Quantum Metrology and Many-Body Physics: Pushing the
Frontier of the Optical Lattice Clockby
Michael J. Martin
B.S., Harvey Mudd College, 2006

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
2013This thesis entitled:
Quantum Metrology and Many-Body Physics: Pushing the Frontier of the Optical Lattice Clockwritten by Michael J. Martin
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
Michael J. Martin, (Ph.D., Physics)
Quantum Metrology and Many-Body Physics: Pushing the Frontier of the Optical Lattice Clock
Thesis directed by Prof. Jun Ye

Neutral atom optical standards require the highest levels of laser precision to operate near
the limit set by quantum fluctuations. We develop state-of-the-art ultra-stable laser systems toachieve a factor of 10 enhancement in clock measurement precision and additionally demonstrate
optical linewidths below 50 mHz. The most stable of these lasers reaches its thermal noise floor of
1 × 10−16 fractional frequency instability, allowing the attainment of near quantum-noise-limited√
clock operation with single-clock instabilities of 3 × 10−16 / τ . We utilize this high level of spectral
resolution to operate a 87 Sr optical lattice clock in a regime in which quantum collisions playa dominant role in the dynamics, enabling the study of quantum many-body physics. With afractional level of precision of near 1 × 10−16 at 1 s, we clearly resolve the signatures of many-bodyinteractions. We find that the complicated interplay between the p wave-dominated elastic andinelastic interaction processes between lattice-trapped atoms leads to severe lineshape distortion,
shifts, and loss of Ramsey fringe contrast. We additionally explore the theoretical prediction thatthese many-body interactions will modify the quantum fluctuations of the system and we find thatin certain parameter regimes the quantum noise distribution exhibits a quadrature dependence.
We further present technological advancements that will permit ultra-stable lasers to operate with
reduced thermal noise, leading to a potential gain of an additional factor of 10 in stability. This
indicates that laser fractional frequency instabilities of 1 × 10−17 are within experimental reach, as
is a fully-quantum-limited regime of optical clock operation.

Dedication

To my parents:
For all that you’ve taught me and for your guidance towards this path.
andIn memory of Don and Jean Martin, who strongly supported my education.

v

Acknowledgements

I would like to begin by thanking my parents Marie and Rob who have always supported mewith their love and kept me on the right track. I would also like to acknowledge my stepmother
Caroline, who played a significant role in my life. I would also like to thank my extended family
for their support and love. Finally, I thank my new family-in-law, the Donahues and Boyds, forwelcoming me with such love and kindness.
My interest in physics began at Corvallis High School, where I was fortunate to take Pat
Canan’s physics course. Mr. Canan was a constant guide and mentor who spent countless hours of
his personal time training myself and friend and classmate Connelly Barnes to pursue a spot on thenational physics team. While we didn’t make the final cut, this experience prepared me extremely
well for the rigors of college and got me hooked on physics.
I am indebted to Harvey Mudd College for giving me the opportunity to attend and for thegreat education I received. I would like to specifically thank Professors John Townsend, Thomas
Donnelly, Richard Haskell, and Peter Saeta. Peter directed my research along with three otherstudents with whom I was fortunate to work: Rachel Kirby, Tristan Sharp, and Shannon Woods.
When I arrived at the University of Colorado for graduate school, I was fortunate to become
housemates with a great group of guys, two of whom were JILA physicists. I am grateful for theguidance of both Seth Rittenhouse and Evan Salim, who helped me orient myself as a first-year
student in the graduate program and gave me invaluable advice. Evan also introduced me to thebetter mountain biking trails in the area, and I have enjoyed, in addition to his friendship, biking
with him over the years that I have been in graduate school. I would also like to thank othervi
Boulder-area friends for their support during my time as a graduate student including Nora Mateland Erinn Looney-Triggs.
During my first year at CU, I met Carly Donahue, whom I would later marry. I have enjoyed
our journey through graduate school together and I am a much happier person thanks to her; I
can’t imagine life without her. I am overjoyed that I will soon be joining her in Pasadena andgrateful for her patience as I finished my work in Boulder.
When I first joined Jun Ye’s group at JILA, I came under the mentorship of graduate student
Seth Foreman. Under his direction, I was able to quickly learn the lab technology and get mybearings, all while having a good laugh now and then. I am grateful to Seth for his extremely
generous time expenditures and cheerful, skilled tutelage. I also worked closely with Marcio Miranda
at that time, and I am grateful for his good-natured assistance and friendship.
Other early co-workers I had the opportunity to work with were Andrew Ludlow, Marty
Boyd, Gretchen Campbell, Tanya Zelevinsky, and Thomas Zanon-Willette. Gretchen served as an
mentor during her time with us in the lab, and I am very thankful for her good-spirited tutelage
and her scientific talent. I am also grateful to Marty for serving as a mentor and friend during ouroverlap in Jun’s group, and he and I worked together on the early details of the JILA-PTB siliconcavity project reported in this thesis. I am also grateful for continued interaction with Andrew
Ludlow, now with the NIST Time and Frequency Division.
Over the last 6.5 years I have been privileged to overlap significantly with a lot of greatlabmates. On the Sr1 experiment, I was very fortunate to work with postdoc Matt Swallows, atalented physicist and good friend. I learned a great deal of physics from Matt through his generous
and thoughtful teaching. I always appreciated the fact that Matt was willing to take the time toshare his encyclopedic knowledge and discuss (crazy) ideas objectively. Perhaps more importantly,
Matt also exemplified the traits of a good scientist and set a standard to which I aspire. Visiting
Fellow and collaborator Jan Thomsen has been an amazing co-worker and friend. I have also greatly
enjoyed working with another Sr1 teammate, Mike Bishof. Mike is a very talented scientist and I
am continually impressed with how easy he makes it all look. Another addition to the Sr team forvii
a year or so was Craig Benko, who eventually joined the XUV comb project. Craig is very talented
and thoughtful, and I am indebted to him for his help with some of the less-than-glamorous aspectsof setting up the Big ULE ultrastable laser, which he rendered in good spirits. A visitor to ourlab, Yige Lin, worked with us on the Sr1 project and made significant contributions and I’m gladfor the time that we had together in the lab. Finally, I have had the pleasure to work with arecent addition to Sr1, Xibo Zhang. Xibo is a very deep thinker and I have enjoyed our scientificdiscussions. With Mike and Xibo at the helm, I have no doubt that Sr1 is on the right path.
In addition to my Sr1 teammates, I have had the pleasure to work with many talented folksfrom the Sr2 project. Sebastian Blatt was one of the Sr labmates with whom I have overlapped thelongest. I always enjoyed talking to him about physics and I have learned a lot from him. I will
miss the feuerzangenbowle he would share at holiday parties. Travis Nicholson has also been in thelab nearly as long as I have. I’ve always enjoyed discussing physics with Travis due to his broad
knowledge, and passion. Ben Bloom and I have also overlapped considerably, and I’ve always
appreciated his rigorous approach and thoughtfulness. Newer additions to Sr2 have been Jason
Williams and Sara Campbell. Jason has been a wonderful co-worker due to his skill, knowledge,
and kindness. Sara is the newest addition to the “Sr gang” and I have no doubt that through her
enthusiasm, energy, and talent the long-term success of the Sr projects will be ensured.
I would also like to thank Dylan Yost, Thomas Schibli, Mike Thorpe, Matt Stowe, MarkYeo, Ben Stuhl, Matt Hummon, Brian Sawyer, Florian Adler, Tom Allison, Arman Cingoz, Steven
Moses, and Kevin Cossel for their assistance over the years. Kevin has been particularly generous
with his time and energy, and I’m indebted to him for several great suggestions, his technicalexpertise, and kind assistance on more than one occasion. We’ve also shared some (somewhat epic)
experiences in the wilds of Colorado that I will certainly never forget!
In developing next-generation ultrastable laser systems, I have been very fortunate to workwith some amazing people at JILA. It has been a great honor and pleasure for me to work with Jan
Hall. Jan is a founder of the field of laser stabilization and has been a driving force in AMO science;
his expertise was invaluable for the development of the laser systems reported in this thesis. Janviii
was also a most gracious labmate—especially when we were making all sorts noise and commotion
in his (nominally) “quiet room” during the installation of the new laser system. I have greatly
enjoyed discussions with Jan ranging from science to life. He is a unique and powerful intellect, notto mention a great person. Mark Notcutt was an early collaborator in designing some of the laser
systems reported here, and a long-term leader in the field of laser stabilization. Lisheng Chen wasa visitor during one summer, and I learned a lot from him in the months that we worked together.
A recent addition to the Ye labs frequency comb and ultrastable laser development team has been
postdoc Wei Zhang. Wei has been an amazingly hard-working and good-natured colleague and I
am confident that he will realize another order of magnitude in laser stability. Recently, we havebegun a collaboration with Markus Aspelmeyer and Garrett Cole to develop new crystalline mirror
coatings, the first results of which are reported in this thesis. I’ve greatly enjoyed the opportunity
to interact with them both.
The silicon cavity PTB/JILA collaboration has yielded exciting scientific results, and I am
also grateful for the opportunity to get to know our colleagues at PTB. I worked very closelywith Thomas Kessler and Christian Hagemann on characterizing the silicon cavity. Thomas andChristian were great hosts, and I thank them for their generosity. It was also a great pleasure towork with Fritz Riehle, Uwe Sterr, and Thomas Legero, and I thank them for the great opportunity
to participate in the collaboration. Other PTB colleagues with whom I enjoyed discussions are
Christian Lisdat, Stephan Falke, Thomas Middelmann, Christian Grebing, Nils Huntemann, Harald
Schnatz, Gesine Grosche, Eckhard Peik, Piet Schmidt, Olaf Mandel, and Sana Amairi.
JILA and NIST are full of remarkably talented scientists, and I am grateful for the interactions
I have had with other groups. Ana Maria Rey has been a wonderful collaborator and teacher over
the years that we have worked with her in trying to understand the intricacies of the inter-atomic
interactions within the lattice clock. Murray Holland, Dominic Meiser have driven the development
cavity QED-inspired ideas employing Sr. James Thompson and Zilong Chen helped develop theideas explored in Chapter 6 of this thesis. I would also like to thank Scott Diddams, Jeff Sherman,
Tara Fortier, Andrew Ludlow (again), Nathan Lemke, and Chris Oates at the NIST Time andixFrequency Division.
I have also benefitted from interactions with visiting scientists over the years. Professor
Long-Sheng Ma has been a recurring and cherished visitor. I have always been amazed by his
subtle and remarkably apt questions regarding our exeperiments. During Roee Ozeri’s visit, wefirst began to discuss the possibility of the atomic spectrum analyzer (Chapter 6). I also enjoyedRana Adhikari’s visit to our lab, and thank him for fruitful discussions. Ingmar Hartl, Axel Ruehl,
and Martin Fermann have taught me much of what I know about Yb:fiber lasers.
I would additionally like to thank the JILA personnel who have been so instrumental (punintended) in making everything work. From the instrument shop Kim Hagen, Hans Green, Tracy
Keep, Tom Foote, Todd Asnicar, David Alchenberger, Ariel Paul, and Blaine Horner have all
played central roles in helping construct apparatus that are central to our experiments. I am
specifically indebted to Kim, Hans, and Todd for their great work in helping me construct the “BigULE” system. From the electronics shop, I have greatly benefitted from the assistance and skill of
Terry Brown, James Fung-a-Fat, Mike Whitmore, Carl Sauer, David Tegart, and Paul Beckingham.
Terry, especially, has shown great patience in answering any questions I might have. I would also
like to thank Diane Moreland for her assistance over the years.
Finally, I would like to thank my advisor, teacher, and guide over these last 6.5 years, Jun
Ye. It should be clear from the list of scientists and coworkers whom I have thanked above that Jun
runs a dynamic, productive, and talented group at the cutting edge of many fields within atomic,
molecular, and optical science. He is a remarkable scientist and I always marvel at his ability tosuccessfully guide so many different projects to such high levels of success. He is also a kind andgenerous person, which is mirrored in the lab atmosphere. For Jun’s commitment to his students’success I am extremely grateful.

My thanks to all!

-MJM, 4/19/2013

Contents

Chapter
1 Introduction

1

1.1

Historical discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Precision spectroscopy and atomic clocks. . . . . . . . . . . . . . . . . . . . . . . .

2

Ultrastable lasers in optical clocks . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3

Characterizing the stability of clocks . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.4

Performance of neutral-atom clocks . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.5

Frequency combs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.6

Beyond timekeeping: Clocks in modern physics . . . . . . . . . . . . . . . . . . . . . 11

1.7

Overview of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.1

2 The 87 Sr optical lattice clock

13

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2

Narrow-line laser cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3

Lattice dipole trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4

2.3.1

Atomic polarizability calculation . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.2

One-dimensional optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Experimental apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.1

Laser cooling at 461 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.2

Narrow line cooling with 87 Sr. . . . . . . . . . . . . . . . . . . . . . . . . . 30

xi

2.5

2.4.3

Optical dipole trap and clock spectroscopy: experimental configuration

2.4.4

Atom state detection and measurement . . . . . . . . . . . . . . . . . . . . . 40

Lattice spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5.1

2.6

. . . 36

Lattice spectroscopy: experimental . . . . . . . . . . . . . . . . . . . . . . . . 45

Limits to atom-laser coherence due to photon scattering . . . . . . . . . . . . . . . . 48
2.6.1

Kramers-Heisenberg formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.7

In situ thermometry and cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.8

Systematic uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.8.1

2.9

Blackbody radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Quantum projection noise and the Dick effect . . . . . . . . . . . . . . . . . . . . . . 61

3 Thermal noise in optical interferometers

67

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.2

Thermo-mechanical noise in physical systems . . . . . . . . . . . . . . . . . . . . . . 68

3.3

3.4

3.2.1

Fluctuation-dissipation theorem and Levin’s “direct approach” . . . . . . . . 70

3.2.2

Brownian motion in optical cavities

Thermodynamic noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3.1

Substrate thermoelastic noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.3.2

Coating thermo-optic noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Total thermal noise contribution to cavity frequency stability . . . . . . . . . . . . . 81
3.4.1

3.5

. . . . . . . . . . . . . . . . . . . . . . . 73

Examples of thermal noise floor . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4 High-precision laser stabilization with optical cavities

90

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.2

Review of optical cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.2.1

Measuring Finesse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.2.2

Pound-Drever-Hall Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

xii

4.3

4.2.3

Sources of lock error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.2.4

Choice of EOM crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.2.5

Active control of residual amplitude modulation . . . . . . . . . . . . . . . . 107

Mechanical design of optical reference cavities . . . . . . . . . . . . . . . . . . . . . . 109
4.3.1

4.4

4.5

Vibration sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Designing two ultrastable lasers for the 10−16 level performance . . . . . . . . . . . . 111
4.4.1

Design of the “Big ULE” cavity . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.4.2

Design of the “JILA cavity” . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.4.3

Mesh size effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.4.4

Thermo-mechanical perturbations . . . . . . . . . . . . . . . . . . . . . . . . 122

A near quantum-limited lattice clock . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.5.1

Experimental details of the Big ULE . . . . . . . . . . . . . . . . . . . . . . . 126

4.5.2

Verification of thermal-noise limited performance . . . . . . . . . . . . . . . . 131

4.5.3

High stability for rapid systematic evaluation . . . . . . . . . . . . . . . . . . 131

4.5.4

Clock comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.6

PTB Si cavity and JILA cavity intercomparison . . . . . . . . . . . . . . . . . . . . . 138

4.7

Crystalline mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.8

Frequency comb development: bridging spectral gaps coherently . . . . . . . . . . . 148

4.9

4.8.1

Ultra-broadband Yb:fiber comb . . . . . . . . . . . . . . . . . . . . . . . . . . 149

4.8.2

Ultimate limits of comb performance . . . . . . . . . . . . . . . . . . . . . . . 151

Outlook and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

5 The optical lattice clock as a quantum many-body system
5.1

5.2

156

Atomic collisions at ultracold temperatures . . . . . . . . . . . . . . . . . . . . . . . 156
5.1.1

Partial wave treatment of atomic interactions . . . . . . . . . . . . . . . . . . 157

5.1.2

Considerations for clock shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

First observations of cold-collisions in an optical lattice clock . . . . . . . . . . . . . 160

xiii
5.2.1
5.3

5.4

Treatment of the clock shift in a many-body framework . . . . . . . . . . . . . . . . 164
5.3.1

Second quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.3.2

Many-body Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

5.3.3

Writing the many-body Hamiltonian with spin operators . . . . . . . . . . . . 169

5.3.4

Adding the atom-laser interaction . . . . . . . . . . . . . . . . . . . . . . . . 172

S-wave model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.4.1

5.5

5.6

5.7

5.8

5.9

An s-wave collisional mechanism in a spin-polarized 87 Sr lattice clock . . . . 162

Concluding remarks regarding s-wave interactions . . . . . . . . . . . . . . . 176

New experimental evidence for p-wave interactions . . . . . . . . . . . . . . . . . . . 177
5.5.1

Inelastic loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

5.5.2

Lineshape distortion and loss of Ramsey fringe contrast . . . . . . . . . . . . 181

p-wave many-body model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.6.1

Collective model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

5.6.2

The collective model and density shifts in a 2D optical lattice . . . . . . . . . 189

Mean-field and perturbative tests of the many-body dynamics . . . . . . . . . . . . . 189
5.7.1

Perturbative tests of the many-body Hamiltonian . . . . . . . . . . . . . . . . 190

5.7.2

Mean-field solution of the collective model . . . . . . . . . . . . . . . . . . . . 192

5.7.3

Density shift in the mean field limit . . . . . . . . . . . . . . . . . . . . . . . 193

5.7.4

Measuring the density shift with a Ramsey interrogation scheme . . . . . . . 194

5.7.5

Describing spectroscopic lineshapes with the mean-field model . . . . . . . . 199

Beyond mean-field effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
5.8.1

Perturbative corrections to the spin model . . . . . . . . . . . . . . . . . . . . 204

5.8.2

Truncated Wigner approximation . . . . . . . . . . . . . . . . . . . . . . . . . 205

5.8.3

Ramsey fringe contrast decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

5.8.4

Quadrature-dependent noise measurement . . . . . . . . . . . . . . . . . . . . 210

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

xiv
6 Measuring laser noise via quantum state manipulation

217

6.1

Introduction to the general formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 217

6.2

Limit of infinitely short pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

6.3

6.2.1

Computing variances from the sensitivity function . . . . . . . . . . . . . . . 221

6.2.2

Computing the variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

6.2.3

Computing the Allan variance

6.2.4

The sensitivity function R (f ) and experimental results . . . . . . . . . . . . 225. . . . . . . . . . . . . . . . . . . . . . . . . . 224

Adding the effects of frequency noise during the pulses . . . . . . . . . . . . . . . . . 227
6.3.1

Sensitivity function for Rabi spectroscopy . . . . . . . . . . . . . . . . . . . . 234

6.4

Mapping the laser spectrum: multi-echo sequences . . . . . . . . . . . . . . . . . . . 235

6.5

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

7 Conclusion

239

7.1

Summary of key results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

7.2

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

Bibliography

244

Appendix
A Coherent atom-light interactions

264

A.1 Derivation of the Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
A.2 Time evolution operator for lineshape calculation . . . . . . . . . . . . . . . . . . . . 266
A.2.1 Rabi lineshape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
A.2.2 Ramsey lineshape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

A.2.3 Dick sensitivity function for Rabi spectroscopy . . . . . . . . . . . . . . . . . 269
A.2.4 Dick sensitivity function for Ramsey spectroscopy . . . . . . . . . . . . . . . 270

xv
B Calculating the dipole moments from measured atomic lifetimes

272

C Spherical tensor treatment of the light shift

276

D The s- and p-wave interaction overlap integrals

280

E Site occupancy distribution

283

E.1 Sites with n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
E.2 Linearity of the density shift and average occupancy . . . . . . . . . . . . . . . . . . 285

xvi

TablesTable
1.1

Common Allan variances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1

Upper states used in calculating the polarizability of the 1 S0 and 3 P0 clock states. . 23

2.2

Scattering rates from 3 P0 into the 3 P fine structure manifold and 1 S0 Rayleigh scattering at the magic wavelength.

2.3

7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Systematic frequency shifts and their uncertainties for the 87 Sr frequency standard(as in Refs. [1, 2]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.1

Summary of the parameters used in the text. . . . . . . . . . . . . . . . . . . . . . . 75

3.2

Mechanical properties of typical optical materials. . . . . . . . . . . . . . . . . . . . 75

3.3

Thermal and thermo-optic/optical properties of common optical materials at room
temperature and at 1064 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.1

Zero-crossing sensitivities obtained from the finite element modeling. . . . . . . . . . 121

Figures

Figure1.1

Coherent atom-light interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Ramsey and Rabi spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

Schematic diagram of a frequency standard. . . . . . . . . . . . . . . . . . . . . . . .

5

1.4

Graphical depiction of the sampling method employed to calculate the Allan deviation of a continuous measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.5

Frequency comb and optical clocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1

Strontium level diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2

Polarizability of the 1 S0 and 3 P0 clock states. . . . . . . . . . . . . . . . . . . . . . . 24

2.3

Experimental apparatus for trapping ultracold 87 Sr . . . . . . . . . . . . . . . . . . . 27

2.4

Hyperfine structure of the 5s5p1 P1 and 5s5p3 P1 levels . . . . . . . . . . . . . . . . . 29

2.5

Schematic depiction of the 689 nm MOT for the F = 9/2 → F = 11/2 transition. . . 31

2.6

Ratio of σ ± absorption as a function of ground state magnetic sublevel, mg .

2.7

Field gradient programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.8

Clock operation with stretched nuclear spin states for first-order Zeeman insensitivity 36

2.9

Optical lattice dipole trap and clock laser pathway . . . . . . . . . . . . . . . . . . . 37

. . . . 32

2.10 The clock laser delivery system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.11 Atomic state detection scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.12 Spectroscopy in a one-dimensional optical lattice . . . . . . . . . . . . . . . . . . . . 43

xviii2.13 Spectroscopic resolution of longitudinal and radial sidebands in a 1D optical latticepotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.14 Spectroscopic lineshape obtained with Rabi spectroscopy performed on the 1 S0 →3 P0clock transition at 429 THz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.15 Rabi flopping as a probe of the spread in Rabi frequency. . . . . . . . . . . . . . . . 46
2.16 Raman scattering rate as a function of wavelength . . . . . . . . . . . . . . . . . . . 51
2.17 Rayleigh scattering rate as a function of wavelength . . . . . . . . . . . . . . . . . . 52
2.18 Experimental measurement of optical pumping due to lattice scattering. . . . . . . . 54
2.19 Temperature extraction and effect of additional cooling . . . . . . . . . . . . . . . . 56
2.20 Two-dimensional lattice geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.21 Ramsey and Rabi lineshapes compared for quantum projection noise calculation. . . 62
2.22 Fourier and time-domain representation of the Dick sensitivity function and its implications for reaching quantum-limited clock performance. . . . . . . . . . . . . . . 64
3.1

Modified Hooke’s law system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.2

Schematic of the relevant optical components for calculating the thermal noise of an
optical cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.3

Thermal noise of cavity A with SiO2 /Ta2 O5 mirrors. . . . . . . . . . . . . . . . . . . 84

3.4

Thermal noise of cavity A with GaAs/AlGaAs mirrors. . . . . . . . . . . . . . . . . . 85

3.5

Thermal noise of cavity A with GaAs/AlGaAs mirrors and a half-wave cap. . . . . . 86

3.6

Thermal noise of cavity B with SiO2 /Ta2 O5 mirrors, T = 124 K. . . . . . . . . . . . 87

3.7

Thermal noise of cavity B with GaAs/AlGaAs mirrors, T = 124 K. . . . . . . . . . . 88

3.8

Thermal noise of cavity B with GaAs/AlGaAs mirrors and an additional half-wave
cap, T = 124 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.1

Schematic of an optical cavity in the standing wave configuration. . . . . . . . . . . 93

4.2

Reflection coefficient and corresponding phase shift of the reflected light incident
upon an optical cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

xix
4.3

Measurement of cavity photon storage lifetime. . . . . . . . . . . . . . . . . . . . . . 97

4.4

Measurement of photon storage time with a swept-frequency optical field. . . . . . . 99

4.5

Pound-Drever-Hall sidebands and error signal.

4.6

LiNbO3 and ADP modulator configurations. . . . . . . . . . . . . . . . . . . . . . . . 105

4.7

Experimental setup for active control of residual amplitude modulation (RAM) in a

. . . . . . . . . . . . . . . . . . . . . 101

waveguide electro-optic modulator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.8

Experimental demonstration of reduced residual amplitude modulation (RAM) . . . 108

4.9

The finite element model considered for the “Big ULE” cavity. . . . . . . . . . . . . 112

4.10 Two dimensional planar fits to offset (a) and tilt (b) as a function of z and d. . . . . 114
4.11 The Big ULE cavity and Zerodur support structure. . . . . . . . . . . . . . . . . . . 115
4.12 The double-layer vacuum system for the Big ULE. . . . . . . . . . . . . . . . . . . . 116
4.13 The Big ULE cavity with its new Zerodur glass support structure. The cavity rests
upon Viton hemispheres (not shown). . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.14 The JILA cavity design and vacuum system.

. . . . . . . . . . . . . . . . . . . . . . 119

4.15 FEA simulation of the “JILA cavity.” . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.16 Effect of mesh size on the FEA-predicted mirror displacement. . . . . . . . . . . . . 122
4.17 Temperature dependence of coefficients of thermal expansion (CTEs) of common
cavity spacer and substrate materials. . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.18 Deformation of mixed-material optical cavities . . . . . . . . . . . . . . . . . . . . . 124
4.19 Net coefficient of thermal expansion for the Big ULE assuming a spacer T0 of 15.4◦ C.125
4.20 Two-stage Pound-Drever-Hall (PDH) locking for laser stabilization with the Big ULE.128
4.21 Effect of the sound-proof box on the optical frequency spectrum. . . . . . . . . . . . 130
4.22 Demonstration of a thermal-noise limited laser at fractional instability of 1 × 10−16
from 1–1000 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.23 Ultra-narrow Rabi lineshapes demonstrating 3-s optical-atomic coherence. . . . . . . 133
4.24 Stability of interleaved measurements for systematic evaluation with Rabi pulse duration of 500 ms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

xx
4.25 Clock comparison scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.26 Clock comparison Allan deviations and pulse timing. . . . . . . . . . . . . . . . . . . 137
4.27 The silicon cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.28 Direct optical beat between the JILA cavity-stabilized laser and the Si cavitystabilized laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.29 Modified Allan deviation of the stabilities extrapolated via the three-cornered-hat
analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.30 Crystalline mirror and cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.31 Comb frequency-transfer and stability of a laser locked to the cavity employing
GaAs/AlGaAs crystalline mirrors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.32 Spectrum of the ultra-broadband Yb:fiber comb and optical beats. . . . . . . . . . . 149
4.33 Spectrum of broadened Yb oscillator and relevant beat notes at 698 nm and 1.54 µm.150
4.34 Experimental test of the intrinsic limits of the an optical frequency comb. . . . . . . 153
5.1

First observation of a density shift with a spin-polarized 87 Sr lattice clock. . . . . . . 161

5.2

Harmonic approximation compared to the true radial profile of the trap. . . . . . . . 168

5.3Interaction motional eigenstates within a single trap site. . . . . . . . . . . . . . . . 169

5.4

Two-particle illustration of the s-wave shift mechanism. . . . . . . . . . . . . . . . . 173

5.5−
Two-particle lineshapes with s-wave interaction energy. (a) U−
12 = 0 (b) U12 = 1Ω−(c) U−
12 = 2Ω (d) U12 = 5Ω. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

5.6

The s-wave density shift and the strongly interacting regime. . . . . . . . . . . . . . 176

5.7

Atom loss as a function of lattice hold time. . . . . . . . . . . . . . . . . . . . . . . . 178

5.8

Two-body loss coefficient γee for spin polarized 87 Sr in 3 P0 |F = 9/2, mF = +9/2i . 180

5.9

Broadened lineshape with 1 s atom-light interaction time. . . . . . . . . . . . . . . . 181

5.10 Low density lineshape vs. high density lineshape with lattice depth consistent withνz = 80 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

xxi
5.11 Narrow line scan (1 s probe time) with 40% typical lattice power and a sample
temperature of 1 µK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
5.12 Comparison of two lineshapes taken under similar atom number conditions of Ntot '5000.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

5.13 Ramsey fringes contrast and two body-loss. . . . . . . . . . . . . . . . . . . . . . . . 185
5.14 Visual depiction of the collective model. . . . . . . . . . . . . . . . . . . . . . . . . . 187
5.15 Temperature dependence of the p-wave interaction energies and their thermal distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
5.16 Spectroscopy of the clock transition in the linear response regime with pulse area
fixed to θ = 0.2π. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.17 Effect of two-body losses on the density shift measurement for two test cases, with
χ = 2π × 0.2 Hz, C = −0.3χ, and Γee = 2π × 0.07 Hz. . . . . . . . . . . . . . . . . . 195
5.18 Timing sequence and example of inter-sequence excitation fraction measurement. . . 197
5.19 Measured Ramsey shift as a function of the scaled excitation fraction f .

. . . . . . 198

5.20 Lineshape (red circles) and normalized total atom number (black circles) as a function of detuning and for decreasing Ω ' π/tR . . . . . . . . . . . . . . . . . . . . . . 200
5.21 Lineshape as a function of experimental parameters. . . . . . . . . . . . . . . . . . . 202
5.22 Interaction effects visible at the sub-Hz energy scale. . . . . . . . . . . . . . . . . . . 203
5.23 Ramsey contrast decay measurement scheme and the failure of mean-field model. . . 206
5.24 Ramsey fringe contrast decay vs. initial pulse area. . . . . . . . . . . . . . . . . . . . 208
5.25 Effect of the (S z )3 term on the Ramsey fringe contrast. . . . . . . . . . . . . . . . . 209
5.26 Spin noise measurement sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
5.27 Bloch sphere evolution as a function of dark time for Ntot = 20. . . . . . . . . . . . . 212
5.28 Spin noise vs. rotation quadrature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
5.29 Magnified spin noise vs. rotation quadrature. . . . . . . . . . . . . . . . . . . . . . . 216
6.1

The spin echo sequence being considered. . . . . . . . . . . . . . . . . . . . . . . . . 219

xxii
6.2

The sensitivity function, r (t) of a four-pulse echo sequence in the limit of infinitely
short π-pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

6.3

Diagnosing the laser spectrum with spectroscopy. . . . . . . . . . . . . . . . . . . . . 226

6.4

The time-domain sensitivity function r(t) for the echo sequence with 80 ms durationand 20 ms π-pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

6.5

Verification of improved laser noise spectrum with a spin echo sequence. . . . . . . . 232

6.6

Overview of the sensitivity functions |R (f )|2 for increasing echo pulse number. . . . 233

6.7

Linear scale plots of the Fourier-space sensitivity functions. . . . . . . . . . . . . . . 235

6.8

Multi-pulse extrapolation of laser noise spectrum. . . . . . . . . . . . . . . . . . . . . 236

6.9

Dick effect-limited laser performance with the newly evaluated laser spectrum. . . . 238

A.1 Rabi spectroscopy lineshape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
A.2 Ramsey spectroscopy lineshape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
A.3 Rabi and Ramsey sensitivity functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 270
D.1 s- and p-wave mode-dependent interaction strength coefficients. . . . . . . . . . . . . 282
E.1 Lattice loading from the red MOT and longitudinal distribution. . . . . . . . . . . . 284
E.2 Distribution of site occupancies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
E.3 Average site occupancy as a function of total atom number. . . . . . . . . . . . . . . 286

Chapter 1

Introduction

Many of the tools that are crucial for optical atomic clocks and high-precision optical spectroscopy played a prominent role in 20th century physics. Likewise, the development of both
microwave and optical frequency standards has been an important driving force in quantum measurement and atomic, molecular, and optical science. The technology utilized in today’s stateof-the-art optical frequency standards has spurred numerous new research directions, including
optical frequency combs, ultrastable optical interferometers, and interacting quantum gases. Here
we provide an introduction to optical frequency standards, their key components, and metrics forevaluating their performance.

1.1

Historical discussion
Precision spectroscopy has played a foundational role in our understanding of modern physics.

Radio-frequency spectroscopy enabled early measurements of nuclear magnetic moments [3, 4], theLamb shift in atomic hydrogen [5], and the accurate determination on an anomalous electron
g-factor [6]. These measurements each represent experimental milestones that spurred theoreticaldevelopment in atomic and nuclear physics as well as quantum optics and quantum electrodynamics.
The pioneering work of Norman Ramsey in developing the method of separated oscillatory fields[7] set the stage for the development of radio frequency (RF) frequency standards by enabling theaccurate and precise determination of resonances in atomic beams. The Cs fountain clocks thatdefine the SI second today operate using Ramsey’s scheme, and the precise RF clocks flying aboard

2
GPS satellites have fundamentally changed the way much of the world navigates.
With the invention of the laser [8], the field of laser-based optical spectroscopy was born,allowing optical measurement of fundamental quantities, such as the Lamb shift [9, 10]. Furthermore, optical analogues of the Ramsey separated fields technique were demonstrated in the opticaldomain [11, 12], representing the first demonstrations of coherent optical spectroscopy and setting
the stage for high resolution spectroscopy of narrow optical transitions in molecules, atoms, andions.
The advent of laser cooling and trapping [13, 14, 15, 16, 17, 18, 19] represents anothergroundbreaking milestone for modern science and has opened up the quantum world to direct
laboratory exploration and manipulation. With laser-cooled atoms and ions, coherence times for
optical (e.g., [20, 21, 22]) and microwave (e.g., [23]) spectroscopy were dramatically improved. Thelong coherence times were a direct contributor to the success of frequency standards based on laser
cooled atoms and ions that emerged from this work, and directly enabled the most precise andaccurate clocks in existence today.

1.2

Precision spectroscopy and atomic clocks
The physical basis for precision spectroscopy and atomic timekeeping is the relationshipbetween energy and frequency, given by Plank’s famous equation relating the frequency of photon
to its energy∆E = hν,(1.1)

where ∆E is the energy between the ground and excited clock states (g and e, respectively), h is
Planck’s constant and ν is the transition frequency (i.e., the frequency of the absorbed or emitted
radiation). Optical and microwave transitions between metastable states in ions and atoms makegood frequency references because they are based on quantum systems whose transition frequencies
depend very directly on the fundamental laws of nature, and should be perfectly reproducible fromone realization to another. In many cases, g is the energetic ground state of the atom or ion,

3
Freq
ue

ncy
det
u

e

nin
g

Excited atomsΩ

guration
Pulse dFigure 1.1: Coherent spectroscopy in a 87 Sr optical lattice clock. The atomic population undergoes
a series of collapses and revivals as a function of laser detuning and pulse duration, demonstratinga coherent regime of spectroscopy.

while e is a highly metastable, excited state whose natural lifetime exceeds all experimentally
relevant timescales. As a result, the techniques of coherent spectroscopy, as first developed by I.
I. Rabi and N. H. Ramsey for nuclear magnetic resonance measurements, are still utilized today,from RF frequencies to ultraviolet optical frequencies. Figure 1.1 depicts an example of coherent
spectroscopy using lattice-trapped and spin-polarized 87 Sr—the 87 Sr optical lattice clock that is
the subject of this thesis. In Chapter 2 we provide a complete description of the 87 Sr optical latticeclock. Briefly, an ultrastable laser is tuned such that the photon energy is nearly matched to ∆E
and the pulse is applied for a given duration. The duration is chosen such that the maximum
population transfer occurs on resonance for a given coupling strength between the laser and atom
called the “Rabi frequency.” For an electric dipole-allowed transition, the Rabi frequency Ω is givenbyΩ = hg|d̂ · E0 |ei/~,

(1.2)

4
Ramsey spectroscopy

Rabi spectroscopy

Time

Time

0.8

Excitation fraction

Excitation fraction

1.0

0.6
0.4
0.2
0.0
1000

500

0

500

Detuning (Hz)

1000

0.8
0.6

FWHM: 5 Hz

0.4
0.2
0.0
-40

-20

0

Detuning (Hz)20

Figure 1.2: Spectroscopic lineshapes obtained via Ramsey and Rabi spectroscopy in a 87 Sr latticeclock. Both techniques are utilized in optical clocks. The temporal width of the features in both
Ramsey and Rabi spectroscopy are inversely proportional to the total spectroscopy time. For
the Rabi spectroscopy lineshape shown at right, the probe time was 160 ms, corresponding to an
expected frequency full-width at half-maximum (FWHM) of 5 Hz.

where d̂ is the dipole operator d̂ ≡ er̂, e is the electron charge, and r̂ is the position operator. HereE0 is the amplitude of the laser electric field. As can be seen in Fig. 1.1, the population of atoms
in e undergoes periodic collapses and revivals depending on the laser detuning from resonance, andthe duration the pulse is applied. This is the hallmark of coherent spectroscopy; the atom-field
coupling is the relevant part of the interaction, while spontaneous emission occurs so infrequently
that it does not play a large role in the dynamics of the population evolution. Figure 1.2 shows anexample of each Rabi and Ramsey spectroscopy utilized in the 87 Sr optical lattice clock. The fit
lineshapes represent the ideal case of an isolated two-level system and are derived in Appendix A.
In order to consider a given ultranarrow resonance in an atom or ion a good candidate
for a frequency standard, the transition of interest should exhibit strong immunity to external
perturbations, such as magnetic fields. Additionally, atoms or ions serving as optical standards are
typically trapped sufficiently tightly that problems such as Doppler broadening and recoil shifts
can be mitigated by the tight trap.1
1

These traps consist of RF Paul traps for ions [24] andIn RF frequency standards, the effect of recoil is negligible and the Doppler effect is vastly reduced.

5

Discriminator

LO

Figure 1.3: Schematic diagram of a frequency standard. The local oscillator (LO) probes the atomic
system used as a frequency reference (solid line). The information gained from this measurement
is used to correct the LO frequency such that it is actively locked to atomic transition.

engineered, standing-wave optical dipole traps at the “magic-wavelength” for neutral atom opticalclocks [25, 26], also known as optical lattices. In addition to these general considerations, thetransition used for the clock must be sufficiently narrow to provide a sharp frequency discrimination
signal. This last condition is met by using multiply forbidden transitions in the clock atoms orions, resulting in extremely long excited state lifetimes (in some cases > 100 s) and correspondingly
very narrow resonance linewidths.
Atomic clocks of all types (microwave, optical trapped ion, and optical neutral atom) all relyon the same principle of operation. An oscillator with good short-term stability, the local oscillator(LO), is used to interrogate a transition in the ion or atomic ensemble as shown in Figure 1.3. The
LO very precisely probes the transition, and the information is then used to correct the LO suchthat it is maintained on resonance.
Optical atomic clocks have now reached unprecedented levels of stability and accuracy. At
the forefront of accuracy, a clock located at the National Institute of Standards and Technology

6(NIST), based on a single aluminum ion and probed via quantum logic spectroscopy now has afractional frequency uncertainty of 8.6 × 10−18 [27]. A second clock based on a mercury ion, also at
NIST, is at the 2×10−17 fractional frequency uncertainty level [28]. Additionally, a Sr+ ion operated
by the National Research Council of Canada has recently attained an uncertainty of 2.2 × 10−17[29]. Finally we note that the PTB Yb+ ion clock has reached an uncertainty of 7.1 × 10−17 [30].
These remarkable advances in ion-based clocks are followed closely by a new class of optical clocks
based on ensembles of ultracold neutral atoms held tightly in optical lattices. The most accurate
at the level of 1 × 10−16 fractional uncertainty [31, 32, 33, 34], with the 87 Sr lattice clock currently
the most accurate of the neutral atom-based clocks. The Sr lattice clock also has the best record ofinternational agreement in measurements against the Cs primary frequency standard of any opticalstandard [35, 34].

1.2.1

Ultrastable lasers in optical clocks

Central to the performance of any frequency standard is the stability of its local oscillator.
The task of the local oscillator is twofold: it must probe the reference transition in the quantum
reference while introducing minimal noise through an aliasing process known as the Dick effect[36, 37], and it must act as a “flywheel” to maintain the short- to medium-term stability of thefrequency reference. These two considerations alone have spurred massive development efforts inultrastable lasers for use with optical standards (e.g., [38, 39, 40, 41, 42, 43]), which stand the mostto gain from improvements in the local oscillator.
From a broader standpoint, optical interferometers are found at the heart of many diverse experiments that perform the highest-precision laser-based measurements, from quantum mechanicalto cosmological scales. Chapter 3 explores key limitations to optical interferometers: the detrimental thermal coupling to the environment arises as a direct consequence of mechanical losses in theinterferometer substrates and coatings, which in turn limit laser stability. Chapter 4 specifically
discusses the development of three ultrastable laser systems for use with optical clocks, including
the construction of an ultrastable laser with fractional instability at 1 × 10−16 from 1–1000 s andNoise type
White
FlickerGν (f )h0
h−1 f −1

Random walk

h−2

f −2

σy2 (τ )7

h02τ

2 log 2h−1
2π 2 h−2 τ
3Table 1.1: Common Allan variances for different noise spectra (as in [45]).

a linewidth as low as 26 mHz.

1.3

Characterizing the stability of clocks
Forming metrics for the stability of atomic clocks is important in order to characterize theirperformance. By performance we mean “how long does it take to make a measurement with agiven statistical uncertainty.” This question is distinct from, but related to, the clock uncertainty.
Systematic uncertainty, by contrast, characterizes the absolute (dis)agreement we expect between two systems due to our limited knowledge of perturbing effects that introduce subtle, systematic differences in the operating conditions of each clock. Ultimately, it is these effects that
limit the accuracy of a frequency standard.
In order to characterize the performance of a frequency standard, the most common andwidely-used metric is the Allan deviation [44]. The Allan deviation is a procedure to characterize thetime-dependence of the noise in a string of frequency measurements. Unlike the standard deviation,
which can become divergent for certain noise spectra, the Allan deviation can characterize data
described by a wider range of underlying noise processes [45]. For a continuous measurement, theAllan deviation is defined as
σy2 (τ ) =1 (ν i+1 − ν i )2
h
i.
2
ν02

(1.3)

Here, each quantity ν i etc. is averaged over the time interval τ at the ith such time interval, as
illustrated in Fig. 1.4. Equation 1.3 represents the expected value of the pair deviation for a given
continuous measurement when it is averaged in bins with width τ .
The power of Eq. 1.3 is that it allows easy identification of the underlying noise process.
Table 1.1 shows the behavior of the Allan deviation for different types of underlying frequency noise

8

ν (t)

ν0

...
...

...
τ
νi

ν i+1

ν i+2

ν i+3

...

Time
Figure 1.4: Graphical depiction of the sampling method employed to calculate the Allan deviationof a continuous measurement. Here, the data is averaged in bins of time-width τ . The data iscentered around frequency ν0 .

(see, e.g., [45]), where Gν (f ) is the single-sided frequency noise power spectral density. For example,
one can easily distinguish between noise spectra of the type Gν (f ) = hα f α for α ∈ {0, −1, −2}.
Here Gν (f ) is the one-sided power spectral density of frequency fluctuations. Throughout thisthesis we will use Gξ (f ) for one-sided spectral densities and Sξ (f ) for two-sided spectral densities
of an arbitrary quantity ξ (e.g., frequency, phase, position, electric field, etc.). In Chapter 6 wewill discuss the Allan deviation and its relation to frequency noise spectrum more thoroughly, andelucidate the relationship between atomic spin observables and the local oscillator frequency noise
power spectral density.

1.4

Performance of neutral-atom clocks
While the accuracy of single ion-based clocks is extraordinary, there is a compelling reason

to pursue in parallel standards based upon ensembles of atoms: signal to noise. Roughly speaking,√making N parallel measurements versus one single measurement should yield a N enhancement of√
the signal to noise ratio (SNR). The N -dependence of the noise—which allows the corresponding√
SNR gain of N — is known as quantum projection noise [46] (discussed in detail in Chapter 2).
The quantum projection noise–limited stability of an optical atomic clock based on a quantity, N ,9of quantum references (neutral atoms or ions) is given by [47, 48]
χ√σy (τ ) =
πQ N

r

Tc
.
τ

(1.4)

Here, χ is a constant of order unity that accounts for the details of spectroscopy and fraction of
atoms excited, and Q is the fractional line quality factor, which for optical standards can be > 1014 ,and is one of the key motivating factors for their development. For even modest input parameters,p
an instability of σy (τ ) < 10−16 / τ /1 s is possible.
Until recently, no neutral atom-based optical standards were able to achieve quantum limited
performance due to broadband laser noise, which ends up contaminating the clock correction signal
through the Dick effect (discussed in Chapter 2). In Chapter 4 we will describe development ofa next generation ultrastable laser local oscillator for the 87 Sr clock, which has allowed the 87 Srclock to operate in the near-quantum limited regime. This development represents an importantmilestone in the history of optical standards.

1.5

Frequency combs
Femtosecond laser-based optical frequency combs are important tools for optical metrology

and have revolutionized the field [49, 50]. With laser media ranging from bulk Ti:Sapphire andoptical fibers to microtoroidal resonators, the frequency comb revolution shows no signs of slowing
down. The spectral coverage of frequency combs has been demonstrated to span the mid-IR to thevacuum ultraviolet [51, 52, 53].
At the heart of a comb’s utility is the equation that describes the optical frequency of a givenmode, νn , asνn = nfrep + f0 .

(1.5)

Here, frep = 1/τ is the comb pulse repetition rate, where τ is the time between successive pulses.
f0 is the carrier envelope offset frequency which arises due to the fact that the group and phasevelocities inside the laser cavity are different and is related to the pulse-to-pulse carrier envelope

10

ν1
f1

∆φ

ν2

frep

... ...

n1

f2

n2

f0

×2

Figure 1.5: Frequency comb and optical clocks. The time-domain picture at left leads to afrequency-domain spectrum of evenly spaced spectral components or “teeth.” Here, a fully stabilized frequency comb is depicted comparing two optical clocks at disparate optical frequencies.
By measuring the RF beats, f1 and f2 , the two optical frequencies can be compared. By measuring
the comb repetition rate, frep , an absolute determination of the optical frequencies ν1 and ν2 canbe made.

phase slippage (∆φ in Figure 1.5) by
f0 = ∆φfrep / (2π) .

(1.6)

In principle, frep and f0 are the comb’s only degrees of freedom when describing the frequency ofa given “tooth” in the frequency domain.
Figure 1.5 depicts a frequency comb as utilized in the direct comparison of optical clocks,
or in the absolute frequency measurement of a clock (see, e.g., [1] for an absolute frequency measurement of 87 Sr utilizing an optical frequency comb). By locking a frequency comb to an opticalsource and stabilizing f0 by the self-referencing technique [54, 50], the comb degrees of freedom arecompletely constrained and directly related to the optical phase of the reference laser. By making
a heterodyne beat with a second laser, the phase of the two optical sources can be directly compared (Figure 1.5b), often across > 100 THz of spectral bandwidth [55, 56]. This technique can be11used to compare optical atomic clocks based upon different atomic species to constrain the driftof fundamental constants [28], and also allows optical frequencies to be measured against primary
frequency standards with sub-Hz accuracy [57, 1, 32, 33, 30, 34]. Paired with frequency combs,
phase-stabilized fiber links [58, 59, 60] allow frequency comparisons of clocks hundreds of kilometers
removed and separated by hundreds of THz of bandwidth.
Even at the single-clock level, frequency combs will remain relevant and perhaps even playa more direct role in clock operation as local oscillator ultrastable lasers are developed over abroad spectral range. We will briefly discuss ongoing development of frequency combs to connect
lasers hundreds of THz apart and to support the 87 Sr optical lattice clock in Chapter 4. In Chapters 3 and 4, we will see some of the motivating factors for developing ultrastable lasers in the
1.5 µm region to connect lasers in the near-IR wavelength region to visible wavelengths.

1.6

Beyond timekeeping: Clocks in modern physics
Because they represent some of the most precise measuring device ever built, optical clocks

are important tools for modern science. Due to their high precision and accuracy, optical clocks are
the ideal devices for measuring spatio-temporal variations in the laws of physics, such as testingfor gravitational coupling to fundamental constants in tests of local position invariance, and thetime invariance of the fundamental constants of nature [61, 35, 34]. Another application of ultraprecise clocks is geodesy. A clock’s sensitivity to the gravitational redshift allows it to be an
extremely sensitive probe to differences in the earth’s gravitational potential due to the gravitational
redshift effect [62]. However, clocks employing many particles are also interesting systems in and ofthemselves from a quantum measurement perspective. Equation 1.4 is valid for uncorrelated atoms,
but in the case that the atoms were maximally entangled (i.e., spin squeezed), the instability would√
be lower by approximately an additional factor of N [63].
Many body quantum systems comprised of quantum gases of atoms [64, 65, 66, 67, 68, 69]
and molecules [70, 71, 72] show promise for realizing novel quantum phases of matters and forthe simulation of intractably complex condensed matter analogues. Alkaline-earth atoms possess

12
SU(N ) symmetry in the nuclear spin degrees of freedom, which could facilitate frustrated quantum
magnetic models [73, 74, 75, 76]. Even “open” quantum systems subject to inelastic losses canenable new opportunities for studying quantum entanglement [77]. Chapter 5 explores many-bodyinteractions between atoms in the 87 Sr optical lattice clock, where spin correlations emerge asa consequence of collective elastic and inelastic p-wave interactions. The correlations manifest
themselves in the decay of Ramsey fringe contrast and as quadrature-dependent quantum noise in
the effective spin degree of freedom encoded in the 1 S0 and 3 P0 clock states.

1.7

Overview of this thesis
This thesis describes the development and implementation of ultrastable lasers for spec-

troscopy in a 87 Sr lattice clock, and for exploring the many-body nature of the interactions between
87 Sr in a one-dimensional optical lattice. We demonstrate a 10-fold improvement in clock stability

over any previous optical clock system, either ion- or neutral atom-based, allowing a 100-fold reduction in the time required for evaluating systematic effects. In Chapter 2, we describe the optical lattice clock system used in this work. Chapter 3 explores thermal noise in optical interferometers—akey limitation in developing stable lasers with performance at the 10−16 level. Chapter 4 describes
the design and implementation of ultrastable laser systems. Chapter 5 explores the many-bodynature of the optical lattice clock, utilizing the 10-fold improved stability of the next-generation
local oscillator laser. Chapter 6 describes a theoretical framework and experimental demonstrationof utilizing the optical clock as a frequency discriminator by utilizing pulse sequences, allowing
unprecedented precision for characterizing the broadband noise of a single ultrastable laser system.

