transfer efficiency
to the broadband red MOT is measured, the usual blue probe beam is used to measure each of theother transfer stages. In the case of a dense 88 Sr single frequency red MOT the density can gethigh enough that collective effects need to be taken into account and waiting for a longer time offlight will give an increase fluorescence signal.