Chapter 2
The 87 Sr optical lattice clock

The Sr lattice clock was the first optical lattice clock to be developed, and is now by far themost prolific and well-characterized neutral atom optical frequency standard. This fact is reflected
in its acceptance as a secondary representation of the second by the Bureau International des Poids
et Mesures (BIPM) and the strong record of international agreement of the optical frequency of
the 1 S0 →3 P0 clock transition in 87 Sr . In this chapter, we review the properties that enable 87 Sr
to be a successful optical lattice clock, the experimental apparatus utilized in the JILA 87 Sr latticeclock, and key limitations to both stability and uncertainty.

2.1

Introduction
The features that make Sr an interesting and productive species for wide-ranging studies inultracold quantum gases also make it a highly suitable candidate for an optical clock. In the 1990s,
the first experiments to laser cool Sr were performed, and were motivated by the novelty of the J = 0
ground state and the potential for precision spectroscopy [78, 79]. As seen from the level structure
diagram in Fig. 2.1, strontium possesses two transitions from the 1 S0 ground state that are suitable
for laser cooling: the broad 1 S0 →1 P1 permits magneto-optical traps (MOTs) with > 109 atoms
at temperatures at the level of 1 mK (which was used in the initial laser-cooling experiments), andthe narrow 1 S0 →3 P1 transition permitting a quantum-limited minimum temperature of ∼ 250 nK[80]. Furthermore, the lowest-lying state of the triplet manifold is metastable with an estimatedlifetime of ∼ 100 s, for 87 Sr, whereas in 88 Sr, the transition lifetime is estimated at nearly 6000 years

14
2
1
0

5p2 3 PJ

32
1

5s5d 3 DJ

32
1

5s4d 3 DJ

5s6s 3 S1

461 nmΓ = 2π × 32 MHz

679 nmnm

5s4d 1 D2688

707 nm5s5p 1 P1

2.9 µm2.7 µm
2.6 µm2
1
05s5p 3 PJ

689 nmΓ = 2π × 7.5 kHz
Clock transition
698 nm5s2 1 S0
Figure 2.1: Strontium level diagram. Solid lines indicate laser-driven transitions. The dashed blacklines are the problematic decay processes from the MOT at 461 nm that must be closed via opticalpumping at 679 nm and 707 nm. The grey boxed transitions indicate the manifolds of primary
relevance for the 3 P0 polarizability at the lattice wavelength and also the DC polarizability. Detailsof these states are given in the main text.

[81]! In both isotopes, therefore, this level can truly be considered metastable, and is thus the 1 S0
→3 P0 transition is suitable for ultra-high precision optical spectroscopy and forms the basis of
the strontium lattice clock. Finally, the alkaline earth-like level structure of the triplet and singlet
manifolds possess sufficient complexity such that they support “magic wavelengths” for the 1 S0
and 3 P0 states, and this was a key component of the first lattice clock proposal based on the 1 S0
→3 P0 transition [82].

15
In a magic-wavelength trap, the AC stark shift of a probe or dipole trapping optical field
perturbs each state equally [26]. Within the context of precision spectroscopy, the presence of amagic wavelength permits the construction of an optical dipole where the transition between themetastable clock states is unperturbed by the trapping field. This permits a very elegant conceptual
simplification of the dynamics of an atom in a magic-wavelength dipole trap. We consider thecoherent superposition|ψi = α|g, ni + β|e, ni.

(2.1)

Here, g represents the 1 S0 ground state, e represents the 3 P0 metastable state and n ≡ {nx , ny , nz }labels the motional state of the atom in a three-dimensional harmonic trap. At exactly the magic
wavelength, the motional and electronic quantum states are decoupled, so that we may instead
write Eq. 2.1 as|ψi = (α|gi + β|ei) ⊗ |ni.

(2.2)

The success of the clock based on the 1 S0 →3 P0 transition relies directly on the decoupling of the
motional and electronic degrees of freedom, such that the spectra obtained in the lattice clock arefree from AC Stark shifts or broadening as a result of thermally-distributed motion of the atoms.
This Chapter describes in detail the apparatus used to cool 87 Sr into a magic-wavelength
optical lattice dipole trap. We will explore the details of the magic wavelength trapping including
limitations due to lattice photon scattering. Details regarding the ultrastable clock laser are given
later, in Chapter 4. In Chapter 4 we also demonstrate operation of the clock in a near-quantumlimited regime.

2.2

Narrow-line laser cooling
As mentioned in the previous discussion, one feature of the level structure of Sr that makes

it appealing for use in an optical standard, as well as for quantum gas research, is the presence ofthe narrow 1 S0 →3 P1 transition. In fact, the exploration of narrow-line cooling in Sr and otherAlkaline earth/Alkaline earth-like atoms represents an important milestone in the history of laser

16
cooling. To date, ∆J = 1 intercombination transitions (i.e. the 1 S0 →3 P1 transition) have beenutilized to trap and cool a large array of species and isotopes of alkaline earth and alkaline earth-likeatoms. Examples include Sr [83, 84, 85], Calcium ([86]), and Ytterbium [87]. Due to the breakdownof LS-coupling towards jj-coupling in heavier atoms [88], the intercombination transitions couplingin Mercury [89, 90] and Radium [91] are well suited for single-stage laser cooling, and have beenused for this purpose, but are too broad to be considered narrow-line cooling transitions. In thissection, we briefly describe some of the important features of narrow-line laser cooling.
The expression “narrow line cooling” is typically used to denote a laser-cooling on a cyclingtransition for which the recoil frequency shift of the transition, given byωrec =

~ωa2
,
2mc2

(2.3)

where ωa is the optical transition angular frequency and m is the atomic mass, is comparable tothe linewidth of the transition that is utilized for cooling. In the case of Strontium, the narrow 1 S0
→3 P1 transition is a nearly ideal transition for laser cooling. For Sr, ωrec / (2π) = 4.8 kHz and isthus comparable to the 7.5 kHz natural linewidth of the 1 S0 →3 P1 transition. In a sense, this is
the ideal situation. For laser cooling on transitions where ωrec . Γ, the minimum temperature is
expected to be in the quantum-limited regime set by the recoil energy [92],kB T = ~ωrec .

(2.4)

At the same time, the peak force of the cooling beams, given by Fmax = ~kΓ/2 and which determines
the velocity capture range, must be large enough to provide robust trapping. Thus, there is
no real benefit to performing traditional optical molasses laser cooling on a transition for whichΓ  ωrec with respect to a transition for which Γ ' ωrec . Specifically, in narrow-line MOTs, Fmax
can be dangerously close to the force exerted by gravity, mg. In the case of strontium, whereΓ = 2π × 7.5 kHz, Fmax ' 16 × mg [85], allowing the atoms to be trapped against gravity. In
Calcium, for example, where the 1 S0 →3 P1 transition has Γ = 2 × 103 1/s, Fmax is only 1.5 × mg,
necessitating the use of an optical quenching technique to artificially increase Γ [86].

17
The nearly ideal parameters of the Sr 1 S0 →3 P1 led to successes in laser cooling with thistransition, yielding temperatures as low as ∼ 250 nK [80] and in-MOT phase-space densities of ∼0.1
[93]. Further work to overcome the detrimental effect of photon rescattering has recently led tothe production of a directly laser-cooled Bose-Einstein condensate (BEC) [94] in 84 Sr, a techniquethat shows great promise in creating a continuous source of BEC (i.e., a continuous atom laser)and facilitating other quantum gas experiments that benefit from fast repetition.

2.3

Lattice dipole trapping
Trapping and manipulating neutral atoms with intense light fields is a well-established andextremely powerful technique [95, 96]. In contrast to magnetic trapping, optical dipole traps can
produce intricate trapping potentials with sub-wavelength precision, the most prolific of which are
optical lattices. In an optical lattice, the interference of two or more several phase coherent laser
beams (or a single retro reflected beam) can give rise to periodic light patterns in one, two, andthree dimensions. Among other uses, the use of such highly periodic potentials can aid quantum
simulation of condensed matter systems [97], and serve as a test bed for exploring quantum phasetransitions [64, 98], frustrated quantum systems [99], quantum Hall physics [100], and quantum
gases and polar molecules in reduced dimensionsionality [101, 71]—and this is only to name a few.
In the context of optical lattice clocks, the periodicity of the potential is actually not a necessary, or even desired feature. Rather, the interferometric nature of the optical trapping field allows
extremely tight local trapping potentials, with trapping frequencies & 100 kHz easily accessible.
This in turn enables spectroscopy in the Lamb-Dicke regime [102, 103, 104], where the recoil energy is much less than the harmonic oscillator zero-point energy of the trap. In the Lamb-Dicke
regime, Doppler broadening effects are well-resolved from the unperturbed carrier transition, while
the recoil momentum is absorbed by the trapping field (we will return to the topic of Lamb-Dicke
confinement in Section 2.5). This is an essential requirement for a precise optical atomic clock,18
given that Doppler broadening of an atomic transition is (see e.g., [88])
∆ν
=
ν

r

8kB T log 2
,
mc2

(2.5)

which corresponds to ∆ν/ν ' 1 × 10−10 for 87 Sr at 1 µK, corresponding to a Doppler-broadened
linewidth of 30 kHz. Of course, Doppler-free spectroscopy techniques have existed since nearly theadvent of the laser. For instance, Doppler-free Ramsey [11, 12] and Ramsey-Bordé spectroscopy
can be performed on narrow-line transitions within atomic beams and MOTs [105, 106], allowing
sub-kHz features to be observed [20] and forming the basis of the calcium neutral-atom standard[107]. However, the natural linewidths of highly forbidden atomic transitions in Alkaline-earths,
such as the ∼1 mHz 1 S0 →3 P0 transition is 87 Sr are still many orders of magnitude below the best
linewidths achieved with free-space Doppler-free techniques, such as Ramsey Bordé spectroscopy.
The role of the optical lattice then is not just to remove the effects of Doppler broadening andthe potential systematics associated with Doppler-free Ramsey-Bordé spectroscopy, but to entera regime where a large fraction of the atoms can be sufficiently localized to be probed at the
one-second timescale.

2.3.1

Atomic polarizability calculation

In general, the utility of dipole trapping as a scientific technique lies in the favorable scalingof scattered radiation, which causes heating and decoherence, to the total depth of the trapping
potential. If we consider an isolated resonance in the rotating wave approximation, and for detunings small compared to the transition’s optical frequency, the ratio of scattered radiation to thedipole potential is [96],
~Γ =

γ
U.
∆

(2.6)

where Γ is the rate of scattered radiation, γ is the decay rate of the atomic transition considered,∆ is the trapping laser detuning (∆ ≡ ωL − ωa ), and U is the induced dipole potential. We notethat although Eq. 2.6 is utilized quite frequently to estimate the photon scattering rate in opticallattice clocks, where often |∆| ∼ ωa , the expression given in Eq. 2.6 fails badly because it does not19take into account the wavelength-dependence of the density of states for the scattered photon andFermi Golden Rule energy scaling, nor does it include the counter-rotating term if we do not makethe rotating wave approximation. Including both these effects for a two-level system yields [96]~Γ = U

ωL
ωa

3 

γ
γ
−
∆ ωL + ωa


,

(2.7)

which can yield fairly accurate results for the ground state scattering of 87 Sr , where the 1 P1
level dominates the scattering. We also note that, while U is straightforward to calculate for a
simple two-level system, there are regimes where no single atomic resonance dominates and thus
determining U requires summing over many atomic levels. The significance of Eqs. 2.6 and 2.7
is that the ratio of trapping strength (good) to rate of photon scattering (bad) gets better andbetter as the detuning is moved farther away from resonance. This is the motivation behind thefar off-resonance trap [95]. We will study the limitations to the Sr lattice clock due to photon
scattering in Section 2.6.
To accurately calculate the trapping potential for a general system, where the two-levelapproximation does not necessarily apply, we begin by considering a monochromatic optical fieldwith a spatial amplitude dependence such that1
1
E (r, t) = E0 (r) e−iωt + E∗0 (r) eiωt .
2
2

(2.8)

The term E0 (r) can further be expressed as
E0 (r) = εE0 (r) ,(2.9)

where ε and E0 (r) are the field polarization and amplitude, respectively. Both of these quantities
are complex. For a monochromatic plane wave with linear polarization, Eq. A.1 reduces to thefamiliar expression E (r, t) = εE0 cos (kx − ωt).
For the electric field given in Eq. A.1, the dipole trapping potential is given by1
1
U (r) = − α (ε) h|E (r, t)|2 i = − α (ε) |E0 (r)|22
4(2.10)

20
Here the h. . .i represents a time average over an optical cycle so that α is defined with respect
to the mean square electric field. This allows consistency with the definition in the case of a dc
electric field [88].
The polarizability is found from second-order perturbation theory [88, 108] as2

X ωik hφi |ε · d̂|φk i .
α (ε) = 2
2 − ω2~ ωikk(2.11)

Here, the polarizability is determined by summing over all dipole-allowed upper states, {φk } whered is the dipole operator that connects the atomic level of interest, φi to upper atomic levels thatare responsible for the induced dipole moment. Here, ωik is the frequency difference between thestate φi and the states {φk }. The dependence on the light polarization follows fundamentally fromthe tensorial nature of the polarizability operator. In fact, we can write α (ε) = ε∗µ εν αµν (repeatedindices are summed), where αµν is the polarizability tensor. As with any cartesian tensor operator,
we can decompose αij into a spherical tensor basis, which will allow us to make some generalstatements about the light- and atom-polarization dependence of the polarizability. Thus, as it is
writte, Eq. 2.11 does not cast the polarizability in its most useful form.

2.3.1.1

Scalar, vector, and tensor light shifts

While Eq. 2.11 is perfectly valid for calculating the dipole trapping potential and light shifts,
the tensorial nature of the polarizability is not apparent, nor can we make any qualitative statements
regarding its behavior. It is the goal of this section to summarize the most useful results regarding
the polarizability tensor. Specifically, we can write [109]U (r) = −Eµ(−) Eν(+) αµν(2.12)

whereαµν = 2X ωik hφi |dˆµ |φk ihφk |dˆν |φi i
.
2 − ω2~ωikk(2.13)The electric field is now expressed with slightly different parameters such that E(+) ≡ E0 (r) /2 =
 (−) ∗E
. In Appendix C, we follow the approach of [108, 109] to reduce the tensorial αµν into

21
contributions of irreducible spherical tensors to derive the standard expression for scalar, vector,
and tensor light shifts (see, eg, [110])α (ε) = αs + iαv (ε × ε∗ ) · ez

m
+ αt
F


3 |ε · ez |2 − 1  3m2 − F (F + 1) 
2F (2F − 1).

(2.14)

Here, we have introduced αs , αv , and αt , the “scalar”, “vector”, and “tensor” polarizabilities,
respectively. In terms of the reduced matrix elements, the polarizabilities are [108, 109]X 2ωnF →n0 F 0 hnF ||d̂||n0 F 0 iαs (ω) =2
2
3~
ω
0F 0 − ω
0
0
nF
→n
nF

2

2

X 2ωnJ→n0 J 0 hnJ||d̂||n0 J 0 i',(2.15)
2
2
3~
ω
0J 0 − ω
0
0
nJ→n
nJ


2
s

1 1 1  ωnF →n0 F 0 hnF ||d̂||n0 F 0 i
X
6F
(2F
+
1)
0

,
(2.16)αv (ω) =(−1)F +F +1
2
2(F + 1) ~ ωnF
F F F 0 
→n0 F 0 − ω
n0 F 02s ωnF →n0 F 0 hnF ||d̂||n0 F 0 i
1 1 2 
X
40F
(2F
−
1)
(2F
+
1)
0

αt (ω) =(−1)F +F
. (2.17)22
(1 + F ) (2F + 3)
~ ωnF
F F F 0 00→n0 F 0 − ωnF

We can relate the reduced matrix element hF ||d̂||F 0 i to hJ||d̂||J 0 i by [111, 108, 109]




0

J J 1
p
F 0 +J+I+1
0
00hF ||d̂||F i = hJ||d̂||J i (2F + 1) (2J + 1) (−1).
F 0 F I (2.18)

Here, the terms in curly brackets are the Wigner 6-j symbols and ωnF →n0 F 0 is the transition
frequency between the state with quantum numbers n and F to the state with quantum numbers
n0 and F 0 . In Appendix B, we describe the procedure for relating the reduced matrix elements
hJ||d̂||J 0 i and hF ||d̂||F 0 i to transition lifetimes commonly reported in the literature.
There are several special cases worth considering for αv and αt . The first interesting scenario
is the case where F = 0, which describes the clock states for 88 Sr. Here, it is immediately clearthat αv = 0 and αt = 0. This matches the intuitive picture of an F = 0 as a state devoid of a
specific orientation—not to mention that there are no mF levels, rendering the very notion of avector or tensor shift invalid. The second case is very similar, but more relevant for this work. Thisis the case where J = 0 but there is nuclear spin I, such that F = I. If we insert the result of
Eq. C.26 for hF ||d||F 0 i into Eqs. C.23–C.25 and perform the sum over F 0 for any J 0 value, we get

22
a number that is very nearly zero for αv and αt as long as the laser detuning is much larger thanthe hyperfine splitting. If we ignore the hyperfine splitting completely, then the result for αv andαt is identically zero. Finally, the last case of interest, and one that is relevant for the 171 Yb clock[32], is the case1 where F = 1/2 and J = 0. Here we see that αt = 0. Again, this makes intuitiveand pragmatic sense because with only two mF levels, the concept of a tensor shift is ill-defined.
Finally, it is worth mentioning that, due to the hyperfine state mixing between the 3 P0
clock state and the 3 P1 and 1 P1 states, the clock state does acquire a small but finite vector andtensor shift. These quantities have been theoretically explored [112] and thoroughly measured by
the SYRTE Sr lattice clock group [110]. As seen in Eq. 2.14, as long as the light polarization iscontrolled with respect to the quantization axis, these shifts can be removed by including them inthe definition of the magic wavelength, or by a post-correction, and should pose no threat to clock
accuracies below 10−17 [110].

2.3.1.2

Numerical results for Sr

The 1 S0 and 3 P0 clock states of 87 Sr both have J = 0, thus based on the discussion of
the previous section the polarizability should only have a contribution for αs . Ignoring hyperfine
structure, it is straightforward to calculate αs for both the 1 S0 and 3 P0 states from Eqs. C.23 andC.26 (see appendix B) asSpecifically,
α3 P0 (ω) = 2π0 ~c3
α1 S0 = 6π0 ~c3X

X ωnF →n0 F 0 AT (n0 , J 0 ) (2L0 + 1)

3
2
2
3
ω̃
0 J 0 ωnF →n0 F 0 − ω
0
0
n
n ,J

(2.19)AT (n0 , J 0 )
.
2
2
ωnF
→n0 F 0 − ω(2.20)

ω20 0
n0 ,J 0 nF →n F

Here, AT (n0 , J 0 ) is the total decay rate of the excited state labelled by quantum numbers n0 and
J 0 , where n0 is shorthand for the electronic configuration and J 0 is the total angular momentum(for all dipole-allowed transitions from the clock states, J 0 = 1); and ω̃n0 J 0 is an effective transition
1
This instance is seemingly covered by the J = 0 rule, which precludes both vector and tensor shifts. It turns
out the origin of the vector and tensor shifts in the clock state arises from hyperfine-induced state mixing between
levels of like F [112]. Thus, for the case of J = 0, F = 1/2, the statement that there is no tensor shift is a rigorous
one, while the statement that for J = 0 levels αv = αt = 0 is not.

States used for 3 P0 scattering rate calculation
State
A-coefficient Detuning from 3 P0
35s6s S1
8.5∗
2.773∗
5s7s 3 S1
1.75∗
4.356∗
3
5s8s S1
0.822
4.984
5s9s 3 S1
0.453
5.303
3
5s10s S1
0.277
5.487
5s4d 3 D1
0.03450.724
5s5d 3 D1
6.1
3.899
3
5s6d D1
2.67
4.781
5s7d 3 D1
1.42
5.192
3
5s8d D1
0.851
5.419
5s9d 3 D1
0.551
5.59
5p5p 3 P1
12.7
3.982
States used for 1 S0 polarizabilityState
A-coefficient Detuning from 1 S0
15s5p P1
19.05
4.089981
5s6p P1
0.186
6.42732
5s7p 1 P1
0.319
7.333611
4d5p P1
1.2
7.76074
5s8p 1 P1
1.49
7.76074
5s9p 1 P1
1.16
8.00391
5s10p P1
0.76
8.16695
5s11p 1 P1
0.488
8.2821223

Table 2.1: Upper states used in calculating the polarizability of the 1 S0 and 3 P0 clock states. The
Einstein A-coefficient is given in units of 107 1/s, and the detuning is given in units of 1015 rad/s(i.e., 10−15 × ∆ω, where ∆ω is the transition frequency). The numerical values are obtained from[112] and references therein, with the exception of the data marked with an asterisk, which is from[113].

frequency from the upper state connecting to 3 P1 that takes into account the relativistic correction
due to the fine structure energy splitting within the 3 P manifold. This latter correction is derived
in Appendix B. With these particularly simple expressions for the J = 0 clock states, we calculate
the polarizability employing the decay rates of a subset of the upper-lying states, given by thevalues in Table 2.1. The results of this calculation are shown in Fig. 2.2. As expected from several
calculations and measurements spanning the history of the Sr clock’s development, we find a magic
wavelength in the vicinity of 813 nm, where the discrepancy between experiment arises from an
incomplete knowledge of the transition rates and also the fact that we do not treat continuum24
Polarizability a.u.

2000

Polarizability (a.u.)

1000

400
350
300
250
200

760 780 800 820 840 860
Lattice wavelength (nm)Polarizability a.u.

0 1000 2000

200

300

500

700

1800
1400
1000
600
495

500
505
510
515
Lattice wavelength (nm)

1000

1500

2000

3000

Lattice wavelength (nm)Figure 2.2: Polarizability of the 1 S0 (blue line) and 3 P0 (red line) clock states in atomic units(a.u.). The insets show two regions where the polarizabilities of the 1 S0 and 3 P0 clock states arematched.

states in this calculation [114, 115]. Interestingly, another magic wavelength crossing occurs in
the vicinity of 505 nm. As conveyed qualitatively by Eq. 2.6, one might expect that using thisas a lattice wavelength would lead to unacceptably large lattice scattering. We will treat photon
scattering from the lattice in Section 2.6. Surprisingly, we find that the magic wavelength at 505 nm
will have acceptable scattering rates for an equivalent trapping frequency, indicating that this may
be an interesting magic wavelength for use with relatively low-power optical sources.

2.3.2

One-dimensional optical lattice

Now that we have calculated the polarizabilities, we consider the specific case of a onedimensional optical lattice formed by a retro-reflected laser beam. We consider a focused input

25
beam with electric field amplitude profile
1
E in (r; t) = E0in (r⊥ ) e−ikz−iωt + c.c.
2

(2.21)

with the radially-dependent amplitudesE0in (r⊥ ) =4P0 −r2 /w02e ⊥ ,π0 cw02(2.22)

where P0 is the total input power. If the beam is perfectly retro-reflected to the same focus, theelectric field profile will be
E (r; t) =1
2 cos (kz) E0in (r⊥ ) e−iωt + c.c.
2

(2.23)and thus according to Eq. 2.10 the lattice potential will be
2

U (r) = −α cos2 (kz) E0in (r⊥ ) .

(2.24)

If we expand the potential along the z-direction at r⊥ = 0 we can treat the potential as a harmonic
oscillator. The longitudinal trap frequency associated with this profile is easily found to be [112]sr
32παP01 αIpeak1=.
(2.25)
νz =
2πw0 λc0 M
λ 0 cM
where Ipeak = 20 c E0in (0)2= 8P0 / πw2 . Similarly, the trap frequency associated with the radialconfinement is given by
1
νr =
2πw02

r

16αP0
1
=
π0 cM
2πw0s

2αIpeak.
0 cM(2.26)

We find that at 813.428 nm, the calculated Sr clock-state polarizability is 2.8×102 a.u. (a.u. stands
for atomic units of polarizability, 1 a.u. = 4π0 a30 ). For νz = 80 kHz, Ipeak = 3.4 × 104 W/cm2
corresponding to P0 = 140 mW for w0 = 32 µm, which is beam waist utilized in the experiment(we describe the lattice setup in Section 2.4.3 and the details of lattice spectroscopy in Section 2.5).

2.4

Experimental apparatus
The experimental apparatus used in this work has been extensively described elsewhere [112,114, 80]. For completeness, we describe the essential details here. The vacuum system used to cool26
and trap 87 Sr is shown in Fig. 2.3. An effusive oven loaded with high-purity strontium is operated
at a maximum temperature of 575◦ C.2 Earlier measurements of the beam flux indicated a beam
flux of 3 × 1011 atoms/s [80]. We measure similar loading rates and MOT sizes, so we estimatethat our current flux is also consistent with this number despite the fact that the oven has been
disassembled and reloaded in the intervening decade. As depicted in Fig. 2.3, a rotary feedthroughactuates a cylindrical atomic beam shutter, which can be used to extinguish the atomic beamduring operation of the experiment if desired. Interestingly, we observe no detrimental effect due tothe atomic beam except that it leads to an additional background signal that must be subtracted.
A total of three 40 L/s ion pumps maintain a pressure estimated in the low 10−10 Torr levelin the main chamber. This represents a four-fold improvement from the previously-reported experimental design [112, 114, 80], as indicated by in-trap lifetime measurements. The main improvement
permitting these significantly lower pressures is the addition of a secondary differential pumping
tube. In the original experiment, there was a single differential pumping tube, separating the ovenfrom the main chamber, and permitting an order of magnitude difference in the pressures. With theaddition of the second differential pumping region, we maintain an almost two order of magnitudedifference between the oven and main chamber, for a vacuum-limited trap lifetime of 8 s [116].
A pair of Anti-Helmhotz coils (not pictured in Fig. 2.3) are placed so that the axial field
direction is oriented vertically. They are capable of producing an axial field gradient of 50 G/cm
for an input current of approximately 60 A. The current in these coils is servo-controlled and capableof being switched at ms timescales. This fast and precise control is utilized to perform programmed
field-ramps for different stages of optical cooling. Small Helmholtz coil pairs oriented along the H1,
H2 directions as labeled in Fig. 2.3 (horizontal direction 1 and 2, respectively) provide fine tuning
to null stray fields in the horizontal plane. A third Helmholtz coil pair in the vertical direction (V)provides fine-tuning of the magnetic field in that direction.
During operation of the experiment, stray magnetic fields are nulled to < 10 mG by coils in
The nozzle (front) part of the oven is operated at 575◦ C, while the back end of the oven is maintained at 365◦ C.
This ensures that the nozzle region does not accumulate Sr and clog.
2

27

Glass cell(vestigial)

H1 axisH Lat.
retro.

40 L/s ion pumpH2 axisTransverse
cooling

LatticeDM

Zeeman slowerOven

ZS beamImaging
viewport
MOT beams

Rotaryshutter

V. Lat.
retro.
Clocklaser inH. Lat.
(optional)

Differential
pumping
apertures

To ionpump

DM
40 L/s ion pump

Figure 2.3: Experimental apparatus for trapping ultracold 87 Sr . A collimated atomic beam isemitted from the oven region and is subsequently transversally cooled by transverse cooling beams
and slowed by a Zeeman slower (ZS). A MOT is formed in the octagonal main chamber (antiHelmhotz coils not depicted). A total of three 40 L/s ion pumps are attached to the apparatus(pump located nearest the oven is not pictured), and each pump is separated from the others
by a differential pumping tube, supporting a nearly two orders of magnitude difference in theoven vacuum pressure with respect to the main chamber. The optical lattice enters from the top
viewport with an angle of approximately 19◦ with respect to gravity and is retroreflected after
passing through the top and bottom viewports of the chamber. The lattice polarization is aligned
along the H2 axis. Dichroic mirrors (DM) in the H1-axis MOT arms permit a secondary horizontal
optical lattice to be overlapped with the vertical lattice.

the H1 and V directions. A residual field of ∼ 500 mG is applied along H2 for spectroscopy, whilea smaller field of (∼ 100 mG) is applied for an optical pumping phase to spin-polarize the atoms if
desired. As a result, H2 is the de facto quantization axis in the experiment.
Optical access is provided at several locations. Immediately after the oven nozzle, a cube with
2 3/4” flanges provides access for transverse cooling beams. The next location in the apparatus with28
significant optical access is the main chamber itself, which is octagonal in shape (in fact, it is calleda “spherical octagon” by the manufacturer, Kimball Physics). Here, two large 6” flange-mounted
viewports allow a wide range of optical access from the top and bottom of the chamber. The top
viewport serves multiple functions, including providing optical access for florescence detection, theoptical lattice, and MOT beams. Almost all of the sides of the octagon also have 2 3/4” flangemounted viewport, utilized for the mot beams and optical imaging, as in Fig. 2.3. The viewportlabelled “imaging viewport” in Fig. 2.3 provides access for an amplified CCD camera. All viewports
are anti-reflection coated for visible wavelengths.
The trapping and cooling of 87 Sr is performed using two transitions for laser cooling. The
atomic cloud is directly cooled into a far off resonance optical lattice comprising a standing waveof laser light at 813 nm. To avoid uncontrolled vector and tensor light shifts—which will befurther described later—the dipole trap polarization is aligned with H2 with a high extinctionratio polarizer (∼104 ). The first transition for laser cooling is the 1 S0 →1 P1 transition at 461 nm,
which is used for both slowing the beam and trapping and cooling 87 Sr in a magneto-optical trap
(MOT). The second transition employed is the narrow 1 S0 →3 P1 intercombination transition,
which permits direct laser cooling to sub µK temperatures. These stages of laser cooling andtrapping are described in detail in the next sections.

2.4.1

Laser cooling at 461 nm

The first stage of laser cooling is performed on the broad 1 S0 →1 P1 transition, which is shownin Fig. 2.1 and the hyperfine structure of which is shown in Fig. 2.4. We note that the presence ofhyperfine structure in 87 Sr complicates the cooling dynamics, and can lead to sub-Doppler cooling[117]. In clock operation, we use the first-stage 461 nm MOT to quickly accumulate ∼ 2 × 105
atoms in 500 ms. This quantity is approximately 10% of the maximum 87 Sr 461 nm MOT peakatom number.
The laser light utilized for the 461 nm MOT is derived via second harmonic generation. An
external-cavity laser diode (ECDL) seeds a tapered amplifier, which provides approximately 1.6 W

Blue MOTF = 7/2

F = 7/2
88

Sr

46.3 MHz
37 MHz15s5p P1
F = 11/2

Sr1352 MHz88

F = 9/2
6 MHz

87

29

Red MOT

46 MHz23 MHz222 MHzSr3

5s5p P1
62.3 MHz

1242 MHz87

F = 9/2

Sr

Stirring

Trapping

Trapping

F = 11/2

Figure 2.4: Hyperfine structure of the 5s5p1 P1 and 5s5p3 P1 levels.

of power at 922 nm. After being spatially filtered via a single-mode fiber, approximately 350 mW ofpower at 922 nm is delivered to two separate doubling cavities employing KNbO3 crystals. Phasematching is achieved via temperature, with an operating temperature of approximately 150◦ C.
Details can be found in Refs. [112, 114]. Each cavity produces approximately 150 mW of power at
461, which is utilized for slowing, cooling, and trapping with the 1 S0 →1 P1 transition, as described
below.
After exiting the oven, the atoms are cooled transversally by elliptical beams tuned 10 MHz
below the 1 S0 →1 P1 F = 11/2 (Fig. 2.3). After the transverse cooling stage, they are subsequently
slowed in a Zeeman slower with a peak field of 600 G, optical power of ∼60 mW, and detuning of-1030 MHz below the 1 S0 →1 P1 F = 11/2 transition. Finally, upon reaching the main chamber theatoms are collected in a MOT operating on the 1 S0 →1 P1 transition. The horizontal MOT beams
have ∼3 cm diameter and the horizontal beams have 8 mW of power, while the vertical beam has
3 mW of power. The field gradient utilized for the 1 S0 →1 P1 MOT (AKA “blue mot”) is 50 G/cm
in the vertical direction.
While the 1 S0 →1 P1 transition is nearly closed, as shown in Fig. 2.1, there is a decay pathway

30
to the metastable 3 P2 level. In order to maintain efficient MOT operation, the 3 P2 level must be
repumped by a separate laser at 679 nm connecting the 5s5s 3 P2 and 5s6s 3 S1 levels. Due to thefact that the 5s6s 3 S1 level employed in the repump scheme can in turn decay to the metastable
3P

0state, an additional repump laser must be utilized to cycle these atoms out of 3 P0 , again

via 5s6s 3 S1 . Since the hyperfine manifolds of the 5s5s3 P2 and 5s6s 3 S1 are quite large, we scan
the repump lasers so that all the hyperfine levels are reached by a single laser. Finally, in orderto prevent dark states from forming, the repump lasers are propagated along both the H1 and V
MOT axes. Ultimately, the repumps make a huge difference, with a factor of ∼40 improvement in
the observed fluorescence, and an order of magnitude increase in the MOT lifetime.

2.4.2

Narrow line cooling with 87 Sr

A significant body of literature exists that explores the qualitative differences and challengesof laser cooling 87 Sr as opposed to its bosonic counterpart, 88 Sr [118, 112, 114]. Here we briefly
summarize some of the key differences and describe the experimental methods used to cool andtrap 87 Sr on the 1 S0 →3 P1 transition.
2.4.2.1

The 87 Sr 689 nm MOT: principle of operation

In 88 Sr, the narrow line cooling transition is an ideal J = 0 → J = 1 cycling transition, whichallows the MOT to operate very efficiently. In the case of 87 Sr , with nuclear spin I = 9/2, one must
consider the hyperfine levels of the 3 P1 level, with quantum number F ∈ {7/2, 9/2, 11/2}. At firstglance, it might seem straightforward to operate the MOT on the 1 S0 (F = 9/2) → 3 P0 (F = 11/2)
transition. However there is a large disparity between the ground state g-factor of gS = −6×10−4 '0 and the excited state g-factors of g7/2 = −1/3, g9/2 = 2/33, and g11/2 = 3/11 [118, 112]. Thismeans that the Zeeman shift in an applied magnetic field is almost exclusively restricted to theexcited state manifold. This has two important implications for the MOT operated on the 1 S0
→3 P1 F = 11/2 transition. First, the differential Zeeman shift of the 87 Sr 1 S0 →3 P1 (F = 11/2)
transition, as depicted in Fig. 2.5, has significant implications for MOT operation. To simplify

31

B>0

F
σ−

σ+

-11/2

r

(−)

-7/2

-9/2

(−9/2)
−

σ

-9/2

1
+

-1/2

-3/2

-5/2

1

σ−

σ+

-5/2

1/2

3/2

5/2

δ

r(−) (−3/2)

σ

-7/2

me gµB B-3/2

Not trapped

-1/2

1/2

3/2

5/2

Figure 2.5: Schematic depiction of the 689 nm MOT for the F = 9/2 → F = 11/2 transition. Dueto the absence of an electronic magnetic moment in the ground state, several mF sublevels are nottrapped. The relative σ + /σ − transition strengths play a role in optical pumping and maintaining
the MOT restoring force as well.

the discussion, we consider only one MOT axis. For a cooling laser detuned below the zero-fieldresonance with detuning δ, if the motion of the atom brings it into a region of the MOT with B > 0,
there are a large number of ground-state mF levels that are not trapped (specifically mF ≥ 3/2).
This is because the increasing magnetic field will tune the Zeeman shift in the “wrong” direction
for these states, and they will sail away without scattering any photons. An example of such a
non-trapped state is depicted with blue arrows in Fig. 2.5. An identical effect prevents the stateswith mF ≤ −3/2 in the region with B < 0 from being trapped (assuming we use the same definitionof quantization axis as before).
A second important implication of the 87 Sr 1 S0 →3 P1 (F = 11/2) Zeeman structure arises
when the atom is brought into resonance by the Zeeman shift. As depicted for the mF = −9/2 statein Fig. 2.5, both σ + and σ − transitions are allowed. An immediate concern is that the position
dependent light force could be artificially small or even nullified by scattering photons from both
MOT beams. However, it turns out that the Wigner-Eckart theorem effectively eliminates this32

F = 11/2(−)(+)

r11/2 (mg )r11/2 (mg )10

r9/2 (mg )

1

1

(±)

(±)

r11/2 (mg )10

F = 9/2

0.1
−9/2

−5/2

−1/2 1/2

5/2

(−)

0.1

9/2

−9/2

mg

(+)

r9/2 (mg )−5/2

r9/2 (mg )−1/2 1/2

mg

5/2

9/2

Figure 2.6: Ratio of σ ± absorption as a function of ground state magnetic sublevel, mg .

concern [118]. We can write the ratio of transition probabilities for a σ − transition with respect toa σ + transition for a general F → F + 1 transition as(−)

rF +1 (mg ) =|hF, mg ; 1, −1|F + 1, mg − 1i|2
2

|hF, mg ; 1, 1|F + 1, mg + 1i|=(2 + F − mg )! (F + mg )!
(2 + F + mg )! (F − mg )!

(+)(2.27)(−)

Similarly, rF +1 (mg ) is defined as the ratio of σ + to σ − transitions, and is related to rF +1 (mg )
(+)

(−)

(−)

(+)

simply as rF +1 (mg ) = 1/rF +1 (mg ). Both rF +1 (mg ) and rF +1 (mg ) are plotted with respect to
mg in Fig. 2.6. Remarkably, for the situation in Fig. 2.5 with B > 0 and mg = −9/2 the ratioof “good” σ − photons absorbed to “bad” σ + photons absorbed is 55 : 1. As seen from Fig. 2.6,
if a state is trappable, the ratio of “good” absorptions to “bad” absorptions is greater than unity.
Furthermore, we can expect that as a result, the atom will be quickly pumped to the stretchedstate, which causes the ratio to be maximized.
As we’ve seen, nature already provided one built-in solution to multi-level alkaline earth
atoms to permit their trapping, in that the relative transition strengths play an essential role inensuring the MOT operates stably. However, the first complication that we briefly discussed, namelythat there are untrapped ground state mg levels, requires more direct intervention. The techniquedeveloped by the Sr group at U. Tokyo was to utilize a second laser, termed the “stirring laser”[118]. In contrast to the trapping laser, which operates on the F = 9/2 → F = 11/2 transition, the33stirring laser operates on the F = 9/2 → F = 9/2 transition. The effect of this second laser is torandomize the ground state mg distribution so that trappable mg states are occupied.
The F = 9/2 → F = 9/2 transition is ideal as a “stirring” transition for two reasons. The
first is that the g-factor is much smaller for the F = 9/2 level meaning that a near resonant laser
will scatter over a larger range of magnetic field. Secondly, and unlike the F = 9/2 → F = 11/2
transition, the ratio of σ + absorptions to σ − absorptions is fairly uniform. To quantify this, we(±)

can define rF (mg ) as in Eq. 2.27. This quantity is plotted in Fig. 2.6, and it is clear that there
is very little variation with mg , meaning that the atoms are indeed efficiently stirred through theground state mg manifold, and also that the pumping slightly favors moving atoms away from theuntrappable mg levels.

2.4.2.2

The 87 Sr 689 nm MOT: experimental details

To operate the 689 nm MOT, we employ two cooling lasers whose use was motivated above:the “trapping” laser and the “stirring” laser. These two systems are phase locked with variablefrequency offset from a master laser, with linewidth of ∼500 Hz and whose central frequency islocked to the 88 Sr 1 S0 →3 P1 (m = 0 → m = 0) transition via a Doppler-free spectrometer basedon a heat-pipe vapor cell with 25 mTorr of Ar buffer gas, which keeps the Sr from coating thewindows. Details can be found in [80, 112, 114]. As depicted in Fig. 2.4, the trapping and stirring
lasers’ frequencies are chosen such that they are near-resonant with the F = 11/2 and F = 9/2
hyperfine states, respectively.
In order to enable efficient transfer from the 461 nm MOT into the 689 nm MOT, three
distinct MOT phases of the 689 nm MOT are utilized. In the first stage, the trapping and stirring
lasers are modulated with a few MHz bandwidth. The modulation parameters are chosen such thatthe spectrum extends from just below resonance with the 9/2 → 9/2 and 9/2 → 11/2 transitions,to several MHz below. The modulation is chosen to provide good overlap with the Doppler profileof the atoms after being pre-cooled in the 461 nm MOT, with final temperatures of ∼1 mK.
In the initial broadband phase of the red MOT, the field gradient is reduced from the 50 G/cm

34

Blue Mot

Field gradient

50 G/cm

Broad. Red

SF Red11 G/cm

4 G/cm
300-500 ms

110 msA-H Off
100 ms55 ms

Figure 2.7: Field gradient programming during the three MOT stages: 461 nm (blue) MOT,
broadband red MOT, and single-frequency (SF) red. After the single-frequency red mot, the antiHelmholt (A-H) coils are turned off to prepare the system for optical pumping and spectroscopy.

employed in the 461 nm MOT, to ∼4 G/cm. This provides good spatial overlap, or mode-matching,
between the blue MOT atomic density distribution and the broadband red MOT. Subsequently,
the field is ramped to ∼11 G/cm over 100 ms. The ramp compresses the MOT in all three spatial
dimensions. When the field gradient reaches 11 G/cm, the broadband frequency modulation isswitched off, and both the trapping and stirring laser are tuned near resonance, to form the final
stage of MOT cooling, the single frequency red MOT. This field ramping procedure and MOTstages are depicted in Fig. 2.7.
The transfer efficiency from the blue MOT is approximately 15%, and results in a sample ofatoms with µK temperatures. When the atoms are cooled in the presence of the optical lattice, anadditional fraction of the atoms in the red MOT are cooled into the lattice. Due to poor spatialmode-matching between the optical lattice and the red MOT density distribution, the transfer
efficiency between the lattice and red MOT is also approximately 15%. Ultimately, we can load
approximately Ntot = 5000 87 Sr atoms into ∼ 100 lattice sites. The distribution of population in
the lattice sites is derived in Appendix F, and on average, atoms share lattice site with ∼ 20 other35
atoms for Ntot = 5000.

2.4.2.3

Nuclear spin-polarizing for spectroscopy

The final step prior to clock spectroscopy on the lattice-trapped sample is to nuclear spinpolarize the 87 Sr so that a single mF level is occupied. This serves two purposes: first it prepares
an ensemble of identical fermionic 87 Sr atoms, meaning that collisional interactions due to the swave channel should be suppressed or completely eliminated, which is important for clock accuracy.
We will discuss the collisional interaction in detail in Chapter 5.Second, spin-polarized atoms all
contribute to the useful spectroscopy signal utilized to operate the clock.
As a consequence of the differential g-factor between the clock ground and excited states,
the clock transition possesses a small magnetic field sensitivity, even for mF preserving transitions
between 1 S0 and 3 P0 are resolvable in a few hundred mG magnetic bias field. Specifically, there isa Zeeman shift between the clock states given by [119]∆νπ = −δgmF µ0 B/h,

(2.28)

where δg is the differential g-factor, µ0 is the Bohr magneton, and the subscript π indicatesthat the formula is for non-mf changing transitions (“π” transitions). The quantity −δgµ0 /h =108.4(4) Hz/G [119]. In order to clearly resolve each nuclear spin sublevel via clock spectroscopy,a ∼500 mG field is applied during spectroscopy.
In order to prepare the atoms in the single nuclear spin state, we employ optical pumping
on the 1 S0 →3 P1 F = 9/2 → F = 9/2 transition with circularly polarized light (both σ + and σ −orientations) along a bias field applied along H2, which is the same axis as the bias field applied
during clock spectroscopy. During the polarizing, a small bias field of ∼100 mG is applied to define
a quantization axis. A liquid crystal wave plate allows us freedom to switch between σ + and σ −polarization. This freedom to switch polarization is important for removing 1st-order sensitivity tomagnetic fields; by alternating between mF = +9/2 and mF = −9/2 states, the average of the twotransition frequencies, as indicated by Eq. 2.28, will be first-order insensitive to magnetic fields [31].

Excitation probability

36
1.0

-9/2

+9/2

0.8
0.6

PID 2

PID 1

0.4

f0

0.2
0.0

Frequency
Figure 2.8: Clock operation with stretched nuclear spin states for first-order Zeeman insensitivity.
Two independent locks are averaged to find the linear Zeeman shift-free center frequency, f0 .

Figure 2.8 shows the principle of operation for the dual-spin clock operation. Two independent
PID locks—one for mF = +9/2 and the other for mF = −9/2—are used to extrapolate the linear
Zeeman shift-free center frequency, f0 .

2.4.3

Optical dipole trap and clock spectroscopy: experimental configuration

As described in the previous section, the atoms are directly cooled from the 689 nm MOT
into the optical lattice, for which the basic principle of operation was described in Section 2.3. Here
we describe the experimental apparatus utilized to create the optical lattice trapping potential.
The lattice optical dipole trap potential is formed by a Gaussian beam retro-reflected uponitself, such that the foci of the input beam and retro-reflected beam coincide. As depicted inFig. 2.9, the input lattice beam is focused to a 1/e2 radius of 32 µm by an input lens with focal
length of 14 cm such that the beam waist lies at approximately the geometric center of the mainchamber. The lattice beam then passes out through the bottom viewport after which it is folded
onto a curved retro-reflector mirror with 20 cm radius of curvature. The longitudinal position of

37

Clock beam

Lattice input
Pol.

Lattice retro-reflector
assembly-1st diffracted order

Clocklaser in0th order retro-reflected

Figure 2.9: Optical lattice dipole trap and clock laser pathway. The lattice is approximately 19◦
from vertical. The focusing lens has a focal length of 14 cm and the radius of curvature of theretro-reflector is 20 cm.

the mirror is translatable, and its position and orientation are chosen such that a maximum of thelattice light is back-coupled into the single-mode from which the lattice beam originates.
The lattice light is produced by an amplified external cavity laser diode. Approximately
35–40 mW derived from a Littrow-configuration laser diode are injected into a tapered amplifier(Eagleyard EYP-TPA-0808), which produces 800 mW of output power. The output of the tapered
amplifier is spectrally filtered with a ∼ 1 nm spectral bandwidth interference filter. An optional
filter cavity with Finesse ∼ 100 can further spectrally filter the output of the tapered amplifier

38
chip. These precautions are taken because tapered amplifier chips are known to possess amplified
spontaneous emission (ASE) spectral pedestals. This uncontrolled stark shift due to these pedestals
has been recognized as an important effect to be controlled in lattice clocks [34]. To date, allpublished systematic evaluations of the JILA 87 Sr lattice clock uncertainty were performed with a
Ti:Sapphire laser, which will not suffer from an ASE problem.
In order to perform clock spectroscopy, the clock laser light is injected through the lattice
retro-reflector with 99% transmission. A focusing lens ensures that the clock laser is concentrated at
the atoms, while the waist of the clock light is kept several times larger than the lattice waist, thus
ensuring that inhomogeneities due to the k-vector spread of a tight focus are minimized (discussed
more fully in Section 2.5).

2.4.3.1

Clock laser path and RF phase chirp cancellation

The clock laser light is derived from an ultrastable laser system in a separate room, which we
describe in Chapter 4. This separate room was chosen for its temperature stability, acoustic noiselevels, and relative calm, from which the name “Jan’s quiet room” is derived. The clock laser lightis delivered to the “distribution center” (Fig. 2.10). In the distribution center, the clock laser lightis amplified by injection locking a second diode with ∼ 500 µW of optical power. The output powerof the injection system is around 35 mW, which provides ample power for multiple experiments, aswell as interface with a femtosecond frequency comb.
After exiting the distribution center, a fiber delivers clock laser light near the experimental
chamber. An acousto-optic modulator (AOM) provides the frequency shift for scanning the clock
laser onto the atomic resonance, as shown in Fig. 2.10. The delivery of pulsed light to the atoms thatis free from switching transients and phase chirp errors is a challenging problem [120]. As shown
schematically in Fig. 2.9, care is taken to cancel phase shifts due to RF-induced heating of theAOM, by referencing the light for the fiber noise cancellation system [121] through the AOM, using
the 0th-order diffracted light. In this way, the fiber noise cancellation system is always engaged,
yet samples approximately the same optical path deviations as the −1st diffracted order, which is39

Distribution center
Inj. lock-1st order
0th
Spec.
AOM

FNE

Clock laser

RF drive
FNE
AOM

From
“Jan’s quiet room”

Figure 2.10: The clock laser delivery system. The ultrastable laser is operated in a separate room
and is delivered via noise-cancelled fiber to the distribution center. In the distribution center,∼ 500 µW of clock laser light are used to injection lock a laser diode. A second “fiber noise eater”(FNE) system ensures that the light is faithfully delivered to the atoms. Importantly, the effect ofRF-heating of the spectroscopy acousto-optic modulator (AOM) is mitigated by including it in thefiber noise cancellation loop.

used for spectroscopy. We have tested that phase chirp-induced frequency shifts can be reduced to−3(8) mHZ, corresponding to a fractional frequency shift of -0.7(2)×10−17 with the AOM diffractingat ∼ 50% efficiency (corresponding to ∼ 1 W RF drive) and with short 10 ms pulses. We expectthat reducing the RF drive and elongating the clock pulses, as typically utilized in clock operation,
can further reduce this shift by an order of magnitude. The 10 ms pulse duration was specificallychosen to emphasize the effect, and an experimentally relevant “cooling off” period of 1 s was used
between cycles. In contrast, we measure a large phase chirp when the fiber noise cancellation plane
is moved before the AOM, thus the cancellation pathway does not “see” the phase chirp. In thiscase, the frequency shift was measured at −129(14) mHz, corresponding to a fractional frequency
shift of 3.0(3) × 10−16 . This indicates that an active cancellation or characterization of the AOM
phase chirp is indeed necessary for a high-accuracy clock.

40
2.4.4

Atom state detection and measurement

In order to detect the fraction of atoms in the metastable clock state, a series of fluorescencemeasurements are performed. As depicted in Fig. 2.11, the first step in the detection process is tocount the number of ground state atoms, Ng . This is achieved by measuring the fluorescence froma strong probe beam that completely overlaps the lattice-trapped atoms and is retro-reflected, sothat each atom cycles many times before being lost. The power in the probe beam is ∼ 30 mW,
which is enough to strongly saturate the transition. On average, we collect & 6 useful photons
per atom in the trap in a photo-multiplier tube (PMT). This quantity already includes the PMT
quantum efficiency. The effect of the probe is to completely remove the lattice-trapped groundstate population. We then repump the excited state atoms to the ground state using the 707 nm
and 679 nm repumping transitions. The ground state population is again counted, but represents
the number of atoms that were in the excited state, Ne . In between the measurements of Ng and
Ne , a third fluorescence measurement characterizes the signal background, Nbg , so that Ne andNg are accurate representations of the lattice-trapped populations. The fraction of atoms in the
excited state can then be calculated from Ne , Ng , and Nbg as
fexc =

2.5

Ne − Nbg.
Ne + Ng − 2Nbg

(2.29)

Lattice spectroscopy
The precision of the 87 Sr optical lattice clock arises from the pairing of an ultrastable laser

with sub-Hz linewidth, with an extremely narrow optical transition, whose fundamental limit is thenatural linewidth of 87 Sr . However, in order to realize the potential of such a precise spectroscopy
system, interaction times between the atoms and probing optical field must be at the 1 s timescale.
While it might be possible to envision a multi-pulse Ramsey-Bordé optical fountain clock, there
are systematics such as the wave-front curvature of the beams leading to first-order Doppler shifts,
that will limit the ultimate accuracy of this technique [107, 122], not to mention that the apparatus
for 1 s interrogation would be quite large and complex.

Count ground state
3

Ng

41

Pump excited
population to ground
3

P0

P0

Count ground state
3

Ne

P0

707 nm and 679 nm
repumping

1

S0

1

S0

1

S0

Figure 2.11: Atomic state detection scheme. After the atoms are placed in a superposition of
1 S and 3 P
0
0 by the clock laser, the state detection proceeds in three steps: (1) the ground state
population is counted (2) the excited state population is moved to 1 S0 via repumping and (3) the
ground state population is again counted.

A better way to realize long-term optical and atomic coherence times is to strongly localize
the atoms. This technique was used to demonstrate recoil-free optical spectroscopy [103] andsubsequent record-breaking line-Q (Q ≡ ∆ν/ν) with a single trapped Hg+ ion, tightly localized ina quadrupole trap [22]. In fact, the Hg+ ion was so localized, that the characteristic length scaleof its thermal motion was much less than the wavelength of the spectroscopy laser. This regimeof spectroscopy is known as the Lamb-Dicke regime, where the characteristic energy scale of theconfining potential is larger than the photon recoil energy.
During free-space spectroscopy, conservation of momentum requires that as an atom absorbs
a photon, it receives an impulse of ~k, resulting in a shift of the frequency of(~k)2
Er = ~ωr =,2m(2.30)

where ωr is the recoil frequency, k = λ/ (2π) and m is the mass of the atom. In general, when an
quantum emitter/absorber is more localized than a wavelength of light, its absorption spectrum
will be strongly modified. For example, a bound atom or ion with quantized motional eigenstates(e.g. harmonic oscillator eigenstates) can only emit or absorb between initial and final quantizedeigenstates, leading to a quantized energy spectrum. Early demonstrations of recoil-free spec-

42
troscopy were performed with Mössbauer spectroscopy, where the recoil momentum is absorbed by
the crystalline lattice of the absorbing or emitting material [123] and can lead to extremely highline Q ' 1015 [124], which has only recently been matched by the 87 Sr lattice clock reported in thisthesis with a Q of 9 × 1014 [125].
Optical lattices used in optical lattice clocks play an analogous role to the crystal lattice inMössbauer spectrscopy in the sense that they can tightly localize the atoms and also absorb therecoil momentum. We consider the case where the optical potential is harmonic, such that
1
1
1
V (x, y, z) = mωx x2 + mωy y 2 + mωz z 2 ,22
2(2.31)

where ωx = ωy = 2πνr , ωz = 2πνz , and νz (νr ) is the longitudinal (radial) trapping frequency given
by Eq. 2.25 (Eq. 2.26). In a one-dimensional optical lattice, as depicted in Fig. 2.12, ωz  ωx ' ωy
The corresponding motional eigenstates of the system are the harmonic oscillator energy eigenstates,
which we label as |~ni ≡ |nx i|ny i|nz i. These will be discussed in detail in Chapter 5.
When performing spectroscopy in a trap with motional eigenstates given by |~ni, there are twoclasses of laser-driven transitions possible. The first is where the motional states are left unchanged,
such that |~ni → |~ni and is referred to as the carrier transition. The second class of transition iswhere one or more motional quantum numbers are changed such that |~ni → |n~0 i, and is referred
to as a sideband transition. The simplest and most experimentally relevant cases are the carriertransition and the sideband transitions for which a harmonic oscillator level is changed by one
vibrational quanta, as depicted in Fig. 2.12. Fig. 2.13 shows the measured sideband transitions
with respect to the carrier transition in both the radial and longitudinal directions. In this way, we
are able to quantify our trap frequencies fairly accurately, which in turn gives us precise knowledgeof the harmonic potential in which the atoms sit.
The motional states of an atom in a harmonic trap will also affect its interaction with aradiation field. The relevant dipole moment for a general transition in a quantized trap is given by[15]
d~n~n0 = h~n|eik·r̂ |~n0 ihg|d̂|ei.

(2.32)

43

Clock laser

Lattice sites

n=2
n=1
n=0

n=2
n=1
n=0

Figure 2.12: Spectroscopy in a one-dimensional optical lattice. The clock probe beam is aligned
along the lattice propagation axis. The frequency of the transition is modified according to thechange in motional quanta. A blue sideband transition excites an additional motional quanta, while
the red sideband transition de-excites a motional quanta.

As a result, we can write the mode-dependent Rabi frequency as a function of the bare Rabi
frequency Ω0 ≡ hg|E · d̂|ei/~ as [15, 126]v
Y −η2 /2 uuΩ~n~n0 = Ω0 h~n|eik·r̂ |~n0 i = Ω0
e j tjn<jj (iηj )∆nj L∆n
ηi2 .
<
n
j
n<
j + ∆nj !

(2.33)

0Here, j ∈ {x, y, z}, ∆nj = n0j − nj , n<
j is the lesser of nj and nj , and ηj is the Lamb-Dicke

parameter in the jth direction. Specifically, the Lamb-Dicke parameter is defined as√
ηj = kj aj / 2
where aj is the characteristic length for the jth harmonic degree of freedom aj =(2.34)p~/ (mωj ).

Finally, kj is simply the probe k-vector projection along the jth direction.
As clearly seen from Eq. 2.34, the Lamb-Dicke parameter scales as the ratio of the harmonic
confinement length to the wavelength of the probe light. One general statement is that if ηj  1,the Rabi frequency for an nj changing transition is suppressed relative to the carrier by a factor of

44
1.0

0.4

a

0.8

νz = 80 kHz

0.3

0.2

0.1

0.0
100

50

0

50

Detuning (kHz)100

Excitation fraction

Excitation fraction

0.5

b

0.6

νr

0.4

450 Hz

0.2

0.0

500

0

Detuning (Hz)500

Figure 2.13: Spectroscopic resolution of longitudinal and radial sidebands in a 1D optical latticepotential. (a) The laser frequency is scanned over 100 kHz on either side of the carrier transition,
revealing ∆nz = ±1 sideband transitions. The asymmetric shape of the sidebands arises fromcoupling between the radial and longitudinal optical potential. (b) Radial sidebands with ∆nr =±1. In order to excite the radial sideband transition, the clock laser was misaligned from the latticeby ∆θ ' 10 mrad.

∆nj

ηj

. Furthermore, in the case where ~n = ~n0 , Eq. 2.34 simplifies significantly, such thatΩ~n = Ω0

Y −η2 /2Y
e j Lnj ηi2 ' Ω0
1 − (nj + 1) ηj2 ,j(2.35)jwhere the final approximation is valid in the limit ηj2  1. Equation 2.35 demonstrates a dependenceof the Rabi frequency on |~ni. In general, for a one-dimensional lattice, the optimal situation is toalign the spectroscopy laser along z (the axis of strong confinement) to minimize radial sideband
excitation and Rabi frequency inhomogeneity.
One final detail to address is the effect of tunneling on lattice spectroscopy. Tunneling in an
optical lattice can give rise to frequency shifts and line broadening due to the band structure of thelattice [127, 128]. In order to circumvent this problem, it was prescribed to work in an accelerated
frame, achieved by tilting the lattice so that a component of gravity lies along the standing wave.
In the so-called Wannier-Stark ladder, tunneling effects are effectively detuned by the ∼ 1 kHzenergy detuning between adjacent sites due to the effect of gravity [127].

45

1.0

Excitation fraction

0.8

0.6

Excitation fraction

0.6
0.4

500 mHz0.2

FWHM: 5 Hz

0.0

0.4

−4

−2
0
2
Detuning (Hz)

4

0.2

0.0
-40

-20

0

20

Frequency (Hz)Figure 2.14: Spectroscopic lineshape obtained with Rabi spectroscopy performed on the 1 S0 →3 P0clock transition at 429 THz with a π-pulse time of Tpulse = 160 ms. Inset: High-resolution scantaken with Tpulse = 3 s. Both scans were taken without any averaging. The ultra-high resolutionof these lineshapes is enabled by the clock laser described in Chapter 4.

2.5.1

Lattice spectroscopy: experimental

When performing spectroscopy on the lattice-trapped atoms, the three main parameters of
relevance are the duration of the clock pulse(s), the number of clock pulses applied, and the intensityof the clock laser beam. The most common form of spectroscopy employed is Rabi spectroscopy.
As seen from the preceding discussion, care must be taken to make sure that the clock probe beam
is well-aligned with the optical lattice so that the atoms are well within the Lamb-Dicke regime.
Ideally, for the 1D optical lattice, the clock probe has no projection along the x or y directions, suchthat ηx = ηy = 0. In principle, there are small deviations from this due to a beam misalignments,
such that, e.g., ηx = k sin (θ) ax . Here, typical misalignments are at the few mrad level.

a

0.6
0.4
0.2
0.0

b

0.8
Excitation fraction

Excitation fraction

0.8

46

1.0

Exc. frac.

1.0

0.6

1.0
0.8
0.6
0.4
0.2
0.0

0

20 40 60
Time ms

80

0.4
0.2

0

50

100

Time ms

150

0.0

0

10

20

30

40

Time ms

50

60

70

Figure 2.15: Rabi flopping as a probe of the spread in Rabi frequency. (a) Rabi flopping witha longitudinally cooled (Tz = 1.5 µK) and radially hot (Tr = 3.0) µK sample. The circles areexperimentally recorded data and yield an effective misalignment of 2.5 mrad, resulting in ∆Ω/Ω =0.02 (b) Series of Rabi flopping curves taken with Tz ' Tr ' 4 µK. The extrapolated effective
misalignment varied from 3 mrad to 8 mrad (∆Ω/Ω between 0.06 and 0.11.) Inset: Theory curvefor the best-aligned case.

Clock spectroscopy is performed by sending a pulse or pulses of clock laser light onto theatoms along the z axis—the direction along which the atoms are strongly confined by the lattice.
In the case of Rabi spectroscopy, the applied pulse is chosen such that the Rabi frequency satisfiesΩTpulse = π.

(2.36)

This results in unity excitation probability on resonance. After the pulse is applied, the excitedstate fraction of the atoms is detected as described in Section 2.4.4. An example of a Rabi lineshapewith τ = 160 ms is shown in Fig. 2.14. The full-width at half-max (FWHM) of the rabi lineshape isrelated to Tpulse by ∆νFWHM ' 0.8/Tpulse , resulting in a 5 Hz FWHM for the 160 ms pulse duration
depicted in Fig. 2.14. The theoretical lineshapes and details of both Rabi and Ramsey spectroscopy
are discussed in Appendix 1.
In order to experimentally measure the ~n-dependence of the Rabi frequency, we performed
several experiments. With our best alignment at Tr = 3 µK and Tr = 1.5 µK (temperature along
the longitudinal and radial directions, resctively), we measured very long Rabi coherence, as afunction of the number of on-resonance Rabi “flops” (i.e. excitation cycles |gi → |ei → . . ., where

47
the population returns to the ground state). In order to provide a theoretical fit for the data,
we calculate a Boltzmann-weighted sum of different evolution frequencies, whose n-dependence isgiven by Eq. 2.35a. We assume a fixed misalignment between the clock probe, which is assumed
to be a perfect plane wave, and the lattice. The data for the cold, well-aligned case, is shown in
Fig. 2.15. The theoretical fit indicates that the effective misalignment is as small as 2.5 mrad, andthat for these conditions, the RMS spread of Rabi frequencies ∆Ω is about 2%. This is a significant
improvement over previously reported results from our group [112, 114]. At hotter temperatures,
we measure much faster decay due to the larger number of motional states that are populated. We
took numerous traces, as shown in Fig. 2.15b, with varying misalignment. Again, at the optimalalignment, we find an effective misalignment of 3 mrad, while the most severe misalignment shown
here is only about 8 mrad. As seen from the figure, probe misalignments are most visible at hotter
temperatures, and we used this to our advantage when trying to optimize alignment between theprobe and the lattice.
While the agreement between the theory and the data shown in Fig. 2.15 is qualitatively
quite good, there are minor discrepancies. There are several reasons for this. The first is that
2-body inelastic losses can affect the evolution of the atomic coherence at longer times (we describe
the inelastic losses in detail in Chapter 5). A second reason is that we assume that the only
wavefront misalignment between the lattice and the clock probe is due to an angular misalignment.
In fact, the probe itself is focused, and the act of focusing the clock probe results in a Gaussian
spread of wave-vectors k. It can be shown that for a probe with 1/e2 intensity radius w0 , themode-dependence of the Rabi frequencies isΩ~n2= πw02 e−ηz /2 Lnz ηz2
Ω0

Z



2
2
2
2
2
dfx dfy e−π (fx +fy )(w0 +ax ) Lnx 2π 2 fx a2x Lny 2π 2 fx a2y .

(2.37)A more complete description of the sidebands would include this effect as well, but for all practical
purposes the measured ∆Ω/Ω ' 0.02 is more than sufficient for high-precision spectroscopy, andwe choose to characterize the misalignment as an “effective” one.

48

2.6

Limits to atom-laser coherence due to photon scattering
The question as to whether optical-atomic coherence in an optical lattice suffers due to photon

scattering is of central concern for optical frequency standards employing optical lattices. In thecase of Sr, some estimates have placed the scattering rate as high as ∼3 Hz for a lattice depthof ∼100 Erec [129], which would already have negative consequences for the clock and precision
spectroscopy reported in this thesis. It is the purpose of this section to calculate the rate of photon
scattering from the two clock states.
We begin by noting that the question of decoherence due to light scatter is a nuanced one,primarily because there are two scattering processes that can occur. The first type of light scatteringis Rayleigh scattering, where a photon is scattered from the lattice beam but the internal state of
the atom does not change. In a recent study of a qubit formed by the ground-state spin of a Be+
ion, it was found that even if the Rayleigh scattering rates of the two qubit states were matched, therelative sign of the Rayleigh scattering amplitude was the important quantity for the decoherenceof the qubit [130]. In a magic wavelength lattice, the AC polarizability and the Rayleigh scattering
rate are intimately linked, as it is the induced dipole that both allows the atom to be trapped andthat causes Rayleigh scattering.
The second kind of scattering is Raman scattering. Here, the “color” of the absorbed photon
and the emitted photon are different, implying that the internal state of the atom has been changed.
In the case of Sr, the only Raman scattering processes that are relevant are 3 P0 →3 P1 and 3 P0
→3 P2 . Of these two, it is the first process that is harmful for the coherence of the clock state,since the 3 P1 state can decay back to the ground state. Thus, this process can be considered
as a modification of the clock state lifetime due to a small but finite optical pumping from thelattice. The 3 P0 →3 P2 simply appears as loss from the excited clock state, and thus should not
significantly affect the coherence.

49
2.6.1

Kramers-Heisenberg formula

In order to calculate all the relevant scattering rates, we begin with the Kramers-Heisenberg
formula3 which describes the cross section for scattering between two internal states of an atom—labelled by a and b and connected via an intermediate state j—as [88]2
0 · d )2 X (ε̂0 · d ) (ε̂ · d )
(ε̂
·
d
)
(ε̂
m
dσ
ja
ja
bj
bj
.
= r02 ωω 03 2 4+dΩ~ eωja − ωωja + ω 0

(2.38)

jHere, r0 is the “classical electron radius” and is given by r0 = e2 /(4π0 mc2 ); ω 0 is the wavelengthof the (energy conserving) scattered radiation; ε̂ and ε̂0 are the polarizations of the incoming andscattered photons, respectively; and ωja is the complex transition frequency between states a andn, given byωja = ωj − ωa + iAT, j /2,(2.39)

where AT, j is the Einstein A-coefficient of the excited state j (inverse the excited state lifetime).
When a = b we have Rayleigh scattering, and ω = ω 0 . When a and b differ, we have Raman
scattering, and ω 0 differs from ω to conserve energy. Here, j is a shorthand notation for all possible
intermediate states connected by the dipole operator. The dipole matrix elements, are given by,e.g.,
dja = hj|er̂|ai,(2.40)

where a, b, and j are shorthand for a specific state, with each described by a term symbol S LJ ,electronic configuration, and angular momentum projection, mJ . Since we consider the clock states,
the initial and final states a and b are restricted to be ∈ {5s2 1 S0 , 5s5p 3 P0 , 5s5p 3 P1 , 5s5p 3 P2 }. To
determine the photon scattering rate, we multiply the cross section by the incoming photon flux
such thatdγIdσε̂, ε̂0 =
×
dΩ
~ω dΩ

2
X  (ε̂0 · dbj ) (ε̂ · dja ) (ε̂ · dbj ) (ε̂0 · dja ) 
Iω 03
=
+
.
ωja − ωωja + ω 0
(4π0 )2 c4 ~3 j

3A good derivation of this formula is found in [131](2.41)(2.42)

50
The interpretation of dγ/dΩ is photons scattered per second into a solid angle dΩ. This quantity
depends on the incoming intensity.
At first glance, Eq. 2.42 looks fairly intractable. After all, we want the total scatter rate
given by the integral over 4π of solid angle, namely
Z
γ=


dγ
ε̂, ε̂0 dΩ.
dΩ(2.43)

However, we can modify Eq. 2.42 to look more reasonable. Focusing on the sum, we note thatX  (ε̂0 · dbj ) (ε̂ · dja ) (ε̂ · dbj ) (ε̂0 · dja ) X  (dbj ) (ε̂ · dja ) (ε̂ · dbj ) (dja ) 
0
=
ε̂
·
+
+
ωja − ωωja + ω 0
ωja − ωωja + ω 0jj

≡ ε̂0 · D.

(2.44)The term D (no subscript) can be thought of as a net induced dipole for the scattered radiation.
Now the angular integral can be performed, and we must also sum over the two final polarization
states which we label as ε̂01 (n) and ε̂02 (n). This can be written asIω 03(4π0 )2 c4 ~3

Z

h
i
2
2
dΩ ε̂01 (n) · D + ε̂02 (n) · D .

(2.45)

It turns out that the integral in Eq. 2.45 is exactly the same equation that appears in the treatmentof spontaneous emission (see problem 4.2 of Ref. [88]), and the final result for γ is given by


Iω 03
8π
2
γ=
|D| ,
(2.46)(4π0 )2 c4 ~3 3
where the term in parentheses is the evaluated integral of Eq. 2.45 and D is defined as
X  dbj (ε̂ · dja ) dja (ε̂ · dbj ) D≡+.
ωja − ωωja + ω 0

(2.47)j

We proceed by noting that in Eq. 2.46, we have a simple expression that depends on |D|2 , which
is defined in Eq. 2.47. The meaning of |D|2 is simply|D|2 = |Dx |2 + |Dy |2 + |Dz |2 =

1
X
q=−1

|Dq |2

Here, Dq is given by
Dq =

X
j


(ε̂ · dbj )(ε̂ · dja )hb|erq |ji+ hj|erq |ai.
ωja − ωωja + ω 0

(2.48)

51

103
103

Scattering rate (1/s)

Scattering rate (1/s)

10

102

2

101

101
100
10−1
10−2
10−3

10

500

550

600

650

700

750

800

Lattice wavelength (nm)

0

10−1
10−2
Inset span

−3

10

200

500

1000

Lattice wavelength (nm)2000

Figure 2.16: Raman scattering rate as a function of wavelength for scattering from 3 P0 into 3 P1
(solid), and 3 P2 (dashed). The lattice peak intensity is 3.4 × 104 W/cm2 , corresponding to νz =80 kHz for an 813.428 nm lattice. The inset (span indicated by a grey bar) shows a detailed viewof two magic wavelengths, including the 813 nm magic wavelength used for clock operation.

At this point, if we assume linear polarization of the input beam (lattice), there is a natural
choice of quantization axis that simplifies things even more. If we choose the quantization axis
to be along the direction of the lattice polarization (also realized in the experiment, conveniently)
then Eq. 2.49 becomesX
hj|er̂0 |ai
hb|er̂0 |ji+ hj|er̂q |ai
Dq =hb|er̂q |ji.
ωja − ωωja + ω 0

(2.49)j

At this point we can use Eq. 2.46 along with Eqs. 2.48 and 2.49 and the treatment in Appendix B
to calculate the scattering rate for a lattice frequency ω and scattered photon frequency ω 0 . In thiscalculation, the sum over intermediate states n is carried out for the states given in Table 2.1 forscattering from 3 P0 and 1 S0 .

52

103

103

Scattering rate (1/s)

102Scattering rate (1/s)

102
10

1

101
100
10−1
10−2
10−3

10

500

550

600

650

700

750

800

Lattice wavelength (nm)

0

10−1
10−2
Inset span

10−3
200

500

1000

2000

Lattice wavelength (nm)Figure 2.17: Rayleigh scattering rate as a function of wavelength for 3 P0 (solid) and 1 S0 (dashed).
The lattice peak intensity is 3.4 × 104 W/cm2 , corresponding to νz = 80 kHz for a lattice at
813.428 nm. As before, the inset (span indicated again by a grey bar) shows a detailed view of two
magic wavelengths, including the 813 nm magic wavelength used for clock operation.Note that theRayleigh scattering rate is the same at the magic wavelength, as expected.

Finally, we extrapolate the lattice laser intensity, I, from the observed sideband frequencies.
For the purposes of this work, we assume that the atoms are at the very bottom of the wells,such that they are subject to the maximum lattice intensity. In principle, the finite temperatureof the sample will reduce the average intensity seen by the atoms, and the fractional reduction isapproximately the same as the ratio of the atomic temperature to the trap depth.
As presented previously, we assume a lattice intensity profile given by
I (x, y, z) =8P0
22
e−2r /w (x,y,z) cos2 (kz) ,2πw (x, y, z)(2.50)

where P0 is the power of a single beam. The longitudinal trap frequency associated with this profile

Results at 813.428 nm.
3 P →3 P
7.3 × 10−2 × (νz /80 kHz)2 [1/s]
00
3 P →3 P
6.4 × 10−2 × (νz /80 kHz)2 [1/s]0
1
3 P →3 P
4.4 × 10−2 × (νz /80 kHz)2 [1/s]0
2
1 S →1 S
7.3 × 10−2 × (νz /80 kHz)2 [1/s]00Results at 505 nm.
3 P →3 P
4.9 × (νz /260 kHz)2 [1/s]
00
3 P →3 P
1.7 × 10−1 × (νz /260 kHz)2 [1/s]0
1
3 P →3 P
8.2 × 10−1 × (νz /260 kHz)2 [1/s]0
2
1 S →1 S
4.9 × (νz /260 kHz)2 [1/s]0053Table 2.2: Scattering rates from 3 P0 into the 3 P fine structure manifold and 1 S0 Rayleigh scattering
at the magic wavelength. The Rayleigh rates agree at the magic wavelengths, as expected. The
Raman process 3 P0 →3 P1 is expected to be the most detrimental to clock coherence. In all relevant
conditions at both magic wavelengths we find that this rate is well below 1 Hz.

is given by1νz =2πw0 λr

32παP0
1
=
c0 M
λ

s

αIpeakc0 M(2.51)


where Ipeak = 8P0 / πw2 . For νz = 80 kHz, Ipeak = 3.4 × 104 W/cm2 corresponding to P0 =140 mW. Using this value for intensity, we find the scattering rate into the 3 P fine structure
manifold from 3 P0 as a function of wavelength as shown in Fig. 2.16., while the comparison of theRayleigh rates of scattering out of 1 S0 and 3 P0 is shown in Fig. 2.17. It is important to note thatno saturation effects are taken into account, so only the far-off resonant segments of this plot are
valid for such high intensity.

2.6.1.1

Numerical values and experimental confirmation

The wavelengths of greatest practical importance are the magic wavelengths near 813 nm
and 505 nm. For both these magic wavelengths, we tabulate the scattering rates for a lattice
depth corresponding to νz ' 80 kHz with λlat = 813.428 nm, which corresponds to 140 mW input
power and a lattice depth of 130 Er (22 µ K). In order to calculate the scattering rate at 505 nm,
we simply assume the same input power (140 mW), but due to the difference in wavelength andpolarizability, νz is significantly larger than for the lattice formed with the equivalent power at

54

Atom number

3000
2500

3

P0

2000
1500
1000

1

S 0 × 10

500
0

0.4

0.6

0.8

1.0

1.2

1.4

Hold time (s)Figure 2.18: Experimental measurement of optical pumping due to lattice scattering. The extrapolated 3 P0 →1 S0 decay rate is due to the both spontaneous decay and optical pumping due to
Raman scattering of lattice photons in addition to spontaneous emission from the metastable clockstate.

813 nm. In Table 2.2, we summarize the result of the scattering rate for 140 mW input power.
The rate will scale with the total input power, and we show the scaling as a function of trap
frequency. We note that for 80 kHz sidebands at 505 nm, only 15 mW of optical power are
required and all scattering rates—especially the Raman rates—are small. Thus, we anticipate thata lattice at this wavelength could be an intriguing possibility, especially for applications requiring
low power consumption. Finally, we note that an important modification must be considered whenthe ensemble average of the scattering rate is desired; instead of using the Ipeak obtained by invertingEq. 2.51, it is necessary to find the RMS intensity averaged over the thermal density-distributionof the atoms. This effect roughly scales as the ratio of the total trap depth to kB Tr and kB Tz . For
Tz = Tr = 3.5 µK, the average scattering rate is 70% of the peak scattering rate given in Table 2.2.
In order to test the accuracy of the scattering rate calculation, we performed a measurementof the 1 S0 population as a function of hold time of the lattice. In order to initialize the system,
we first excited close to 100% of the atoms to the 3 P0 state with a clock pulse and subsequently
removed residual ground-state atoms by using the 461 nm probe beam as a method to “blow-away”
ground state atoms. We observed that the ground state population grew at a rate proportional
to the excited state population. Due to the effects of two-body inelastic collisions between 3 P0

55
atoms (described in Chapter 5), the excited state population also decayed while the ground state
population was being populated due to 3 P0 →3 P1 Raman scattering, and we took this effectinto account. By fitting the rate of accumulation in 1 S0 , we were able to extrapolate that theatoms in 3 P0 were decaying with Γe→g = 5.0(5) × 10−2 1/s. We expect this to be consistent
with the temperature corrected 3 P0 →3 P1 Raman rate in Table 2.2, of ΓRaman= 4.4 × 10−2 1/s.
e→gInterestingly, while there is fairly good agreement between these numbers, there is one effect that
we have not taken into account: the natural lifetime of 87 Sr due to spontaneous emission. The
3P

Spont
1
−3 1/s, and thus there is
0 → S0 natural decay rate has been calculated as Γe→g = 9(3) × 10

Raman = 5.3(3) × 10−2 1/s and the measured
good agreement between the predicted ΓSponte→g + Γe→g

Γe→g = 5.0(5) × 10−2 1/s.
Finally, we note that this general technique of measuring decay to 1 S0 should permit an
accurate evaluation of the 87 Sr natural lifetime. By measuring the optical pumping rate as afunction of lattice intensity, the zero-intensity decay rate could be extrapolated, representing ameasurement of the natural lifetime. This measurement would benefit from the fact that its analysis
will not depend heavily upon the details of the scattering rate calculation.

2.7

In situ thermometry and cooling
As described in the previous section, the atomic temperature can have direct and noticeable

impact on degree of dephasing between atoms during the laser excitation process. As we will see
in Chapter 5, knowledge of and control of excitation inhomogeneity is important for characterizing
density-dependent frequency shifts. Furthermore, the atomic density itself is dependent on thetemperature, which is also important for characterizing density-dependent effects. In this section,
we describe the techniques used for measuring the temperature of the atomic sample in the longitudinal direction (Tz ) and in the radial direction (Tr ), as well as additional cooling we can perform
on the lattice-trapped atoms.
In order to extract Tz , the red and blue longitudinal sidebands are compared in size. Since
atoms with nz = 0 cannot lose any further motional quanta, atoms in the longitudinal ground do

0.4

56

a

0.25

Tz = 2.1 µK
Tz = 3.3 µK

0.3

νz = 80 kHz

0.2

0.1

Excitation fraction

Excitation fraction

0.5b

Tr = 2.1 µK
Tr = 4.6 µK

0.20
0.15
0.10
0.05
0.00

0.0
100

50

0

Detuning (kHz)

50

100

100

0

50

50

100

Detuning (kHz)

Figure 2.19: Temperature extraction and effect of additional cooling. (a) Uncooled (red) and cooled(blue) longitudinal sideband structure. (b) Uncooled (red) and cooled (blue) Doppler-broadened
clock transition. The black line is a Gaussian fit to each temperature condition.

not contribute to the red sideband, thus creating a temperature-dependent asymmetry between the
red and blue sidebands. This is in contrast to the other motional states with nz > 0 that contributeto each sideband equally4 . It can be shown that the ratio of the sideband areas is related to thetemperature as [126]
total
σred
e−E0 /(kB Tz )
P
=
1
−
.
total
−Enz /(kB Tz )σblue
nz =0 e(2.52)

Here, the sum is taken over the approximate number of bound states in the longitudinal direction,
which is approximately 6 for a typical 80 kHz lattice. The mode dependent energy, Enz , includes
the effects of quartic distortion of the cos2 (kz) potential and is given by [126]1νrec 2
Enz = hνz nz +−nz + nz + 1 ,2
2(2.53)

where νrec = ωrec / (2π), where ωrec is the recoil frequency as defined by Eq. 2.30. In order to findTz , we simply numerically solve Eq. 2.52 for Tz .
The technique we employ to find Tr is more straightforward. A secondary clock laser beam
pathway along the H2 axis (see Fig. 2.3) probes the lattice-trapped atoms along the weakly-confined
axis, resulting in a Doppler-broadened transition profile. We fit the doppler width of the profile
4

From Eq. 2.33 it might be expected that there would be an asymmetry between the excitations of each sideband.
Experimentally, we utilize pulses that “over-drive” the transition, meaning that the excitation saturates to a constantvalue due to the dephasing explored in the previous section

57
using a Gaussian distribution Pe (ν) ∝ exp [−(ν − ν0 )2 / (2σ)] where [88]rσ = ν0kB T.
mc2(2.54)

Typical values for the frequency width σ are 15–30 kHz, representing temperatures in the 1–4 µK
range.
In order to reduce the temperature of the lattice-trapped 87 Sr , we employ two techniques.
In order to cool the longitudinal temperature, we employ additional cooling on the lattice-trapped,spin-polarized sample using MOT on the 1 S0 →3 P1 , F = 9/2 → F = 11/2 transition, and whose
frequency is tuned to optimize the longitudinal temperature. During this sideband cooling stage,
the optical pumping beam and bias magnetic field used in the spin-polarizing procedure are left
on, ensuring that the atoms cycle primarily on the mF = 9/2 → mF = 11/2 transition and do not
depolarize. Since the optical pumping beam (along H2) and the MOT beam along this same axis
compete, we must disable this MOT beam pathway during the cooling. In order to cool the atoms
in the polarizing transverse dimension, we optionally apply an additional cooling beam along theimaging axis, which cools the atoms transversally along that axis. Figure 2.19 shows the effect ofthese extra cooling steps on the measured Tz and Tr .

2.8

Systematic uncertainty
The 87 Sr optical lattice clock was the first neutral atom standard to surpass the Cs primary

frequency standard in its reported systematic uncertainty. In a series of measurements in 2007–2008,
we characterized many relevant clock systematics of the Sr lattice clock below the 10−16 opticallattice clock by comparing the 87 Sr clock frequency with the NIST Ca clock [31]. The comparison
was facilitated by the use of an ultrastable octave spanning frequency comb [132] in order to connect
the 87 Sr clock frequency to a transfer laser at 1064 nm. The transfer laser was transmitted througha noise-cancelled ∼ 3.5 km fiber to NIST [55] where a second frequency comb allowed comparison
with the Ca clock, which served as a stable reference at intermediate timescales. Due to slow
drifts in the Ca clock, data was typically taken in an interleaved fashion, with 100 seconds per

Contributor
Lattice Stark (scalar/tensor)
Hyperpolarizability (lattice)BBR Stark
AC Stark (probe)
1st order Zeeman
2nd order Zeeman
DensityLine pulling
Servo error
2nd order Doppler
Systematic total

Correction (10−16 )
-6.5
0.1
54.0
0.2
0.2
0.36
3.8
0
0
0
52.1

Uncertainty (10−16 )
0.5
0.1
1.0
0.1
0.2
0.04
0.5
0.2
0.5
0.01
1.36

58

Table 2.3: Systematic frequency shifts and their uncertainties for the 87 Sr frequency standard (asin Refs. [1, 2]). The largest contributor to clock uncertainty, the black-body radiation (BBR) shift,
is marked in red.

condition (e.g., high-density vs. low-density operation). The net result of the evaluation is shownin Table 2.3. Using the same fiber for microwave frequency transfer, we measured the 87 Sr latticeclock against the NIST F1 primary frequency standard, and obtained [1]νSr = 429, 228, 004, 229, 873.65(37) Hz.

(2.55)

We note that this value is in good agreement with all previous and subsequent measurements[133, 33, 134, 34].
In the intervening four years, the optical lattice setup was changed from an injection-locked
Ti:sapphire system to an all-solid state one. Additional improvements to the lattice were madeat that time, allowing better overlap between the input and retro-reflected lattice beams, thusincreasing the longitudinal trap frequency from ∼ 40 kHz, to ∼ 80 kHz. This change greatly
facilitated the collisional physics studied in Chapter 5 of this thesis, but will require a re-evaluationof systematics. However, as we show in Chapter 4, a new ultrastable laser system permits evaluationof a systematic through interleaved measurement (as utilized in [135]) on a very rapid timescale. In
contrast to the result presented in [135], where data records spanning several weeks were required
for systematic evaluation at the 1 × 10−17 level, the new laser system presented in Chapter 4 ofthis thesis allows for similar measurement precision to be obtained in 30 minutes. Thus, one could

59

Figure 2.20: Two-dimensional lattice geometry. Here, the single-site occupancies are mainly restricted to zero, one, or two atoms.

envision doing a complete clock systematic evaluation at the 10−17 level in one day if averaging
time were the only impediment to such a measurement!
As will be shown in Chapter 5, a high-density lattice clock does not make the best frequency
reference due to the strong interactions between atoms in the optical lattice. The current experimental configuration is best suited for experimental studies of collisional physics due to thehigh operating density. However, we have still been able to evaluate the density shift to the low
2.4×10−17 fractional frequency uncertainty in the current experimental configuration with a reduced
operating atom number of Ntot = 1000. In the meantime, a second 87 Sr clock under development
at JILA employs a much lower-density operating condition by utilizing a buildup cavity for thelattice, allowing a large reduction of density. This second system has been able to demonstrate adensity shift uncertainty for operation with 1000 atoms of 8 × 10−19 [43].
A second approach to reducing the density shift that we employed was to confine the atoms
in a 2D lattice, depicted schematically in Fig. 2.21. Here, a reduction of the density shift to the1 × 10−17 was observed, which was made possible primarily by the single-site occupancy combined

60with strong interactions for the particles in doubly- and triply- occupied sites [135]. The additional
operational complexity of the 2D lattice is not favorable for robust clock operation, but this work
was an important demonstration of methods for reducing density shifts by reducing the chance anatom resides in a multiply-occupied site. A three-dimensional lattice would be a natural extensionof this technique, and we note that 3D lattice geometries have also been realized with bosonic
88 Sr [136].

For the remainder of this thesis, we restrict ourselves to the one-dimensional latticegeometry. In this configuration, the single site occupancies are large enough to constitute quantum
many-body systems, and we study them in this context in Chapter 5.
We note that a significant exploration of all lattice-induced shifts has been presented in [110],
where effects such as hyperpolarizability (two-photon-induced stark shifts which scale with latticeintensity as I 2 ) and shifts due to magnetic dipole and electric quadrupole transitions [137], which√scale as I. The conclusion of Ref. [110] is that lattice systematics can be controlled at below the1 × 10−17 level for the lattice depths we employ in the clock.
2.8.1

Blackbody radiation

As seen from the systematic uncertainties listed in table 2.3, black-body radiation is thelargest contributor the 87 Sr clock uncertainty (highlighted in red), and this is the case for all 87 Sr
lattice clocks currently at the 1 × 10−16 uncertainty level [31, 33, 34]. The frequency shift arises
due to room-temperature radiation coupling to the AC-polarizability of the clock states. To firstorder, however, the black body radiation (BBR) depends primarily on the DC polarizability of theclock states as [138, 139, 115]1h∆νBBR = − hE 2 iT ∆α (0) [1 + η (T )] .
2

(2.56)

Here, hE 2 iT = (8.319 V/cm)2 (T /300 K)4 is the average squared electric field magnitude for agiven temperature; ∆α (0) = αe (0) − αg (0) is the differential DC polarizability, where αe (0) andαg (0) are defined in Eqs. 2.19 and 2.20, respectively; and η (T ) is the dynamic correction, whichtakes into account shifts due to corrections beyond the DC polarizability. Until recently, lack61
of knowledge of several oscillator strengths utilized for the calculation of αe (0) had limited thetheoretical uncertainty of the room-temperature BBR correction at 7 × 10−17 [140].
Two experiments have successfully measured ∆α (0) in Yb [139] and Sr [115] optical lattice clocks. In the case of Sr, a discrepancy between the theoretical Sr blackbody frequency correction calculated in Ref. [138] and the measured value was discovered by Middelmann et al.
[115]. The discrepancy amounts to an additional fractional frequency correction of −1.8 × 10−16
at 300 K. Additional theoretical work has resolved the discrepancy [141], indicating the measured values should supersede the original calculation.

For Sr, the measurements of Middel-mann et al. indicate ∆α = 4.07865(11) × 10−39 C m2 /V, resulting in a static BBR shift of
4
1
− 2h
hE 2 iT ∆α (0) = −2.130186(60)× 300T K Hz. Additionally, Refs. [115] and [141] agree regarding1
the contribution of the dynamic polarizabilities to the frequency shift as − 2hhE 2 iT ∆α (0) η (T ) =−0.1477(23) × ( 300T K )6 Hz (Middelmann et al., [115]) and −0.1492(16) × (T /300)6 Hz (Safronova
et al., [141]).
The measurements have essentially eliminated the systematic uncertainty associated withknowledge of the static polarizability of Sr. However, the total BBR uncertainty quoted in Table 2.3
includes two contributions: the now-irrelevant uncertainty in ∆α (0) and the uncertainty in the
BBR environment experienced by the atoms. The latter effect still contributes the largest andmost difficult-to-characterize systematic shift to the 87 Sr lattice clock. A 1 K uncertainty in thetemperature induces a 7×10−17 systematic effect. This reflects the unfortunate fact that Sr has thelargest BBR shift coefficient of all lattice clock atomic species. Future work will undoubtedly focus
on providing a well-characterized, or possibly cryogenic [142] environment for the lattice-trapped
atoms.

2.9

Quantum projection noise and the Dick effect
One of the strongest motivating factors for developing optical lattice clocks is the potential for

extremely high stability due to the large signal-to-noise ratio provided by the parallel interrogationof > 103 atoms. For a clock operated at average excitation fraction given by p, the number of atoms

a

0.8
0.6
0.4
0.2
0.0

2

62

1.0

Excitation probability p

Excitation probability p

1.0

1

0

1

2

b

0.8
0.6
0.4
0.2
0.0

2

∆ν × Tpulse (Ramsey curve only)

1

0

1

2

∆ν × Tpulse

Figure 2.21: Ramsey (dashed) and Rabi (solid) lineshapes compared for quantum projection noise
calculation. (a) Here, the ratio of Rabi to Ramsey pulse times is 1.6, such that the full width at
half maximum (FWHM) of the Rabi and Ramsey lineshapes are equivalent. In this scenario, theslope at p = 0.5 is nearly identical. (b) Tpulse is the same for both traces. For a given Tpulse , the
FWHM of the Ramsey lineshape is a factor of ' 1.6 narrower.

found in the excited state follows a binomial distribution, assuming the atoms are uncorrellated(i.e., the system is not spin-squeezed). It then follows that the variance of the measured excitation
fraction, (see Eq. 2.29) is2
δfexc≡ Var [fexc ] =p (1 − p).
N(2.57)

This relationship between the noise and the total atom number is known as quantum projection
noise. In fact, measuring quantum projection noise is the method we employ to calibrate theconstant of proportionality between our fluorescence detection and the actual total number ofatoms in the lattice.
Equation 2.57 is a key result for noise in optical lattice clocks: as the number of atoms
increases the noise decreases. However, this is only part of the story. One must convert the atomic
noise into frequency noise by use of the spectroscopic lineshape. Typically, clocks are operated
with p = 0.5, as depicted in Fig. 2.21, where the slope of the excitation probability with respect tofrequency is maximized. From the analytic shapes of the Rabi and Ramsey lineshapes, it can be63
shown that
σyRabi (τ ) '∆ν Rabi√
3.03ν0 Ntotr∆ν Ramsey√πν0 Ntotr

σyRamsey (τ ) =Tcycle,τ(2.58)

Tcycle,τ(2.59)

where Tcycle is the experimental cycle time, ν0 is the frequency of the transition, and ∆ν Rabi(∆ν Rabi ) is the FWHM of the spectroscopic feature used for feedback. ∆ν Rabi and ∆ν Rabi are
given by∆ν Rabi ' 0.80/Tpulse ,(2.60)∆ν Ramsey = 1/ (2Tdark ) ,(2.61)

where Tdark is the Ramsey free evolution time. Thus, even for a modest spectroscopy time of 200 ms
p
and only Ntot = 1000, quantum-projection noise limited clock operation is still 1 × 10−16 / τ /1 s
for Rabi spectroscopy.
One roadblock to benefiting from the SNR afforded by thousands of atoms is broadband laser
noise, which ends up contaminating the error signal through the Dick effect. The Dick effect is aprocess through which a periodic clock interrogation with spectroscopic dead time writes noise onto
the correction signal, degrading long term stability [36, 143]. For example, for every 1 s of time
per clock interrogation cycle a neutral atom system might spend cooling and trapping atoms in an
optical lattice, the time during which is spectroscopy is performed might only by 100 ms. Thus,
there is an inevitable dead time between spectroscopy sequences, resulting in a periodic samplingof the laser phase noise, and leading to aliasing of higher-frequency laser noise, deteriorating thestability.
Specifically, it can be shown that the Dick effect-limited Allan deviation due to the aliasing
mechanism is given by [37]1σy2 (τ ) =τ∞X|R (m/Tc )|2 Gν (m/Tc )m=1|R (0)|2

ν02.

(2.62)

Here Tc is the clock cycle time, including both atom loading and spectroscopy time; R (f ) is given64a|R (f )| / |R (0)|

2

1
0.01

106

r (t)

2

104

108

T0

0

1010
0.1

1

10

100

1000

Fourier frequency (Hz)2  1015
Rabi

1  10σy (τ = 1 s)

15

Ra

ms

5  1016

ey

b
QP

N

QP

N

2  1016
1  1016
5  1017
2  1017
0.01

0.02

0.05

0.10

0.20

0.50

1.00

Spectroscopy time T0 (s)Figure 2.22: Fourier and time-domain representation of the Dick sensitivity function and its implications for reaching quantum-limited clock performance. (a) Fourier sensitivity for Rabi (red)and Ramsey (blue) corresponding to the sensitivity functions shown in the inset. Here the time T0
corresponds to Tpulse (Tdark ) in the case of Rabi (Ramsey) spectroscopy. For the frequency-domainplots, T0 = 200 ms. (b) Here, the loading time is 1 s, such that the cycle time Tc = 1 + T0 . We plot
the quantum projection noise-limited instability for Ntot = 1000 for both Rabi and Ramsey spectroscopy (red and blue, respectively) and the Dick-limited instability for lasers with σy (τ ) = 10−15(dashed black/grey) and σy (τ ) = 10−16 (solid black/grey). The black (grey) lines are using the
Rabi (Ramsey) Dick sensitivity function to calculate the Dick-limited
performance. In all cases thepinstability has been extrapolated to 1 s, so that σ (τ ) = σ (1) / τ /1 s.

65
byR (f ) = F [r (t)] ,(2.63)

where F [. . .] is the Fourier transform operation; and Gν (f ) is the laser frequency noise power
spectral density. The function r(t) describes the spectroscopic sensitivity to a phase shift in thelocal oscillator laser and is defined through [36]
δfexc =

1
2

Z T0
dt∆ω (t) r (t) ,(2.64)

0where ∆ω (t) is time varying laser detuning (i.e., noise) and δfexc is a fluctuation about the expected
excitation fraction due to laser noise. The upper limit of the integral is simply the duration of thespectroscopy, which is assumed to start at t = 0. We will return to a detailed discussion ofsensitivity functions in Chapter 6. For the moment it is sufficient to give g (t) for Ramsey and Rabi
spectroscopy. The Rabi sensitivity function is given by
r

Rabi


(t) = −

∆0 Ω20
Ω3






Ω
Ω
sin (Ωt) − sin π+ sin(π − tΩ0 ) ,Ω0Ω0where generalized Rabi frequency Ω is related to the bare Rabi frequency Ω0 by Ω =(2.65)pΩ20 + ∆20 ,and ∆0 is a static laser detuning from the transition chosen so that p = 0.5. The Ramsey sensitivity
function is given byr

Ramsey(t) ' rect

t
Tdark


− 1/2 .

(2.66)A derivation of Eq. 2.65 is given in Appendix A. In Chapter 6, we additionally show that Eq. 2.66
is a simple case of the class of sensitivity functions that arise from spin-echo sequences with fast
π/2- and π-pulses. Figure 2.22a shows the time and frequency-domain behavior of these functions.
Figure 2.22b shows the implications of the Dick effect on the long-term stability of an opticalfrequency standard. Here, two different flicker-noise limited local oscillator spectra are considered,representing a laser with σy (τ ) = 1 × 10−15 , as originally used in the 87 Sr lattice clock [40], and alaser with σy (τ ) = 1 × 10−16 , which is the thermal noise floor of the ultrastable laser reported inthis thesis and described in Chapter 4. As seen from Fig. 2.22, only the laser with σy (τ ) = 1×10−16

66
is not limited by the Dick effect over wide range of spectroscopy times, and furthermore the overall
clock performance is better by an order of magnitude. Here, flicker noise is chosen because itcorresponds to the frequency noise caused by thermal noise in optical interferometers, and is the
subject of the next chapter.
As the preceding discussion indicates, laser stability is a crucial aspect of an optical atomic
clock. Through the Dick effect, the stability of the local oscillator laser always impacts the stability of the clock. Furthermore, the short term stability of lasers greatly impacts the amountof time required to make systematic evaluations via interleaved measurements. Here, a factor of
10 improvement in laser stability yields a 100-fold reduction of the measurement time necessary,
reducing month-long measurement campaigns to mere hours. In the next two chapters, we explore
some challenges to improving laser stability, and present three separate laser systems that perform
at or near the 1 × 10−16 level of instability.

Chapter 3

Thermal noise in optical interferometers

The goal of this chapter is to introduce the concepts of thermally-driven noise in (opto)mechanical systems in a qualitative manner; we then proceed into detailed equations that describe
noise in optical interferometers. A careful consideration of the sources of noise is necessary fordeveloping state of the art ultrastable lasers, and we use the detailed formalism summarized here
to motivate current highly successful optical cavity designs, as well as potential future ultrastablelaser systems with instability at the level of 1 × 10−17 .

3.1

Introduction
Optical interferometers are found at the heart of experiments that probe the physical world,from quantum mechanical [144, 145, 146] to cosmological scales [147]. As introduced in Chapter 2,cavity-stabilized lasers are essential components of the most precise optical atomic clocks [43] andtheir performance directly impacts the ultimate stability of clocks employing thousands of atoms.
The mechanism that currently sets the stability limit for interferometers—from µm to km lengthscales—is detrimental thermal coupling to the environment. This coupling is a direct consequenceof mechanical losses in the interferometer substrates and coatings and causes a mechanical displacement which directly impacts the optical phase. In the case of Fabry-Pérot interferometer-basedlaser stabilization, the length stability of the interferometer directly impacts the frequency noise of
the stabilized laser, leading to frequency noise that is directly proportional to the fractional length

68change of the interferometer as1
∆ν
∆L
=−
.
ν
L

(3.1)

Here, L is the length of the Fabry-Pérot interferometer and ν is the central frequency of the laser
stabilized to it.
In this chapter, we describe the mechanism by which both Brownian noise and thermodynamical temperature fluctuations in the interferometer components become effective length fluctuationsof optical interferometers. We emphasized that understanding these mechanisms is important for
designs which mitigate the effect of thermal noise. We also describe a second thermal noise mechanism that arises due to statistical temperature fluctuations in the bulk of optical substrates andcoatings. This in turn will also drive length changes in the interferometer through thermal expansion and the thermo-optic effect. In Chapter 4 we will extensively discuss systems developed
utilizing these insights.

3.2

Thermo-mechanical noise in physical systems
According to the fluctuation-dissipation theorem, the thermally-driven displacement fluctu-

ations of a system are directly proportional to the admittance of the system such that the singlesided displacement power spectral density for any mechanical degree of freedom with dissipations
is [148, 149, 150, 151]
Gx (f ) =kB T |< [Y (f )]|
.
π2f 2

(3.2)

Here, < [Y (f )] is the real part of the systems complex admittance, which is defined as
Y (f ) = V (f ) /F (f ) ,(3.3)

where V (f ) is the velocity response to an applied force F (f ) in the Fourier domain.
Since Gx (f ) depends directly on Y (f ), the displacement noise in a system depends directlyon the type of mechanical damping model that is employed. In the case of bulk mechanical samples,
1

In Chapter 4 we rigorously derive the relationship.

69(1 + iφ) k

MFigure 3.1: Modified Hooke’s law system. Here the addition of the loss angle φ creates adisplacement-dependent dissipation such that the energy dissipated in one period of motion is
2πφ and thus Q = 1/φ.

such as mirror substrates and optical coatings, a type of damping called “internal damping” applies,
which can be thought of as a modification of Hooke’s law such that [150]F (f ) = − (1 + iφ) kX (f ) .

(3.4)

Here, φ is the loss angle, which parameterizes the energy dissipated in the material. It is straightforward to show that the fraction of energy dissipated in a cycle is 2πφ, such that Q = 1/φ, where
Q is the material quality factor.
Equation 3.4 represents the simple case where we consider one mechanical degree of freedom.
For the mass on a spring depicted in Fig. 3.1, it is easily shown that
2πif
.
(3.5)
(1 + iφ) k − m (2πf )2p
In the low-frequency limit, where 2πf  k/m and φ  1, < [Y (f )] → 2πf φ/k and thus for the
Y (f ) =simple mass-on-a-spring system considered in Fig. 3.1, 
2kB T φGx (f ) →
.
πf
k

(3.6)

In Eq. 3.6 lie all of the essential considerations for a mechanical system driven by Brownian noise:the low-frequency limit of the noise is 1/f in its frequency dependence, it is proportional to theloss angle φ (and thus inversely proportional to Q) and T , and it is inversely proportional to thematerial stiffness k.
While the case of a mass on a spring is instructive for building a basic understanding ofthermally driven fluctuations in a mechanical system with one degree of freedom, the components

70of optical systems, such as mirrors, are much more complicated. The mechanical displacement ofa bulk object such as a mirror can be decomposed into normal modes, each of which are effective degrees of freedom with a generalized displacement amplitude. Early attempts to apply thefluctuation-dissipation theorem of Eq. 3.2 to such systems relied upon a decomposition into these
normal modes, whose frequencies were numerically calculated [152]. In this approach, the beamprofile is weighted by the normal mode displacement amplitude to properly evaluate the opticalphase shift due to a given mode. In general, this sum-over-modes method should work, but it iscomputationally intensive and converges slowly [152, 151]. To solve these issues a new approach
was developed, and is the subject of the next section.

3.2.1

Fluctuation-dissipation theorem and Levin’s “direct approach”

In 1997, Levin published a landmark work describing a new approach to calculating the effectof Brownian motion on the stability of optical interferometers [151]. At the heart of this methodlies the use of a generalized coordinate. It can be shown that the quantity which affects the opticalphase of a reflected beam with electric field amplitude profile E (x, y) ∝ ψij (x, y) at the mirror’ssurface is [152]Z∆φ =|ψij (x, y)|2 k · ~uS (x, y) dxdy.

(3.7)

Here ∆φ indicates the resulting optical phase shift cause by ~uS (x, y) (and is distinct from the loss
angle φ), the mirror displacement evaluated at the mirror surface; and k is the laser propagation
wave-vector with |k| = 2π/λ. It is assumed that the laser is propagating in the z-directions. The
mode profile is normalized such thatZ∗
ψij (x, y) ψkl(x, y) dxdy = δik δjl ,(3.8)

where δ is the Kroeneker delta. The indices i and j could in principle label Laguerre-Gauss modes
for the case of a spherical mirror resonator, or other solutions to the paraxial Helmholtz equation forother types of boundary conditions (e.g. the case of conical or Mesa beams [153]). For the remainderof this Thesis, we will consider only Gaussian beams (i.e. the TEM00 mode of a spherical-mirror

71
optical resonator), as these are the lowest-order stable solution in a resonator formed by sphericalmirrors, such that|ψ (x, y)|2 = |ψ00 (x, y)|2 =222e−2(x +y )/w(z) ,πw (z)(3.9)

where w (z) is the beam waist at the mirror under consideration.
The phase shift ∆φ of Eq. 3.7 can be used to define an effective displacement—essentiallythe mirror displacement weighted by the beam intensity profile—asx = ∆φ/ |k| .

(3.10)

Thus, even though the mirror itself is a complicated object with many relevant degrees of freedom,
it is thus possible to reduce this to a single relevant coordinate, x, via Eq. 3.10.
Levin’s technique allows direct computation of the displacement of this coordinate via thefluctuation-dissipation theorem. The whole point of the calculation is to calculate the behavior of< [Y (f )] at frequencies well below the lowest-frequency resonant mode. However, it is not clear
what the generalized force should be with respect to the generalized coordinate. The approach that
Levin developed was to define a generalized force via a pressure of the form
P (x, y; t) = F0 |ψij (x, y)|2 cos (2πf t) ez .

(3.11)

Here, f is the Fourier frequency of the oscillating pressure. In this way, the total hamiltonian for
the generalized force and generalized displacement is
H = −F0 cos (2πf t)Z|ψij (x, y)|2 ez · ~uS (x, y) = −F0 cos (2πf t) x

(3.12)

which shows that the generalized force enters the Hamiltonian as in the case where only a single
mechanical degree of freedom was considered.
The next step of Levin’s procedure is to calculate < [Y (f )] from the generalized force anddisplacement. Here, the admittance is now defined with respect to the generalized variables. For
the admittance defined in Eq. 3.3, it is easily seen that that the average dissipated power (by which

72
we mean the total energy lost in one cycle divided by the cycle time) is2
1
1
Wdiss = <{F0 × [F0 Y (f )]∗ } = F02 < [Y (f )] .
2
2

(3.13)

Thus, we can quite simply solve for the generalized < [Y (f )] such that [151]< [Y (f )] =4πf Umax φ
2Wdiss.
2 =|F0 ||F0 |2(3.14)

Here, the last equality simply relates the dissipated energy to the maximum elastic energy stored,
Umax , via the loss angle. This follows directly from the modified Hooke’s law of Eq. 3.4. Finally,
by applying Eq. 3.2 to the case of generalized displacement we obtain Levin’s result [151]kB T |< [Y (f )]|2kB T φ 2Umax
Gx (f ) =
=
,
π2f 2
πf
|F0 |2(3.15)

which is anologous to the simple spring case of Eq. 3.6, where Umax = |F0 |2 / (2k).
The final step of the procedure requires a “simple” calculation of Wdiss for the generalized
force given by Eq. 3.11 via a calculation of Umax for a static generalized force. The use of a staticforce is allowed as long as the frequency is well below any resonant frequencies of the system. In
principle this can be done numerically very straightforwardly [155]. However, analytic solutions forthe case of a Gaussian beam profile on a cylindrical mirror have been obtained [151, 156]. Finally,
we note that the calculation method is easily extended to include the surface contribution due to athin dielectric coating film [156]. As it turns out, this aspect of the mirror is absolutely crucial toconsider, as the loss angle can be from one to four orders of magnitude larger than the bulk mirror
substrate, depending on the substrate material.
Before proceeding further, we can make some qualitative statements. As in Eq. 3.6, we seethat the noise of the generalized coordinate x will scale linearly with the temperature and loss
angle, while it will scale inversely with the material stiffness, which is parameterized by an effective
spring constant k̃ = |F0 |2 / (2Umax ). In the case of the bulk substrate, which can be thought of as
an infinite half-plane, dimensional analysis indicates that k̃ ∝ w0 E, where E is the bulk Young’s
modulus and w0 is the beam 1/e2 intensity radius.
2

This computation is similar to time averages performed in electrodynamics. See, e.g., Jackson [154] for a
derivation of the time average formula for oscillatory fields.

73
L

Rspacer

Thermal noiserspacer

Figure 3.2: Schematic of the relevant optical components for calculating the thermal noise of an
optical cavity. The mirrors substrates (blue), optical coatings (thin red line), and spacer (tan) all
contribute to thermally-driven displacement noise at the mirror surface.

3.2.2

Brownian motion in optical cavities

As seen in the previous section, a powerful formalism exists for evaluating Brownian thermal
noise in optical components. In the case of a cavity, the components of concern are the spacer,
mirror substrate, and mirror coatings, as depicted in Fig. 3.2. In each case, the thermally driven
fluctuations of the effective displacement are calculated as outlined in the previous section. While
we do not go into the details of the calculation, we note that in the specific case of a Gaussian beam,
it entails calculating the stored energy in an infinite half-plane (which approximates the mirror)
when subject to a Gaussian pressure profile. It has been found that the one-sided power spectral
density of position fluctuations (in units of m2 /Hz) arising from each of the cavity components at

74
Fourier frequency f are3 [156, 157, 155]
2kB T 1 − σ 2
φs ,π 3/2 f w0 Es
1
2kB T 1 − σ 2dcoating√Gx(f ) = 3/22
π Es Ec (1 − σc2 ) (1 − σs2 )π f w0 Esh
× Ec2 (1 + σs )2 (1 − 2σs )2 φk + Es Ec σc (1 + σs ) (1 + σc ) φk − φ⊥io
+ Es2 (1 + σc )2 (1 − 2σc ) φ⊥ ,
φspacerL
2kB T.
Gspacer=x22
f π 2 Rspacer− rspacer
Espacer
Gsubstrate(f ) =
x(3.16)(3.17)(3.18)

Some of the parameters used in these equations are defined in Table 3.1. We note that for the
purposes of calculations in this Thesis, we assume that the coating parallel loss angle (φk ) and thecoating perpendicular loss angle (φ⊥ ) are equivalent.
Two qualitative remarks can be made at this point, the first of which is that both thesubstrate and coating displacement noise power spectral densities, given by Equations 3.16 and 3.17,
respectively, do not depend on the length of the cavity. This is due to the fact that the fluctuations
are localized to the mirror surface and this property can be exploited in order to reduce frequency
noise. By increasing the cavity length, the fractional length fluctuations decrease, resulting ina substrate and coating thermal noise-induced optical frequency noise spectral density that is
proportional to 1/L2 . Secondly, while the spacer thermal noise contribution scales with length, and2. Thus, for a spacer with constant aspect ratio, the spacer contribution to theinversely with Rspacer

total thermal noise in fact decreases with length. This is due to the fact that to convert the power
spectral densities to fractional frequency fluctuations, they must be divided by L2 . Additionally,
longer optical cavities typically have larger radii in order to maintain favorable mounting geometry,
meaning that as L increases, so too does Rspacer .
One clear avenue to decrease this noise limit is to decrease the material losses, lower thetemperature, or do both. However, the situation is not so simple. For example, the loss angleof fused silica—a commonly-utilized mirror substrate material—begins to increase sharply at temperatures below ∼ 250 K, ultimately suffering an almost four orders of magnitude increase before
3

There has recently been an important correction to the spacer contribution [155].

w0
d
φs
φ⊥(k)
Es(c)
σs(c)
αs(c)
κs(c)
Cs(c)
sub(coat)ωc
Rspacer (rspacer )Definition of parameters
Beam 1/e2 intensity radius
Coating thickness
Substrate loss angle
Coating loss angle perpendicular (parallel) to substrate
Substrate (coating) Young’s modulus
Substrate (coating) Poisson ratio
Substrate (coating) coefficient of thermal expansion
Substrate (coating) thermal conductivity
Substrate (coating) heat capacity
Substrate (coating) cutoff frequency
Spacer (spacer central bore) effective radius

75

Table 3.1: Summary of the parameters used in the text.
Substrate and spacer materials
Materialφ
E (GPa)Fused silica (SiO2 )10−6 –10−8 72
ULE glass
1.7 × 10−5 68
Silicon
10−8
130–190∗
Sapphire3 × 10−9400∗Dielectric multilayers
Materialφ
E (GPa)SiO2 /Ta2 O2 Bragg stack
4 × 10−4
110−5GaAs/Al92 Ga8 As Bragg stack 2.5 × 10
85∗

σ
0.17
0.18
0.05–0.35∗
0.29∗
σ
0.2
0.32∗

Table 3.2: Mechanical properties of typical optical materials. Values are obtained from [163, 164,
156, 165, 157, 166, 167]. ∗ Orientation-dependent

it levels off at 50 K, completely eliminating the benefit of operating at these temperatures [158].
However, crystalline materials such as sapphire [159], calcium fluoride [160], and silicon [158] offer
the benefits of low thermal expansion and low loss angle at cryogenic temperatures. Unfortunately,
typical coating loss angles exhibit an approximate factor of 3 increase at cryogenic temperatures[161], which offsets some of the gains of operating at low temperatures. We note in passing that
there is an active search for low-loss coating materials or dopants to reduce the mechanical loss ofthe existing coating technolgy [162]. It should also be noted that it becomes increasingly difficult
to shield cryostat vibrations at very low temperatures, due to the large cooling powers required.
A powerful and ground-breaking approach to the coating noise problem—which currently

76
sets the ultimate noise floor of ultrastable optical cavities [155, 168, 43]—is to use a fundamentallydifferent coating material for the high-reflectivity dielectric coating. The standard material is analternating structure of λ/4 layers of SiO2 and Ta2 O5 . However, as seen in Table 3.2, the loss angleof a SiO2 and Ta2 O5 is large [169] compared to fused silica, a commonly-utilized substrate material.
Indeed, for a typical optical cavity employing fused silica mirror substrates, a ULE glass spacer,and SiO2 /Ta2 O5 coatings, & 80% of the Brownian thermal noise power spectral density resultsfrom the excess mechanical loss of the conventional coating. The benefit of SiO2 /Ta2 O5 coatings istheir extremely low optical loss, which is at the few ppm level [170]. However, recent breakthroughsin engineering both high-reflectivity and low-mechanical-loss structures for use in optomechanics
have lead to simultaneously high optical- and high mechanical-quality microresonators employing
AlGaAs heterostructures[171]. As seen from the mechanical quality factor listed in Table 3.2, theloss angle of AlGaAs structures is at least an order of magnitude lower than SiO2 /Ta2 O5 coatings.4
With a recent demonstration of optical losses of order ∼ 15 ppm [172], these coatings promise anorder of magnitude reduction of coating thermal noise. Experimental developments along this front
will be discussed further in Chapter 4.
Finally, it is worth mentioning that a final alternative approach to reduce Brownian coating
and substrate thermal noise relies not on reducing the temperature or utilizing novel materials, butinstead on using specially shaped beams, such as mesa, conical, or higher order Laguerre-Gauss
beams [153]. This has the effect of better averaging the position fluctuations of the mirror surface
due to thermal noise. While one could simply envision working near the stability edge of an opticalcavity with typical spherical mirrors to create larger mode-areas on the mirrors, the input pointing
stability requirements become more stringent in these regimes. Specially shaped beams also havetheir own challenges as well; controlling the manufacturing process to create satisfactory mirror
profiles and alignment (including linear displacement) in small-scale optics used in optical cavities
is difficult, limiting the optical quality [173].
4

These measurements of the loss angles of AlGaAs were performed on free-standing micro-scale structures. We
present evidence in Chapter 4 that verifies that the loss angle is still valid when applied to a macroscopic coating
bonded to a substrate.

77

3.3

Thermodynamic noise
Brownian motion is not the only thermally-driven source of effective length fluctuations in

optical interferometers. For any thermodynamical system comprising a bulk material, there aretemperature fluctuations whose magnitude is given by [174]hδT 2 i =kB T 2.
ρCV(3.19)

Here ρ is the material density, C is the heat capacity, and V is the volume over which the temperature fluctuations are considered. In materials for which the coefficient of thermal expansion (CTE)and the thermo-refractive coefficient are non-zero, these statistical temperature fluctuations cancouple to the reflected phase of the optical field by two mechanisms. The first involves both thethermo-refractive and thermal expansion response of the coating itself and is know as thermo-optic
noise. Thermo-optic noise is usually restricted to describing only the coating. The second mechanism involves the thermal expansion of the substrate alone and is known as substrate thermo-elastic
noise.
In the theoretical treatment of both thermo-optic and thermo-elastic noise, the general approach is to calculate the effect of the bulk fluctuations as described by Eq. 3.19 on the effective
degree of freedom, x, as introduced in Section 3.2.1. The general approach is to solve the heat
equation in the mirror with a Langevin driving term that reproduces equation 3.19 for an arbitrary
volume [174]. The solution for the heat equation then acts as a driving term for elastic expansion,and in the case of thermo-optic noise, the thermo-refractive effect. The final solution for the spectral density of fluctuations is the net result of this process. In the case of thermo-optic noise, thethermo-refractive effect and the elastic displacement must be added coherently, because they sharethe same driving temperature fluctuations [175].
Substrate thermoelastic and coating thermo-optic noise have been studied as a noise sourcefor gravitational wave detectors [176, 177, 175]. There are two Fourier frequency regimes in theanalysis of thermo-optic noise. The first is where the thermal diffusion length scale is smaller thanthe laser spot size, allowing an averaging effect to take place. This regime is known as the adiabatic

78
limit and only applies to time-domain Fourier frequencies f that satisfy 2πf  ωc where the cutoff
frequency, ωc , is given by
ωc =

2κ
.
w02 ρC(3.20)

Here, κ is the thermal conductivity and w0 is the intensity 1/e2 radius of the laser beam. Dueto the small beam sizes (∼ 100 µm) and interest in the frequency noise spectrum all the way to
DC in cavity-stabilized laser systems, one must be aware that ωc is typically in the 1 Hz range.
Thus, consideration of thermo-optic noise in the semi isothermal regime, at Fourier frequencies
2πf < ωc , is necessary for a complete picture of the various contributions to the frequency noise ofcavity stabilized lasers. Typically the coating cutoff frequency and the substrate cutoff frequency
are different.

3.3.1

Substrate thermoelastic noise

To date, many optical cavities have employed mirrors made from ULE substrates [40, 178].
As a result, consideration of substrate thermoelastic noise is not necessary for these systems, as thematerial coefficient of thermal expansion (CTE) is close to zero. This approximation has also been
made in the case of fused silica substrates [179]. In fact, while alarming predictions for substrate
thermoelastic noise can be obtained by extrapolating the high–frequency behavior of fused silica
thermoelastic to 1 Hz-level frequencies, using the appropriate expression for the low-frequency
behavior verifies that the substrate thermoelastic noise is at least an order of magnitude below the
Brownian noise of the substrates and coatings in the case of fused silica. For other materials suchas sapphire at room temperature, this will not be the case.
It has been shown that the one-sided power spectral density of mirror length fluctuations dueto the substrate is [174, 176]
4
KB T 2 w0
GTE,sub(f ) = √ αs2 (1 + σs )2J [Ω (f )] .
xκsπ(3.21)

79
Here, Ω (f ) = 2πf /ωcsub , and J [Ω] is given byr Z ∞ Z ∞2u3 e−u /2
2
h
i.
dv
du
J [Ω] =
2
π3 0
2
2
2
2
3
−∞
(u + v ) (u + v ) + Ω(3.22)

While the integral can be evaluated numerically, it is more instructive to calculate thermal noisein the low and high frequency limits. Specifically,√
KB T 28 2 2TE,sub, Ω1
αs (1 + σs )2 √Gx→
3π
2πf κs ρs Cs
16
KB T 2 κs
GTE,sub→ √ αs2 (1 + σs )2, Ω  1.
xπ(2πf ρs Cs w0 )2(3.23)(3.24)

These equations indicate that at low frequencies, thermoelastic noise rises less rapidly than extrapolated from the high-frequency behavior. Qualitatively, this effect can be explained as a crossoverfrom the regime where the thermal diffusion length is smaller than the spot size to one where itis larger. Thus, this change in behavior can be thought of as an averaging effect that is no longer
valid at low frequencies [176, 177].

3.3.2

Coating thermo-optic noise

A second way that optical cavities are sensitive to thermodynamic temperature fluctuations
are through a pair of correlated mechanisms present in the mirror coatings: thermorefractive andthermoelastic effects, collectively called thermo-optic noise. Typical dielectric mirrors are made of a
stack (also known as a Bragg stack) of alternating high- and low-index materials whose thicknesses
are chosen such that the optical path length in each layer is λ/4, where λ is the vacuum wavelengthof the optical frequency of interest. For ultra-low loss coatings comprised of alternating quarterwave layers of SiO2 , and Ta2 O5 , 20 pairs of such layers will give a transmission loss of 1 ppm (see,e.g., [180]). For such a mirror, it has been shown that the typically opposite signs of these coherent
mechanisms reduces their impact and the total effect can be written as [175]


Cc 2
TO
Gx (f ) = G∆T (f ) ᾱc d − β̄c λ − ᾱs d.
Cs(3.25)The term in parentheses is the coherent sum of thermoelastic and thermorefractive effects; thethermo-optic noise. The parameter ᾱc (ᾱs ) is the effective coating (substrate) coefficient of ther-

80
mal expansion, and β̄c is the effective coating thermo-refractive coefficients, as defined in [175].
Modifications to the mirror Bragg stack, such as depositing an extra 1/2 λ cap on the surface (a“half wave cap”), can be used to modify β̄c and thus in principle the thermo-refractive noise canbe coherently cancelled or reduced via Bragg stack engineering [181, 182]. Table 3.3 lists thermal
and thermo-optic parameters for relevant materials.
As shown in Ref. [183], the averaged thermodynamic fluctuations that contribute to coating
thermo-optic fluctuations coupled through the CTE areZZ
8KB T 2 κ ∞ 2πk⊥ dk⊥ ∞ dkz
GTO
(ω)
=
∆T
(ρc Cc ) 0(2π)2−∞ 2π a42
kz2 + k⊥2 2e−k⊥ r0 /2
22
kz2 + k⊥+ ω2

1
1 + kz2 l2

(3.26)

p√
Here, r0 = w0 / 2, a = κ/ (ρc Cc ) and l is the optical penetration depth in the coating and istypically < 1 µm. The rightmost term in parentheses is only present in the treatment of thermorefractive noise [183, 165], and is not present for thermoelastic coating fluctuations. However, since
the penetration depth, l is much smaller than w0 . Ignoring this term only introduces an error oforder l/w0 . This expression can therefore be written√

2
GTO∆T (ω) =2KB T 2[K (Ω) + O (l/w0 )] ,πκω0(3.27)#√
u2 + iΩ√.
u4 + Ω2(3.28)

withK (Ω) = <"Z∞

−u2 /2du ue
0Here, Ω = ω/ωccoat , where ωc is the cutoff frequency defined in Eq. 3.20 applied to the coating
material properties. Two important limits for K (Ω) are
r

π
, Ω1
2
1
K (Ω) → √ , Ω  1
2Ω
K (Ω) →

(3.29)(3.30)

Thus, the two corresponding limits for the temperature fluctuations are
2KB T 2GTO
, Ω1
∆T → √
πw0 κc√2 2KB T 2
, Ω1
GTO
√
∆T →
πw02 κc ρcCc ω(3.31)(3.32)

Material
Fused silica (SiO2 )ULE glass
Ta2 O5
Silicon (124 K)
Sapphire
GaAs
Al92 Ga08 As

α [ppb/K]
2.6 × 103∼0
3.6 × 103∼0
6 × 103
5.7 × 103
5.2 × 103β [1/K]
8 × 10−6NA
1.4 × 10−5NANA
3.66 × 10−4
1.79 × 10−4

κ [W/m-K]
1.4
1.31
33
6
46
55
70

ρ × C [J/m3 -K]
1.6 × 106
1.7 × 106
2.1 × 106
8×105
3.1×106
1.8 × 106
1.8 × 106

n
1.45
1.48
2.06
3.96
1.77
3.48
2.98

81

Table 3.3: Thermal and thermo-optic/optical properties of common optical materials at room
temperature and at 1064 nm. The symbols are α, the thermal expansion; β, the thermo-refractive
coefficient; κ, the thermal conductivity; ρ × C, the per-volume heat capacity; and n, the index
of refraction. Values are obtained from [165, 175, 184]. For GaAs the material properties were
found from [185, 186, 187, 188, 189]. Full temperature dependent material values for many of thematerials listed above can be found in [184]. The thermo-refractive coefficients of ULE, Sapphire,and silicon are not applicable to the discussion because they are not used as coating materials.

A final correction comes about due to the finite thickness of the coating. We do not go into
detail here, but for large coating thicknesses, the coherence between the thermo-optic and thermorefractive effects is reduced at higher Fourier frequencies. The effect is parameterized by a functionTOΓ (f ) as described in Ref. [175], such that GTO
x (f ) → Γ (f ) Gx (f ).

3.4

Total thermal noise contribution to cavity frequency stability
The total thermally-driven (effective) displacement noise is given by
Gtot
x =

X
L,R

GTO
x +

X

GTE
x +

L,R

X
L,R

Gsubstrate+x

X

Gcoating+ Gspacer,xx(3.33)

L,R

where the sum over left and right (L, R) takes into account that the beam waist is potentiallydifferent at the left and right mirrors. The noises directly sum because they are statisticallyindependent.
Converting the total length fluctuation power spectral density, Gtot
x (f ), into optical frequency
deviations can be accomplished by use of Equation 3.1, which directly relates fractional lengthchange to frequency fluctations. We obtain
2 tot
2
Gtot
ν (f ) = ν0 Gx (f ) /L ,(3.34)

82
where L is the length of the cavity and ν0 is the laser’s optical frequency.

3.4.1

Examples of thermal noise floor

We consider several test cases utilizing two different types of cavity materials and two different
types of coatings: SiO2 /Ta2 O5 and GaAs/Al92 Ga8 . We perform calculation utilizing Eqs. 3.21,
3.25, 3.33, and 3.34. In order to calculate the Allan deviation from the power spectral densities, atransfer-function approach is used (see, e.g., [45] Eqn 5.12). Two spacer geometries are considered
due to their relevance for future experiments. Plots of the calculated Allan deviations and frequency
noise power spectral densities are shown on pages 84–89. The two cavities considered are• Cavity A: The spacer 25 cm in length, cylindrical in shape, with a diameter of 10 cm.
The spacer material is ULE glass, while the mirror substrates are fused silica glass, eachwith 1 m radius of curvature (mirror radii set the beam size on each mirror, as we show in
Chapter 4). The operating wavelength is 1064 nm and we consider operation at 300 K.
• Cavity B: The spacer is 22 cm in length, shaped as a tapered cylinder 8 cm in diameter at its widest point. The spacer material and mirror substrates are both made fromsingle-crystal silicon, and the mirror substrates each have 1 m radius of curvature. This
design is identical to the one considered in [168] (aside from the slightly different mirror
radii we utilize here for clarity). The operating wavelength is 1542 nm and the operating
temperature is 124 K, so that the Si coefficient of thermal expansion is nulled as in [168].
As seen from Figs. 3.3–3.8, the use of crystalline materials at cryogenic temperatures canresult in a gain of an order of magnitude of stability for thermal-noise limited performance of acavity-stabilized laser. Also, the use of a coating half-wave cap is an effective strategy to reduce
the thermo-optic noise in the mirror coatings. The thermo-optic noise manifests itself at higher
frequencies, and thus does not affect the long term laser stability. However, frequency noise in thisband is still important for reducing the noise in atomic clocks, as described in Chapter 2, and also
for gravitational wave detectors [147].

83

3.5

Conclusion
In this Chapter, we explored the relevant thermal noise sources for optical cavities. As shownin the results plotted in Figs. 3.3–3.8, the cavity materials are an important factor in determining
the magnitude and character of the noise. The summary given in this chapter should hopefully serve
as a useful reference for future cavity designs. In the next Chapter, we describe high-precision laser
stabilization using optical cavities. Using the insights developed in this chapter, we will describe
the construction of four ultrastable laser systems, three of which operate at the level of 1×10−16 . A
fourth system is described in which the new coating material described in this chapter—AlGaAs—istested, verifying its very promising optical properties and paving the way for lasers with thermal
noise-limited instability at 1 × 10−17 .

84

2  1015

a

1  1015
Allan deviation

5  1016
2  1016
1  10















16





















5  1017
2  1017
1  1017
0.1

1

10

Averaging time s

0.1

100

b

0.01
0.001

2

GΝ f Hz Hz

1000

104
105
106
0.01

0.1

1

10

Frequency Hz

100

1000

104

Figure 3.3: Thermal noise of cavity A with SiO2 /Ta2 O5 mirrors, T=300 K. The Bragg stack is
assumed to be 20 high/low index pairs for a coating thickness of 6 µm. The beam radius at eachmirror is w0 = 360 µm. (a) Allan deviation of a laser stabilized to this cavity configuration. (b)Frequency noise power spectral density due to thermo-optic noise in the mirror coatings (blue),thermo-elastic noise in the mirror substrate (red), and Brownian noise (black). The orange curveis the sum of all these noise sources, while the green curve is the sum of only the thermo-optic
and thermo-elastic noise. In this case, the coating contributes ∼ 85% of the total Brownian noise.
Here, the substrates contribute 13% of the Brownian noise, the coatings 84%, and the spacer 2%.

85

2  1015

a

1  1015
Allan deviation

5  1016
2  1016
1  1016







5  10







17





















2  1017
1  1017
0.1

1

10

100

Averaging time s

1000

0.01

b

GΝ f Hz2 Hz

0.001
104
105
106
107
0.01

0.1

1

10

Frequency Hz

100

1000

104

Figure 3.4: Thermal noise of cavity A with GaAs/AlGaAs mirrors, T=300 K. The Bragg stack is
assumed to be 40 high/low index pairs for a coating thickness of 7 µm. The beam radius at eachmirror is w0 = 360 µm. (a) Allan deviation of a laser stabilized to this cavity configuration. (b)Frequency noise power spectral density due to thermo-optic noise in the mirror coatings (blue),thermo-elastic noise in the mirror substrate (red), and Brownian noise (black). The orange curveis the sum of all these noise sources, while the green curve is the sum of only the thermo-optic andthermo-elastic noise. In this case, the coating only contributes ∼ 25% of the total Brownian noise.
Here, the substrates contribute 64% of the Brownian noise, the coatings 24%, and the spacer 12%.

86

2  1015

a

1  1015
Allan deviation

5  1016
2  1016
1  1016



5  10











17





















2  1017
1  1017
0.1

1

10

Averaging time s

100

0.01

1000

b

2

GΝ f Hz Hz

0.001
104
105
106
107
0.01

0.1

1

10

Frequency Hz

100

1000

104

Figure 3.5: Thermal noise of cavity A with GaAs/AlGaAs mirrors and a half-wave cap , T=300 K.
The Bragg stack is assumed to be 40 high/low index pairs for a coating thickness of 7 µm. The
beam radius at each mirror is w0 = 360 µm. (a) Allan deviation of a laser stabilized to this cavity
configuration. (b) Frequency noise power spectral density due to thermo-optic noise in the mirror
coatings (blue), thermo-elastic noise in the mirror substrate (red), and Brownian noise (black).
The orange curve is the sum of all these noise sources, while the green curve is the sum of only thethermo-optic and thermo-elastic noise. The half-wave cap dramatically reduces the thermo-optic
noise contribution. Again, the substrates contribute 64% of the Brownian noise, the coatings 24%,and the spacer 12%.

87

2  1015

a

1  1015
Allan deviation

5  1016
2  1016
1  1016
5  1017



































2  1017
1  1017
0.1

1

10

Averaging time s

0.01

100

1000

b

104

2

GΝ f Hz Hz

0.001

105
106
107
0.01

0.1

1

10

Frequency Hz

100

1000

104

Figure 3.6: Thermal noise of cavity B with SiO2 /Ta2 O5 mirrors, T = 124 K. The Bragg stack is
assumed to be 20 high/low index pairs for a coating thickness of 9 µm. w0 = µm. The beam radius
at each mirror is w0 = 415 µm. (a) Allan deviation of a laser stabilized to this cavity configuration.
(b) Frequency noise power spectral density due to thermo-optic noise in the mirror coatings (blue),thermo-elastic noise in the mirror substrate (red), and Brownian noise (black). The orange curveis the sum of all these noise sources, while the green curve is the sum of only the thermo-optic andthermo-elastic noise. Over 99% of the Brownian noise arises from the coatings.

88

2  1015

a

1  1015
Allan deviation

5  1016
2  1016
1  1016
5  1017


2  1017

1  10







17

0.1





1











10

Averaging time s







GΝ f Hz Hz



100



1000

b

104

2



105
106
107
108
109
0.01

0.1

1

10

Frequency Hz

100

1000

104

Figure 3.7: Thermal noise of cavity B with GaAs/AlGaAs mirrors, T = 124 K. Since the AlGaAs
parameters are not well-known, extrapolated values based on the literature were used in the calculation. The Bragg stack is assumed to be 40 high/low index pairs for a coating thickness of 10 µm.
w0 = µm. The beam radius at each mirror is w0 = 415 µm. (a) Allan deviation of a laser stabilized
to this cavity configuration. (b) Frequency noise power spectral density due to thermo-optic noisein the mirror coatings (blue), thermo-elastic noise in the mirror substrate (red), and Brownian
noise (black). The orange curve is the sum of all these noise sources, while the green curve is thesum of only the thermo-optic and thermo-elastic noise. The thermal fluctuation noise is dominatedby the thermo-optic noise, as the temperature is chosen to be near the CTE null for silicon, which
also nulls the substrate thermo-elastic noise. Even with the high-Q crystallin coatings, 98% of the
Brownian noise is due to the coatings.

89

2  1015

a

1  1015
Allan deviation

5  1016
2  1016
1  1016
5  1017
2  1017







1  1017
0.1





1











10

Averaging time s







GΝ f Hz Hz



100



1000

b

104

2



105
106
107
108
109
0.01

0.1

1

10

Frequency Hz

100

1000

104

Figure 3.8: Thermal noise of cavity B with GaAs/AlGaAs mirrors and an additional half-wave
cap, T = 124 K. Since the AlGaAs parameters are not well-known, extrapolated values based on
the literature were used in the calculation. The Bragg stack is assumed to be 40 high/low index
pairs for a coating thickness of 10 µm. The beam radius at each mirror is w0 = 415 µm. (a) Allan
deviation of a laser stabilized to this cavity configuration. (b) Frequency noise power spectral
density due to thermo-optic noise in the mirror coatings (blue), thermo-elastic noise in the mirror
substrate (red), and Brownian noise (black). The orange curve is the sum of all these noise sources,
while the green curve is the sum of only the thermo-optic and thermo-elastic noise. The thermal
fluctuation noise is dominated by the thermo-optic noise, as the temperature is chosen to be near
the CTE null for silicon, which also nulls the substrate thermo-elastic noise. Again, the addition of
the half-wave cap dramatically reduces the thermo-optic contribution. Again, even with the high-Q
crystalline coatings, 98% of the Brownian noise is due to the coatings.

Chapter 4

High-precision laser stabilization with optical cavities

As discussed in Chapter 2, ultrastable lasers are central to the operation of atomic clocks.
Importantly, the most precise optical lattice clocks in existence today are limited by the noiseof these lasers, which arises from thermally-driven fluctuations of the optical cavities to which
they are stabilized. This chapter gives a detailed description of the technology required for highprecision laser stabilization and describes the development of three ultrastable laser systems that
perform at the level of 1 × 10−16 . Additionally, we describe a new coating technology and present
experimental results that show that the 1×10−17 level of laser stabilization is within reach. Finally,
we briefly summarize key results with optical frequency combs, as they will play an essential rolein transferring the stability of ultrastable lasers that operate in the near-IR wavelength region to
the visible spectrum.

4.1

Introduction
Optical cavities are extremely useful devices in laser-based research. Within the context ofprecision measurement, they enable tests of the laws that govern the macroscopic structure of theuniverse, embodied in the search for gravitational waves [147]. At the other end of the length scale,cavity-stabilized lasers are powerful tools for precision spectroscopy that probes nature at thequantum mechanical level, through tests quantum electrodynamics [190]. Furthermore, cavities
enable high-sensitivity broadband spectroscopy [191], which has practical applications in trace
gas sensing; exploration of new light-matter interaction regimes in cavity QED [192]; tests of

91
fundamental physical principles including relativity [193, 194, 195], local position invariance [196],
and the time invariance of the fundamental constants of nature [61]; and nonlinear optics, including
coherent light build-up for studies of extremely nonlinear effects [51, 52]. In general, optical cavities
have become indispensable tools at the heart of many modern experiments.
As discussed in Chapters 1 and 2, cavity-stabilized laser systems have enabled the development of highly precise frequency standards based on neutral atoms [197, 198, 199, 200, 31, 32] andtrapped ions [201, 202, 203, 28]. In recent years, several ion-based standards [57, 27, 30, 29], andneutral atom clocks [31, 32, 33] have surpassed the fractional frequency uncertainty of the primarycesium frequency standards that define the SI second [204, 205].
Accuracy is not the only benefit of optical standards; ultrastable lasers paired with ultranarrow atomic transitions in the optical domain have allowed the realization of optical clocks thatare orders of magnitude more stable than current microwave-based frequency references, with thep
87 Sr clock now operating at 3 × 10−16 / τ /1 s [43], and which we discuss in this chapter. With
increased stability, highly precise measurements of intricate physical effects can be made in shortperiods of time. For example, in Chapter 5, we will explore collisional effects between ultracold at
the 10−17 level. This precision can now be attained within a few thousand seconds due to a 10-foldimprovement of the Sr clock laser, which we describe in detail in this chapter.
As discussed in Chapter 3, thermal fluctuations set the ultimate limit to the stability of agiven interferometer. The desire to further improve the stability of optical clocks will continue todrive advances in cavity-stabilized laser systems, and the next milestone will be a laser with thermal
noise-limited fractional stability at 10−17 . In this chapter, we also discuss exciting new results thatindicate that this regime should be within reach.
We begin this chapter by reviewing the basics of optical cavities and Pound-Drever-Hall(PDH) laser frequency locking. We then proceed to a detailed discussion of cavity design considerations, which are applied to several systems.

92

4.2

Review of optical cavities
A basic optical cavity is formed by an array of two opposite-facing mirrors (Figure 4.1). For

high-precision frequency stabilization, these mirrors are typically held apart by a rigid spacer andare kept under vacuum to eliminate a varying intra-cavity index of refraction due to air. Althoughmore complicated cavity geometries exist, including ring-type cavities (e.g. a cavity formed froma triangular mirror configuration) where the optical field is a running wave, we consider only thisbasic configuration as it is most common for precision frequency stabilization. For an incident laserof electric field amplitude Einc , the steady-state electric field inside such a cavity, ESS , is obtainedby enforcing the condition√
ESS = ESS eiϕ R + Einc T .

(4.1)

Here, ϕ is the round-trip phase accumulated by the light, T is the mirror intensity transmission
coefficient and R is the corresponding reflectivity. In the absence of mirror absorption and scatter,
it is possible to relate the magnitude of the R and T coefficients by R + T = 1. However, we choose
to allow for the real-world situation, where mirror losses are influential, by leaving these distinct.
The cavity phase shift, ϕ, can be re-written in terms of the cavity length, L, and the laser’s opticalfrequency, ω = 2πν, as1
ϕ=2Lω
+ ϕ̃m,n .
c(4.2)

The term ϕ̃m,n is due to an additional mode-dependent diffraction phase term and is given by
ϕ̃m,n = −2 (m + n + 1) ∆ζ.

(4.3)

Here, the indices m and n label higher-order transverse modes (e.g., TEM01, TEM11, TEM02,etc.), while for the TEM00 (Gaussian) mode, m = n = 0. ∆ζ is the differential Gouy phase, which
is given by∆ζ = tan−1 (z2 /z0 ) − tan−1 (z1 /z0 )
1

(4.4)

We also note that the cavity length, L, includes the effects of optical field penetration into the mirror coating,which typically requires a correction to the physical length on the order of an optical wavelength. For macroscopic
cavities, this effect is negligible, but it becomes important for cavities whose size is of the order of an optical wavelength[180].

93

R1

R, T

R, T

R2

ESS

Einc

Et

w0
Er
z1

z=0

L

z2

Figure 4.1: Schematic of an optical cavity in the standing wave configuration. The mirror amplitudereflectivity and transmission coefficients are given by r and t, respectively. ESS is the steady-stateelectric field for a given incident field, Ei . Some of Ei is transmitted as Et , while some is reflected
as Er . For simplicity, we assume the mirror properties are identical.

where z2(1) is the position of right (left) mirror and z0 is the Raleigh range of the stable mode within
the optical cavity, whose waist is defined to occur at position z = 0. In general, for a resonatorcomprised of a pair of opposite-facing, concave, spherical mirrors, the Rayleigh range is given by[206],z02 =d (R1 − L) (R2 − L) (R2 + R1 − L),
(R2 + R1 − 2L)2which is related to the beam waist at the center of the cavity as
r
λz0
w0 =
,
π
where, as usual, w0 is the 1/e2 intensity radius. It can also be shown that [207]s
L (R2 − L)z1=−z0
(R1 − L) (R2 + R1 − L)

(4.5)

(4.6)

(4.7)

and similarly
z2
=
z0

s

L (R1 − L),
(R2 − L) (R2 + R1 − L)allowing us to find the beam waist at the end mirrors ass
z1(2) 2w z1(2) = w0 1 +
.
z0

(4.8)

(4.9)

94
Commonly, R1 = R2 = R0 and thus∆ζ = 2 tan−1rL
2R1 − L!
.

(4.10)

By solving Equation 4.1 for the steady-state field, we find that the transmitted field ampli√
tude, given by Et = T ESS is
Et (ω)
eiϕ/2 T=.
Einc (ω)
1 − eiϕ R(4.11)

Similarly, the reflected field, Er , is given by
Er (ω)
1 − eiϕ (T + R).
≡ R (ω) = r
Einc (ω)
1 − eiϕ R(4.12)

The cavity reflection transfer function, R (ω), is plotted in Fig. 4.2. As can be seen from Equations 4.11 and 4.12, the transmission (reflection) is maximized (minimized) when the round trip
phase is a multiple of 2π. When this condition is met, the cavity is said to be on resonance. This
results in the resonance condition



∆ζ c
νq = q + (1 + n + m),
π 2L

(4.13)

where q is the longitudinal mode index, L is the distance separating the mirrors, c is the speed oflight, and ∆ζ is given by Eq. 4.4. We note that there are geometrical configurations of the cavity
such that ∆ζ is a rational multiple of π. In these configurations, it is easy to see that there will
be accidental frequency degeneracies of modes. One such example is the confocal geometry, whereR0 = d. Here, every even mode (i.e., m + n is even) is frequency degenerate with the TEM00
mode. Another example is R0 = 2d, which results in degeneracy for every third higher-ordermode. It should be stressed that these accidental frequency degeneracies should be avoided, as
exact degeneracies are unlikely, but line-pulling due to near-degeneracies is a possibility and canbe problematic.
From Eq. 4.13, it is obvious that the longitudinal modes are spaced in frequency by c/2L, aquantity known as the free spectral range (FSR). By analyzing the denominator of Eq. 4.12, thewidth of the cavity resonance in units of intensity (i.e. the width of a dip in |R|2 ), denoted as

95

Reflection coefficient

1.0

R

FWHM

0.5

FSR

0.5

Phase (rad)

0.0

Phase

0.0

-0.5
-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

FWHM

Figure 4.2: Reflection coefficient and corresponding phase shift of the reflected light incident uponan optical cavity. If there are no mirror losses, there is a discontinuity in phase as the reflected
light drops to zero. Here, a small mirror loss term has been included, causing the reflection dip to
not reach zero.

∆νFWHM , is related to the FSR by
√
∆νFSR
πr
π R
π
F=
=
=
'
,
2
∆νFWHM
1−r
1−R
1−R(4.14)

where the last equality is accurate if F > 100. This ratio, F, is known as the cavity finesse and, as
Eq. 4.14 shows, depends only on the mirror reflectivity.

4.2.1

Measuring Finesse

In order to measure the finesse of an optical cavity, one approach is simply to scan a tunable
laser across the cavity resonances as depicted in Fig. 4.2 and measure the width of the resonancedips. However, for very high finesse cavities, this is not a feasible approach as the resonance dip
can be extremely narrow with respect to the FSR and even with respect to the laser linewidth.

96
The general approach used to circumvent this problem is to use a time-domain measurement.
Quite generally, if an oscillatory system loses energy at a rate proportional to the amount of energy
stored, then the associated time dynamics are exponentially decaying such that U (t) = U0 e−t/τ ,where U (t) is the energy stored. Many physical systems can be described in this way. We thenassociate a field, u (t), with the stored energy U (t), such that U (t) = |u (t)|2 . The time dynamics
of u (t) are then [154]u(t) = u0 eiω0 t−t/(2τ ) ,(4.15)and the transfer function of the system is given by|H (ω)|2 ∝1.
1 + 4 (ω − ω0 )2 τ 2(4.16)

Thus, the FWHM of the resonance in frequency is given by∆νFWHM = 1/ (2πτ ) .

(4.17)The decay constant τ is related to the quality factor, Q, as
Q = ν0 /∆νFWHM = ω0 τ.

(4.18)The Q-factor can also be understood as the rate of stored energy loss divided by ω0 .
In an optical cavity we can calculate τ very easily. We assume that in a time L/c, the stored
energy will have been reduced by (1 − R). This implies that the intensity loss rate is given by
1/τ = (1 − R)c/L such thatdI
(1 − R)c=−I
dt
L(4.19)

We can thus calculate the finesse by noting that ∆νFWHM = c/(2LF) = 1/(2πτ ). We thus find
that
F=

πcτ
.
L

(4.20)

Equation 4.20 implies that finding the photon decay time of the optical cavity allows adetermination of the cavity finesse, assuming the length is well-characterized. One way to measureτ is to stabilize a laser to the optical cavity under test and then very quickly switch off the power.

97|Et | (arb. intensity)

2.0

1.5

2

1.0

0.5

0.0
-200

0

200

400

600

Time (µs )Figure 4.3: Measurement of cavity photon storage lifetime. Here, a laser was actively stabilized tothe optical cavity and its power abruptly extinguished. The red solid trace is the intensity ringdown
data with a exponential fit (dash-dotted line). The agreement between the measured data and theexponential fit is excellent. Also shown is the combined response time of the detector and opticalpower switching (dashed grey line). The photon storage time is found to be τ = 81.3(1) µs.

This method assumes that the the laser stabilization has already been achieved—making thismethod technologically complicated—but the results are exceedingly easy to interpret. Figure 4.3
shows an example of such a procedure. Here, a laser was locked to a high-finesse optical cavity oflength L = 39.4 cm (which we will later describe in detail) and a ringdown time of τ = 81.3(1) µs
was measured. This results in a finesse of F = 194, 000.
In many instances, it is desirable to make a finesse measurement without the added complications of locking. An alternative approach is to sweep the laser frequency across the cavity
resonance and record either the transmitted or reflected time-domain signal. However, when thelaser stays near the cavity resonance for a duration that is comparable to the photon storage time,interference effects can emerge which make interpreting the signal difficult. Specifically, the field
inside the optical cavity given by E (t) can be described by a simple differential equation for alinearly swept laser [208] asdE= − 1 − iν̃t0 E + iη.
0dt(4.21)

98
Here t0 = t/τs is the time normalized by the field ringdown time (as opposed to the intensity√
ringdown time) where τs = 2τ . The constant η is the drive rate, given by η = T FEi /π (here T
is the input mirror transmission as in Fig. 4.1). The sweep rate is parameterized by ν̃, whereν̃ =2Fdω̇τs,πc(4.22)

with the laser frequency sweep rate denoted by ω̇ (in units of radians/s2 ). Here, t0 = 0 when thelaser is on resonance with the cavity.
While Eq. 4.21 is trivially solved numerically, it turns out that closed-form solutions exist
for any initial condition. In the limit that the laser sweep starts far off resonance and we ignore allother cavity modes, the solution for the reflected and transmitted power areβ|Er (t)| = |Ei | 1 − √
ν̃
2

2

r

π −t0 +iν̃t02 /2−i/(2ν̃) √
e
+ 2iD
2i



i + t0 ν̃
√
2iν̃

 2
(4.23)

andβ 2 |Ei |2|Et (t)| =
ν̃

r

2

√
π −t0 +iν̃t02 /2−i/(2ν̃)+ i 2De2

i + t0 ν̃
√
2iν̃

2
.

(4.24)

Here, D (x) is the Dawson integral2 , and β is the cavity contrast parameter, given by
β = T / (1 − R) .

(4.25)

On cavity resonance, the fraction of transmitted power is β 2 while the fraction of reflected poweris (1 − β)2 . The closed-form solutions to Eq. 4.21 facilitate extracting τ from swept-laser ringdown
measurements. As an example of a swept-laser cavity finesse measurement, Eqs. 4.24 and 4.23 wereused to fit time-domain transmission and reflection signals from two different optical cavities. As
seen in Fig. 4.4, the agreement with the data is excellent.
The reason T and R typically differ is due to absorption or scatter in the mirror coatings,
leading to extra losses beyond those due to light exiting the cavity through transmission (see e.g.,[180] ). Measurements of β and cavity finesse allow T and R to be extracted. In steady-state, the2

The Dawson integral is D (x) = e−x

2

Rx
0

2

eξ dξ and is built into Mathematica as the DawsonF function.

Cavity 1, reflectionb

Cavity 2, transmission22|Er | (arb. intensity)a|Et | (arb. intensity)

99

-50

0

50

Time (µs )

100

-20

-10

0

10

Time (µs )

20

30

40

Figure 4.4: Measurement of photon storage time with a swept-frequency optical field. (a) Timedomain signal of the reflected optical power from a swept ringdown measurement of the cavity
presented in [40]. Here the photon storage time is 14 µs, indicating a finesse of 180,000 (b) Sweptfrequency transmission signal of the cavity presented in [56]. Here, the photon storage time is 10 µs,
indicating the finesse is 280,000.

power that enters the cavity is related to the power leaving the cavity and the total input power,
P0 by


Pin = P0 1 − (1 − β)2 and Pout = P0 β 2 .

(4.26)

We can solve for β as
β=2.
1 + Pin /Pout(4.27)

Thus, a simple measurement of Pin , Pout , and cavity photon storage lifetime is sufficient to determine
the mirror properties.

4.2.2

Pound-Drever-Hall Locking

As seen in the previous section, an optical cavity defines a series of narrow resonances in thefrequency domain. A common way to stabilize a laser to such a resonance is through a frequency
modulation locking technique. The most commonly used and successful frequency modulation
technique for laser stabilization is the Pound-Drever-Hall (PDH) stabilization scheme [209], where
the frequency modulation is performed at a much higher frequency than the cavity linewidth.
There are several reasons for the widespread adoption of PDH locking. First, there are no100
restrictions upon the phase modulation frequency, as long as it is larger than the cavity linewidth.
A higher modulation frequency gives the lock immunity to common laser amplitude noise offsets
and also permits the use of resonant electro optic modulators (EOMs). Additionally, in the PDH
scheme, the lock bandwidth is not restricted by the cavity linewidth, allowing extremely narrowcavity resonance features to provide high-bandwidth stabilization.
Given the reflection spectrum of Fig. 4.2, one could imagine that a very simple locking scheme
would be to slowly frequency modulate the probing beam and use this signal to determine whereon the reflection dip the laser frequency lies. However, one main reason not to take this approachfor high-finesse cavities is that laser relative intensity noise (RIN) increases at lower frequencies,
in addition to electronic noise floors. These amplitude fluctuations would contaminate the locking
signal, affecting the lock stability. Additionally, the lock bandwidth would be severely limited.
One way around this noise problem is to increase the laser modulation frequency well above
the cavity bandwidth and beyond the spectral region with significant laser amplitude noise. In thisconfigurations, it is best to think of the laser electric field distributed as shown in Fig. 4.5. Whenthe modulation sidebands are well outside the cavity transmission bandwidth, the PDH locking
configuration [209].
Several reviews of PDH locking and laser feedback control theory exist [210, 211, 212, 213].
Here, we briefly discuss the important results of the PDH locking technique. In order to measure the PDH error signal from an optical cavity, the laser must first be phase modulated. A
phase modulated signal can be decomposed into a carrier and sidebands using the Jacobi-Anger
expansion [214]:
E0 e−iω0 t−i∆ϕ sin(Ωt) = E0 J0 (∆ϕ) e−iω0 t
+ E0

∞
X

h
i
Jn (∆ϕ) e−i(ω0 +nΩ)t + (−1)n e−i(ω0 −nΩ)t .

(4.28)

n=1

Here, the term ∆ϕ is the phase modulation (PM) depth and Ω is the PM frequency. From thisexpansion, it is clear that the first order sidebands are 180 degrees out of phase (as are all odd
order sidebands), as pictured in Figure 4.5. When the PM frequency is well outside the cavity

101

a

J0(∆ϕ)

1.0

0.5

J1(∆ϕ)J1(∆ϕ)

0.0

-0.5
-1.5

-1.0

-0.5

0.0

0.5

1.0

b

Error signal (arb. linear)E-field (relative)

1.5

1.5

-0.4

-0.2

0.0

0.2

0.4

FWHM

Figure 4.5: Pound-Drever-Hall sidebands and error signal. (a) Electric field amplitude in Fourier
space in the presence of phase modulation of depth ∆ϕ. (b) Pound-Drever-Hall error signal as afunction of laser detuning from the cavity resonance.

bandwidth, these sidebands are sufficiently detuned from the cavity resonance such that they arepromptly reflected from the cavity unaffected. The carrier, which is near the cavity resonance, isaffected by the complex response of the cavity (as shown in Figure 4.2 and given in Equation 4.12)
and interferes with the PM sidebands upon reflection.
This interference term can be explored by assuming that the PM sidebands are completelyreflected and finding the time-dependent reflected optical power, Pref (t), when the carrier is nearresonance. Making use of only the first order sidebands of Equation 4.28,
Pref (t) =1
2
Ec, ref + Es e−iΩt − Es eiΩt2

= Pc, ref + 2Ps − 2Im {Ec, ref Es∗ } sin (Ωt) + 2Ω terms.

(4.29)

Here, Ec, ref (Pc, ref ) is the reflected electric field amplitude (power) in the carrier and Es (Ps ) is
the reflected electric field amplitude (power) in the sidebands. Keeping everything that oscillates
at Ω or below, and making use of Equations 4.28 and 4.12,
Pref (t) = P0 J02 (∆ϕ) |R|2 + 2J12 (∆ϕ)
+ 4P0 J0 (∆ϕ) J1 (∆ϕ) Im {R} sin (Ωt) .

(4.30)

102
When the carrier is less than a cavity linewidth from resonance,
Im {R} '

−2βF
δν
∆νFSR

(4.31)

Here, δν is the laser detuning from cavity resonance given by δν = νlaser − νcavity and β the cavity
contrast parameter defined in Eq. 4.25. The term in equation 4.30 that oscillates as sin (Ωt), isthus given byDδν sin (Ωt) ,(4.32)

8βP0 J0 (∆ϕ) J1 (∆ϕ),∆νFWHM(4.33)

where we have used the definition
D≡−

along with the relationship ∆νFSR /F = ∆νFWHM to derive Equation 4.33.
Equation 4.32 gives the component of optical power that oscillates at the phase modulation
frequency. For small detunings, the amplitude is linear in δν, and can thus be used to lock the laser
to the optical cavity after the optical power has been detected on a photodiode and demodulated.
The degree to which the amplitude changes for a given detuning is characterized by the parameter
D, which, as should be expected, varies inversely with cavity linewidth and is proportional to the
product of the zero and first-order Bessel functions. In passing, we note that this can be used todefine an optimal modulation depth, given by ∆ϕ = 1.08. By measuring this oscillating RF signal,and demodulating by mixing in the proper quadrature at the frequency Ω, a linear control signal
can be obtained with which to feedback upon the laser frequency.
While Equation 4.32 describes the behavior of the error signal near the cavity resonance, one
may go a step further and include the frequency dependence of Im{R} to calculate the shape ofthe error signal over a broader range, as shown in Figure 4.5 [211]. When this detail is included,
it can be shown that the frequency response of Eq. 4.31 needs to be multiplied by a single–pole
low-pass filter function with a corner frequency equal to the cavity half–width in order to provide
a better approximation of the system’s frequency response. This effect can be compensated for byappropriate servo design, such that the lock bandwidth need not be limited by the cavity linewidth.

103
Physically, this low-passing effect represents a transition from a regime in which the cavity issensitive to frequency fluctuations of the laser to one in which the error signal is proportional tophase fluctuations of the laser. As expected, this transition occurs at the cavity half-width.

4.2.3

Sources of lock error

There are two important considerations when frequency stabilizing the laser. The first isthat the reference cavity optical length must be as stable as possible and will be discussed later.
However, an equally important question is whether there are any effects that can prevent a laserfrom precisely tracking the reference cavity resonance, due to either technical or quantum effects.
The most fundamental but least important source of error in cavity locking systems is quantum noise (see e.g. [215, 210]). The optical power spectrum of shot noise on the light at the detector
is given by the single-sided power spectral density
GP = 2hνPopt 2

W /Hz .

(4.34)

Thus, assuming β = 1, the expected frequency noise due to shot noise in the most ideal case (andwith an ideal demodulator) is
Gν =

2
 2

hν∆νFWHM
GP
=
Hz
/Hz
,
ηD2
16ηP0 J02

(4.35)

where η takes into account the detector quantum efficiency and ν = c/λ is the optical frequency.
Substituting in the very modest parameters λ = 1 µm, P0 = 10 µW, η = 0.5, and ∆νFWHM =10 kHz, the very low shot noise floor of Sν = 4 × 10−7 Hz2 /Hz can be achieved. This can be related
to the locked laser linewidth by∆νlocked = πGν ,(4.36)

where it is assumed that Gν is white noise. For the parameters given above, this results in alocked linewidth of 1 µHz. Thus, for high-finesse cavities, the shot noise locking limit is far below
any cavity locking result, even in experiments designed to exclude other technical and thermal
effects [215]. This indicates that in practical situations, “technical” effects are most important.

104
Residual amplitude modulation (RAM) is a term that collectively describes a variety of effects
that induce amplitude modulation at the phase modulation frequency. For example, temperaturedependent parasitic etalons within the optical system can induce RAM which is then demodulated
along with the cavity error signal. This causes an offset to be introduced in the locking system.
Other effects that can cause RAM are typically related to the phase modulation device, most often
an electro-optic crystal-based (e.g. LiNbO3 , ADP, KDP) modulator. For instance, stress-induced
birefringence in crystal devices can rotate the principle crystal axis, creating not only an electricfield-dependent phase shift, but also a corresponding polarization rotation, which can be distributed
across the optical wave front.
While in many cases a temperature-controlled electro-optic (EO) crystal has low enoughRAM for acceptable performance, a certain degree of success has been achieved by using a DC
electric field on crystal-based EO modulators to actively servo the RAM [216]. However, beam
pointing deviations can cause slightly different regions of the crystal to be sampled, causing thephase of the RAM to shift, and limiting the effectiveness of the active system.
It is important to note that the effect of RAM on laser stability is reduced for cavities withnarrower resonances due to the fact that a given fractional change in RAM results in a smallerchange in frequency for a narrower resonance. Thus, for a given cavity length, higher cavity finesse
is always desirable to help mitigate RAM-induced line pulling.

4.2.4

Choice of EOM crystal

In many cases, limiting RAM is of central importance for high-precision locking. As new
materials continue to be developed to lower the fundamental thermal noise limitations to cavity
locking (see Chapter 3 for a detailed description of thermal noise in interferometers), RAM-induced
lock error will continue to be an important factor in high-precision laser stabilization. Here we
discuss strategies for mitigating EOM-induced RAM.
Crystal symmetry plays a big role in the properties of an EOM. Lithium niobate and lithium
tantalate are uniaxial crystals that both belong to the trigonal 3m crystal group, while ammonium

105

LiNbO3

ADP

z

x

y

z

x

z

y y

Figure 4.6: LiNbO3 and ADP modulator configurations. In the case of ADP, the beam that receives
the phase modulation (horizontally polarized in this figure), walks off from the vertically polarizedcomponent.

dihydrogen phosphate (ADP) and potassium dihydrogen phosphate (KDP), also uniaxial, belong to
the tetragonal 42m crystal group. While a full discussion of the crystal symmetries and implications
for electro-optic modulation of light is beyond the scope of this section, we note that in general
the electro-optic coefficients—the coefficients that modify the crystal impermiability tensor in thepresence of an applied field—depend directly on the type of crystal symmetry. For example, ther33 coefficient in trigonal 3m crystals—which produces an index change for light polarized along
the extraordinary axis with an applied electric field along the same axis and is widely used in
LiNbO3 -based EOMs—is zero for tetragonal 42m crystals, such as ADP.
We begin our discussion of the most conceptually simple of the two classes of crystals considered, the trigonal 3m crystal group, to which LiNbO3 and LiTaO3 belong. For light polarizedalong the z-direction and with a modulation electric field also applied along z, the change of theindex of refraction can be shown to be [207]1
ne (E) ' ne − n3e r33 Ez .
2

(4.37)

Here, ne ' 2.2 is the index of refraction of the extraordinary axis (z-axis). This configuration is also

106
shown in figure 4.6. For the specific case of LiNbO3 , the r33 coefficient is approximately 30 pm/V
and is the largest electro-optic coefficient among all of the LiNbO3 EO coefficients [217].
One of the most important intra-EO sources of RAM is a polarization-rotation at the drive
frequency that gets converted to RAM by polarization-sensitive optics in the PDH system lateron [216]. Specifically, if there is any misalignment between the incoming light polarization andthe LiNbO3 extraordinary axis (z)-axis, there will be a RAM effect. Temperature fluctuations
can couple to stress-induced birefringence to make the z-axis a moving target. Thus, temperature
control and a very high quality, carefully aligned input polarizer are necessary for reducing thetime-dependent RAM in LiNbO3 crystals.
It turns out that ADP largely addresses the issue of polarization rotation. Due to the different
crystal symmetry of tetragonal 42m crystals3 the EO coefficient of interest is r41 and the crystal
should be cut as in Fig. 4.6. It can be shown that the index change for the ray polarized along they 0 axis is [218]1ny0 (E) ' ny0 + n3y0 r41 Ex ,2

(4.38)

where ny0 is given by
n =
y0

√


2

1
1
+ 2
n2y
nz

−1/2
' 1.5.

(4.39)In the case of ADP, r41 is approximately 25 pm/V [218], thus it is competitive with LiNbO3 interms of phase modulation.
Due to the qualitatively different orientation of the axes for phase modulation, ADP canserve as its own polarizer. As shown in Fig. 4.6, the polarization component along the y 0 axis has
a walk-off due to the fact that this axis is not a principle axis. On the other hand, the (unwanted)
polarization component along x has no walk-off due to the fact that x is a principle axis. The angle
between the y 0 polarization k-vector and the x polarization k-vector is approximately 1.8◦ [219].
This means that the two polarization components can be spatially separated after transmission
through the crystal, as depicted in Fig. 4.6, effectively eliminating the polarization rotation effect.
3

The reader should be aware that the orientation of ADP discussed here is not common in the literature due tothe fact that a different configuration is commonly utilized for amplitude modulation.

107
Input pol.

Waveguide EOM

Output pol.

Vrf
Vdc

Detect RAM

RF PD

LO

Figure 4.7: Experimental setup for active control of residual amplitude modulation (RAM) in a
waveguide electro-optic modulator. The RAM is detected using the same local oscillator as is used
to drive the crystal.

4.2.5

Active control of residual amplitude modulation

As discussed in the previous section, RAM can crucially depend on the polarization alignment
into crystals of the trigonal 3m family. In general, it is possible to control the polarization alignment
by careful control of the input light polarization, and to mitigate temperature-dependent stressinduced birefringence. Wong and Hall [216] devised a further step, based upon applying a DC
electric field in addition to the RF drive, in order to actively cancel the polarization-rotationinduced RAM. Specifically, they show that for misaligments between the crystal extraordinary axis
and the input and output polarizers, given by angles θ1 and θ2 , respectively, there will be RAM
that scales as [216]∆P (t) = A0 sin (Ωt) sin [∆φ (Edc )] .

(4.40)

A0 = −P0 sin (2θ1 ) sin (2θ2 ) J1 (∆ϕ) ,(4.41)

Here

where P0 is the input power, and Ω and ∆ϕ are the same quantities as utilized in Eq. 4.28; and∆φ (Edc ) is the phase shift between the ordinary (o) and extraordinary (e) polarization components
within the crystal. This phase shift depends on the crystal length, the crystal temperature, the difference between ne and no , and importantly the applied electric field. The electric field dependence

108
-4

Hz

2000
Locked0-2000

Free-running

10

-13

10

-14

10

Locked

-15

10

-4000
0

5000
Time (s)bFree-running-12-8
Allan Deviation

Frequency offset (Hz)

0

a

4000

10000

0

10

1

10

2

3

10
10
Averaging Time (s)410Figure 4.8: Experimental demonstration of reduced residual amplitude modulation (RAM).(a)RAM-induced frequency offsets in a lock. Here, the free running lock baseline signal was converted to frequency by the known slope of the cavity frequency discriminant. Engaging the RAM
servo reduces the effect to the few-Hz level. (b)Allan deviation of the extrapolated RAM-induced
frequency offset.

was exploited in Ref. [216] to actively maintain ∆φ (Edc ) = 0.
Equation 4.41 suggests that if θ1 = θ2 can be enforced, there will be no RAM. While freespace modulators can come close to attaining this condition if temperature-dependent effects arecarefully controlled, it rules out a fairly attractive class of modulators: waveguide-based EOMs.
Waveguide EOMs can provide high modulation depth for very low applied voltage, allowing easy
optimization of cavity locking systems. However, fiber based devices do not offer precise control of
the input polarization due to manufacturing difficulty of aligning the polarization axes of the fiber
to crystal.
Despite this difficulty, we have had great success employing the technique of Wong and Hall
to null RAM-induced frequency noise. Modern fiber-based waveguide EOMs have the advantagethat the waveguide defines a precise interaction region. As a result, waveguide modulators are freefrom beam-pointing fluctuations that can be converted to RAM. We attribute the success of thistechnique to this key difference and we implemented RAM servos in several optical cavity systems,
most notably those described in Ref. [168]. For one cavity, the RAM servo allowed the cavity
resonance to be tracked with a fractional imprecision less than a part in 105 . Figure 4.8 shows

109
an example of such a RAM servo in action, where the frequency offset of the non RAM-controlledcase would be at the kHz level. Engaging the RAM servo increases the stability by two ordersof magnitude. Since the RAM servo technique is insensitive to RAM beyond the RAM detection
photodiode, it is quite possible that the level of stability of the data shown Fig. 4.8 was limited inpart by optical etalons.

4.3

Mechanical design of optical reference cavities
Despite the considerable challenges present in building a high-quality cavity locking systemthat is free from residual frequency offsets, these effects are not what limit most high-finesse opticalcavity frequency references. Instead, perturbations to the length of optical reference cavity are
ultimately what limit the frequency stability. These perturbing effects fall into two categories:
mechanical and thermal perturbations that are not fundamental, i.e. that are non-statistical inorigin; and fundamental statistical fluctuations in the cavity spacer, substrate, and coatings thatarise from their contact with a thermal reservoir at room (or cryogenic) temperature.
In this section, we discuss non-statistical perturbing effects and describe methods for their
mitigation. These mechanical effects can be divided into two categories: those caused by vibrations(accelerations) that structurally deform the cavity, and those that couple through the coefficient ofthermal expansion (CTE) of the cavity materials. Use of finite element analysis to optimize cavity
geometries and choice of materials has drastically reduced and elucidated these effects.

4.3.1

Vibration sensitivity

Although optical cavity mirrors and spacers are typically made out of an extremely rigid
substance, such as ultra low expansion glass (ULE), the length stability requirements are stringent
for sub-Hz lasers. As can be seen from Equation 4.13, the fractional frequency change of a cavity
resonance is directly related to the fractional length change by
∆ν
∆L
=−
.
ν
L

(4.42)

110
A world record for cavity stabilization—and one that stood for over a decade—was set in 1999 with
1-second stability at 3 × 10−16 , and employed a very impressive vibration isolation scheme in orderto keep the cavity length constant: the entire optical table was suspended on giant rubber bands,
resulting in a highly-effective seismic isolation system [38]. Although the cavity length in Eq. 4.42
is the effective length sensed by the optical field, which is averaged over the mirror surface, it isstill astounding that the length stability needed for a sub-Hz laser is sub-fm (10−15 m)—the lengthscale of the proton radius! It should thus come as no surprise that length stability at or below onepart in 1015 takes significant engineering effort.
One approach that significantly reduces the dependence on vibration-isolating structures isto design the cavity spacer such that mirror-spacer system is insensitive to vibrations. An intuitive
way to achieve this is to mount the cavity at its midplane in the vertical direction [39]. In thisway, the top and bottom mirrors move equal amounts when subject to vibrations along the verticalaxis. However, one can only get so far exploiting intuitive geometry for the simple reason thatthe support structure breaks perfect vertical symmetry. Thus to finalize any cavity design finite
element analysis (FEA) must be employed [220]. This technique can be applied to a variety ofcavity geometries and tailored to a specific design goal, such as insensitivity in a specific direction,
and was employed to design the vertical cavity reported in [40], and which is still operating in thelab as of the time of this thesis. Further improvements of the technique have yielded even better
results for a vertical design, with vibration sensitivty at the level of 10−11 / m/s2 [221]. These
results are quite good given that the highest grade commercial isolation platforms can give isolation√
performance at the 50 ng/ Hz level, resulting in a vibration-limited frequency noise performance√of order 10 mHz/ Hz for the sensitivities exhibited by modern cavities in the visible spectrum.
In principle, one is not restrained to vertical configurations. In fact, there may be good reason
to choose a horizontal configuration, especially if it is expected that the majority of vibrations will
be in the vertical direction. Other motivating factors include structural stability, especially forlarger cavities, and the experimental ease of access for horizontal geometries. Most importantly, inthe horizontal configuration, the coupling of vertical accelerations to deviations along the optical

111
axis is reduced by the Poisson ratio, representing roughly an 80% reduction in sensitivity. However,horizontal accelerations can still couple into mirror displacement, although the inherent symmetry
in the two horizontal axes limits this effect.

Designing two ultrastable lasers for the 10−16 level performance

4.4

In this section, we discuss the design of two ultrastable optical cavities for laser stabilization
using finite element analysis. In both cases, the ULE glass spacers were pre-existing (i.e., they were
not ordered expressly for the purpose of creating a design with insensitivity to vibrations). As aresult, the geometry was fixed, permitting fewer degrees of freedom to explore in the modeling.
Despite this constraint, we find support configurations that yield immunity to vibrations.

4.4.1

Design of the “Big ULE” cavity

The cavity lovingly nicknamed the “Big ULE” is one of the largest ultrastable cavities to be
implemented in a force-insensitive configuration [2]. This cavity first joined our lab in the early
2000s to fulfill another role. Jason Jones et al. employed it for direct frequency comb stabilization,and its length chosen such that the free spectral range matched the repetition rate of a Ti:Sapphirelaser [222]. At 39.4 m in length, it is over six times as long as the well-known (American) “football”shaped cavity of Ref. [199]. Its mass is over 20 kG. This author can assure the reader that every
moment transporting this cavity, often with nothing more than a steady grip, was a religious
experience!
In order to perform finite element modeling, we consider a symmetric half-cavity shown in
Fig. 4.9. The support strategy was chosen to give the feet a uniform surface on which to sit, yetstill be easy to machine. The support strategy that we decided to implement was based on millingvertical shafts with a flat support surface into the sides of the cavity (see Fig. 4.9a).
Additionally, we wanted to take a risk and use fused silica (FS) mirrors. As seen from thediscussion in the previous chapter, fused silica has a much lower loss angle that ULE, and thus canlower the thermal noise floor of a cavity by about a factor of two over an equivalent design with

b

c

112

z=80.36 mm

d=53.37 mmaFigure 4.9: The finite element model considered for the “Big ULE” cavity. It is important tonote that the teflon legs shown here are no longer in use. (a) One half of the cavity was used in
the modeling due to the axis of symmetry in the midplane (b) Front view of the model used andoptimal support depth, d. (c) Top view of the model, and optimum support axial displacement z.

ULE mirrors. The risk comes in the mismatch of coefficients of thermal expansion (CTEs) of thetwo materials. In order to mitigate the problems arising from the mismatch, we included ULE ringsin the design, which were assumed to be contacted to the back surface of the FS mirrors. This had
been shown to tune the the thermal expansion properties in a way that helps offset the differencein CTEs [223] (we will discuss this in more detail in Section 4.4.4.1). We note parenthetically, that,while the ULE rings were included in the mechanical modeling, their effect is not very significant,and shifts the zero crossing points by only ∼ 0.1 mm.
4.4.1.1

Finding the optimal support strategy

In order to find the optimal support strategy, we modeled the cavity’s response to verticalaccelerations using finite element analysis (FEA). The modeling goal was to eliminate both themirror tilt and the mirror displacement in response to vertical accelerations. The two parametersthat were tuned to find the vibrationally insensitive points are shown in Fig. 4.10. They represent
the position of the support surfaces with respect to the lower surface of the spacer (d), and the
distance of the support surfaces to the end of the spacer (z). In this modeling, teflon rods wereassumed as the support structure. During the modeling, the distance from the optical axis to the113bottom of the foot was kept at 105 mm to fit into the vacuum assembly. We note that the teflonrods were later abandoned after the first implementation of the cavity because they led to high-Q
resonances (in Chapter 6, we will see how these resonances manifested themselves in atom-based
measurements).
We performed simulations for varying z and d and recorded the simulated tilt (as measured
along the mirror vertically) and average mirror displacement. We assumed linear coupling between{z, d} and, the tilt and displacement (given by θ and ∆, respectively), such that     ∆   a11 a12   z   ∆0 =× +.
θ
a21 a22dθ0(4.43)

More compactly, this can be written
Y = AX + Y0 .

(4.44)

When the displacement vector, Y, satisfies Y = 0, the optimum parameter set is given by
X = −A−1 Y0(4.45)The planes shown in Fig. 4.10 are fits to a total of 11 different simulations, each for a different
combination of z and d. The extrapolated optimal support points are found, via Eq. 4.45, to bez = 80.36 mm and d = 53.37 mm. Figure 4.11a shows the final support configuration of the Big
ULE cavity, where the problematic teflon has been replaced. Since a soft material between the
optical cavity and the support structure is always necessary, we used Viton hemispheres with aradius of ∼ 1 cm (these can be obtained from McMaster-Carr, part 3645K7). Figure 4.11b showsthe exaggerated deformation due to the acceleration of gravity in the vertical direction for the
optimal support configuration.
Once the optimal support points were chosen, we checked the sensitivity to horizontal accelerations via FEA. Due to the presence of horizontal symmetry, we expected that the displacement
sensitivity should be zero, and it was (to within error). However, the mirror tilt due to horizontal
accelerations was quite large. We show mirror displacements due to accelerations in the vertical andhorizontal directions in Fig. 4.11c. If the optical mode forms at the center of the mirror, horizontal

114

a

b

Figure 4.10: Two dimensional planar fits to offset (a) and tilt (b) as a function of z and d. The
data points are obtained via finite element analysis, and a planar surface was fit to them.

accelerations would not couple to the net optical path length change. However, if the optical mode
forms (or if the mirrors are optically contacted) a few mm away from the geometric center of thespacer, a residual sensitivity of 10−9 /g is possible, where g is the acceleration of gravity. In fact,
this is closer to the value we estimate for the sensitivity to vibrations in the horizontal direction,and this seems to be the dominant mechanism by which vibrations perturb the optical path length.
Figure 4.12 (page 116) shows the Big ULE installed safely within its double-vacuum enclosure.
The outer rough vacuum is kept below 10 Torr to provide thermal isolation from the outside
world and promote temperature homogeneity over the surface of the inner vacuum chamber. The
measured time constant between the inner vacuum chamber and the cavity is 1.6 days, and theactively controlled temperature of the inner chamber maintains sub-mK stability over the 1 hour
timescale, and few-mK stability temperature stability over the course of a day. Figure 4.13 (page117) shows the new support structure and heat shield of the Big ULE. This new support structure
was installed after we discovered the problems with the original support structure.

115

a

c
2

-12

2x10

-12

0x10

2x10

-10

0x10

-10

-10

-4

-2

0

2

4

-2x10

2

-12

-2x10

Frac. x sens. (1/ms )Frac. y sens. (1/ms )b

Position along mirror (mm)Figure 4.11: The Big ULE cavity and Zerodur support structure. (a) The final design of the Big
ULE removed the teflon legs in favor of a Zerodur support shelf. The cavity rests on Viton hemispheres (not pictured). (b) Exaggerated cavity deformation in response to a vertical acceleration
and when held at the optimal points via finite element analysis (FEA). (c) FEA-predicted mirror
displacement per acceleration as a function of vertical (left) and horizontal (right) displacement
along the mirror surface when the cavity is supported optimally.

116

2

4

6

8

60

3000
2500

50
40

2000

30

1500

20

1000

10

500

0

0

200

400

3

Drift rate (Hz/sec)

Vacuum chamber temp (C)Time (days)
00600x10Time (s)Figure 4.12: The double-layer vacuum system for the Big ULE. The inner vacuum chamber isactively temperature stabilized by Peltier elements (visible at right) to the level of several mK(sub-mK) per day (hour). The outer vacuum is kept below 10 Torr to provide thermal isolationfrom the outside world and promote temperature homogeneity over the surface of the inner vacuum
chamber. A vacuum level of 2 × 10−8 Torr is maintained in the inner chaber with a 75 L/s ion
pump. The measured time constant between the inner vacuum chamber and the cavity is 1.6 days.
Inset: Measurement of the thermal time constant of the system. A large temperature change wasapplied to the system and the optical beat of the laser stabilized to the Big ULE was measured
with a second stable laser system.

117

Figure 4.13: The Big ULE cavity with new Zerodur glass support structure. The cavity rests upon
Viton hemispheres (not shown). Due to the change in support structure, we constructed a newheat-shield box out of polished aluminum to replace the copper one shown in Fig. 4.12.

118
4.4.2

Design of the “JILA cavity”

In order to support the testing of a new cavity constructed from monocrystalline silicon incollaboration with PTB,4 which we discuss in Section 4.6, we set out to design a second cavity
that would perform at the low 10−16 level. The name, “JILA cavity” comes from its time at PTB,
as this was an easy way to differentiate it from the other systems there. Since this system was
supposed to be transportable, we attempted to make the design as compact as possible. The basic
design is shown in Fig. 4.14 and uses a support scheme similar to that of [224]; the spacer is ahorizontally mounted cylinder, with a diameter of 10 cm and a length of 25 cm. Support shelves,
whose exact positions were chosen via FEA, extend the entire length of the spacer with four support
points defined by small Teflon cylinders. In order to save space, the support scheme involving sucha lip along the edge of the cavity allows the cavity to rest on a rigid rectangular support structure
made of ULE glass through which the bottom half of the cavity hangs. The vacuum system isthus considerably smaller than the Big ULE, and is compact enough so that it can fit onto a single
Minus-K commercial isolation platform, as shown in Fig. 4.14d. As with the Big ULE, the mirrors
are fused silica and the ULE rings have been added.

4.4.2.1

Finding the zero crossing point

By tuning the depth of the cuts and z-position of support points, we extrapolated the bestvalues from a total of 9 FEA simulations. We found that the data had a quadratic distortion due
to the fact that changing the position of the support shelf removes material along the entire lengthof the cavity, changing the effective cross section. We therefore included couplings up to quadratic
4

Physikalisch-Technische Bundesanstalt, Germany’s standards institute. http://www.ptb.de.

119

b

c

d = 7.18 mm

z = 49.13 mmadFigure 4.14: The “JILA” cavity design and vacuum system. (a) Model used for performing finite
element analysis (FEA). Only 1/4 of the total cavity is shown, as there are two planes of symmetrythat were exploited for the FEA modeling. The support is a small teflon cube. (b) Head-on viewof the model. As depicted here, the support shelf optimal distance with respect to the optical axis
was d = 7.18 mm. (c) Top-down view of the cavity and optimal z displacement of 49.13 mm.
(d) Cavity installed inside its vacuum chamber. The inner heat shield is made of copper and its
temperature is actively controlled with four Peltier on which the heat shield directly rests. Thermal
contact between the heat shield and the Peltier elements is facilitated with indium foil. The cavity
is supported by four small teflon supports which rest upon a secondary ULE support structure. The
support structure rests upon the heat shield with four additional teflon stand-offs. The entire system
including ion pump was small enough to rest on a single commercial Minus-K platform.

120

b
12

40

10

30

8

20

Mirror tilt (nrad)

Mirror displacement (pm)

a

10
0
10
20
30
40

6
4
2
0
2
4

50
5
51

6

50

z-po

sitio 49
n (m

7

m)

8

48
47

9

pth

Cu

e
td

m
h(

m)

9

6
8
47

8

47.5

48

48.5

z-pos

7
49

ition (

49.5

mm)

50

6
50.5

51

5

t

ep

td
Cu

)

mm

(
hh

Figure 4.15: FEA simulation of the “JILA cavity.” Here, a fit up to quadratic order was used todescribe the mirror (a) displacement and (b) tilt as a function of the support positions.

order in the fit to the FEA data, such that the displacement surface is described by


 z 




 
 

 h    ∆0  ∆   a11 a12 a13 a14 a15  
2 

=
×
.
 z +θ
a21 a22 a23 a24 a25
θ0
 2 
 h 




hz

(4.46)

The fits to the FEA data are shown in Fig. 4.15. The quadratic dependence on the support positions
is clearly seen.
As before, we can write Eq. 4.46 in a more compact form: it becomes
Y = AX + Y0 .

(4.47)

Since A is not a square matrix, we can’t invert it as before (we are still only solving for two unknown
quantities), but rather find the family of solutions for
AX = −Y0 .

(4.48)

Only one of the solutions is physical, and the the extrapolated values for the vibrationally insensitive
mounting are z0 = 49.13 mm and h0 = 7.18 mm.

Tilt
Displacement (fractional)

Big ULE
5.5 nrad/mm-g
1.3 × 10−10 /mm-g

Nezrick
4.3 nrad/mm-g
1.0 × 10−10 /mm-g

121

Table 4.1: Zero-crossing sensitivities obtained from the finite element modeling. Note that thez-position affects almost exclusively the mirror tilt, while the depth of the support affects almost
exclusively the displacement. All values in this table are quoted against these major contributors
for the parameter (tilt or displacement) of interest.

4.4.2.2

Sensitivity compared to the “Big ULE”

Having two sets of modeling results allows a comparison of how sensitive the zero-crossings
are to placement errors of the support points. Using the linear models from both the JILA cavity
and the Big ULE modeling (in the case of the JILA cavity, we use the linear coefficients of themodel), these sensitivities can be expressed at 1 g and per mm of placement error. This is relevant
for defining how accurately the support points must be defined. These quantities are listed inTable 4.1. We find that in general the Big ULE and JILA cavity have very similar sensitivity toplacement errors of the support points, and even 1 mm of placement error is quite tolerable, atleast in the FEA. Probably a more insidious problem is that the FEA assumes complete symmetry
amongst all support points; it is easy to imagine that three of the four support positions aresupporting the weight. This is why we use soft materials between the support surfaces and therigid support strucure in both the Big ULE and the JILA cavity.

4.4.3

Mesh size effect

As a final test of the modeling, we wanted to make sure that the FEA did not have a strong
dependence on the discretization mesh (as in Ref. [220]). The mesh size was limited by this author’s
laptop RAM, which was 2 GB (at the time of this writing already this value sounds obsolete). In
the case of the Big ULE, the optimal support positions were extrapolated based upon data obtainedwith a 3 mm mesh. The red curve in Fig. 4.16 shows that this extrapolation is quite good, witha fractional cavity length change on the order of 1 × 10−12 /g. However, changing the mesh tosmaller values perturbs this result, meaning that the optimum values of z and d in the limit of zero

30
25
20
15

12

- L/L x 10 / g

10122
Mesh size:
2.25 mm
2.5 mm2.75 mm
3 mm
3.25 mm
3.5 mm

5
0
-5
-10
-15
-20
-25
-30
-35
-40
-6

-4

-2
0
Position along mirror (mm)24Figure 4.16: Effect of mesh size on the FEA-predicted mirror displacement. The optimal placements
were calculated with a mesh size of 3 mm (red curve). The smallest mesh size of 2.25 mm is closer
than 2.5 mm mesh results, indicating a turning point in the mesh size effect.

mesh size (which we hope would then be equivalent to reality) are something different. However,by reducing the mesh size to a minimum of 2.25 mm, we can see that the perturbation does notincrease monotonically and that at this smallest mesh size it is even heading back in the right
direction. This could indicate that our residual error based on mesh size alone could be of order1 × 10−11 /g. This is still quite small compared to what are considered “good” sensitivities toacceleration, typically on the order of 10−10 /g. From this we conclude that numerical errors due
to mesh size effects will not be a dominant source of error in the modeling.

4.4.4

Thermo-mechanical perturbations

The most common cavity spacer and mirror substrate materials for ultrastable reference
cavities are ULE, fused silica (FS), sapphire, and silicon. The CTEs of these materials are shownin Figure 4.17. Due to large room-temperature CTEs, the latter two materials have primarily beenused at cryogenic temperatures [225, 226, 227].
In general, a cavity made of a uniform-CTE material will experience a fractional length

ULE

50
0

2

Slope = 1.7 ppb/K

-50

4
2
0
-2
-4

4000

CTE (ppb/K)

CTE (ppb/K)

a

123

5000

CTE (ppb/K)

520
480
440
400

Fused silica

3000
2000

0

10

-20

-10

0

10

T (C)

20

30

40

50

Si

20

1000
2

Slope = 17 ppb/K

0

-100

b

Sapphire

0

50

100

150

200

250

300

350

T (K)

Figure 4.17: Temperature dependence of coefficients of thermal expansion (CTEs) of common cavity
spacer and substrate materials. (a) CTEs of fused silica and ultra low expansion glass (ULE) [228]near room temperature. The gray band on the ULE curve represents the manufacturer’s stateduncertainty in the zero-crossing temperature. (b) CTEs of sapphire [229, 230] and silicon [231].
The inset shows that silicon has a second zero-crossing 17 K in addition to the zero-crossing at
123 K.

change given by
1∆L/L = α(T )δT + α0 (T )δT 2 + O δT 3 ,2(4.49)

where α(T ) is the CTE at the operating temperature, T ; α0 (T ) is its first derivative; and δTare small temperature variations. Ideally, one operates a reference cavity around a zero-crossing
temperature of a material’s CTE, TZC , defined as
α (TZC ) = 0.

(4.50)In the case of sapphire, which has no zero crossing, optimal operation is at cryogenic temperatures
which reduce the CTE to an acceptable level. Taking Equation 4.49 in the vicinity of a zero crossing
and substituting in α0 (TZC ) = 1.7 ppb/K2 for ULE, it can be seen that the temperature needs tobe stable below 1 mK for stability at the 10−16 level. While this is in principle quite difficult,the large thermal mass of the ULE spacer tends to limit temperature effects to longer time scales.
Mechanical coupling from the vacuum chamber itself, not the CTE of the spacer material, can also
induce temperature-dependent frequency shift. Thus, care should also be taken to mechanicallydecouple the cavity support structure from the chamber. In the case of the Big ULE and JILA

124

Figure 4.18: Deformation of mixed-material optical cavities. These images show the qualitative
effect of the additional ULE ring as first demonstrated in [223]. At left, the effect of the CTE
mismatch is seen by the bowing outward of the fused silica mirror in response to an increase intemperature. At right, this effect has been partially mitigated by the addition of a ULE ring.

cavity, this was accomplished by using Zerodur or ULE baseplate structures that were only weakly
coupled to the cavity via Viton or Teflon. In this way, the baseplate support serves as a mechanicalbuffer between the cavity and the aluminum vacuum chamber.

4.4.4.1

Numerical thermo-mechanical modelingIn the case of a cavity with a mirror substrate made out of a different material than thespacer, extra complications arise. The two materials are typically optically contacted very firmly,
such that the mismatch of CTEs causes an auxiliary mechanical effect, effectively causing the mirror
substrate to bend. This modifies the effective CTE, αnet , such that [228]αnet = αspacer + 2δR
(αmirror − αspacer ) + Γ.
L

(4.51)

Here, R is the mirror radius and L is the cavity length. The term δ describes thermo-mechanicalstresses coupling into length change and Γ accounts for deviations from the ideal model [228].
Thermo-mechanical finite element analysis can be used to find δ for a given cavity geometry, such

125
Tc = 20◦
Tc = 20◦ (no ring)
Tc = 10◦

Composite CTE (ppb)

20
0
20
40
60
0

5

10

Temperature (C)1520

Figure 4.19: Net coefficient of thermal expansion for the Big ULE assuming a spacer T0 of 15.4◦ C.
The addition of a ULE ring raises the temperature of the CTE zero-crossing of the BIG ULE (solid
red vs. solid black). We find a dependence of the zero-crossing temperature on the cavity assembly(i.e., the temperature at which the mirrors/rings are optically contacted to the cavity) temperature(solid black vs. dashed black). The assumed assembly temperatures, Tc are given in the legend. For
the case where the contact is performed at 20◦ C, the ULE ring raises the zero-crossing temperature
to 16.9◦ C.

that αnet can be found and minimized by operating at the new zero-crossing temperature [228, 223].
Finally, it is worth noting that there has been success tuning the zero cross point by contacting an additional piece of ring-shaped ULE glass to the back of the mirror substrate, effectively“sandwiching” the substrate between two equivalent CTE materials. By varying the parameters of
the ring-shaped piece of glass, one can engineer a more favorable T0 [223].
In order to calculate the effect of the coupling indicated by Eq. 4.51, we performed simulationsof the thermo-mechanical deformation of the Big ULE. With the help of Rich Fox at NIST, wewere able to measure the Big ULE spacer’s CTE zero-crossing temperature as T0 = 15.4 K using a
speed of sound measurement technique. We then used this information to model the net thermal
expansion of the Big ULE spacer with fused silca (FS) mirrors. Figure 4.18 shows the deformation
with and without the extra addition of a ring due to a temperature change. As explored in [223],

126
we find that the net CTE of the ULE-FS mixed-material cavity is shifted colder. When we add a
ULE ring to the FS mirror, we find that T0 shifts to higher temperatures. Additionally, we findthat the temperature at which the cavity is assembled is a crucial parameter. In general, the net
zero-crossing gets pulled in the direction of the temperature at which the cavity was assembled.
This general effect is illustrated in Fig. 4.19. We ultimately assumed that the assembly temperature
for the Big ULE was 20◦ C, which led to a T0 of 16.9◦ C, and this is the temperature at which we
maintain the Big ULE enclosure.
We note in passing that the JILA cavity spacer had a very cold T0 ∼ 0◦ C. We could not
be sure if this result was reliable, since the speed of sound measurement had not been tested inthat regime, but subsequent measurements with the JILA cavity at PTB indicated that its T0 was
colder than 5◦ C, which is inconvenient but not insurmountable.

4.5

A near quantum-limited lattice clock
In this section, we describe how the Big ULE cavity has allowed the 87 Sr lattice clock(s) at

JILA to demonstrate the highest precision of any neutral atom clocks in the world. Crucially, the1 × 10−16 near-thermal noise limited performance of the Big ULE has enabled the 87 Sr lattice clock
to operate close to the quantum projection noise (QPN) limit (QPN in lattice clocks is discussed
in detail in Chapter 2). Furthermore, we demonstrate that single-system systematic evaluations—typically carried out with an interleaved measurement scheme—can resolve systematic effects at
1 × 10−17 fractional uncertainty with only 1000 s of measurement time.
4.5.1

Experimental details of the Big ULE

We discussed the support strategy and mounting structure of the Big ULE cavity in Section 4.4. Here, we describe the specific details regarding the Pound-Drever-Hall stabilization,
environmental isolation, and delivery of the stable light to the 87 Sr lattice clock.
While the finite element analysis was performed to optimize the vibration sensitivity of the
Big ULE, we find that a residual sensitivity to accelerations remains at the level of 1 − 2 × 10−9 /g127
in the horizontal direction. The vertical direction seems to be an order of magnitude smaller, but
this is difficult to measure with precision. While this is somewhat far from the design goal of
low 10−10 /g, this value still reflects favorably on the technique; even though the cavity is almost
seven times larger than the “football” cavity design, the vibration sensitivity is only 2–3 times aslarge. However, a vibration sensitivity at this level requires the best vibration isolation that iscommercially available, as well as good acoustic shielding.
In order to provide the highest levels of vibration isolation, we placed the double-layer vacuum
chamber that contains the Big ULE onto an actively stabilized optical table (Herzan, model 350S).
The accelerations of the top surface are actively sensed and cancelled at Fourier frequencies as low
as 2 Hz via piezo-actuated legs. Remarkably, the system can support very heavy loads, and in
the current configuration the maximum capacity including the optical breadboard is 600 kG. We√
measure that the system is capable of reducing the vibrations to the level of 50 ng/ Hz to Fourier
frequencies of approximately 20 Hz, above which it appears the acoustic environment begins toaffect the measurement.
The laser that we eventually stabilize to the Big ULE cavity (see Fig. 4.20 for a schematic
overview of the setup) is an external cavity laser diode (ECDL) comprising an anti-reflection laser
diode (Opnext/Oclaro part HL7001MG) in the Litmann-Metcalf external cavity configuration [232].
The output power is approximately 40 mW. We lock this laser first to a pre-stabilization cavity
using the Pound-Drever-Hall (PDH) locking techniques as in [40], which narrows the laser spectrum
to the sub kHz level. Originally we utilized a small portion of the laser output power for the lock,but recently we altered the configuration so that nearly all the laser power is now used for the
pre-stabilization lock. The mirrors for the prestabilization cavity are ultra-high finesse mirrors(F > 100, 000), and they are spaced only ∼ 1 cm apart so that the linewidth of the prestabilization
cavity is not so narrow as to affect the ECDL lock to it. Of the 30 mW that arrives at theprestabilization cavity, & 10 mW of optical power is transmitted through it when the cavity islocked. The transmitted light is used for locking to the Big ULE cavity; the linewidth of the prestabilization cavity (∼ 100 kHz) allows extra filtration of the optical spectrum past this frequency.

128
Actively stabilized platform

Prestab. cav.
length (slow)
ECDL

λ/4

PDH det.

AOM freq.
(fast)

Int. servo det.

LD current
and external
cavity length

AOM
ADP
EOM

Temperature-controlled devices:

AOM drive power

λ/4-ECDL diode-ECDL baseplate-EOM crystal

To Sr distribtion
center-Inner vacuum
chamber of Big ULE

Figure 4.20: Two-stage Pound-Drever-Hall (PDH) locking for laser stabilization with the Big ULE.
A fast loop locks the external cavity diode laser (ECDL) to the prestabilization cavity (lock details
not pictured). The transmitted light double-passes an acousto-optic modulator (AOM) before
traveling to the actively stabilized platform. The PDH lock to the Big ULE employs an ammonium
dihydrogen phosphate (ADP) electro-optic modulator (EOM). Part of the ∼ 100 µW sent to the big
ULE is sampled for active control of the intensity arriving at the Big ULE. A 30 m noise-cancelled
fiber delivers ∼ 500 µW of power to the Sr distribution center, which is described in Chapter 2.

The laser frequency actuation for the lock to the pre-stabilization cavity is performed via feedback
to the laser diode current and the external cavity length via a piezo-actuated mirror. We achieve
close to a 2 MHz lock bandwidth, which allows robust tracking as the pre-stabilization cavity lengthis tuned by a piezo-actuated mirror.
After the prestabilization stage, the laser light goes through a double-pass acousto-optic
modulator (AOM) before entering a single-mode fiber that delivers it to the actively controlled
optical table and onto a secondary platform near the level of the Big ULE optical axis. We findthat the double-pass configuration introduces minimal amplitude noise onto the light as the AOM

129
frequency is tuned over several MHz.5

The fiber collimator is chosen (based on the known mode

field diameter within the optical fiber) such that the properly sized beam waist forms at ∼ 50 cm
from the fiber tip without the need of additional mode-matching optics. The Big ULE has one
flat mirror, which serves as the input mirror, and one curved mirror, which serves as the output
mirror. The beam waist on the input mirror is 330 µm. As shown in Section 4.2.1, the finesse ismeasured to be 194,000, resulting in a 2 kHz cavity resonance full-width at half-maximum and thus
a resonance Q-factor of 2 × 1011 . As we describe in Chapter 6, our estimate for the locked-laser
linewidth is 26 mHz, indicating that we have achieved a lock precision of nearly 1 part in 105 .
The phase modulation sidebands are provided by an ADP-crystal electro-optic modulator.
We find that the ADP modulator produces no noticeable residual amplitude modulation (RAM) for
the reasons discussed in Section 4.2.4. We do find it necessary to include an optical isolator in frontof the ADP modulator, as parasitic etalons can form between the crystal face and other opticalelements. As a further step to reduce RAM, we stabilize the crystal temperature. We measure that
the RAM and electronic drifts induce lock baseline noise below the level of the expected thermal
noise floor of the Big ULE from Fourier frequencies of 10−2 Hz to 100 Hz.
We engage the second-stage PDH lock by feeding back to the double-pass AOM with a highbandwidth voltage-controlled oscillator (Mini-Circuits ZX95-78-S+, chosen for its large modulation bandwidth). In order to limit potential optical-power to frequency-offset conversion (i.e., theinstance where a laser power change gets converted to a frequency jump), we actively control thepower of the light being delivered to the Big ULE for frequency stabilization by additionally feedingback the the RF drive of the double-pass AOM. We do not find that the frequency stability has asensitive dependence on the optical power being delivered to the Big ULE for PDH locking, but
we typically keep it in the 100–200 µW range so that the PDH photodetector is near photon shotnoise limited (as explained in Section 4.2.3, fully reaching the shot noise limited regime is overkill).
A portion of the light prior to the EOM is diverted towards a noise-cancelled, 30 m, polarization5

In fact, the double-pass AOM setup is so robust that the AOM alone can lock the free-running laser diode to
the Big ULE cavity for tens of seconds before the ECDL drifts out of acquisition range.

Frequency noise Hz

Hz 

130

2.0
1.0
0.5
0.2
0.1
1

2

5

10

20

Fourier frequency Hz

50

100

Figure 4.21: Effect of the sound-proof box on the optical frequency spectrum. The purple trace
was a frequency measurement against the football cavity and is limited by that systems thermal
noise floor at low frequencies (dashed red line). The dashed black trace is a model spectrum for
the Big ULE system in the presence of the acoustic noise. The brown trace is the frequency noise
spectrum after the installation of the sound-proof box. It is evident that the addition of thisenclosure provides & 20 dB of isolation at frequencies above 50 Hz. We attribute the small, sharp
resonances at 22 Hz, 30 Hz, and 42 Hz to acoustic resonances within the “quiet room.”

maintaining, single-mode optical fiber. This fiber delivers light to the injection lock in the Sr“distribution center,” and this system, including the details of how light is delivered to atoms, isdiscussed in Chapter 2, Section 2.4.3.1.
As a final step to isolate the Big ULE from the world, have constructed a large (40 × 60 × 30 )acoustic enclosure around the actively controlled optical breadboard. We utilized multiple layersof damping material, including a high-density polymer and large, textured sound absorbing panels.
This enclosure serves to drastically reduce acoustic noise > 50 Hz by & 20 dB. Figure 4.21 shows thebefore-and-after optical frequency noise spectra when comparing the big ULE to the old clock laser(i.e., the football cavity), indicating that the large noise peak in the vicinity of 50–100 Hz is almost
completely eliminated. Small, sharp resonances remain at 22 Hz, 30 Hz, and 42 Hz. Remarkably,
we attribute these to acoustic resonances inside “Jans quiet room” (this is the name for room in131which the clock laser resides), as the frequencies agree remarkably well with the expected resonant
frequencies corresponding to the dimensions of the room. This leads the author to believe that
small rooms are optimal for high-precision work, so that the room resonant frequencies can be
more easily isolated by acoustic enclosures!

4.5.2

Verification of thermal-noise limited performance

As a test of the laser’s performance, we locked the clock laser to the 87 Sr transition frequency
as described in chapter 2. We record the frequency corrections necessary to keep the laser onresonance with the Zeeman-insensitve frequency of the average 87 Sr |F = 9/2, mF = ±9/2i clocktransition. These corrections are equivalent to to the difference between the laser frequency andthe 87 Sr clock frequency with a constant drift and offset removed. The in-loop error signal averages
down quickly indicating that at the 10 s and beyond timescale the measured fluctuations should
correspond with the laser frequency fluctuations. As seen in Fig. 4.22, the laser stability is at thelevel of σy (τ ) = 1 × 10−16 , which is consistent with the theoretically predicted value of 7 × 10−17 .
In order to test the frequency resolution of the system, we scanned out ultra-narrow Rabilineshapes. A composite scan of such lineshapes is shown in Fig. 4.23 (page 133). We observethat a line-broadening that is correlated with the atomic density, which we explore in detail in
Chapter 5. At the lowest densities and using a 3 s Rabi pulse, we are able to repeatedly resolve
500 mHz lineshapes in a single scan, representing a line Q of 9 × 1014 , representing a significant
milestone in optical lattice spectroscopy.

4.5.3

High stability for rapid systematic evaluation

As seen in the previous section, the Big ULE performs at the level of thermal noise. One
exciting application of such a high-precision system is the possibility to quickly measure a systematic
shift via interleaved measurements. The procedure is to operate two independent locks, timed suchthat their measurements are interleaved with one-another (e.g., lock 1, lock 2, lock 1... etc.) andsuch that two separate experimental conditions are utilized for each lock. For instance, lock 1

132

1  1016
5  1017Frequency correction (Hz)Allan deviation

5  1016

1  1017
5  10

18

0.3
0.2
0.1

0.0
 0.1
 0.2
 0.3
0

200

400

600

800

1000

Measurement number

1  1018

1

5

10

50 100500 1000

Averaging Time
Figure 4.22: Demonstration of a thermal-noise limited laser at fractional instability of 1 × 10−16
from 1–1000 s. The laser performance is measured by analyzing the frequency corrections necessary
to keep it on-resonance with the 87 Sr clock transition (blue). The in-loop error signal is shown intransparent red, indicating that at time scales > 10 s, the applied correction signals track the clock
laser frequency, thus demonstrating that the clock laser performs with stability of 1 × 10−16 . The√
dashed black line is a 1/ τ guide to the eye, indicating that the in-loop error signal averages down√
faster than 1/ τ . (Inset) Time-domain measurement of the clock laser frequency via the 87 Sr
lattice clock. Each point represents approximately 2 s. Removing the clear quadratic drift resultsin sub 10−16 performance.

133

0.4
0.3

0.6
Excitation fraction

Excitation fraction

0.5

0.2

0.4

500 mHz

0.2
0.0
−4

−2
0
2
Detuning (Hz)

4

0.1
0.0
−6

−4

−2
0
Detuning (Hz)2

4

Figure 4.23: Ultra-narrow Rabi lineshapes demonstrating 3-s optical-atomic coherence. The lineshape shown in the main figure is a composite linescan comprised of a total of 20 scans. The
750 mHz linewidth is broadened from the Fourier-limited value of 400 mHZ by density-dependent
effects. We explore these effects in detail in Chapter 5. Inset: an extremely low density single scanwith 3 s probe time. The linewidth is 500 mHz and was repeatable as long as the atomic density
was kept extremely low. Here, the line Q ' 1015 , representing a milestone in optical lattice clock
spectroscopy.

134
might be a high density condition while lock 2 corresponds to a low density condition. The rapidity
with which the difference between these two locks can reach a given measurement precision sets
the time scale for measuring systematics in the optical lattice clock. For instance, in the work
reported in [135], many days’ of averaging were required to make measurements at the level of1 × 1017 . However, with the stability of the Big ULE-stabilized clock laser, such measurements canbe made at the 1 hour timescale. To test the ultimate performance of the techniques, we operated
two independent locks to the same atomic peak (i.e., spin state) and compared the stability of theirfrequency difference. In order to obtain the highest precision, we used a Rabi pulse time of 500 ms.
We compared the central frequency of lock 1 to lock 2 such that∆ν (t) = ν2 (t + ∆τ ) − ν1 (t) .

(4.52)To remove biases from residual first- and second-order drift, we used a three-point string approachas in [233, 135], as this is the method used for systematic evaluations and is necessary for eliminating
systematic errors due to linear and higher-order frequency drifts of the local oscillator. As shownin Fig. 4.24, the precision of a systematic evaluation measurement can reach 1 × 10−17 at 1000 s.
√
The faster than 1/ τ averaging in Fig. 4.24 indicates that the measurement has not yet reached√the Dick effect-limited noise floor, at which point it will average down with a 1/ τ dependence.

4.5.4

Clock comparison

At the same time that the Big ULE cavity stabilized laser system was being perfected, asecond 87 Sr lattice clock came to life in the Ye lab. This second system, lead by Travis Nicholson
and Jason Williams, is based on a similar design to the system discussed in Chapter 2 (we will
henceforth refer to the system that is described in Chapter 2 and that is the subject of this thesis
as “Sr1”; we will call the Nicholson/Williams system “Sr2”). The main difference between Sr1and Sr2 is the lattice geometry; Sr2 has a cavity-enhanced horizontal lattice with a 160 µm beam
waist, such that the trapping frequencies are 100 kHz in the longitudinal direction and 120 Hzin the radial direction. Due to the fact that overlapping the red MOT and the lattice with theAllan deviation

5  10135
16

1  1016
5  1017
1  1017
5  1018
1  1018

1

5

10

50 100500 1000

Averaging Time
Figure 4.24: Stability of interleaved measurements for systematic evaluation with Rabi pulse duration of 500 ms. A three-point string was used to analyze the data. A new value of δν is calculatedwith the point string analysis every experimental cycle, thus the first few points on the Allan deviation are low due to correlations between subsequent point-string filtered data points. At longer
averaging times the correlations are minimal and the Allan deviation represents the true stabilityof lock 1 compared to lock 2.

lattice perpendicular to gravity—as opposed to along it—loads the atoms much more diffusely along
the lattice, the densities of Sr2 are approximately a factor of 100 lower than those of Sr1, whencombined with the reduced radial trap confinement. The better spatial overlap with the red MOT
afforded by the lattice geometry, Sr2 is also capable of loading a factor of 10 more atoms into thelattice than Sr1. We note that the results discussed here can be found in [43].
In order to test the relative stability of the clocks, we used the laser stabilized to Big ULE as
the clock laser for both systems. As depicted in Fig. 4.25, Sr1 and Sr2 each have an independent
AOM to shift the frequency onto resonance with each clock, with frequencies f1 and f2 , respectively.
The difference frequency between the systems is derived from the record of f1 and f2 . As demonstrated in Ref. [234], frequency comparisons between two clocks that sample the same frequency
noise should be immune to the Dick effect. Thus, in order to test the level of instability due to theDick effect, we performed the experiment with the clock pulses applied synchronously and a with a

136

Sr1

Sr2
f1

f2

Figure 4.25: Clock comparison scheme. Two independent frequency shifts are applied to the clock
laser to steer it to each clock’s central frequency. As schematically depicted here, the density of
Sr2 is much lower than Sr1 due to both the wider lattice waist and the longitudinal distribution of
atoms.

timing offset of 10 ms longer than a single pulse duration. Additionally, in order to limit sensitivity
to density-dependent shifts, Sr1 operated with 1000 atoms, while Sr2 operated with 2000 atoms.
In order to eliminate first-order sensitivity to magnetic fields both clocks were operated using thetwo-spin-state procedure as detailed in Chapter 2. Additionally, Sr1 operated at approximately
50% the usual lattice power, such that νz = 55 kHz. This was necessitated by the addition of acavity in the lattice beam to act as a spectral filter, which reduced the optical power, but it also
served to better control time-varying density-dependent effect, as the density was decreased withlower trap depth corresponding to νz = 55 kHz.
We find that the Dick sensitivity function for comparing two clocks that sample the samelaser frequency with a timing offset is related to that of a single clock as derived in Chapter 2 asR(2) (f ) → 4 sin2 (πf T0 ) R(1) (f ) ' 2R(1) (f ) .

(4.53)

Here, R(1) (f ) is the sensitivity function considered for the single clock case, and R(2) (f ) is theclock comparison case. The most notable change is the sin2 (πf T0 ) term, which accounts for thefact that the laser noise is coherently sampled with a time delay and can be derived using theFourier shift theorem. If the time delay is large enough, this term is rapidly oscillating and we canAllan deviation

Asynchronousb

T0

1  1016
5  10137

a

5  1016

Sr1

17

Sr2
Synchronous1  1017
Sr1

5  1018

Sr2

1  1018

1

5

10

50 100

Averaging time s

500 1000

Figure 4.26: Clock comparison Allan deviations and pulse timing. (a) Allan deviations of the clock
difference frequency with with synchronous (blue data points) and asynchronous clock interrogation(red data points). The solid black line is the predicted quantum projection noise floor for themeasurement, and the red dashed line is the Dick limit for a laser with thermal noise floor of
1×10−16 . Here, the total Rabi pulse time was 160 ms. (b) Timing diagrams for asynchronous (top)and synchronous (bottom) interrogation of the two clocks. For the data shown in (a), T0 = 170 ms.

replace it with its average (as is done for the rightmost expression in Eq. 4.53). This represents
the limit where the two clocks are operated with independent lasers. We find that the inclusionof the sin2 (πf T0 ) term affects the predicted Dick noise limit with respect to the assumption of
uncorrelated clocks, at the level of 5%. Thus, the unsynchronized comparison is a robust indicatorof independent clock performance.
As shown in Fig. 4.26, the synchronous interrogation averages as σ sync. (τ ) ' 3×10−16 /pτ /1 s,while the asynchronous one averages down with only slightly larger instability, at σ async. (τ ) 'p
4 × 10−16 / τ /1 s. This indicates that the Dick noise plays only a small role in the clock instability. From these data, we were able to demonstrate an independent clock comparison (i.e.,
asynchronously operated clock comparison) to within a factor of 2 of the expected quantum projection noise limit. This represents an important milestone in the history of lattice clocks and isan order of magnitude improvement over the best comparison of ion clocks, which have attained

138√
measurement precision of σy (τ ) = 2.8 × 10−15 / τ [27]. Thus, a given frequency measurement
would take a factor of nearly 100 less averaging time when performed with the 87 Sr lattice clock
vs. a single-particle clock! This turns a solid month of measurement into an afternoon.
We note that at the level of 1 × 10−16 , many other noise sources besides laser noise arealso possible, including fast bias magnetic field noise (which creates its own Dick effect) and othertechnical sources of noise. Finally, we note that the laser is not expected to be thermal noise
limited for all Fourier frequencies. In Fig. 4.22, we demonstrate that the long-term behavior of theclock laser is consistent with 1/f noise that produces a flicker-floor limited instability at 1 × 10−16 .
However, we will see in Chapter 6 that simple spectroscopic tests indicate the presence of white noiseof Gwhite(f ) = 3.3 × 10−3 Hz2 /Hz, in addition to the flicker floor of f × Gflicker(f ) = 1.3 × 10−3 Hz2ννthat corresponds to a σ (τ ) = 1×10−16 instability. We will show the presence of additional coherent
noise features, which do not affect the Dick floor significantly for the operating conditions considered
here, so we save discussion of these features for Chapter 6. The addition of the extra white noise
floor, which we rigorously verify in Chapter 6, is sufficient to completely describe the measured
instability in the clock comparison of Sr1 and Sr2.

4.6

PTB Si cavity and JILA cavity intercomparison
As discussed in the previous section, improvements in ultrastable lasers can have a huge im-pact on optical lattice clocks. This is embodied in the reduction of clock systematic measurement
times from weeks to mere hours and the attainment of near QPN-limited clock performance. However, the reader may have gotten the feeling that conventional technology is approaching its limits;
after all, the Big ULE is nearly 40 cm in length, so to gain another factor of 10 improvement, one
would need a 4 m cavity! There are other tricks one can play, including operating in near-concentricor near Fabry-Pérot (i.e., flat-flat mirror combination) geometries, but the fact remains: gaining afactor of 10 is extremely challenging. Clearly novel technologies are needed.
One such avenue opened in 2007 in collaboration between JILA (Ye Lab) and PTB. The
idea was to use single-crystal silicon as a cavity spacer and mirror substrate material. Silicon has

a

b

139

Figure 4.27: The silicon cavity. (a) Silicon cavity with inset showing the Si crystal lattice structure.
The green surfaces on the mirrors are from the anti-reflection coatings applied to the Si. (b) The
PTB-designed cryogenic vacuum system and support strategy. The cavity is held vertically andis supported by three symmetrically placed Teflon posts. An outer heat shield is cooled to 124 K
within the vacuum envelope, while a second heat shield increases the thermal time constant between
the temperature-controlled outer shield and the optical cavity to 10 days.

several advantages. Firstly, the loss angle is 1–2 orders of magnitude smaller than fused silica anddoesn’t suffer increase at lower temperatures as does fused silica. Secondly, the Young’s modulus isalso larger than that of glass by about a factor of two, which reduces thermal noise. Additionally,
the stiffness of the material, or ratio of Young’s modulus to density E/ρ is about a factor of
2.5 larger than ULE or fused silica, indicating that one should expect a comparable decrease invibration sensitivity. Finally, the coefficient of thermal expansion (CTE) has two zero-crossings, ofwhich the one at higher temperature is at 124 K. At the zero crossing, the cavity length should be
extremely stable in the long-term as the crystalline nature eliminates the material creep observedin amorphous glass.
The original FEA simulations for the cavity were performed at JILA and subsequently PTB,
with Lisheng Chen performing the finite element analysis on the JILA side. After several years,
the design work culminated in the construction of a vibrationally insensitive high-finesse opticalcavity comprised of a monocrystalline silicon spacer and optically contacted silicon mirrors with
SiO2/Ta2O5 coatings. The spacer geometry is shown in Fig. 4.27a. In order to exploit the crystal

140

a

b

Figure 4.28: Direct optical beat between the JILA cavity and the Si cavity. (a) The time-domain
signal of the optical beat mixed to near DC exhibits a ∼ 10 s coherence time. (b) The power
spectrum of the optical beat between the two systems displays a repeatable 50 mHz linewidth(different colors correspond to separate measurements).

symmetry, the cavity is oriented with the Si h111i crystal axis pointed vertically. The mirrors
are contacted with the crystal axes nominally aligned, so that the entire ensemble simulates asingle-crystal. The cavity is supported on Teflon legs, each along one of the three equivalent h110i
directions [168].
To reach the CTE zero-crossing, PTB colleagues designed a novel low-vibration cryostat tohouse the silicon cavity and cool it to 124 K in-vacuum (the apparatus is shown in Fig. 4.27b).
The cryostat uses nitrogen gas boiled off from a liquid nitrogen dewar to cool a heat shield inside
the system’s vacuum chamber, achieving sub-mK stability on an outer thermal shield within thevacuum envelope. The time constant between this envelope and the cavity is 10 days, indicatingthat fast-temperature fluctuations will be heavily suppressed [168].
In order to test the performance of the silicon cavity, we transported the “JILA cavity,”

141

T. L.
JILA
Silicon-15

mod σy (τ )

10

-16

10

0.1

1

τ (s)

10

100

Figure 4.29: Modified Allan deviation of the stabilities extrapolated via the three-cornered-hat
analysis. Here, the modified Allan variance is used to reduce sensitivity to higher frequency phasenoise and better reveal the thermal noise floor.

described in Section 4.4.2 of this thesis, to Braunschweig, the charming German town on whose
outskirts PTB is located. With a predicted thermal noise floor 2 × 10−16 , this cavity would beclose to the predicted noise floor of the Si cavity of 7 × 10−17 . The cavity was installed in its
vacuum chamber and, despite the presence of non vacuum-compatible Peltier elements and wires(we drew inspiration from Ref. [178]), we were able to pump the system to the level of 10−7 Torr
and were able to cool the cavity to ∼ 10◦ C with the in-vacuum Peltier elements. Any attempt
to reach colder temperatures induced the system (which was inside a sound-proof and rather wellthermally-insulated box) to go into thermal run-away.
An early setback for the JILA cavity was the discovery that the finesse of the mirrors was
only 80,000. In fact, a similar problem was observed with the Si cavity for the TEM00 mode,

142
but fortuitously the TEM01 mode had a finesse of 240,000. Unfortunately for the JILA cavity,
we had no such luck with higher order modes. This meant that any residual amplitude modulation would create a proportionally larger frequency offset, due to the small slope of the cavity
frequency discriminant. However, both cavities employed waveguide modulators with the active
RAM cancellation system whose development we described in Section 4.2.5. When both systems
were working successfully, which took two weeks of hunting down parasitic effects, we were able toobserve a 50 mHz beat between the two systems, and near 10 s optical coherence times, as shown in
Fig. 4.28, and which represents a record for a measured optical linewidth between two independent
lasers. This corresponds to a coherence length beyond 1 × 109 m, the length scale of the proposed
Laser Interferometer Space Antenna (LISA). This result additionally demonstrates the success of
the RAM servo; in order to achieve a linewidth of 50 mHz, the JILA cavity RAM cancellation servo
permitted lock precision sufficient to split the cavity resonance better than 1 part in 105 . In thefield of cavity stabilization, this level of lock precision is among the best reported (see, e.g., [215]).
We additionally note that the intercomparison between the JILA cavity verified the successof the finite element analysis. By shaking the Minus-K platform on which the JILA cavity sat andmeasuring both the frequency of the beat signal and magnitude of the accelerations, we found thatthe vibration sensitivity was < 5 × 10−10 /g.
In order to isolate the performance of the silicon cavity, we included a third cavity-stabilized
laser system in the measurement. This laser was designed and constructed by PTB colleague
Thomas Legero (thus we will call this laser “T. L. cavity” system) and was designed to be transportable. It is similar to the system described in [235]. Due to its smaller size and ULE mirrors,
the thermal noise floor of this cavity is 6 × 10−16 .
We performed three simultaneous beat measurements between all two-way permutations
amongst the Silicon cavity-, JILA cavity-, and T. L. cavity-stabilized laser systems. The beats
were performed at a common location connected by noise-cancelled fibers to all three systems.
We analyzed the beat record using the “three-cornered-hat” method [236]. Assuming uncorrelated
noise between three laser systems, the instability of one of the systems at a given time can be143calculated from the other two as
r 

1 2
2 − σ2 ,σi =σij + σikkj
2

(4.54)

where i, j, and k can represent any laser we listed above: silicon, JILA, or T. L. In order toremove the possibility of correlations amongst the lasers, we used the extended technique describedin Ref. [237]. The result of this analysis is shown in Fig. 4.29. The Si cavity-stabilized laser
has extrapolated performance at the level of 1 × 10−16 from 0.1 s to 10 s, showing the promiseof cryogenic cavities. As mentioned previously the expected thermal noise floor of the Siliconcavity in 7 × 10−17 , and which is entirely dominated by the coating (> 99.8%). The JILA cavity√
also performed at a near-thermal noise limited level from 0.1 s to 2 s. The τ rise evident in
the JILA cavity’s stability in Fig. 4.29 is indicative of random walk frequency noise. We believethat this originates from material stresses in the machined ULE support structure for the JILA
spacer. We have mitigated this problem in the Big ULE cavity (where it had also initially been
evident) by using large, soft Viton hemispheres to only weakly couple the baseplate and cavity.
Future implementations of the JILA cavity would also benefit from this support strategy. Anotherpossible reason for the rapid increase the Allan deviation of the JILA cavity is the fact that thetemperature was maintained about 10◦ C above the nominal zero-crossing temperature for thatspacer, leading to a possible first-order temperature sensitivity of approximately 1.8 × 10−8 /K.
Finally, we note that subsequent tests of the long-term stability of the Si cavity are very
promising, already demonstrating the best long-term stability of any passive optical etalon. This
verifies the expectation that a crystalline material should be free of the “creep” that seems to be
intrinsic to amorphous materials. For instance, the big ULE cavity is elongating at approximately
1 nm/year (or 5 × 10−17 m/s), as extrapolated by the predictable day-to-day frequency drift.6

In

contrast, the measured long term frequency stability of the silicon cavity indicates that its longterm drift is below 200 µHz/s [168], indicating that the mirrors are stable with respect to each otheron the scale of 1 fm over the course of the day. Paired with an atomic clock, this level of stability
6

It is remarkable to think that one can measure these extremely slow velocities with atomic clocks. This would
also seem to dispel the myth that the ripples in old windows are caused by amorphous creep!

144
could yield an extremely precise Kennedy-Thorndike test of relativity due to the unprecedented
combination of short- and long-term stability in an optical resonator [238].

4.7

Crystalline mirrors
The results of the previous section serve to demonstrate that a cryogenic system can perform

at the highest levels of stability, approaching the level of the best conventional cavities, such as the“Big ULE” cavity demonstrated in Section 4.5. However, the improvement in laser stability due tooperating at 124 K alone is less than a factor of two. The real gain from silicon, which has better
material properties, but requires operation at 124 K, is that the substrate thermal noise is almost
entirely removed, leaving only the coating contribution. Indeed, for the silicon cavity described in
the previous section, > 99.8% of the thermal noise is due to the coating itself. This implies thateven modest improvements in coating loss angle will result in significant advances in laser stability.
In this section, we report such an advance, where we extend the use of high-Q crystalline materials
to the high-reflectivity mirrors themselves.
The desire to engineer cavities with components possessing the lowest possible mechanical loss
shares connections with the field of quantum optomechanics, where coupling to a thermal bath via
mechanical losses can impede measurement precision and quantum coherence. [171, 144]. As mentioned in Chapter 3, Section 3.2.2, heterostructures based on GaAs and Alx Ga1−x As semiconductormaterials (the subscripts stand for doping concentrations) have demonstrated high mechanical Q’s
when used as isolated optomechanical structures, with Q-factors of up to 40,000, corresponding toa loss angle of 2.5 × 10−5 [167]. Thus, crystalline mirrors made from a multilayer GaAa/AlGaAs
would be a powerful new technology for high-precision laser stability.
Luckily the group of Markus Aspelmeyer along with colleague Garrett Cole were alreadythinking along those lines, and provided us with the opportunity to test the feasibility of their
coating technology—which works so well at the nanoscale—in macroscopic mirrors. They developed
a technique to remove a film of of multilayer GaAs/Al0.92 Ga0.08 As from its growth substrate andto directly bond it to a separate substrate of any other material via optical contacting. In this way,

145

a

b

c

Figure 4.30: Crystalline mirror and cavity. (a) GaAs/Al0.92 Ga0.08 As crystalline mirror bonded toa fused silica mirror substrate. (b) Completed cavity 3.5 cm in length employing two opticallycontacted GaAs/Al0.92 Ga0.08 As mirrors for operation at 1064 nm. The cavity spacer is Zerodur (c)Optical ringdown signal with a swept input field. The extrapolated finesse is 150,000.

the film can be grown as a crystal, and the crystalline properties transferred to a host substrate,
which could be crystalline silicon or sapphire, or even fused silica glass. Figure 4.30a shows acompleted mirror crystalline mirror on a fused silica substrate that was constructed using thisbonding method. With this technique established, all that remained was to test the optical andmechanical properties of such a composite mirror.
In order to test the optical and mechanical properties of the mirrors, a new postdoc, Wei
Zhang, and this author constructed the cavity as shown in Fig. 4.30b. Here, the cavity spacer material is Zerodur glass, which was chosen due to its convenience (i.e., we already had the spacer in thelab). The length is 3.5 cm. The operating wavelength was 1064 nm, which is important given thatwith the semiconductor material interband optical absorption can be an issue, although in principle
mirrors constructed with AlGaAs could work to 650 nm with different doping concentrations thanthose utilized for the 1064 nm mirrors [172]. The mirrors with the transferred crystalline coatings
are optically contacted to the spacer. This particular cavity has a long history of making thermal
noise measurements in the Ye lab, and was utilized in some of the first measurements to explore
the effect of cavity material properties on thermal noise [239]. The first measurements that we
performed with this crystalline-mirror cavity were to test the optical quality factor. The cavity

146

a

Noise measurement

frep
f1

... ...

n1

f2

n2

698 nm Clock laser
GaAs/AlGaAs cavity

f0

b

10-14

10-15
1

10

100

10-1

Power spectral density [Hz2/Hz]

Allan deviation

10-13

×2

c

10-2

10-3
1

Averaging time [s]

10

100

Frequency [Hz]Figure 4.31: Comb frequency-transfer and stability of a laser locked to the cavity employing
GaAs/AlGaAs crystalline mirrors. (a) The optical frequency comb is stabilized by locking it to
the clock laser at 698 nm and locking the carrier envelope offset frequency, f0 , to a RF reference.
(b) Allan deviation of the measured beat between the crystal-mirror cavity and the comb. (c)Frequency noise power spectral density of the beat at 1064 nm (grey line). The contribution ofBrownian noise is indicated by the dashed blue line. The solid red line shows the contribution of
all thermal noise, including substrate thermo-elastic and coating thermo-optic.

was placed inside a temperature controlled vacuum enclosure, and we performed optical ring-down
measurements. It was exciting to see measure a finesse of 150,000 (Fig. 4.30c). This, in fact, wasexactly consistent with measurements of the absorption, scatter, and transmission losses, whichwere measured to be 13.5 ppm, 4 ppm, and 4 pmm, respectively [172].

147
The second test we performed on the cavity was to measure the stability of a laser locked
to it. The laser source was a non-planar ring oscillator (NPRO) YAG, with excellent short-term
stability, corresponding to a free-runnning linewidth of ∼1 kHz. The NPRO was stabilized to the
crystalline cavity. Notably, this setup also employed a waveguide modulator and RAM servo for the
PDH lock, as described in Section 4.2.5. In order to measure the stability of the crystalline-mirror
cavity-stabilized laser, we compared it to the Big ULE clock laser described in Section 4.9 andwhose performance at the level of 1 × 10−16 was verified in Section 4.5.2. In order to span the
spectral gap, we employed the Yb:fiber frequency comb described in [240]. This setup is depicted
schematically in Fig. 4.31; the Yb comb is locked to the clock laser at 698 nm using fast actuation
to an intra-cavity EOM, a piezo-mounted semiconductor saturable absorber mirror (SESAM), andto a fiber-stretching drum piezo. The carrier envelope offset frequency, f0 , is also stabilized using
an AOM external to the laser cavity.
Figures 4.31b and 4.31c show the measured Allan deviation at 1064 nm, and the frequency
noise at 1064 nm, respectively. At 1 s, the measured Triangle deviation is σyΛ (1 s) = 1.05(1)×10−15 .
The reason we measure Triangle deviation is because we use an Agilent 53131A counter in “timearming” mode, which corresponds to a Λ-type sensitivity function, as defined by Dawkins et al.
[241]. By summing the thermal noise contributions of all the components as detailed in Chapter 3,
we find that given the measured σyΛ (1 s) the loss angle of the bonded AlGaAs mirrors is φAlGaAs =0(4) × 10−5 , where the error bar includes an assumed 10% error in the measured σyΛ (1 s), and 20%
errors in the loss angles of the relevant materials, namely Zerodur and fused silica.7
Figure 4.31c shows the measured frequency noise power spectral density. The contribution ofBrownian noise alone (dashed blue line) is compared to the total thermal noise, including thermooptic and thermo-elastic noise, and the measured data. Importantly, we begin to see the effect ofthermo-optic noise in this measurement, although it is not well resolved. This noise is at high enough
Fourier frequencies that it is not a significant problem for optical clocks, but might have important
We utilized the standard loss angle for fused silica of φF S = 10−6 . If instead we used φF S = 10−7 , theextrapolated loss angle of the crystalline coating is φAlGaAs = 4(4) × 10−5 .
7148
implications for LIGO, as the spectral region where thermo-optic noise dominates is LIGO’s relevant
detection band. As discussed briefly in Chapter 3, the addition of a half-wavelength cap of AlGaAs
will potentially drastically reduce the contribution of thermo-optic noise. Thermo-elastic noise will
be eliminated by eventually operating a cavity with Si mirrors at the CTE zero-crossing, such thatthe coefficient of thermal expansion of the mirror substrates is zero.
In conclusion, these measurements indicate that AlGaAs coatings are the future of highprecision laser stabilization and interferometry. AlGaAs heterostructures bonded to substrates canretain the low mechanical losses of free-standing structures. With a silicon cavity with AlGaAs
mirrors, one could expect a thermal-noise limited laser stability at the level of 1 × 10−17 using
the already-established cryogenic silicon system and AlGaAs crystalline coatings. Thus, insteadof the daunting task of improving the thermal noise limitations of conventional cavities, the allcrystalline-cavity approach should allow rapid attainment of the next factor of 10 improvement inlaser stability.

4.8

Frequency comb development: bridging spectral gaps coherently
As seen from the measurement of the crystalline-mirror cavity performance in the previous

section, optical frequency combs will play a central role in the direct operation of atomic clocks. To
date, frequency combs have been used primarily to connect clocks of disparate optical frequencies[31, 242] or for optical frequency measurements e.g., [1] (or even to reverse the process for ultrastablemicrowave generation from ultrastable optical resonators [243]). However, the recent developments
in creating ultrastable lasers based on cavities constructed from crystalline mirrors have restricted
the wavelengths over which ultrastable lasers can operate. A frequency comb [244] (or perhaps
second harmonic generation if the appropriate wavelength is obtained) is required to connect theultrastable laser, which may operate in the telecom spectral region, to the optical clock transition ofinterest. In this section, we summarize some key tests of comb stability that have been performed.

149

Intensity (dB)ν − 2ν

a0simulated spectrum
measured spectrum

-40

PWL

ZDWL

-80

600

1000

800

1200

1400

1800

1600

Wavelength (nm)Power (dB)

-20

-40

-20

0

20

Fourier frequency (MHz)0single detector
balanced detectors

csingle detector

d

-20

-20
-40

0

Power (dB)single detector
balanced detectorsbPower (dB)

0

-20

0

20

Fourier frequency (MHz)

-40

-20

0

20

Fourier frequency (MHz)Figure 4.32: Spectrum of the ultra-broadband Yb:fiber comb and optical beats. (a) Measured
optical spectrum of the comb supercontinuum (solid line) and simulated spectrum based upon the
nonlinear fiber input pulse parameters (dashed). (b) Optical beat with a CW laser at 698 nm. (c)Optical beat with a CW laser at 1020 nm. (d) Optical beat with a CW laser at 1.54 µm. Plots (b–d) indicate varying levels of amplitude noise. This is evident from the beat signal-to-noise without(solid) and with (dashed) balanced detection. No excess amplitude noise was visible at 1.54 µm
while at 698 nm the noise floor rose by 20 dB in the absence of balanced detection, indicating
significant amplitude noise at this wavelength.

4.8.1

Ultra-broadband Yb:fiber comb

To take advantage of the relative robustness, compactness, and a gain medium that does notneed to be pumped with lasers that cost O($100, 000), we extended a collaboration with IMRA
America Inc. to include medium power Yb:fiber laser systems. Already, a 10 W system from IMRA
had shown extremely low-noise operation [245]. With the new system, we did not need 10 W opticalpower; instead we set out to see if we could connect the telecom and the visible wavelength regions
in a single step. We briefly describe the system and results here.
The frequency comb comprised Yb-fiber similariton oscillator [246] and linear chirped pulse
amplifier. Starting with 80 fs 1.4 nJ pulses at a repetition rate of 150 MHz, we broadened thespectrum to 1.5 octaves (Fig. 4.32) in a single piece of suspended-core highly nonlinear fiber [247].
As seen in Fig. 4.32a, the spectrum was highly featured. We found that these features were well

150Power (arb. lin.)

698 nm

-20

-10

-20

20

0

Locked f0 beat

-10

-40

-20
-30

-60

0
10
Frequency (Hz)1542 nm

LockdB

Optical power (dB)0Beat at 1542 nm

1.5 Hz

600

700

800

-40

900

-20
0
20
Frequency (kHz)

1000

40

1100

1200

1300

1400

1500

1600

Wavelength (nm)Figure 4.33: Spectrum of broadened Yb oscillator and relevant beat notes at 698 nm and 1.54 µm.
The self-referenced (bottom inset) comb is locked to a 698 nm sub-Hz linewidth laser and the beatnote with a second ultra-stable laser at 1.54 µm is shown (top inset).

described by the theory of supercontinuum generation in optical fibers, including the phenomena ofsoliton fission (responsible for the two “bumps” in the spectrum at right) and self-phase modulation.
As shown in Figs. 4.32b–4.32c, balanced detection of optical heterodyne beats between the comb andCW laser sources indicated that certain spectral regions of the supercontinuum had large amounts ofamplitude noise, but no phase noise. We were able to explain this phenomenon through modeling of
the supercontinuum generation process, and found that quantum-seeded input noise is responsible
for the presence of amplitude noise. While we do not go into detail here, these measurements
also indicated that the higher fraction of Raman gain in the fiber played an important role inmaintaining phase coherence [56].
In order to test the broadband comb coherence, we performed the experiment depicted inFig. 4.33. With the carrier envelope frequency, f0 , stabilized, we locked the broadened comb to
the “football cavity”-stabilized clock laser at 698 nm [40] and analyzed the beat signal with a151second cavity-stabilized laser at 1542 nm. We found that the comb supports the coherent transfer
of optical phase from 698 nm to 1542 nm at or below the level of 1.5 Hz linewidth, consistent withthe expected performance of the 1.5 µm laser. This represents the largest spectral gap directly
spanned between two ultra-stable lasers by a frequency comb to date. While this may not be thebest way to connect lasers that are separated by over an octave of optical spectrum, it does serve
to define a boundary of what is possible.

4.8.2

Ultimate limits of comb performance

In the previous section, we described an experiment where the test of comb coherence waslimited by the ultrastable lasers used as references. Here, we describe an experiment performed
with an octave spanning Ti:Sapphire frequency comb, where the octave-spanning comb locked toa reference laser and compared the same laser’s second harmonic. Thus, the test was of the comb
properties alone, since the reference laser’s noise was common to its first and second harmonic.
Viewed alternatively, this measurement represents a precision test of second harmonic generation[248].
We begin by describing the noise sources that the experiment was designed to test. A complete
description of the frequency of a comb mode, including the possibility of mode-dependent noise, isgiven byνn (t) = nfrep (t) + f0 (t) + n (t).

(4.55)

Here n is the index labeling the harmonic of the repetition rate, frep (t); f0 (t) is the carrier envelope
offset frequency; and n (t) accounts for mode-dependent frequency noise that is not correlated by
the mode locking process. It is the terms n (t) that we set out to test.
Perfect mode-locking occurs when n (t) → 0 in Eq. 4.55. To test this residual comb error, weconsider the case where we phase lock the heterodyne beat between comb mode n and a continuous
wave (CW) laser, fb,n , and f0 to radio frequency (RF) references. The comb thus ideally acquires
the optical phase information of the reference laser with added locking errors, δfb,n (t) and δf0 (t),due to finite servo gain and shot noise. However it is possible to check for any residual noise

152
described by n (t) by comparing the second harmonic of the CW reference laser to the comb. This
yields an “out-of-loop” (OL) beat frequency
fb,2n (t) = 2fb,n − f0 − δf0 (t) + 2δfb,n (t) + 2n (t) − 2n (t).

(4.56)The term 2n (t) − 2n (t) accounts for relative noise introduced across the spectral gap n → 2n due
to imperfect mode-locking and is indistinguishable from technical noise. In principle, the locking
error due to finite servo gain given by δflock (t) = −δf0 (t) + 2δfb,n (t) can be estimated from thein-loop error signals, excluding consideration of shot noise.
The experiment was performed as schematically shown in Fig. 4.34a; we utilized the octave
spanning Ti:sapphire described in [249] and which is similar to the system described in [250]. Part ofthe spectrum is used to lock f0 , while the remaining optical spectrum was rebroadened to an opticaloctave centered near 750 nm using photonic crystal fiber (PCF) and combined with a Nd:YAG nonplanar ring oscillator (NPRO) at 1064 nm. The co-propagating comb and 1064 nm CW light passed
through a temperature stabilized periodically poled lithium niobate (PPLN) crystal, frequency
doubling a portion of the 1064 nm CW light to produce 0.5 mW of 532 nm CW light while
overlapped with the comb. The co-propagation of the light paths was important for reducing
differential-path noise. A λ/2 at 1064 nm, λ at 532 nm wave plate overlaps the 532 nm CW andcomb light polarization, while ensuring no 1064 nm comb light is doubled. The RF signal generated
by the beat between the comb and ∼ 1 mw of CW power at 1064 nm was used to stabilize the comb
mode at νn (n ' 3 × 106 ). The resulting beat between comb mode 2n and CW light at 532 nm was
the out-of-loop signal.
The frequency stability beat at 532 nm was analyzed. In Fig. 4.34b we show the frequency
stability, which represents the comb’s intrinsic ability to transfer optical stability. As seen fromthe figure, the comb performs better than any laser currently in operation, and should be sufficientfor next generation ultrastable lasers that operate with fractional instability of 1 × 10−17 . We
furthermore measured the optical linewidth of the beat. In order to amplify the phase noise, we
employed a step-recovery diode to create the 10th harmonic of the beat signal. The power spectrum

153

a

Self-referenced octave-spanning comb
lock1150 nm

575 nmX2

2nf rep + f 0

10

ν 1064PCF

locknf rep + f 0

f b ,1064

f b ,532

X2

ν 1064

Analyze

MISER

-17

b

-2

10

-3

c

244 uHz

10
2

-18

Vrms

Allan Deviation

10

MISER and2nd harmonic
Broadened comb

f0

10

2ν 1064

-4

10

-5

10

-6

10

-19

-7

10

10

100

1000

Averaging Time (s)

0.98

1.00

1.02

Fourier frequency (Hz)Figure 4.34: Experimental test of the intrinsic limits of the an optical frequency comb. (a) Experimental setup. An self-referenced octave-spanning frequency comb is re-broadened to an opticaloctave with a piece of photonic crystal fiber (PCF). It is phase locked to monolithic isolated singlemode end-pumped ring laser (MISER, also known as NPRO). The second-harmonic of the MISER
co-propagates with the comb light. The heterodyne beat between the miser second harmonic andthe comb, fb,532 , is the out-of-loop test signal. (b) Allan deviation of fb,532 . (c) Fourier power
spectrum of the 10th harmonic of fb,532 with a ∼ 250 µHz measurement bandwidth.

154
shown in Fig. 4.34c is the 10th harmonic of the beat measured with a fast Fourier spectrum analyzer
employing digitization record nearly 1 hour long, resulting in a 244 µHz measured linewidth. If weassume a white frequency noise power spectrum, then this indicates that the first harmonic could
be as narrow as 2.4 µHz. Thus, as far as we can tell, the comb is a perfect device to the extent
that technical sources of noise can be mitigated. These results are described detail in Ref. [132].

4.9

Outlook and conclusion
While this chapter has focused entirely on optical cavities as a means for precision laser

stabilization, we mention in passing that other techniques may contribute to ultrastable opticalsources for atomic clocks and precision spectroscopy. Lasers stabilized via spectral hole-burningin Eu3+ :Y2 SiO5 represent a promising new technology for which the fundamental limits should
be at the 10−17 level of stability [251]. Active and passive systems employing clock atoms as acontinuous source for either active lasing [252] or spectroscopy [253] should be free of thermal noise
contributions, as they rely solely on the atoms themselves for spectral discrimination.
As discussed in Chapter 2, the stability of current state-of-the-art neutral atom clocks is
directly tied to the thermal noise present in ultrastable cavity mirror coatings and substrates. In
this chapter, we demonstrated that with careful cavity design and high precision locking, we haveobtained near-QPN limited clock performance with a spectroscopy time of 160 ms. While this is asignificant milestone, achieving QPN-limited operation with an order of magnitude more atoms and
1 s probe times still leaves a large space for future improvements in ultrastable laser technology.
While reducing the experimental dead time alone is also a promising step [254], reducing the thermal
noise-induced laser frequency noise has had, and will continue to have, the largest immediate impact
on neutral atom clocks. The new crystalline mirror technology we have demonstrated leaves us very
confident that another factor of 10 improvement is now within reach.
In the context of systematic evaluation, the current stability of the Big ULE cavity-stabilized
laser would allow a full clock evaluation at the level of 10−17 in one day, assuming that statistical
averaging were the sole limiting factor. As we will see in the next chapter, the improvement

155
brought by the Big ULE cavity has already allowed rapid exploration of density-dependent effects
in the 87 Sr lattice clock. Thus, as we will show, in addition to making better clocks, ultrastablelasers become precise tools for probing inter-atomic interactions at the sub-Hz energy scale andperforming quantum state manipulation between optically separated quantum degrees of freedom,or optical qubits.

Chapter 5

The optical lattice clock as a quantum many-body system

It has long been expected that the use of identical fermions in ultracold neutral atom clocks
would eliminate density-dependent frequency shifts entirely [255] . In 2008, it was found that evenin a 87 Sr lattice clock at µK temperatures, a significant frequency shift remained, creating, atthat time, a mystery as to the collision mechanism. In this chapter, we explore the fundamentally
many-body collisional interactions between identical fermionic atoms in the optical lattice clock.
Through precise spectroscopic tests, we elucidate the quantum many-body aspects of the clock anduncover the dominant role of the p-wave interaction channel for many-particle interactions in the
optical lattice. We present the first observations of the emergence of spin correlations between theclock’s optically-separated spin degrees of freedom that arise from the effective spin-spin interactions
between 87 Sr atoms in the optical lattice.

5.1

Atomic collisions at ultracold temperatures
At ultracold temperatures, matter behaves quantum mechanically. It is the goal of thissection to briefly review the basic implications of this statement. Here, we summarize the results ofthe partial wave theory of quantum collisions, which allows collisions to be characterized according
to their angular momentum and vastly simplifies theie treatment. We also discuss treatment ofinter-atomic interactions using pseudopotentials.

157
5.1.1

Partial wave treatment of atomic interactions

Collisions at ultralow temperatures are simplified due to their quantum mechanical nature.
One can represent the effect of an inter-atomic scattering event on the asymptotic wave functionof a scattered particle as [88]
ψ (r → ∞) ∝ e(ik·r) + f (k, θ, φ)

eikr
.
r

(5.1)

The asymptotic wave function is the sum of two components. The first component is the incident
plane wave, while the second term represents the outgoing wave with an angular dependence thatreflects the details of the interaction potential. The differential cross section for identical particles
is related to this angular dependence as [88]dσ
= |f (k, θ, φ) ± f (k, π − θ, φ + π) |2 .
dΩ

(5.2)

Here the “+” and “-” correspond to the case of identical bosons or fermions, respectively, and they
are a direct consequence of the necessary (anti)symmetrization of the outgoing wave function forbosons (fermions). This distinction between the bosons and fermions will be crucial in determining
by which partial waves bosons and fermions can interact.
Assuming an isotropic interaction potential (i.e., the potential has no angular dependence),
we can write a general azimuthally symmetric wave function in the limit of r → ∞ψ (r → ∞) =X

Cl (k) Pl (cos θ)lsin [kr − lπ/2 + δl (k)]
,
kr

(5.3)

for which the angular dependence of the outgoing scattered wave can be expressed as [256]f (k, θ) =1 X
(2l + 1) Tl (k) Pl (cos θ) .
2ik

(5.4)

l

The quantity Tl (k) relates the asymptotic phase shifts of the scattered partial waves, δl (k), to thecross section of Eq. 5.2 and is related directly to the asymptotic phase shift of the scattered wavebyTl (k) = 1 − exp (2iδl ) .

(5.5)

158
An important consequence of the form of Eq. 5.4 is that we can write Eq. 5.2 as a sum ofcross sections for a specific partial wave, and this property follows directly from the orthogonality
of Pl ’s. Additionally, due to the parity of the Pl ’s, the cross section is identically zero for odd(even) partial waves in the case of identical interacting bosons (fermions). We can express thesestatements asboson
σtotal=Xσl ,fermionσtotal=l even

X

σl .

(5.6)

l odd

More emphatically, we can now state that identical bosons (fermions) only interact via even(odd) partial waves. This is a crucial difference between bosons and fermions in ultracold
quantum gas experiments.
At ultralow temperatures, and for the case of a van der Waals-type interaction, the collisional
phase shift is proportional to k 3 for partial waves with l ≥ 1 [256]. In the specific case of s and p
wave interactions (l = 0 and l = 1, respectively), the phase shift varies as [256]δl ∝ k 2l+1 ,for l ∈ {0, 1}.

(5.7)

For collision energies well below the centrifugal barrier, the true interaction potential can be replacedby an approximation of the atomic interaction that depends only on the relevant scattering length[257, 65]. For s- and p-wave interactions, which are the most significant contributors to ultracold
collisional interactions, the scattering lengths are defined as− tan [δ0 (k)],k→0ka = lim− tan [δ1 (k)]
,
k→0
k3

b3 = lim(5.8)

where a is the s-wave scattering length and b3 is the p-wave scattering volume.
Using definitions of Eq. 5.8, one can construct artificial potentials that encapsulate the lowenergy limit of the scattering [258, 259, 260]. This technique is extremely powerful because it allows
a parameterization of the low-energy interaction via a single number, and reduces the extended
inter-particle interaction potential to a more tractable point-like potential. This sort of treatment
is valid in the limit where the collisional energy is well below the centrifugal barrier and there are

159
no accidental resonances with bound states [65]. For s-wave interactions, the potential is given by
Vsαβ (r) =4πaαβ ~2
δ (r) .
m

(5.9)

Here, r = r1 − r2 , where r1(2) is the position of atom 1(2). This is the potential employed in
the prolific Gross-Pitaevski equation for the description of single-mode Bose-Einstein condensates[261]. The subscript αβ indicates that the interaction can be dependent upon the internal statesof the colliding particles.
Similarly, the p-wave interaction potential can be expressed as [260]Vpαβ (r) =2π~b3αβ ←− ∂3→−∇ r 3 r3 δ (r) ∇ r .
m∂r(5.10)

Here r ≡ |r1 −r2 | and the “←” or “→” symbols over the gradient operator indicate that it acts only
to the left or right, respectively; the subscript on the gradient operator indicates that it acts on thecenter of mass coordinate, r = r2 − r1 . Higher order expressions for partial waves in a Cartesianbasis become cumbersome due to the fact that the angular dependence is more naturally expressedin spherical harmonics. Luckily, our consideration of partial-wave interactions in the 87 Sr clock willstop with p-wave—but surprisingly will still include s-waves, even for identical fermions.

5.1.2

Considerations for clock shifts

A direct consequence of Eq. 5.8 is that in the ultracold regime, collisional phase shifts based
upon s-waves can dominate over p-wave phase shifts as the collisional momentum goes to zero (i.e.
k → 0). Specifically, the ratio of the phase shifts is
lim |δ1 (k) /δ0 (k) | = (kb)2 b/a.

k→0

(5.11)

This implies that the ratio of p-wave collisions to s wave collsions becomes small assuming kb isalso small. A similar statement can be reached by considering the action of the pseudopotentials of
Eqs. 5.9 and 5.10 in Fourier space; in the case of the p-wave interaction, the two gradiant operators
will result in a k 2 scaling.

160
This relative scaling of both the scattering lengths and interaction strengths is a well-known
phenomenon which, for example, limits the efficacy of evaporative cooling for gases of ultracold
identical fermions [262]. For thermal gases, k 2 ∝ kB T via the equipartition theorem. Thus for µK
temperatures|δ1 (k) /δ0 (k) | ∝ T.

(5.12)

This property of p wave energy shifts led to the conclusion that fermionic species of atoms make more
ideal frequency references than their bosonic counterparts due to their insensitivity to collisional
shifts at ultralow temperatures [255].

5.2

First observations of cold-collisions in an optical lattice clock
Five years ago, as part of the effort to develop the world’s first neutral-atom clock withfractional uncertainty at 1 × 10−16 , we engaged in a measurement campaign in collaboration with
the NIST Time and Frequency division, where the short-term stability of a Ca neutral atom clock[197] was used to evaluate frequency shifts in the JILA Sr lattice clock [31, 114]. In this work,
we used a two-state spin mixture, comprised of the |F = 9/2, mF = ±9/2i spin states. A
modest fractional frequency shift of −8.9(8) × 10−16 (-480 mHz frequency shift) was measured at
an operating density of ρ0 ∼ 1011 cm−3 . In the case of a density shift measurement in the presenceof a “spectator” atom, we can write the shift due to s-wave interactions [263, 264, 265]∆ν =2~(ãeg − agg ) ρ.
m

(5.13)

Here, ãeg represents the effective scattering length for collisions in an incoherent mixture of e andg atoms in different nuclear spin states. In the case considered here, there is an equal population
in |F = 9/2, mF = +9/2i and |F = 9/2, mF = −9/2i, such that, ρ = ρ0 /2. Using the measured
frequency shift, we find that ãeg − agg ' −99a0 , where a0 = 5.29 × 10−11 m. Combining this resultwith the known 87 Sr g-g internuclear spin-state scattering length of 97a0 [266], we can deduce thata rough estimate for ãeg ' −2a0 . Here, the error in this determination of the scattering length is

161

3 µK
1 µK

Shift [Hz/ρ0]

1.5
1.0
0.5
0.0

0.2

0.4
0.6
Ground State Fraction

0.8

Figure 5.1: First observation of a density shift with a spin-polarized 87 Sr lattice clock. Hereρ0 ∼ 1011 /cm3 .

quite large due to the uncertainty in the absolute density calibration, and also because it neglects
other relevant interactions on which we will further elaborate.
Although the uncertainty due to operating with a two-state mixture was small, the seemingly
favorable ratio of s-wave to p wave interaction strengths in an ultracold regime implied by Eq. 5.11
indicated that spin-polarized 87 Sr should eliminate s-wave collisions, which should dominate overthe p-wave interactions, thus eliminating the shift entirely [198]. However, our first experiments
to measure the density-dependent frequency shift with spin-polarized 87 Sr found that the shift did
not disappear as expected [31, 265].
The measurement of the density shift for the identical particles was performed by opticallypumping to the |F = 9/2, MF = 9/2i spin state. A fast clock pulse with a large Fourier width was
employed for driving the g → e transition for the atoms in the |F = 9/2, MF = 9/2i spin state
only. Subsequently, all remaining ground state atoms were removed with a clean-up pulse resonantwith the 1 S0 → 1 P1 transition. This ensures that the atomic polarization into the stretched spin

162
state is highly pure. The clock was then operated with an 80-ms π-pulse from e → g. Despite
operating in this highly spin-purified configuration, a shift was still observed. As can be seen in
Fig. 5.3, the magnitude was similar to the spin-mixture case and has a nontrivial dependence on
the final excitation fraction p (as defined in Chapter 2), which was varied by changing the widthof the lock points on the de-excitation lineshape.
The measurement of these frequency shifts generated a fair amount of puzzlement. If swave interactions were forbidden and p-waves should have been frozen out, how can there be
frequency shifts in the clock? The consideration of this question led to the discovery of an interestingmechanism that permits s-wave interactions to occur in spin polarized ultracold 87 Sr , although it
will end up being only part of the story.

5.2.1

An s-wave collisional mechanism in a spin-polarized 87 Sr lattice clock

The basic idea behind the presence of s-wave shifts relies on the fact that the clock excitation isslightly inhomogeneous, as discussed in Chapter 2. This means that two fermions initially preparedin the same quantum state have the potential to evolve differentially as they are (de)excited fromg → e (e → g). To illustrate this, we consider two hypothetical colliding atoms in quantum states|ψ1 i and |ψ2 i given by|ψ1 i = α|gi + β|ei

(5.14)|ψ2 i = γ|gi + δ|ei

(5.15)and such that the factors α, β, γ, and δ are time-dependent. The total wavefunction is initializedwith both atoms in the ground state, which we denote as |ggi. However, in order for the total
wavefunction to be anti-symmetric under exchange, as is required for fermions, we need to include
the lattice motional states labeled by |n1 i and |n2 i. When the clock laser excites the atoms, |ψi

163√
dynamically evolves from |ψ0 i = |ggi ⊗ (|n1 n2 i − |n2 n1 i) / 2 as√
|ψ0 i → (|ψ1 ψ2 i|n1 n2 i − |ψ2 ψ1 i|n2 n1 i) / 2(5.16)√√= |ψ − i ⊗ (|n1 n2 i + |n2 n1 i) / 2 + |ψ + i ⊗ (|n1 n2 i − |n2 n1 i) / 2
|
{z
}
|
{z
}
even

(5.17)

odd

Only the part with the even dependence on the spatial functions n1 and n2 can collide via s-wave
interactions. Since the wavefunction for fermions must be antisymmetrized, the electronic part of
the wave function also carries a symmetry under particle exchange labeled by the “+” or “-”. It is
straightforward to show thathψ − |ψ − i = |αδ − γβ|2 = 1 − |αγ ∗ + βδ ∗ |2 ,(5.18)

which is in general nonzero only when α 6= γ and β 6= δ. Furthermore, we can identify this quantity
as the zero-range second order correlation function often encountered in the mean-field theory ofquantum gas interactions, G(2) (0). In a final step, we can then apply this G(2) (0) to the mean fielddensity shift formula and get a time-dependent frequency detuning [265, 126],∆ν (t) =2~a−eg (2)
G (t) (ρg − ρe ) .
m(5.19)

Here, the minus sign in a−eg indicates that this collision is occurring between the antisymmetricelectronic total wavefunction [73] and ρg(e) is the atomic density of atoms in the ground (excited)state.
While Eq. 5.19 gets a lot right, there are aspects of the interaction that it misses entirely. The
main problem with the mean-field energy of Eq. 5.19 is that it is, for lack of a better expression, themean of the interaction energy. Specifically, it is only the part of the wavefunction with |ψ − i thatfeels the s-wave interaction, yet we proceeded as if |ψ + i also feels the mean energy shift. While thisis a good approximation if |ψ − i and |ψ + i are not well-resolved (i.e., the mean field shift is muchsmaller than the Rabi frequency), this condition is not necessarily satisfied in the clock. If we reallywant to treat the problem right, we must keep track of the evolution of the entire wave function; inthis better approach |ψ − i would feel the full interaction, while |ψ + i would feel no interaction—and164
we will not take the mean. Also, in this example, we used a two-particle wave function, so thismeans keeping track of only a few states. The problem with generalizing this approach to theensemble is the exponential growth of complexity as we wish to treat the many-atom system. Thisis the subject of the next section.

5.3

Treatment of the clock shift in a many-body framework
Strongly correlated quantum many-body systems have become a major focus of modern

science. Researchers are using quantum-degenerate atomic gases [64, 65, 66, 67, 68, 69], ultracold
polar molecules [70, 71, 72], and ensembles of trapped ions [267, 268] to realize novel quantum phasesof matter and simulate complex condensed matter systems. In particular, the SU(N ) symmetry in
the nuclear spin degrees of freedom of fermionic alkaline earth atoms may allow implementation of
unconventional frustrated quantum magnetic models including spin-orbital interactions [73, 74, 75,76]. Degenerate gases of alkaline earth(-like) atoms have been also achieved [269, 270, 271, 272],
but have not yet reached the levels of entropy required to observe these novel magnetic phases iscurrently not feasible in ultracold atom experiments.
In this section, it is our goal to unite these novel quantum systems with the collisional physics
within the 87 Sr lattice clock. Atomic interactions in optical lattice clocks were first studied in thecontext of density-dependent frequency shifts that contribute to the clocks’ systematic uncertainty[265, 273, 135]. In the previous section we presented a potential mechanism by which s-wave shifts
can enter into the clock, and that even at this basic level the mean-field shift treatment might
have shortcomings. Here we explore a powerful formalism to address clock shifts arising frompartial wave interactions in the clock. Following the experimental discovery of the 87 Sr clock shift
for identical fermions, the first work to pioneer this many-particle approach to treat s-wave shifts
was from A. M. Rey and co-workers [274]. The two-body physics of the interacting particles was
elucidated by K. Gibble [275]. A description similar to that of [274] was subsequently explored byZ. Yu and C. Pethick [276].

165
5.3.1

Second quantization

In order to develop the formalism for a many-body system of interacting fermionic atoms, we
appeal to the substantial machinery already developed to treat interacting electrons in condensed
matter systems, although the techniques we employ can be used for interacting bosons as well. We
start by replacing the wavefunction with an operator, ψs (x) → ψ̂s (x), such that [277]
hx0 s0 |ψ̂s† (x) |0i = δss0 δ x − x0 .

(5.20)

Here, ψ̂s† (x) creates an atom at position x from the vacuum ket0 and the label s signifies its internal
quantum state. In our case, these “spin” states will actually signify e (↑) or g (↓), where we haverestricted our consideration to a single nuclear spin state, leaving only the electronic and motional
degrees of freedom.
We can decompose ψ̂s† (x) and its Hermitian conjugate ψ̂ (x) in any orthonormal basis set asψ̂s† (x) =X

φ∗i (x) ĉ†i,s .

(5.21)

i

Its conjugate, ψ̂s (x) can be written by conjugating Eq. 5.21. Here, the spatial functions φi (x)are the basis functions and ĉ†i,s and ĉi,s are the creation and annihilation operators for mode i,
respectively, with the spin denoted by s. Correspondingly, the number operator is given byn̂i,s = ĉ†i,s ĉi,s .

(5.22)

An important property of ψ̂s (x) is that it obeys (anti)commutation relations that depend on bosonic(fermionic) nature of the particles described by the field. Since we wish to describe fermionic atoms
in the clock, we therefore have{ψ̂s† (x) , ψ̂s†0 x0 } = δ x − x0 δss0 .

(5.23)

Here, {. . .} represents the anticommutation operation, defined as {α̂, β̂} ≡ α̂β̂ + β̂ α̂. It is important
to note that these relations are responsible for enforcing the necessary symmetries of the wavefunction and, as a direct consequence, they will play a direct role in determining by which partial waves
the atoms are permitted to interact.

166
5.3.2

Many-body Hamiltonian

We are now prepared to write the two-atom interaction potential. The standard expression
for decomposing an interaction potential into the second-quantized basis is [277, 278]
V̂ =

1
2

Z

d3 x d3 yXψ̂s† (x) ψ̂s†0 (y) vss0 (x, y) ψ̂s0 (y) ψ̂s (x) ,(5.24)

ss0

where V̂ is the inter-atomic interaction component of the total Hamiltonian and vss0 (x, y) is thetwo-body spin-dependent inter-particle interaction. We can therefore write the Hamiltonian of theentire interacting system as
Ĥ = Ĥ0 + V̂,(5.25)

where Ĥ0 is the single-particle Hamiltonian given by
Ĥ0 =

XZ
|s

d x ψ̂s† (x)
3




Z
i
h
~2 2
~ωa
−
∇ + V (x) ψ̂s (x) +d3 x ψ̂↑† (x) ψ̂↑ (x) − ψ̂↓† (x) ψ̂↓ (x) ,
2m
2
{z
}
{z
} |

motional

electronic(5.26)
where ωa /(2π) is the clock frequency. At this point it is beneficial to select a basis for ψ̂ (x) that
diagonalizes Ĥ0 (see Eq. 5.21). For the case of atoms confined in a harmonic potential
111
V (x) = mωx2 x2 + mωy2 y 2 + mωz2 z 2 .
2
2
2(5.27)The eigenstates are therefore eigenstates of the three-dimensional harmonic oscillator parameterizedby ni ≡ {nx,i , ny,i , nz,i }, such thatφi (x) = φni,x (x) φni,y (y) φni,z (z) ,(5.28)

where φnξ,i (ξ), with ξ ∈ {x, y, z}, is an eigenfunction of the one-dimensional harmonic oscillator.
With this choice of basis, the “motional” part of Ĥ0 in Eq. 5.26 is a constant of motion, and wewill therefore neglect it. This is justified because the energy scale of Ĥ0 is comparable to the trap
frequencies, which are much larger than the interaction energy scale. The remaining part of Ĥ0 is
the electronic energy between ↑ and ↓, with the clock transition frequency of ωa /(2π) = 429 THz.

167
Thus, Ĥ0 becomes
Ĥ0 →

~ωa X
(n̂↑,i − n̂↓,i ) .
2(5.29)i

We now turn our attention to the inter-atomic interaction characterized by V̂. For an arbitrary
interaction potential vss0 (x, y), we can write0ss † †
Uijkl
ĉi,s ĉj,s0 ĉk,s0 ĉl,s .

(5.30)

d3 x d3 y φ∗i (x) φ∗j (y) vss0 (x, y) φk (y) φl (x)(5.31)

V̂ =

XX
ss0

ijkl

Here [277],
0

ss
Uijkl
=

1
2

Z

It is clear from the form of Eq. 5.30 that we won’t get very far without some simplifications. Luckily,
the trap frequencies ωx , ωy , and ωz (2π × 450 Hz, 2π × 450 Hz, and 2π × 80 kHz, respectively) arelarger than any other scale in the problem, so we will first consider processes only which conserve
total motional quanta. We also assume that ωz is so large that it is frozen out (i.e., nz = 0),
so we only consider modes labeled as n = {nx , ny }, where nx and ny are the harmonic oscillator
quantum numbers in x and y. Conservation of total motional quanta still leaves a large numberof possible combinations of the creation and annihilation operators. As an example we consider
two colliding particles with initial motional states n1 = {n1,x , n1,y } and n2 = {n2,x , n2,y }. In a
perfectly harmonic trap, a range of final motional states are permitted, with motional quantum
numbers given by n0 1 = {n1,x + l, n1,y + m} and n0 2 = {n2,x − l, n2,y − m}. In the case thatthe potential is perfectly isotropic (i.e., ωx = ωy ), the final states n0 1 = {n1,x + l, n1,y + m} andn0 2 = {n2,x − m, n2,y − l} are also permitted. As a first step, we assume a small but finite difference
between ωx and ωy so as to preclude the latter process. However, even with ωx 6= ωy , there arestill a great many final states allowed for a given initial state due to the range over which l and m
can vary. Given that the thermally averaged values for nx and ny ∼ 100, this means that l and m
could span > 100 final states each, meaning that for the average collision between two particles,
there are > 10, 000 final states into which to scatter!
Luckily, there is a solution to this problem, and it arises from the two sources. The first isthat the values of the mode-dependent coupling constants given by the overlap integral of Eq. 5.31

Trap depth168∼ 20 µK

2 µK

Radial position

Figure 5.2: Harmonic approximation compared to the true radial profile of the trap.

tend to be much larger for the cases where motional quantum numbers are preserved or swapped[274]. The second is that for the pseudo-harmonic trap formed by the Gaussian laser profile, we
can solve for the perturbed energies for radial eigenmodes labeled by nx and ny including quartic
distortion as
E (nx , ny ) = ~ωx nx + ~ωy ny +



~2 
3 n2x + n2y + 4nx ny + 5 (nx + ny )24mw0

(5.32)

Here, the first two terms are the usual harmonic oscillator eigen-energy for modes withquantum numbers nx and ny . The second term comes from the quartic distortion due to thegaussian beam profile, with 1/e2 intensity radius w0 . Figure 5.2 shows the range of validity of theharmonic trap approximation in the radial direction. For 2 µK temperatures, the approximation
is fairly good, but the quartic corrections are still significant. With one exception (the case wheren0 1 = n2 and n0 2 = n1 ), the quartic distortion of the trap breaks the symmetry and enforcesl = m = 0. Thus, we are left with only two possible energy-conserving final states:(1) n0 1 = {n1,x , n1,y } and n0 2 = {n2,x , n2,y }: The trivial case. No modes are changed.
(2) n0 1 = {n2,x , n2,y } and n0 2 = {n1,x , n1,y }: Exchange of both motional eigenstates.
Thus, in general, for atoms i and j with initial motional states ni and nj we have two processes:{ni , nj } → {ni , nj } (direct) and {ni , nj } → {nj , ni } (exchange). This allows us to vastly simplify

169

Vee
Veg
Vgg

φm (x).
..
φ.. n (x).
φ2 (x)
φ1 (x)

Figure 5.3: Interaction motional eigenstates within a single trap site. The functions φ (x) areharmonic oscillator motional eigenstates. The interaction strengths, depicted schematically as Vgg ,Veg , and Vee are mode- and spin-dependent.

Eq. 5.30 as
V̂ →

XX
ss0

0

ss † †
ĉi,s ĉj,s0 ĉj,s0 ĉi,s +
Uijji

XX
ss0

ij

|

{z

direct}|0ss † †
Uijij
ĉi,s ĉj,s0 ĉi,s0 ĉj,s .

(5.33)

ij{zexchange}

Equation 5.33 represents a key result for the treatment of the many body shift. We will find that0

ss will further allow simplification of the problem. Additionally,
permutation symmetries of Uijji

connections with SU (2) angular momentum algebra will allow significant conceptual simplificationof the problem, and allow us to make connections with the quantum theory of angular momentum.
This is the subject of the next section.

5.3.3

Writing the many-body Hamiltonian with spin operators

A particularly straightforward way to form combinations of the creation and annihilation
operators for spin e(↑) and g(↓) particles that obey the SU(2) spin algebra commutation relations

170
are [277]
Ŝ =

1X †
ĉ ~σss0 ĉs0
2 0 s

(5.34)ss
Here, Ŝ ≡ Ŝ x , Ŝ y , Ŝ z and ~σ = (σ x , σ y , σ z ), where σ x , σ y , and σ x are the Pauli matrices. A generalproperty of operators constructed as in Eq. 5.34 is that they obey the commutations relationhi 1X 
ĉ†s σ i , σ j ss0 ĉs0 .
Ŝ i , Ŝ j =2 0(5.35)

ss

In the specifice case of S, we havehiŜ i , Ŝ j = iijk S k .

(5.36)As seen from Eq. 5.36, the operators Ŝ x , Ŝ y , and Ŝ z map directly onto the spin operators forspin-1/2 systems. If we write them out fully, they areŜ x =Ŝ y =
Ŝ z =

NX(ĉ†↑ ĉ↓ + ĉ†↓ ĉ↑ )/2,

n=1
N
X

(5.37)(ĉ†↑ ĉ↓ − ĉ†↓ ĉ↑ )/2i,(5.38)(ĉ†↑ ĉ↑ − ĉ†↓ ĉ↓ )/2,

(5.39)

n=1
N
X

n=1

We will now attempt to use these operators to simplify Eq. 5.33. We begin by writing Eq. 5.33
in a slightly different wayV̂ =XX
ss0 i6=j


 XX




ss0
ss0
Uijji
ĉ†i,s ĉi,s ĉ†j,s0 ĉj,s0 −Uijijĉ†i,s ĉi,s0 ĉ†j,s0 ĉj,s .

(5.40)

ss0 i6=jHere, we’ve made use of the anticommutation relations for the fermionic operators to rearrange theoperators into pairs enclosed by parentheses. Finally, we will relate the interaction energy terms
00ss and U ss to one another. Specifically, for the s- and p-wave interactions considered in Eqs. 5.9
Uijji
ijij
0

ss will have opposite parity under exchange of a given pair of ij → ji.
and 5.10, respectively, Uijji

Thus, we can write
1
1 ss0
ss0
Uijji
= Ueg
(1 − δss0 ) + Vij
,
2 ij
2
1
1 ss0
ss0
Uijij
= Ueg
ij (1 − δss0 ) − Vij .
2
2(5.41)(5.42)

171
Equations 5.41 and 5.42 define U and V. The interaction term U (V) are even (odd) under motional
state exchange (i.e. i → j and j → i). As a direct consequence, the corresponding spin states musthave the opposite parity under exchange of s and s0 . Therefore, we drop the superscript on U
and simply call it U−
ij because it only applies to colliding atoms in an antisymmetric electronic
superposition, of which there is only one possibility for two colliding atoms. Here, the (1 − δss0 )term serves to emphasize that the even-parity terms for which s = s0 cancel in Eq. 5.40. This issimilar to the simple s-wave case considered before, where s-wave interactions between the identicalfermions were only able to occur in the eg channel.
Inserting the potentials Eqs. 5.9 and 5.10 into Eq. 5.31, we find
2 Z
4πa−
eg ~
eg
Uij =

d3 x |φi (x)|2 |φj (x)|2mZ
212πb3ss0 ~2
ss0~ j (x) − φj (x) ∇φ~ i (x)d3 x φi (x) ∇φ
Vij=m(5.43)(5.44)A discussion of the mode-dependence of these is given in Appendix D.
We can rearrange Eq. 5.40 by making use of the spin operators of Eqs. 5.37–5.39, to arriveat the result
V̂ =

X
i6=j

Jij⊥ Ŝi · Ŝj + χij Ŝiz Ŝjz + K
Cij  zij
Ŝi + Ŝjz +,2
4(5.45)

whereeg
Jij⊥ = Vij
− Ueg
ij ,

(5.46)ggegee
χij = Vij
+ Vij
− 2Vij,(5.47)gg
ee
Cij = Vij
− Vij,

(5.48)ggeg
ee
Kij = Vij
+ Vij
+ Vij
+ Ueg
ij .

(5.49)

Interestingly, this Hamiltonian links the spin dynamics of interacting thermal fermions at µK
temperatures to those of two-mode Bose-Einstein condensates (BEC) and it has been shown boththeoretically [279, 280, 281, 282] and experimentally [283, 284, 285] to give rise to non-trivial manybody correlations and quantum noise-squeezed states. It is also relevant in trapped-ion quantum
simulation experiments [286, 267, 268].

172
5.3.4

Adding the atom-laser interaction

At this point, we have a general Hamiltonian that describes two-level fermions that are
subject to both s- and p-waves shifts. Writing out the full Hamiltonian, we have so far developed
yieldsĤ = ~ωaX
iŜiz +X
i6=j

Jij⊥ Ŝi · Ŝj + χij Ŝiz Ŝjz + K
Cij  zij
Ŝi + Ŝjz +.
2
4(5.50)

The first term is from the electronic energy separation between g and e given in Eq. 5.29. Here,
we have simply expressed it in our new spin language.
In order to account for the laser excitation, we add the atom-laser interaction to this Hamiltonian. We treat the laser field classically, and the atom-laser Hamiltonian is given by [274]Ω0ĤI = −2Z

d3 xψ̂↑† (x) ei(k·x−ωt) ψ̂↓ (x) + H.C.

(5.51)

Here, k is the laser propagation vector, ω is the laser angular frequency and “H.C.” indicatesthe hermitian conjugate of the previous expression. We note that the minus sign in front simply
represents a specific phase convention for the probe laser electric field, and we keep it in order tobe consistent with the literature. The mode-dependent Rabi frequency Ωi is given byZΩi = Ω0d3 x φ∗i (x) eik·x φi (x) .

(5.52)

By moving to a frame co-rotating with the laser, we finally obtain [125]Ĥ = −δXiŜiz +XiΩi Ŝix +X
i6=jJij⊥ Ŝi · Ŝj + χij Ŝiz Ŝjz + K
Cij  zij.
Ŝi + Ŝjz +2
4(5.53)

Here, δ ≡ ω − ωa . Equation 5.53 is the key result of this section. It expresses the relevant
elastic many-body dynamics in a spin language that makes broad connections to quantum magnetic
systems.

5.4

S-wave model
In the previous section, we derived a powerful machinery to treat collisional shifts in an

optical lattice clock. It is the goal of this section to develop an intuitive understanding of the173
Triplet manifold

|ee
√

Ω/ 2

s-wave interacting
singlet√−∆Ω/ 2(|ge√(|ge + |eg ) / 2√|eg ) / 2

√
∆Ω/ 2

∝ U−

√
Ω/ 2
|gg

Figure 5.4: Two-particle illustration of the s-wave shift mechanism.

s-wave shift mechanism only. Eventually, we will present evidence that shows that, despite thebeautiful mechanism that allows s-wave shifts in a fermion clock, s-waves are not a dominantcollisional channel in the one-dimensional optical lattice clock if the excitation is performed in avery homogeneous manner. This has coincided with significant experimental progress in the last
several years, resulting in the extremely small inhomogeneities reported in Chapter 2, and allowing
peak atomic excitation fractions of nearly 100%. However, we will find that the s-wave interactions
will still play an important auxiliary role in the many-body dynamics.
To begin, we consider Eq. 5.45 with only s-wave interactions so that Vijeg = Vijgg = Vijee = 0.
This was the original context in which many-body shifts were considered in the lattice clock [274].
In this limit, Eq. 5.45 reads [274]
V̂ =

X
i6=j



−Ueg
Ŝ
·
Ŝ
−
1/4
.
i
j
ij

(5.54)

This result is particularly easy to understand if we appeal to the simple picture we developed
earlier employing the even and odd spatial and electronic wavefunctions of Eq. 5.17. There, theelectronic wavefunction evolved via inhomogeneous excitation into a symmetric and antisymmetricpart, corresponding to antisymmetric and symmetric spatial wavefunctions. There, we enforced that
only the part of the wavefunction with a symmetric spatial component was allowed to “feel” the s-174
Excitation fraction

1.0
0.8

1.0

(a)

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0
−60

−40

−20

0

20

40

60

Excitation fraction

(c)

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0
−60

−40

20

40

60

0
20
−20
Detuning (Hz)

40

60

−20

0

1.0

1.0
0.8

0.0
−60

(b)

−40

0
20
−20
Detuning (Hz)

40

60

0.0
−60

(d)

−40

−
Figure 5.5: Two-particle lineshapes with s-wave interaction energy. (a) U−
12 = 0 (b) U12 = 1Ω (c)−−U12 = 2Ω (d) U12 = 5Ω.

wave collision. We then applied this energy shift as a mean-field energy shift for the g → e transition.
Likewise, in Eq. 5.54, we see that the interaction energy depends on the dot-product of the two spinsof atom i and j. Since these spin operators directly map onto a spin-1/2 system, we can easily seethat there are two possible values for Ŝi · Ŝj , namely 1/4 and −3/4, corresponding to the S = 1 and
S = 0 total spin eigenstates of the total spin operator Ŝi + Ŝj . We see then that only the state with√
S = 0 (the spin-singlet |S = 0, mS = 0i = (|gei − |egi) / 2) interacts with energy Ueg
ij , while the√three possible states of the triplet, |S = 1, mS = −1i = |ggi, |S = 1, mS = 0i = (|gei + |gei) / 2,
and |S = 1, mS = 1i = |eei are non-interacting.
A relevant question to ask is what causes the excitation of the singlet state? As an example,
we consider the case of the spin Hamiltonian of Eq. 5.53 with only s-wave interactions for twoparticles. We then havezĤ = −δ Ŝtot
+

2
X
i=1



2
Ωi Ŝix + −Ueg
Ŝ
−
2
.
tot
12

(5.55)

175
The Hamiltonian can be written in a slightly different way




z
x
2
Ĥ → −δ Ŝtot+ ΩŜtot+ ∆Ω Ŝ2x − Ŝ1x − U−
12 Ŝtot − 2 ,(5.56)

z = Ŝ z + Ŝ z and Ŝ
where Ω = (Ω1 + Ω2 ) /2, ∆Ω = (Ω1 − Ω2 ) /2, Stot
tot = Ŝ1 + Ŝ2 . If Ω1 = Ω1 , then1
2

we see that the hamiltonian can be written in terms of total spin operators, namely


z
x
2
Ĥ → −δ Ŝtot+ ΩŜtot
− U−
Ŝ
−
2
.
tot
12

(5.57)

Figure 5.4 schematically depicts this simple two-particle picture. In this instance, the initially non
s-wave interacting |ggi (|S = 1, mS = −1i in the spin parlance) state satisfies Ŝ2tot − 2 |S =1, mS = −1i = 0. Furthermore, since Ŝ2tot commutes with the other total spin operators, it is aconstant of motion and Ŝ2tot − 2 is always equal to zero. Thus inhomogeneities in the atom-clock
laser interaction (e.g., Ω1 6= Ω2 ) are key to the s-wave shift mechanism.
We can gain further insight if we project the Hamiltonian of Eq. 5.55 into the basis formed
by |S = 1, mS = −1, 0, +1i and |S = 0, mS = 0i. In this basis, the hamiltonian becomes [287]


√

√



Ω/ 2 ∆Ω 2 
δ
0


√
√ 


0
−δ
Ω/ 2 −∆Ω 2 Ĥ → √ Ω/√2
Ω/ 2
0
0




√
√
−
∆Ω 2 −∆Ω 2
0
2U12(5.58)

Figure 5.5 shows the solution of the two particle Hamiltonian given above in the presence of−increasing U−
12 . For U12 ' Ω, we have the situation that the interacting spin-singlet is not well-resolved from the central carrier. However, when U−
12  Ω, the interacting singlet emerges as a
sideband, leaving the carrier nearly unperturbed. This is immediately a qualitative difference fromthe mean-field model of Eq. 5.19, and shows the necessity of a true many- (or in this case two-)body model for the s-wave interaction. Specifically we see that in the limit U−
12  Ω, we cantreat the term ∆Ω Ŝ2x − Ŝ1x as a perturbation and find that the |ggi and |eei levels are shifted−∝ ∆Ω2 /U−
12 . In the N -atom case, this effect can be generalized to the condition U N  Ω.

176Shift of peak (mHz)

80
60
40
Strongly interacting regime

20
0
0

2
4
6
8
Scaled interaction strength 2N U− /Ω

10

Figure 5.6: The s-wave density shift and the strongly interacting regime. The inset shows theregion of the lineshape considered.

With the insights gained from the 2-atom case in mind, we re-write the Hamiltonian of
Eq. 5.54 as [274]zxĤ = −δ Ŝtot+ ΩŜtot+

Here, it is the term

P

i

X
i


X eg 
Uij Ŝi · Ŝj − 1/4 .
Ωi − Ω Ŝix −(5.59)i6=jΩi − Ω Ŝix that perturbatively couples the maximally-symmetric nons-wave interacting manifold to the family of states with Stot = N/2 − 1 that do interact via swaves [274]. However, in the strongly interacting limit, defined for the N -atom case as N U−
12 Ω, excitation of the non fully-symmetric states via inhomogeneity is suppressed via a protectiveenergy gap. Furthermore, the coupling to the interacting manifold is governed by the degreeof inhomogeneity in the system—in this specific case excitation inhomogeneity parameterized by
2P∆Ω2 = i Ωi − Ω / (N − 1).
5.4.1

Concluding remarks regarding s-wave interactions

In the preceding discussion, we explored some of the non-trivial aspects of the s-wave interaction in a lattice clock of initially identical fermions. Most importantly, a criterion for strong
interactions was identified: the strongly interacting regime is reached when the Si · Sj term in the
Hamiltonian becomes large enough to block excitation of the interacting state. As an illustration

177of this principle, we plot the analytical solution to the shift of the peak of Rabi lineshape as derived
by Rey et al. [274] in Fig. 5.6. The peak of the resonance is maximally shifted when 2N U ' Ω. In
the strongly interacting regime, the line center as defined by the peak becomes less affected as U
increases.
These theoretical results were sufficient to explain all phenomena that had been experimentally observed, including the 2009 density shift [31, 265], reduced density shifts in a strongly
confining 2D lattice [135], and resolved interaction sidebands in the 2D lattice clock [288]. However,this is not the end of the story with interaction in the 87 Sr lattice clock.

New experimental evidence for p-wave interactions

5.5

The results of the previous section were well-established when the new laser came on line
(see Chapter 4 for a description of the new laser, also known as the “Big ULE” cavity.). With thepromise of 10-fold improved stability the prospect of a high-resolution probe of the density shift
via sub-Hz spectroscopy was a reality. However, we began to see evidence that was not consistent
with the predictions of the model employing solely s-wave interactions. Here we describe the initial
experimental evidence for beyond s-wave collisional physics.

5.5.1

Inelastic loss

Inelastic losses between atoms in the 3 P0 state of 88 Sr were measured in 2009 by two separate
groups [289, 290]. The collision rate is important for determining the feasibility of producing BEC
using metastable states (3 P0 or 3 P2 ), whose elastic collisional properties are potentially morefavorable for evaporative cooling. In these experiments, loss coefficients of γee = 4.0 ± 2.5 ×
10−12 cm−3 s−1 in a quasi two-dimensional trap [290] and 1.9 ± 1.2 × 10−11 cm−3 s−1 in a threedimensional trap [289] were measured. In Ref. [290], large elastic effects on the clock transitionin 88 Sr were also observed, including a strong collisional dephasing and a density shift an order ofmagnitude larger than for 87 Sr .
These collisions had not been observed in 87 Sr and it had been assumed that the large

178

Atom number 103

4

3

5

a




exp fit 

2Body fit
 Pure g atoms

 Pure e atoms



2













1

0



0

2

b



4



e Atom number 103

5

3

6

8

Lattice hold time s

105050 eg mix

Pure e data

Pure e fit







2










1

 

4



0




0

1

2

3

Lattice hold time s

4Figure 5.7: Atom loss as a function of lattice hold time. Atoms in 1 S0 have a lifetime of 8 s,
and the time-dependence is exponential, indicating that the primary loss mechanism is residual
gas collisions. Atoms is the 3 P0 |F = 9/2, mF = +9/2i state decay an order of magnitude more
quickly. The time-dependence of the decay is consistent with that of a two-body inelastic loss
mechanism, and clearly does not fit an exponential decay model.

centrifugal barrier of 75 µK for 3 P0 [81] should strongly suppress such collisions. Our previous
experimental tests with 87 Sr had not shown any loss from 3 P0 consistent with two-body decay due
mainly to the dominant residual gas collisional lifetime of 2 s in the experimental chamber.
After adding an additional differential pumping stage on the oven-side of the experiment (seeFig. 2.3 Chapter 2), we were able to increase the vacuum lifetime to 8 s. This allowed us to clearly
resolve the effect of 2-body collisions in the excited state. As shown in Fig. 5.7a, the inelastic decay
proceeds approximately an order of magnitude more rapidly than the residual gas collision limitedcase. The equation for the density decay due to two-body inelastic collisions in the presence of abackground one-body loss due to residual gas collisions, with a rate Γ, isdn= −Γn − γee n2 .
dt

(5.60)

Here, n is the density and γee is the two-body loss term. By integrating over the spatial density
profile, the excited state population within a single trap site (labelled with the index i) as a function

179of time is given by [290]Ni (t) =Ni (0) exp (−Γt)
.
1 + Ni (0) γee / π 3/2 Γwr2 wz [1 − exp (−Γt)](5.61)

Here, Ni (0) is the initial population within a trap site i at t = 0, wr(z) is the radial 1/e2 density
radius in the radial (axial) direction. Explicitly, the density profile is 3/2exp −2 x2 + y 2 /wr2 − 2z 2 /wz
2,ni (x) = Niπwr2 wz(5.62)

for which the density-weighted average density is given by
ni =

1
Ni

Z

d3 x [ni (x)]2 =Ni.
π 3/2 wx wy wz(5.63)

We fit the loss for atoms initially prepared in e with the model of Eq. 5.61 averaged over the site
occupancy. As seen in Fig. 5.7a, the model fits the data extremely well, while at the same time
ruling out any purely single-particle decay mechanism, as might be caused by photon scatteringfrom the lattice, which we also independently ruled out (see Chapter 2).
In order to test for the presence of an eg inelastic loss channel, as measured for the case of
88 Sr by Lisdat et al., we created a mixture of coherently prepared e and g atoms by using a π/2

laser pulse on resonance and measured the resulting decay. Figure 5.7b shows the result of a puree a sample and a 50/50 e/g mixture with an equivalent number of atoms in e (i.e., the sample
initially started with twice the number of atoms, 50% of which were in the excited state). We wereunable to resolve any inelastic loss process between e and g, which is evident from the excellent
agreement between the data and theoretical curves for the mixture state as compared to the puree mixture.
To further probe the loss dynamics we took inelastic loss data under varying atomic temperature and in the presence of a second nuclear spin state. In order to explain the observationof inelastic loss in the identical, non-spin mixture fermi sample, a p-wave model was necessary,
because the atoms were unquestionably prepared in a pure quantum state. Our theory colleagues(G. Quéméner et al.) were able to use a model with two fit parameters: the unknown phase shift ofthe short range potential, and the probability that an atom is lost during its penetration into theΓee H10-12cm3secL

180

20

æ
æ

15
æ

à

à

àæ

ææ

10

à

æ

à

à
æ

5

Polarized
Stretched

à

0
0

1

2

3

4

5

Temperature HΜKL

6

7

Figure 5.8: Two-body loss coefficient γee for spin polarized 87 Sr in 3 P0 |F = 9/2, mF = +9/2i
and, in a separate experiment, a statstical mixture of 3 P0 |F = 9/2, mF = +9/2i and 3 P0
|F = 9/2, mF = −9/2ishort-range regime. It was found that the loss probability, pls , was significantly below unity, with abest-fit value of pls = 0.4 for the short-range regime of 30 a0 . The same parameters described theinelastic loss rate in the spin-mixture case, giving us confidence in the model, and also reproducing
the s-wave losses observed in 88 Sr. Importantly, the relatively inefficient loss at short ranges meansthat the s- and p- wave inelastic loss channels are comparable in magnitude, whereas normally swave losses would be expected to dominate by nearly two orders of magnitude [116].
As shown in Fig. 5.8, the agreement between the theory and the temperature dependent loss
rates for the pure spin-state and spin-mixture cases is quite good. The generalization for Eq. 5.61
for the case with the spin mixture isdnα= −Γnα − γee n2α − Ksdist + Kpdist nα nβ .
dt(5.64)

Here, nα and nβ represent the densities of the two components of the gas. The additional twobody loss coefficients, Ksdist and Kpdist , represent s- and p- loss channels for two-body inelastic
collisions between the two different spin states, which are distinguishable. This can be simplified
somewhat due to the decoupling of the electronic and nuclear degrees in 87 Sr , and so we expectthat Kpdist = γee /2 [291]. The parameters we theoretically extract for 87 Sr are ultimately γee =Excitation fraction

181

0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00

−10

~5 Hz

−5

0

Detuning (Hz)5

10

Figure 5.9: Broadened lineshape with 1 s atom-light interaction time. The observed linewidth ofapproximately 5 Hz is more than a factor of five broader than expected from the Fourier linewidth,
and the asymmetry effectively excludes laser noise.

T × (4 ± 2)10−6 cm3 /s/K and Ksdist = T × (4 ± 2)10−6 cm3 /s/K. The agreement with the measured
loss rates as a function of temperature is quite good. The simple model also reproduced the 88 Sre-e inelastic loss rate reported by Lisdat et al. [290].

5.5.2

Lineshape distortion and loss of Ramsey fringe contrast

The observation of a p-wave inelastic loss mechanism was an important additional piece of
information to add to our knowledge of the nature of the interactions between 87 Sr in an opticallattice. This observation was made directly possible by an increased vacuum lifetime. Similarly,
with the new laser described in Chapter 4, the “Big ULE,” we were prepared to make frequency
shift and spectroscopy measurements with an order of magnitude increase in precision. In these
initial investigations, we found signatures of elastic interactions that also deviated from the s-wave
picture that had previously been developed.
Upon installing the new laser, we decided to perform a demanding test of the optical-atomic
coherence of the system. We also set out to beat the long-standing record of 2 Hz spectroscopic

182

Excitation fraction

0.6
0.4

820 mHz

0.2
0.0
−10

−5

0
Detuning (Hz)5

10

Figure 5.10: Low density lineshape vs. high density lineshape with lattice depth consistent withνz = 80 kHz (purple circles). The blue line is a Lorentzian fit. The data from Fig. 5.9 (black
circles) is shown for comparison.

features with coherent spectroscopy in 87 Sr , which had stood for around half a decade [40]. Asa first test, we performed Rabi spectroscopy with a π-pulse duration of 1 s, which should resultin a Rabi lineshape with full width at half-maximum (FWHM) of 0.8 Hz. Instead, we observed
the lineshape shown in Fig. 5.9 with a FWHM of approximately 5 Hz! Here, the temperature wasapproximately 4 µK and the density was ρ = 6 × 1011 cm−3 . Another striking difference was thatthe peak excitation fraction, which should theoretically be unity, was nearly an order of magnitude
lower than expected. The data in Fig. 5.9 is comprised of approximately 20 individual scans of theline which were subsequently centered upon one another and binned using a bin size of 250 mHz.
This has the effect of allowing a high resolution lineshape to be formed in a way that is insensitive
to slow laser excursions. Linear laser drifts were removed to the 5 mHz/s level, but to be sure there
was no biasing due to scan direction, we altered the sign of the frequency stepping for the laser
within a given scan.
We then reduced the atom number by an order of magnitude, such that the density was

Excited State Fraction

0.6
0.5
0.4
0.3

Excited State Fraction

183
0.8

4

2

0

0.6

2

4

0.6

890 mHz

0.4

0.4

−4

0.2

870 mHz

0.2

0.2
0

0.8

−2

0

2

4

0

Frequency (Hz)

0.1
0.0

−4

−2

0

2

Frequency (Hz)Figure 5.11: Narrow line scan (1 s probe time) with 40% the typical lattice power and a sample
temperature of 1 µK. Here, despite the density of ρ = 1 × 1011 /cm3 , a narrow line is recovered.
(Inset) Representative single-scan with no binning or averaging.

ρ = 5 × 1010 cm−3 . Under this condition, the line dramatically narrowed to nearly the expected
linewidth, with a FWHM of 820 mHz. The linescan is show in Fig. 5.10 along with the high-density
data for reference. Also shown is a Lorentzian fit to the data, which indicates an asymmetry similarin shape to the high-density case, but far less pronounced.
In a final test, we pre-cooled the sample using Doppler and sideband cooling (see Chapter 2,
Section 2.7) and subsequently reduced the lattice intensity to 40% of the typical operating intensity
using an adiabatic ramp, which had the simultaneous effect of lowering the density, temperature,and total atom number such that the final temperature was 1 µK and the density reduced toρ = 2 × 1011 cm−3 . Under this condition, the line was also narrowed to near the expected lineshape,
as shown in Fig. 5.11. For both this condition and the low-density condition described previously,
the data was processed and binned as described for the high-density and high-lattice case.
From the three experimental conditions that we explored, we were able to deduce that thedominant line-broadening mechanism was density. This is an important distinction because some
calculations of the lattice photon scattering rate had estimated scattering rates as high as ∼3 Hz

1.2

200 ms

1.2

a

b

1

Excitation fraction

Excitation fraction

1

184

750 ms0.8~30% loss
0.6
0.4

0.8
0.6
0.4

0.2

0.2

0

0

−0.2
−50

−40

−30

−20

−10

0

10

20

30

Detuning (Hz)

40

−0.2
−30

−20

−10

0

10

20

30

Detuning (Hz)Figure 5.12: Comparison of two lineshapes taken under similar atom number conditions of Ntot '5000. The upper data (black circles) is the total population scaled by the off resonant atom
number. The lower traces (red circles) are plots of excitation vs laser detuning. These lineshapes arecomposites formed from approximately 20 scans each that have been centered upon one another. (a)The probe (and resonant π-pulse) time is 200 ms. The peak excitation of the line is approximately
60% of that of the non-interacting case. (b) The probe (and resonant π-pulse) time is 200 ms. The
peak excitation of the line is approximately 20% of that the non-interacting case. Here, almost no
inelastic loss is observed, as can be seen from the discrepancy between the upper data points andthe horizontal line indicating the loss from the 200 ms case. The line is also significantly broadened
beyond the ∼ 1 Hz linewidth expected for a 750 ms probe time.

for a lattice depth of ∼120 Erec [292]. However, in addition to ruling out lattice photon scattering
as a contributing mechanism to line broadening, as described above, we have rigorously—both
experimentally and theoretically—ruled this mechanism out entirely for all lattice depths currently
accessible to us (see Chapter 2, Section 2.6).
Beyond the general observations discussed above, the lineshapes gave us initial evidence thatthe s-wave dominated picture of interactions might not be the end of the story. One mechanism thatwas initially considered was the effect of two-body inelastic loss. However, the observed broadening
was much larger than could be accounted for by simply inelastic collisions alone. Furthermore,
the two-body loss under conditions with the most severely broadened lineshape under the longest
π-pulses was minimal, while the two-body loss for shorter π-pulses was increased. The difference
between a 200 ms π-pulse and a 750 ms π-pulse is shown in Fig. 5.12, where in the case of the

185

Excitation fraction

0.6

Pulse1 = 20 ms
Pulse2 = 20 ms
Evol. time = 200 ms

0.5
0.4
0.3
0.2
0.1
0.0

-40

-20

0

Detuning (Hz)20

40Counts/1000

0.7

1.8
1.6

b

1.4
1.2
1

Excitation fractionaCounts

1300
1200
1100
1000
900

0.8
0.6
0.4
0.2
0

0.2

60

40

20

0

20

40

60

Detuning (Hz)Figure 5.13: Ramsey fringes contrast and two body-loss. (a) Ramsey lineshape observed using theparameters listed in the inset of the figure. A marked decrease in the contrast was observed (bottom
trace, red), along with a net atom loss when near resonance (top trace, blue curve; 1 count= 5 atoms.
(b) Density matrix model fit to the Ramsey lineshape with ee inelastic losses as the only collisional
mechanism. The ee loss rate was tuned to match the net observed atom decay. The effect on the
Ramsey fringe constrast is minimal.

latter pulse duration, the two-body loss has been almost completely inhibited.
Finally, we observed strong decoherence of Ramsey fringes during a Ramsey experiment (seeAppendix A for a description of Ramsey spectroscopy) where two-body losses were responsible for∼ 20% losses of total atom number while the fringe contrast decayed to only 30% of the theoreticalvalue of unity. These experimental data are shown in Fig. 5.13a. Under a typical Ramsey experiment, it can be shown that for small two body losses, the fringe contrast should decay quadraticallywith fraction of atoms lost via excited state two-body loss if this is the only inelastic loss channel.
A numerical solution to the optical Bloch equations is shown in Fig. 5.13b for the case of only
inelastic two-body loss. For the appropriate two-body decay coefficient, the fringe decay is so smallthat it is hardly apparent.
These strong pieces of evidence ultimately led us to believe that there was a blockademechanim of elastic origins at work. Since the effect on the carrier was so dramatic, we could
rule out s-wave elastic collisions, which should be reduced in strength with respect to the carrierby ∆Ω/Ω, which is  10% for all experimental conditions probed, as shown in Chapter 2. The

186
natural way to include beyond s-wave effects is via the next partial wave: p-waves. p-wave shifts
were observed in a neutral atom clock based on fermionic 171 Yb [273, 293], which operates at close
to an order of magnitude higher temperatures than the 87 Sr lattice clock. However, these results
were another important indicator that p-wave interactions can be important, even for collisionalenergies well below the centrifugal barrier.

p-wave many-body model

5.6

As discussed in the previous section, the s-wave model fails to describe the qualitative featuresof the spectroscopic lineshapes, even when inelastic p-wave collisions are included. Therefore, it isnatural to seek answers from the next available partial wave. In this section, we will apply successive
simplifying approximations to the p-wave many-body model, which allows good agreement between
the observed lineshapes and density shifts in the linear response regime. We find, however, thatcertain experimental observables require that beyond mean-field effects within the p-wave theory areretained in order to agree with experiment. Ultimately, these measurements allow us to benchmark
the effectiveness of describing the system via a many-body spin-Hamiltonian, opening the door to
exploring many-body effects and entanglement between optical degrees of freedom in lattice clocks.

5.6.1

Collective model

We begin our consideration of elastic p-wave effects in the clock by considering the Hamiltonian of Eq. 5.53, given byĤ = −δXiŜiz +XiΩi Ŝix +X
i6=jJij⊥ Ŝi · Ŝj + χij Ŝiz Ŝjz + K
Cij  zij
Ŝi + Ŝjz +.
2
4(5.65)

Here, the mode-dependent constants are given by Eqs. 5.46–5.49. The p wave component of the
αβinteractions parameters is characterized by the Vij
terms. As shown in Appendix D, the mode-

dependence of the s- and p-wave coefficients is smoothly varying. In the case of the p-wave interaction parameter for two-dimensional traps, the thermal average is independent of the temperature[125].

187

J = N/2 − 1

J = N/2

m=2
m=1
m=0

Neγee

J = N/2 − 1

Figure 5.14: Visual depiction of the collective model. The many-body Hamiltonian has eigenstatescomprised of maximally symmetric superpositions (Dicke states, for which S = N/2) of electronic
ground and excited states, depicted as purple shells. Because of the weak inhomogeneities in the
coupling strengths, the maximally symmetric manifold is coupled to the next lowest manifold with
S = N/2 − 1, depicted as a nested blue shell. However, this coupling is prevented by an energy~ ·S~ term, which is depicted by an
gap present in the many body Hamiltonian as a result of the J ⊥ S
offset of the two manifolds in this figure. Two-body inelastic losses (dashed black arrow) connect
maximally symmetric manifolds of S → S − 1, and thus are not a strong decoherence mechanism.

Motivated by the weak dependence of the interaction parameters on the mode, we make asimplifying assumption that will be verified by the experimental evidence subsequently presented.
We replace all the interaction parameters by their mode-averaged quantities given by, e.g.,
C=

X
i6=j

Cij / [N (N − 1)](5.66)

188
and the averaged laser excitation term is
Ω=

1 X
Ωi .
N

(5.67)i

Here, the sums are averaged over thermally populated distributions of the harmonic oscillator
states, {|ni i}. Using the averaged quantities, Eq. 5.65 becomes 2XĤ = −δ Ŝ z + ΩŜ x + J ⊥ Ŝ · Ŝ + χ Ŝ z + CN Ŝ z +ˆij .
|{z}ij(5.68)

collective

Here, the operators without mode subscript are collective operators such thatŜ {x,y,z} ≡X{x,y,z}Ŝi.

(5.69)i

We have also moved all the residual mode-dependence to the last term of Eq. 5.68. This term
is a perturbative parameter that is small compared to the collective part of the Hamiltonian. It
is interesting to note that the collective part of the Hamiltonian in Eq. 5.68 maps onto that of a
Bose-Einstein in a double-well potential [294, 295].
The collective Hamiltonian of Eq. 5.68 is nearly diagonal in the Dicke state basis, |J, mi,J ∈ {N/2, N/2 − 1, ...} [296, 297] in the case that there is no laser excitation, e.g., Ωj → 0. Fully
symmetric Dicke states are formed by the fully symmetric permutations of ground and excited state
atoms subject to the constraint S z |N/2, mi = m. Specifically [297],
−1/2 X Y
YN!
|N/2, mi =|gin
|ein .
(N/2 + m)! (N/2 − m)!


S n∈S

(5.70)

n∈S/

Here, S represents the set of all permutations for N atoms with the total number of atoms in the
excited state given by Ne = N/2 + m.
When an ideal rotation is applied via a spectroscopy pulse, the prepared state is given by a
product state of single particle states given by|θ, φiN =NY iφ/2esin (θ/2) |ein + e−iφ/2 cos (θ/2) |ginn=1

(5.71)

189
The Bloch states can be written as sums of the Dicke states as|θ, φiN =N/2
X
m=−N/2

−1/2N!
(N/2 + m)! (N/2 − m)!

×eimφ sinN/2+m (θ/2) cosN/2−m (θ/2) |N/2, mi(5.72)The states |θ, φiN can easily be prepared via fast excitation such that Ω  N {Vee , Veg , Vgg }. Thisis the case for Ramsey spectroscopy.

5.6.2

The collective model and density shifts in a 2D optical lattice

As mentioned in Chapter 2, Section 2.8, a two-dimensional lattice (which creates quasione-dimensional trap sites) was employed to enter the strongly interacting regime (i.e., whereatom-laser and atom-atom interactions are energetically comparable), at the expense of reducing
the occupancy to one or two atoms per site [135]. Here p-wave effects were not considered andthe excitation inhomogeneity was much higher [135, 288]. Within the many-particle collectivemodel, the role of s-wave interactions in the absence of excitation inhomogeneity is limited to an
enlargement of J ⊥ , thus enhancing the gap protection and enforcing the validity of the collective
spin Hamiltonian.

5.7

Mean-field and perturbative tests of the many-body dynamics
In this section we consider only the collective part of the Hamiltonian of Eq. 5.68. This isa significant assumption, but one we can validate through experiment. These approximations areimportant for the beyond mean-field treatment of the model as well. Theoretically, there are several
aspects of the collective Hamiltonian that justify its use. They are(1) The atoms are initialized in the fully symmetric |J, −Ji state and will remain in the fully
symmetric manifold under laser excitation as long as ∆Ω  J ⊥ .
(2) In a quasi-2D trap, χ and C are highly insensitive to the sample temperature T for a fixedatom number, given that the linear growth of the p-wave interaction with T is compensated

190

0.4

b

Occurences

ss
∝ Vij

0.3
0.2
0.1
0.0

7000

a

0

2

4
6
T (µK)

8

10

0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ss
∝ Vij

Figure 5.15: Temperature dependence of the p-wave interaction energies and their thermal distribution. (a) Temperature-dependence of the average p-wave interaction parameters. They exhibit
a large insensitivity to the temperature due to the quasi-2D confinement. (b) Histogram of the
distribution of p-wave interaction parameters at 5 µK. In both cases a Boltzmann distribution of
the populated modes was used [125].

by a corresponding 1/T decrease of the density. The validity of this assumption in shownin Fig.. 5.15a.
(3) The interaction parameters have a weak dependence on the thermally populated modes iand j, which is reflected on the fact that that their thermal distributions are peaked about
their averages (see Fig. 5.15b) [125].
(4) There is a protective gap given by the J ⊥ Ŝ · Ŝ term, which should be larger than the peg
wave interactions due to the s-wave term in Jij⊥ , where Jij⊥ = Vij− Ueg
ij . This energeticallyPforbids J → J − 1 due to the inhomogeneous term in Eq. 5.68, ij ˆij , which is  J ⊥ .

5.7.1

Perturbative tests of the many-body HamiltonianIf the system is excited such that only a small fraction of the atoms are promoted from theground to exited states, the shift of the line center in a single trap site isi∆νLRR' (C − χ)(Ni − 1),(5.73)

191
0.06

a

0.05

0.04

Excitation fraction

Excitation fraction

0.05

0.06

0.03
0.02
0.01
0.00
0.01
10

b

0.04
0.03
0.02
0.01
0.00

5

0
Detuning Hz

5

10

0.01
10

5

0
Detuning Hz

5

10

Figure 5.16: Spectroscopy of the clock transition in the linear response regime with pulse area fixed
to θ = 0.2π. (a) Spectroscopy in the linear response regime with tR = 250 ms. The high atom
number curve (red triangles) is shifted to lower detuning with respect to the high frequency curve(teal blue circles). The low atom number is Ntot ' 2000 and high atom number is Ntot ' 5000 (b)Spectroscopy in the linear response regime with tR = 500 ms. The low atom number is Ntot ' 1500
and high atom number is Ntot ' 4000.

where the index i labels the individual site. The total shift in the linear response regime is giveniby an average of ∆νLRR
over the site occupancy distribution. The site occupancy distribution is

described in Appendix E.
In order to probe the linear response regime experimentally, we reduced the clock laser probe
intensity such that for a given probe time, the pulse area was θ ≡ ΩtR = 0.2π. We then tooklineshape measurements under four conditions. In the first condition, we set the pulse duration
to tR = 250 ms and took scans with low (Ntot ' 2000) and high (Ntot ' 5000) atom number
conditions. These data are depicted in Fig. 5.16a. We then increased the total probe time totR = 500 ms, but decreased the probe intensity so that, as before, θ = 0.2π. This longer-probe
time data is shown if Fig. 5.16b. Under this second condition, the low atom number was Ntot ' 1500
and the high atom number was Ntot ' 4000.
Importantly, as seen in Fig. 5.16, the data support the simple description of the shift given
by Eq. 5.73. Here, the theoretical lineshapes are formed by averaging the linear-response shift over192
the average site occupancy distribution as described in Appendix E. Two important features are
immediately evident. The first is that the line center is clearly shifted, even with the weak excitation.
This is in clear contradiction with the s-wave-dominant picture of the density shift, where sidebands
create a line-pulling effect, and should be negligibly small for such weak excitation. The second isthat the line centers are shifted approximately the same, even though the probe time is different
by a factor of two. We note that this statement is not completely rigorous because the total atom
numbers in both cases, in addition to the difference in atom number between the high and low
conditions, were not exactly the the same. However, the fact that there is no dramatic difference
between 250 ms data and the 500 ms data reproduces the universal behavior we expect of the shift
in the limit of weak, perturbative probing of the linear response. The data is fit according to acomposite lineshape formed of Lorentzians shifted according to Eq. 5.73 and weighted by the atom
numbers distribution. The data are consistent with a C − χ = −0.3(1) Hz.
5.7.2

Mean-field solution of the collective model

Even in its vastly simplified form, the collective Hamiltonian of Eq. 5.68 still represents aformidable calculation. In the absence of the laser driving (i.e. Ω = 0), the Hamiltonian is diagonalin the maximally symmetric Dicke state [296] basis, and exact solutions for the time dynamics havebeen obtained [280]. However, in the presence of a laser drive, the dynamics of the collective spinmodel are complicated, but numerically tractable [125]. It is our goal to analyze the dynamics
in a simplified regime. Here, we will find several experimental observables that are well-describedby using a so-called mean-field treatment of time evolution so-called because it considers only theevolution of field operators by making the simplifying assumption that no correlations form.
We begin by defining new operators based upon the previously-defined spin operators. The
first two operators are the number of atoms “spin-up” (i.e. in the excited state) and conversly“spin-down,” given byN̂↑ = Ŝ z + N/2,

N̂↓ = N/2 − Ŝ z .

(5.74)

193
The second operators we define are the canonical angular momentum raising and lowering operators
formed by linear combinations of Ŝ x and Ŝ y , and which are given byŜ ± = Ŝ x ± iŜ y .

(5.75)

We find the time evolution of these operators under the action of the collective Hamiltonian via
the Heisenberg equation of motion and obtainhi∂ ±Ŝ = i H/~, Ŝ ±∂t= ±ΩS z ± i [−δ + C (N − 1)] Ŝ ± ± iχ Ŝ ± Ŝ z + Ŝ z Ŝ ±(5.76)and∂∂N̂↑ = − N̂↓ = iΩ Ŝ + − Ŝ − /2.
∂t∂t(5.77)We then replace operators by their expectation values, thus removing quantum fluctuations
associated with operators and arising from their commutation relations. This will be our operational definition of mean-field dynamics. We will use the convention that hÔi ≡ O. To
the mean field limit to the dynamics of the many-body equations of motion we additionally add ainelastic two body loss term (as in Refs. [290] and [116]), Γee , to properly account for the inelastic
loss described previously. The mean-field equations of motion become


∂ ±
Ω z
Γee
z
S ' ±i S ± i −δ + 2χS + CN ± i
N↑ S ± ,∂t22∂N↓ = iΩ S + − S − /2,∂t∂N↑ = iΩ S − − S + /2 − Γee N↑2 .
∂t(5.78)(5.79)(5.80)

Without loss (i.e., Γee → 0), these equations behave as if there is an effective magnetic field-liketerm in the hamiltonian, with magnitude given by B (N ) = N C + 2χSz .

5.7.3

Density shift in the mean field limit

We begin our consideration of Eqs. 5.78–5.80 by treating the case of Ramsey spectroscopy,
which provides a clean way to decouple the time evolution. Specifically, during the dark time,

194
analytic expressions for Ne (t) and S ± (t) can be found and then used as driving terms for Eq. 5.78.
We start by noting that the general expression for S ± is given byS±(t) = S0± exp




Z τ
±
±i dt u (t) ,(5.81)

0

where


Γee
u± (t) ' −δ + 2χS z (t) + CN (t) ± i
Ne (t) .
2(5.82)

From this expression, the frequency shift with respect to the non-interacting case is easily inferred.
Specifically,
1
∆ω =
τ

Z τ
dt [2χS z (t) + CN (t)] = 2χS z + CN .

(5.83)

0Here, S z and N are the time-averaged values of S z and N , respectively. Thus, only knowledge
of S z and N is required in a given measurement to find χ and C, and knowledge of the specifictime-dependence of these quantities is not strictly necessary. Stated another way,∆ω = N 2χ f − 1/2 + C .

(5.84)

Here, f is the average number-weighted excitation fraction and is given by f ≡ N f /N . This value
is approximately equal to the time-averaged value of the time-dependent excitation fraction, f (t)as used in Ref. [293].

5.7.4

Measuring the density shift with a Ramsey interrogation scheme

As can be seen from Eq. 5.84, the shift in Ramsey spectroscopy should be linear with f as
long as the mean-field expression is valid. This suggests a very straightforward way to extract theinteraction parameters: by performing Ramsey spectroscopy using a modified sequence, one canvary f by varying the initial Ramsey pulse area, θ1 , while keeping the second pulse area, θ2 fixed
to be θ2 = π/2 and thus maximizing fringe contrast and therefore measurement precision. This
was the general approach employed in Refs. [273] and [293], and was the first conclusive proof of
p-wave collisions in a fermionic 171 Yb optical lattice clock.

0.69
0.68

High

f high

0.0

 0.1

low

f high

hig

0.67

b

h

0.1

Low

f low

Shift (Hz)

0.70

195

annulment
error

a

0.71

f

0.2

0.66
0

5

10

15

20

25

Initial atom number

30

 0.2
0.60 0.62

0.64

0.66

f low

0.68

0.70

f

Figure 5.17: Effect of two-body losses on the density shift measurement for two test cases, with
χ = 2π × 0.2 Hz, C = −0.3χ, and Γee = 2π × 0.07 Hz. “High” corresponds to an initial single-site
atom number of 20, while “low” corresponds to an initial number of 10. (a) Atom-number weighted
average excitation fraction, f , as a function of initial atom number with an initial excited state
fraction of 72%. Due to the nonlinearity of the two-body loss, f depends upon the initial atom
number. (b) Annulment of the density shift between “high” and “low” atom number conditions.
The annulment between the two conditions is achieved as desired, but the shift with respect tovanishing density is still finite due to the influence of the two-body on f between the two conditions.

Due to the two-body inelastic loss, however, measuring the Ramsey shift in a self-consistent
way is potentially fraught with peril. To illustrate the problems, we consider a Hypotheticalmeasurement, with a “high” atom condition, corresponding to an initial site occupancy of 20 atoms,
and a “low” atom condition, corresponding to an initial site occupancy of 10 atoms. The idea is
to use the relative shift between the high and low condition to extrapolate χ and C. However, as
shown in Fig. 5.17a, the two-body loss couples total atom number to f , such that when the atom
number is varied between the high and low conditions, f is inadvertently varied.
A given density shift measurement is obtained by comparing the Ramsey fringe central frequency under the high density condition to the low density case, such that∆ωmeas = ∆ωhigh − ∆ωlow = 2χN high f high − p∗ − 2χN low f low − p∗ .

(5.85)

Here, N high and N low represent the ensemble- and time-averaged single-site occupancy for the high196
and low density conditions, respectively. The explicit goal of our measurement is to extract thezero-crossing fraction, p∗ , which can be written as a function of χ and C (see Eq. 5.84) and also
the nonlinear parameter χ. By forming a new weighted quantity of f high and f low we can achieve
this goal. We define a new average quantity of the excitation fraction, which represents a weighted
average of both average excitation fraction asf≡

N high f high − N low f low
N high − N low(5.86)

such that it is algebraically true that∆ωmeas = 2χ N high − N low
f − p∗ .

(5.87)

Equations 5.86 and 5.87 are important because they are universal as long as Eq. 5.84 is valid. It also
gives us an unambiguous way to plot the data. By defining f in this manner, which is a derived
quantity rather than a directly measured one, we have removed the effect of the cross-coupling
between N high(low) and f high(low) .
One important conclusion of the analysis presented above is that a measurement of a nullshift between two different density conditions for a given initial excitation fraction does not imply
that the total shift is zero at this initial excitation fraction, nor does it give an accurate value for p∗ .
This is true even when the fraction of total atoms lost is only at the level of several percent. To see
this, we assume that χ = 2π × 0.2 Hz, C = −0.3χ, and Γee = 2π × 0.07 Hz, resulting in p∗ = 0.65
(these will turn out to be very nearly the measured values for 87 Sr ). Next, the measurement isperformed using an initial “high” initial site occupancy of 20 and an initial “low” site occupancy of
10. This will yield f = p∗ = 0.65 for a properly nulled shift, while on the other hand f high = 0.67 andf low = 0.70. Thus, if we were to choose the “high” conditions as our operating point, the remaining
shift with respect to the zero-density limit would be ' 2χN high × 0.02, which represents a shiftof 4 × 10−16 . This is a huge error and nearly four times larger than the entire clock uncertainty.
Interestingly, if we had instead chosen to use the “low” conditions as our operating parameters(leading to a frequency error of ' 2χN low × 0.05) we would still have exactly the same frequency

1 2

1 2

a

1 2

Atom decay detection
L L S S M M

Density modulation

High
Low

1 2 3 4 5 6 7 8 9 10 1112 13 14 15 16 17 18 19 20 21 22

Sequence number (modulo 22)

197

0.90

Excitation fraction

1 2

2 PID Lock

b

0.85
0.80

Low

0.75

High

0.70
0.65
0.60

Ramsey time window
0

20

40

60

80

100

120

Time after first Ramsey pulse (ms)140

Figure 5.18: Timing sequence and example of inter-sequence excitation fraction measurement.
(a) Timing diagram for the multiplexed Ramsey density shift measurement. Two independent
PID locks are utilized to measure the differential frequency between the high and low density
condition. After four locking cycles, which are alternated for each PID, an atom–decay measurement
is performed. During this measurement, three delays are utilized to extrapolate the two-body loss
curve: short (S), medium (M), and long (L). (b) Example of experimental data for the measurementof 2-body losses during the Ramsey measurement sequence. The low and high density conditions
are fit with two-body loss curves. Error bars are standard deviations and therefore represent the
scatter of the data, not the uncertainty of the mean.

error of 4 × 10−16 . This in fact must be true, since the shift was indeed measured as a relative null
between the high and low operating conditions. Figure 5.17b illustrates this mechanism, whereby
the relative shift is nulled but the total shift is not. The conclusion we must draw from thisexample is that we can only use density shift measurements to extrapolate the proper operating
fraction of the clock, given by f = 0.65. In the presence of two-body loss, one cannot directly rely
on a measured null shift alone to determine either the location of the true zero-crossing lies, andcare must be taken when using such a measurement as proof that the clock has achieved a target
operating uncertainty.
We performed a modified density shift measurement to extract the density shift as a functionof f . The technique is based upon that used in Ref. [135]. As depicted schematically in Fig. 5.18,
we operate two independent locks to the atoms and modulate the relative density between them.
The frequency difference between periodically modulated conditions is recorded. After four pairs of

198

Shift for Ntotal = 4000 (Hz)

4

2

0

−2

p∗

−4
0.0

0.2

0.4

0.6

0.8

1.0

f
Figure 5.19: Measured Ramsey shift as a function of the scaled excitation fraction f . The verticalerror bars are standard errors calculated from the statistical scatter of the measured shift coefficient. The horizontal error bars are the standard error of f calculated from separate measurement
sequences inserted into the density shift measurement sequence. Fluctuations in f mainly arise
from fluctuations in the total atom number. The reduced χ2 statistic for the linear fit is 1.3.
As described in the text, this measurement yields p∗ = 0.640(6), χ = 2π × 0.21(4) Hz, and thus
C = −0.280(12)χ.

high-low measurement cycles, we devote six experimental cycles to measuring the two-body losses
during the Ramsey experiment. Briefly, this decay-measuring cycle proceeds with a single pulse ofarea θ1 , which is identical to the first pulse of the Ramsey sequence. The time between this pulse
and the atom number measurement phase is varied, allowing excited state atom decay as a function
time to be measured. We then fit the decay curves, which consist of three points taken at eachdensity condition, with two-body decay lineshapes. This allows us to calculate the the relevant
quantities, namely f , N high , and N low .

199
The results of the Ramsey density shift measurement are shown in Fig. 5.19. As seen fromthe data, the linear dependence of the shift predicted by Eq. 5.87 agrees extremely well with theexperimental observation, over the full range of excitation fractions. From these measurements we
extract p∗ = 0.640(6). The extracted slope is measured with comparable fractional precision, butconverting the experimentally measured shift parameter to χ introduces extra uncertainty due tothe uncertainty in atom number calibration and axial intra-site distribution. If we fix the atom
number such that the total time-averaged atom number is Ntotal = 4×103 , then the slope plotted inFig. 5.19 is 6.96(15) Hz. For the same value of Ntotal , the ensemble-averaged single-site occupancy
will be N ' 16.3 (see Appendix E). This yields χ = 2π × 0.21(4) Hz. Here the uncertainty comesmainly from the atom number and site distribution uncertainty. Finally, knowledge of p∗ allows us
to relate C to χ, with C = −0.280(12)χ.
As a final note, it is interesting to consider exploiting the measured zero-crossing to cancel density-dependent shifts as proposed in [293]. With a relatively short three-day sequence ofmeasurements, resulting in the data shown in Fig. 5.19, we are able to locate the zero-crossing tobetter than 1%. If we operate with a total number of atoms Ntotal = 1 × 103 , then the resulting
frequency uncertainty is 2.4 × 10−17 . However, other effects such as repump efficiency and spinpolarization efficiency would need to be considered before firmly claiming a final number. It is niceto know however, that even under very high-density operating conditions there are no fundamental
impediments to operating with a density shift contribution at the 10−17 level. In fact, a day offocused density shift measurements could push the statistical uncertainty of the zero-crossing point
below 1 × 10−17 , if a shift uncertainty at this level were to become necessary.
5.7.5

Describing spectroscopic lineshapes with the mean-field model

In the presence of a laser drive, the many-body system has two relevant energy scales, which
represent a boundary between a laser-drive-dominated regime, where Ω > N χ and an interactiondominated regime, where Ω < N χ. This type of transition phenomenon is observed in the dynamics
of double-well BEC systems, which are described by a Hamiltonian equivalent to the collective

Excitation fraction

200
1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

Excitation fraction

−0.2
−50

−40

−30

−20

−10

0

10

20

30

40

50

−0.2
−50

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2
−50

−40

−30

−20

−10

0

10

Detuning (Hz)

20

30

40

50

−0.2
−50

−40

−30

−20

−10

0

10

20

30

40

50

−40

−30

−20

−10

0

10

20

30

40

50

Detuning (Hz)Figure 5.20: Lineshape (red circles) and normalized total atom number (black circles) as a functionof detuning and for decreasing Ω ' π/tR . In all plots, N ' 5000. (a) tR = 100 ms (b) tR = 200 ms(c) tR = 300 ms (d) tR = 750 ms

Hamiltonian of Eq. 5.68 and can lead to phenomena such as excitation blockade or macroscopic
self-trapping [294, 298, 295]. In principle, this transition could be accompanied by a non-trivial
shift in the behavior from a system that is well-described by a mean field treatment to one in which
the true quantum many-body nature of the eigenvalues becomes important.
In order to probe the spin model in the presence of a laser coupling term, we performed
Rabi spectroscopy under a variety of atom numbers and coupling strengths, characterized by Ω.
Figure 5.20 shows both the measured spectroscopic lineshape and atomic loss curves in detail for

201
Ntotal ' 5 × 103 . Here, each lineshape is a composite of 10 individual scans that have been centeredatop one another. The theoretical fits are numerical solutions of the mean-field evolution given
by Eqs. 5.78–5.80 and are computed for a given initial atom. The results are averaged over thesite-occupancy distribution (Appendix E). In order to account for the effect of laser noise within agiven scan and also between scans, a phenomenological dephasing term, L, has been added to thetime evolution of S ± , such that [116]



∂ ±
Γee
Ω z
z
S ' ±i S ± i −δ + 2χS + CN ± i
N↑ + L S ± .
∂t2
2(5.88)

We find that values of L = 2π × 0.4 Hz and Γee = 2π × 0.075 Hz work well for these particular
data. With subsequent improvements we made to the laser, it is likely that L will be found to be
reduced in future measurements of this type. Additionally, while the laser-atom interaction time
is nominaly fixed such that ΩtR ' π, due to the difficulty of experimentally verifying a ΩtR = π,
the equality is only approximate, and in the numerical simulations the pulse area is left as a freeparameter, which can vary from at the level of 10%. As seen from Fig. 5.20, the quantitative
agreement between the theory and data is quite good, including the predicted total atom loss when
the spectroscopy laser is on resonance.
A more thorough display of the different interaction regimes is shown in Fig. 5.21. In thisfigure, we show lineshapes obtained by varying the ratio of χN to Ω in two ways. In the first
approach, we keep Ω fixed at 2π × 2.5 Hz while varying the total atom number. At the lowest two
atom numbers, χN < Ω. For the largest three atom number conditions, χN > Ω. In the second
approach, we instead vary Ω while keeping the atom number fixed at 5 × 103 . In this latter case,all but the curve with tR = 100 ms are in the regime where χN > Ω, with the last curve satisfyingχN /Ω ' 6. Even in this extreme interacting regime, with χN /Ω ' 10, the model does a good job
describing the lineshape. In such a strongly interacting regime, effects analogous to macroscopic
self-trapping [294] are playing a role in many of the trap sites as shown in Fig. 5.21.
Finally, we note that density-dependent effects are visible in the lineshape even for very low
densities/atom numbers of < 10% of those typically utilized. Using a 2 s rabi pulse, we scanned

202

1.4

1.4

1.2

Excitation fraction

1.2

1.0

1.0

om
t
A

b
m
nu

1.1
1.0

e

ob
Pr

0.7

tim

750 mse

500 ms

0.4

0.8

0.8

0.6

Theoreticalprediction

er

0.2

400 ms0.6

300 ms0.4

0.4

200 ms

0.2

0.2

0.0

0.0

100 ms

0
0

50
50

100
100

150
150

200
200

250
250

300
300

Detuning (Hz)Figure 5.21: Lineshape as a function of experimental parameters. Each curve is a composite
of multiple scans that have been centered atop one another and the data subsequently binned.
The curves are offset in both the vertical and horizontal directions for visual clarity. Left side:
Lineshape as a function of density (normalized by ρ0 ), with a π-pulse time of 200 ms. Right side:
Lineshape for ρ ' ρ0 as a function of probe time. Theoretical curves, obtained with the mean-field
treatment including loss, are shown as dashed black lines and agree well with the measurements.

out the lineshape shown in Fig. 5.22a. The linewidth of this composite lineshape is approximately
750 mHz. Even at the sub-Hz energy scale, the interactions manifest themselves as a clear asymmetry in the lineshape. As shown in Fig. 5.22b, we have verified that as the atom number decreases,
the observed linewidth approaches the ideal Fourier-limited value for a 2 s Rabi pulse. At the very
lowest densities and with a 3 s Rabi pulse, we observe repeatable linewidths of 500 mHz (Inset,
Fig. 5.22a), representing a record for coherent atomic spectroscopy and a line Q of nearly 1015 .

203

0.3
0.2

0.4

1.2

500 mHz

0.2
0.0
−4

−2
0
2
Detuning (Hz)

4

0.1

1.0
0.8
0.6
0.4

0.0
−6

b

1.4

FWHM (Hz)

0.4

a

0.6
Excitation fraction

Excitation fraction

0.5

Fourier limit

0.2
−4

−2
0
Detuning (Hz)

2

4

0.0 0

100

200 300
400
Atom number

500

600

Figure 5.22: Interaction effects visible at the sub-Hz energy scale. (a) Composite lineshape obtainedat an operating density of 0.1ρ0 with a 2 s clock probe, where ρ0 is the density obtained for 5 × 103
atoms, and is given by ρ0 = 5(2) × 1011 cm−3 . The line shape shows a clear distortion due to thedensity-dependent interaction at the sub-Hz energy scale. (Inset) Extremely low density scan (ρ = 5 × 10−2 ρ0 ) with a 3 s clock probe, demonstrating the ultimate frequency resolution of thesystem. This is a single scan thus no binning or averaging was applied. (b) measured linewidth asa function of Atom number in the low-density regime for the scans compiled to create the figure
at left. As seen from the data, the density-dependence of the linewidth extrapolates to the 0.4 Hz
Fourier limit for the 2 s Rabi pulse used.

5.8

Beyond mean-field effects
In the previous section, we explored two experimental observables that were well-describedby the mean-field approximation of the collective spin Hamiltonian. For the Ramsey density shift
and Rabi lineshapes, spin correlations that may develop during the time-dependent evolution do
not manifest themselves in the measurement. Furthermore, we expect that in the case of Rabi
spectroscopy, the strong elastic “blockade-like” excitation-inhibition as well as two-body losses
prevent these correlations from forming. However, this was useful for us; from these measurements
we were able to extract the relevant interaction parameters for the collective spin model.
In this section we consider two experimental observables that do have a sensitivity to quantum correlations, and thus require a treatment beyond the mean-field approximation. These two
observables are the Ramsey fringe contrast, and the quadrature dependence of the spin noise. We

204find that in order to accurately treat the Ramsey fringe contrast decay, we must include higher
order corrections due to virtual motional excitations. We furthermore discuss a method to capture
the beyond mean-field dynamics of the model using the truncated Wigner approximation. These
theoretical methods are briefly discussed in the next two sections. We then describe the measurements that confirm the observation of quantum many-body effects that necessitate a beyond
mean-field description.

5.8.1

Perturbative corrections to the spin model

In the derivation of the collective Hamiltonian, we disregarded mode-changing terms, where
the final occupied motional states are different that the original ones (see Section 5.3.2), because
they were energetically disallowed by the trap anharmonicity. We recall that the only allowed
processes were {ni , nj } → {ni , nj } (direct) and {ni , nj } → {nj , ni } (exchange). However, theinclusion of virtual processes introduce higher-order corrections of order (S z )3 . A similar effecthas been observe in interacting bosons, where these processes manifest themselves as an effective
three-body interaction [299].
The additional terms can be included using second order perturbation theory. In orderto proceed, we separate the spin system into two manifolds comprised of spin wavefunctions andmotional states. The first manifold is what we considered originally, namely that no mode-changing
collisions have arisen and the total energy is conserved. Specifically, we write this manifold as[125, 300]
|Φ~Σσ~n i = |σn1 , σn2 , . . . , σnN i,

E0tot ≡

N
X

Enj .

(5.89)

j=1

Here, the σnj terms label the spin in the g-e basis for the particle in the nj motional state. The
second manifold, which is connected to the first via the many body Hamiltonian, is|Ψ~Υσ~k i = |σk1 , σk2 , . . . , σkN i,

Ek̃tot ≡NX
j=1

Ekj 6= E0tot .

(5.90)

Since these two manifolds are connected by a virtual excitation process, these can be included via

205
second-order perturbation theory. The effective Hamiltonian, Ĥ (2) , is given by
(2) Σ
hΦ~Σ
|Φσ~0 i = −σ~n |Ĥ~nΥ ihΨΥ |Ĥ|ΦΣ i
X hΦ~Σ
~σ~σ~
σ~n |Ĥ|Ψ~
σ~0
k

k

E tot − E0tot
k̃

σ~k

~
n

,

(5.91)

When Ĥ S2 is written in terms of spin operators and is projected into the collective Dicke manifold
as above, it can be shown [125, 300] that it gives rise to terms proportional to Ŝ z ,(Ŝ z )2 and (Ŝ z )3 .
Among those corrections, the first two can be absorbed into the spin Hamiltonian given by Eq. 1,
and are included in the measured quantities χ and C. Thus, only the third correction proportional
to (Ŝ z )3 gives rise to additional corrections.

5.8.2

Truncated Wigner approximation

In order to solve fo the many-body dynamics beyond the mean-field level, several approaches
can be used. We note that, in general, solutions to the elastic part of the collective Hamiltonian
are well-known (e.g., [280]). The difficulty comes when including the inelastic two-body loss, which
requires a master equation approach. We note that under certain approximations, the masterequation approach restricted to solving the collective part of the dynamics governed by Eq. 5.68,
while computationally intensive, is still tractable [125, 300]. A master equation approach for solving
Eq. 5.65 in the presence of inelastic losses for the experimentally relevant atom numbers is notpossible due to the exponential scaling of the Hilbert space. Here, we describe a flexible andpowerful approach to treating beyond mean-field effects in the spin model.
For a spin system with N  1, we can write [301, 300]hÔ (t)i 'Z

d3 S0 W (S0 ) OMF (t) ,(5.92)

where OMF (t) is the time-dependent mean field solution for the operator expectation value, andW (S0 ) is the Wigner function for the spin system’s initial state. This forms an approximation
known as the Truncated Wigner Approximation (TWA) [301]. For
S0 ≡ (S0x , S0y , S0z ) =N(sin θ1 , 0, − cos θ1 )2

(5.93)

tal counts after206

aRamsey0.8sequence
0.6

τ

θ1

0.4

c

1.0

π/2
2π/3
π/2
π/3
2π/9

0.8

0.2
56.8680

2.0

x10

3

Contrast

0.0

56.8690

Excitation fractionb
C (τ2 )C (τ1 )

0.6
0.4
0.2
0.0
0

200

400

600

800

Ramsey time (ms)Phase

Figure 5.23: Ramsey contrast decay measurement scheme and the failure of mean-field model.
(a) The sequence employed to measure the contrast decay. An initial pulse tips the Bloch vector
by an angle θ (the pulse area). (b) Examples of contrast extraction from measured excitation
probabilities. The red points are an example of contrast data taken with very little dark time,
while the blue line is taken with longer dead time. Even though the laser phase noise washes
out the sinusoidal nature of the fringe, its amplitude is still resolvable. (c) Failure of the mean
field model (dashed lines) to match the experimentally observed Ramsey fringe contrast decay as afunction of Ramsey time τ . The legend indicates the initial pulse area used for the measurement.

(i.e., the initial state is prepared in the x-z plane) W (S0 ) is given by [301, 300]
W (S0 ) =

2
πN



y 2

2

δ (S0z cos θ1 − S0x sin θ1 + N/2) e−2(S0 ) /N e−2(S0 sin θ1 +S0 cos θ1 ) /N .
zx(5.94)

We note that if we wish to compute products of operators, such as the case where Ô = Ô1 Ô2 , thatEq. 5.92 still applies, with OMF (t) = O1MF (t) O2MF (t).
This approach successfully describes the evolution of the expectation values of spin operators 2hŜ x,y,z i as well as products of spin operators such as h Ŝ z i, hŜ x Ŝ z i etc. in the limit where√N χτ  1 [125, 300].

207
5.8.3

Ramsey fringe contrast decay

Many-body effects can manifest themselves as periodic collapses and revivals of the Ramsey
fringe contrast, which reflect the quantized structure of the many-body spectrum. Specifically, fora fixed N and neglecting both the losses and the (Ŝ z )3 term, the amplitude of the Ramsey fringe√
evolves as Z N −1 with Z 2 ≡ 1 − sin2 (θ1 ) sin2 (χτdark ). At short times (i.e., N χτdark  1), the2 /2 [125]. On the contrary, the mean-field model
contrast is approximately 1−(N − 1) χ2 sin2 (θ1 )τdark

at the single-site level (with fixed N ) predicts no decay of the Ramsey fringe contrast, as a magnetic
field B(N ) can only lead to a pure precession of the collective Bloch vector. By summing over theatom distribution among lattice sites and properly treating two-body loss during the Ramsey darktime, the mean-field model does show a decay of the contrast, as shown in Fig. 5.23c. However,this decay is associated mainly with dephasing arising from different precession rates exhibited bysites with different N and, as seen in Fig. 5.23c, fails badly for θ1 > π/2. This dephasing effect ismost relevant for low excitation fractions where B(N ) is large and the inelastic loss is suppressed.
In order to observe beyond mean-field effects, we performed measurements of the Ramsey
fringe contrast for different experimental conditions, as illustrated in Fig. 5.23. The initial pulse
z i. Here,area θ1 = ΩTR , was chosen such that 0 < θ1 < π, controlled the initial value of hŜtot
z is the sum of Ŝ z over the ∼ 100 relevant sites, such that −N /2 ≤ hŜ z i ≤ N /2, where
Ŝtot
tot
tot
tot

Ntot is the total number of atoms loaded into the lattice. After applying a pulse of area θ1 , weallowed the system to evolve for time τ . We then applied a final pulse of area π/2 and measured
the resulting excitation fraction as a function of the optical phase of the second pulse relative to thefirst pulse. The Ramsey pulse durations were < 6 ms, satisfying Ω  N χ, to suppress interaction
effects during the pulses. We applied the final π/2 readout pulse with a variable relative opticalphase of 0◦ – 360◦ . We recorded the fraction of excited atoms as a function of the readout phase,
which constituted the Ramsey fringe.
For τ & 100 ms, there is a significant additional random phase added due to the frequency
fluctuations of the ultrastable clock laser. A given excitation fraction measurement pi will yield pi =

a

1.0

0.6
0.4
0.2
0.0
0

0.6
0.4
0.2

200

400

600

0.0
0

800

Ramsey time (ms)Contrast

Contrast

0.4

0.2

0.2

600

0.0
0

800

Ramsey time (ms)

0.4

0.2

0.2

400

600

Ramsey time (ms)

400

600

800

f

0.6

0.4

200

200

0.8

2π/3
π/2

0.6

0.0
0

3π/4
2π/3
π/2

1.0

Contrast

Contrast

0.8

dRamsey time (ms)

e

1.0

800

0.6

0.4

400

600

0.8

0.6

200

400

1.0

3π/4
2π/3
π/2

0.8

200

Ramsey time (ms)

c

1.0

0.0
0

2π/3
π/2
π/3
2π/9

0.8

Contrast

Contrast

1.0

2π/3
π/2
π/3
2π/9

0.8

208

b

800

0.0
0

2π/3
π/2
200

400

600

800

Ramsey time (ms)Figure 5.24: Ramsey fringe contrast decay vs. initial pulse area. The pulse area values for the dataand corresponding theory are given in the legends of the plots. Error bars represent the statisticalerror of each contrast measurement, and thus do not account for systematic drifts that occur overthe course of the experiment. The solid lines (top panels) are the many-body calculations, while
the dashed lines (bottom panels) are using the mean-field approximation of the theory. The manybody model and the mean-field approximation agree in the limit of small initial pulse area (i.e.,
Bloch vector polar angle), but disagree for pulse areas & π/2. This is an important confirmation of
the dominance of many-body effects in this parameter regime. (a, b) νZ = 80 kHz, νR = 450 Hz,
and Ntot = 4000; (c, d) νZ = 65 kHz, νR = 370 Hz, and Ntot = 4000; and (e, f ) νZ = 80 kHz,νR = 450 Hz, and Ntot = 1000.

Contrast

209
1.0

With (S z ) 3 {

2π/3
π/2

0.8

Without (S z ) 3 {

2π/3
π/2

0.6
0.4
0.2
0.0
0

100

200
300
400
Ramsey Time (ms)

500

600

Figure 5.25: Effect of the (S z )3 term on the Ramsey fringe contrast. Its inclusion improves agreement between the measured fringe contrast and the theory.

C sin2 (∆φi ), where C is the contrast and ∆φi is the ith realization of the both deterministically andrandomly varied laser phase. Here we have assumed that the contribution of quantum projection
noise is small in comparison, which is true for all experimental conditions we explore. By analyzing
Var [p] = C 2 /8, and assuming a uniform distribution of ∆φi , we obtain the contrast in a way thatis insensitive to the laser noise, yielding a powerful technique for exploring many-body effects.
We explored three distinct experimental conditions to rule out single-particle decoherence
mechanisms and thoroughly test the model. The first condition represents the typical operating
parameters of the lattice clock, with Ntot = 4×103 and νZ = 80 kHz. In the second case, we reduced
the lattice intensity such that νZ = 65 kHz, which results in a reduction of the density by a factor of∼1.8. Finally, we maintained νZ = 80 kHz but reduced the atom number to Ntot = 1 × 103 . Under
all conditions, the predictions of the collective many-body model, calculated using the TWA as
described in Section 5.8.2, are supported by the experimental observations (Figs. 5.24a, 5.24c, and5.24e). The inclusion of the (Ŝ z )3 correction improves the theory-experiment agreement, especiallyfor pulse areas θ1 > π/2 and for the high-density conditions, as shown in Fig. 5.25. While the need210
for the (Ŝ z )3 was rigorously derived (see Section 5.8.1), its magnitude was left as a fit parameter.
We also observe a striking breakdown of the mean-field model for θ1 & π/2 where many-bodycorrections are expected to be dominant.

5.8.4

Quadrature-dependent noise measurement

The frequency shift, lineshape, and Ramsey fringe contrast are quantities that all depend on
the first-order expectation values of the spin operators hŜ x,y,z i. We now turn our attention to the
distribution of quantum noise, which depends on the second-order moments of the spin operators, 2e.g., h Ŝ x i − hŜ x i2 , hŜ x Ŝ z + Ŝ z Ŝ x i − 2hŜ x ihŜ z i, etc. Given that the form of the Hamiltonian in
Eq. 5.68 is known to produce squeezed and entangled states [280], the distribution of the quantum
noise becomes a compelling measurement to probe physics beyond the mean field. However, until
now, optical clocks have lacked the requisite laser precision to probe the quantum noise distribution.
The recent advance in the Sr clock laser has allowed access to this regime.
To minimize single particle decoherence effects and dephasing due to the distribution of site
occupancies, we added a spin-echo pulse to the Ramsey sequence. As a result, the sensitivity to
low-frequency laser noise was reduced at the expense of increased sensitivity to high-frequency
laser noise. With atoms initialized in |gi, we followed the pulse sequence shown in Fig. 5.26 tomanipulate and measure the spin noise of the many-body state. The theoretical evolution of thespin distribution using the TWA is shown in more detail for evolution times of 0 ms–60 ms is shownin Fig. 5.27. For each value of the final rotation angle, representing a specific quadrature in whichz i/N
we measure the spin noise, we repeatedly recorded hŜtot
tot via measurements of the final atomic

2
z
2 −
excitation fraction after the Ramsey sequence. From the data, we determine σ 2 ≡ h Ŝtot
i/Ntotz i2 /N 2 by analyzing the pair variance for successive measurement of hŜ z i/N . We note thathŜtottot
tot
tot

the quantum limit of σ 2 is important for defining the ultimate stability of lattice clocks [43]. For
an ideal coherent spin state of the entire ensemble, the standard quantum limit of σ 2 is given by
2 = p (1 − p) /N , where p is the probability of finding an atom in the excited state, and can
σsql
tot

be estimated as p = hŜ z i/Ntot + 1/2.

211

t=0

x̂

τdark/2π/2−x̂τdark/2

π

ẑ

ẑ

ŷ

φ

ẑ

x̂

x̂

−ŷ

ŷ

−x̂

−ŷ

Figure 5.26: Spin noise measurement sequence. An initial pulse prepares a coherent state along ŷ,which then evolves for τdark /2. An echo pulse then rotates the many-body state 180◦ about −x̂.
After and additional evolution time of τdark /2, a final pulse rotates the state about −ŷ and the spinnoise is measured. The many-body state depicted here represents the spin evolution of a 20 atom
ensemble in a single trap site with τdark = 40 ms.

We performed quadrature-dependent spin noise measurements, as described above, at a given
target atom number Ntot = 1 × 103 or Ntot = 4 × 103 . During the course of these measurements, wetypically observed slow fluctuations of the atom number on the order of ±10% as we operated theexperiment and measure spin noise over the course of ∼10 hours. In most instances, these fluctuations were negligible due to the normalization techniques we employ as part of clock spectroscopy.
However, the atomic spin noise depends directly on the atom number, and a slowly varying atom
number could result in unintended systematic biases. Specifically, spin noise for the coherent spin√state typically considered in optical clocks scales as 1/ Ntot . Thus, the deviations in atom number
can cause variations on the order of ±5% in the measured spin noise. Ideally, these fluctuations are
randomly distributed and should not result in interpretation as a false-positive for non-trivial spinnoise correlations. In the unlikely possibility that these fluctuations were correlated with a specificmeasurement quadrature, they could cause a spurious phase shift in the spin noise minimum. We

212

20 ms 40 ms
0 ms
60 ms
0 ms
60 ms

Figure 5.27: Bloch sphere evolution as a function of dark time for Ntot = 20. Due to the two-bodyinelastic loss, the Bloch vector shrinks in length as a function of hold time. The lines represent the
mapping of a contour of the spin probability distribution within the truncated Wigner approximation.

thus take care to analyze the data in a way that is immune to this potential bias. In the following
section, we describe our process for removing the variability in the spin noise data due to a slowly
fluctuating total atom number.

5.8.4.1

Data analysis methodology for measuring quadrature-dependent noise

z i/N
A given measurement of hŜtot
tot is accomplished by independently measuring Ne(g) , thenumber of atoms in the excited (ground) state after a single Ramsey experimental sequence (seez i /N i , by measuring the ith value ofChapter 2, Section 2.5). We determine its ith value, hŜtotitoti ) and obtain
Ne(g) (which we denote as Ne(g)zihŜtotii /Ntot=Nei− 1/2.
Nei + Ngi

(5.95)z i, denoted {hŜ z i , . . . , hŜ z i , . . . hŜ z i } , we estimate
From the jth set of measurements of hŜtot
tot 1
tot i
tot nj jz )2 i/N 2 − hŜ z i2 /N 2 using a pair variance, such thatσj2 ≡ h(Stot
tot
tot
tot
i=nj

σj2 =

2
X
1
z
z
hŜtotii+1 − hŜtotii .
2 (nj − 1)i=1

(5.96)

213
For white noise, the pair variance is a good estimator for the standard deviation [45], while remaining insensitive to noise processes that only manifest themselves on long time scales. The number ofmeasurements in a set, nj , was typically nj ' 80. For a given measurement quadrature, we average
the results of many such measurement sets to produce one experimental data point.
In order to maintain insensitivity to slow fluctuations in atom number between sets j and j 0 ,we consider the standard expression for quantum noise for the case of a coherent spin state, σsql ,which is expected in the absence of many-body interactions. The explicit goal is to remove any
mechanism by which the trivial case—where the spin noise is described by σsql —can mimic the
many-body effect we predict from the theory. We calculate the jth value of σsql as


j
σsql

2

j
= pj (1 − pj ) /Ntot,(5.97)




n
n
n  
where pj = Mean {Ne1 / Ne1 + Ng1 , . . . , Nei / Nei + Ngi , . . . , Ne j / Ne j + Ng j }j . We additionally consider a technical noise term, which represents the effect of intrinsic technical detection noise,given by ∆sj . This noise is characterized by a separate measurement. The detection noise accounts
for 10% of the observed noise at typical low atom numbers, while at high atom number it is only
∼ 1% of the observed noise, and is therefore negligible. It is quadrature-independent in all cases.
2jFrom the σj2 , we subtract the atom-number-dependent σsql
such that2jσ˜j 2 = σj2 − σsql
− ∆s2j .

(5.98)

Here, σ˜j 2 represents only the effects of non-trivial spin noise and laser noise.
The many-body theory for a given measurement condition is calculated at fixed atom number. To facilitate comparison with the many body theory, we add a noise term back to σ˜j 2 that
2 for the mean atom number over the entire data set, σ 2 . We emphasize that
corresponds to σsql
sql
2 is a constant number, with no quadrature dependence. The many-body theoretical prediction
σsql
2 . Ultimately, the net
is calculated based upon the same mean atom number used to calculate σsql

effect of this process is to remove the variability due to slow fluctuations in atom number, but toretain the part of the noise that departs from σsql due to both laser noise and many-body effects.

214
5.8.4.2

Observation of spin-correlations

We performed measurements under three conditions in order to probe the time evolution of
the quantum noise distribution: τdark = {20 ms, 40 ms, 60 ms}. For these dark times, we used
π-pulse times of TR = {10 ms, 20 ms, 20 ms}, respectively. We found that utilizing long π pulses
reduces the sensitivity to spurious high-frequency components of laser noise. As shown in Fig. 5.28,
for Ntot = 1 × 103 , the quantum noise contribution to the spin noise is comparable to that of thelaser noise. However, with Ntot = 4 × 103 , the laser noise is responsible for a larger fraction of thenoise in repeated measurements of hŜ z i. There are qualitative differences between the low and highatom number cases; for example, for Ntot = 4 × 103 with τdark = 20 ms and 40 ms, we observeda phase shift for the minimum of the spin noise. We have verified that in the limit of small τdark ,the phase shift is no longer prevalent. To compare the predictions of the full many-body masterequation with the experiment, we add the effect of laser noise in quadrature with the calculated spinquantum noise. We additionally find it necessary to fully treat the effects of interactions during thelaser pulses, which can still be accomplished within the TWA formalism. As shown in Figs. 5.28
and 5.29, the theory predicts a direction and magnitude of the phase shift of the noise minimumthat agree with the experimental observations. The theory also predicts significant spin noise forrotations near ±90◦ due to an effect analogous to anti-squeezing. Despite the high sensitivity tolaser noise at this rotation quadrature, the measurements of the total noise are consistent withtheory.

5.9

Conclusion
In this Chapter, we have explored the many-body nature of interactions between atoms in the

87 Sr lattice clock as revealed in the density shifts, lineshape distortion, and Ramsey fringe contrast

decay. We have shown evidence that spin-correlations develop during the Ramsey spectroscopy via
measurement of a quadrature-dependence of the spin noise. Interestingly, these spin correlations
still manifest themselves in the presence of the excited state loss mechanism. In a sense, the lattice

215

1.5

a

b

c

d

e

f

σ 2 × 103

1.0

0.5
0.0
1.5

σ 2 × 103

1.0

0.5
0.0
2.0

σ 2 × 103

1.5
1.0
0.5

0.0 ◦
− 135 − 90◦ − 45◦0◦Rotation

45◦

90◦ 135◦ − 135◦ − 90◦ − 45◦0◦Rotation

45◦

90◦

135◦

Figure 5.28: Spin noise vs. rotation quadrature. In all plots, the dashed line is the pure laser
noise extracted from a fit to the data. The solid line is the laser noise plus the full many-body
prediction of the spin noise. This full theory is simultaneously fit to both the low and high atom
number curves to extract the commohn laser noise for a given dark time. Vertical arrows indicate
significantly phase-shifted minima in the experimentally measured spin noise, consistent with thepredictions of the many-body theory. Plots (a, b): Spin noise for Ntot = 1×103 and Ntot = 4×103 ,respectively, with an interaction time of τdark = 20 ms and a π-pulse time of 10 ms. A clear phaseshift is observed near the origin for plot (b), the high atom number case. (c, d): Spin noise for
Ntot = 1 × 103 and Ntot = 4 × 103 , respectively, with τdark = 40 ms and a π-pulse time of 20 ms. In
plot (d), a clear phase shift is also observed near the origin. (e, f ): Spin noise for Ntot = 1 × 103
and Ntot = 4 × 103 , respectively, with τdark = 60 ms and a π-pulse time of 20 ms.

216

0.5

0.0

0.5

b

σ 2 × 103

σ 2 × 103

a

− 45◦

0◦

Rotation

45◦

0.0

− 45◦

0◦

Rotation

45◦

Figure 5.29: Magnified in spin noise vs. rotation quadrature. These plots show the regions marked
with arrows in Figs. 5.28b (a) and 5.28d (b)clock can be considered an “open” quantum system and can thus open up experimental explorationof the intricate competition between coherent interactions and dissipation as a means to enhance andengineer many-body correlations—an emerging theme in modern quantum science [302, 303, 304].
The capabilities demonstrated here, such as near-quantum limited spin resolution, will develop
into important tools to explore the rich interplay between many-body effects and dissipation inalkaline-earth atoms confined in optical lattices, and possibly other systems such as polar molecules[70, 71, 72, 305, 306]. Furthermore, as a matter of practical importance, these techniques could
be used to generate interaction-induced spin squeezing in an optical lattice clock [77]. Although
the investigation described here is restricted to nuclear-spin-polarized gases, exploration of similar
many-body effects in a clock making use of additional nuclear spin degrees of freedom is a promising
first step towards the investigation of SU(N ) orbital magnetic models in ultracold atoms and future
experiments are already being prepared to explore SU(N ) physics using the precision of the 87 Sr
optical lattice clock.

Chapter 6

Measuring laser noise via quantum state manipulation

It is the goal of this chapter to present a general theoretical treatment of laser noise inmulti-pulse spin echo sequences. This analysis yields information concerning the laser frequency
noise power spectral density. The consideration of the effect of noise during these operations canbe important for quantum state manipulation, as well as for pulse sequences utilized to study
many-body effects as in the previous chapter.

6.1

Introduction to the general formalism
In Section 5.8.4 of the previous chapter, we utilized the spin echo technique, which allows asample of inhomogeneously broadened quantum systems to “rephase” after an appropriate periodof time evolution. This technique was first demonstrated in the context of nuclear magnenetic
resonance spectroscopy [307]. In the case of the lattice-trapped 87 Sr atoms, the different density
shifts between each lattice site introduced an effective inhomogeneous broadening that the spinecho was able to remove, thus leaving only the nonlinear part of the Hamiltonian. The spin echo
technique represents a simple example of the powerful technique of dynamic decoupling, where suchecho pulses can be used to decouple a quibit from a noisy bath [308, 309, 310, 311, 312, 313]. If thebath itself is a quantity to be measured, such as in the case of magnetometry, where the “bath”
might be a time-varying magnetic field, such dynamical decoupling pulse sequences can be used
to measure field amplitudes at specific Fourier components [314, 315] and even phase quadratures[316].

218
In this chapter, our goal is not to measure the background fluctuations, but rather to measure
the frequency spectrum of the local oscillator laser; we expect that in comparison, any backgroundfluctuations (e.g., magnetic field noise) will be small compared to the frequency noise of the clock
laser. The techniques we will use draw direct analogy to dynamical decoupling and spin manipulation because the Hamiltonian for a two-level atom in the presence of a detuned laser field can be
written (see Appendix A)
1
1
Ĥ = ~Ω (r) σ̂φ − ~∆ (t) σ̂z .
2
2

(6.1)

In this Chapter, we treat the atoms as non-interacting two-level systems governed by the above
Hamiltonian, which maps directly onto the case of a spin-1/2 qubit in a magnetic field. This is
the case considered in, e.g., [311]. Here, the laser detuning ∆ (t) = ωL (t) − ωa takes the placeof the magnetic field term (often called an “effective” magnetic field). The Rabi frequency is themagnitude of the interaction between the atoms and the laser field, and φ is the relative phase of thelaser field. We can change this phase by applying deterministic phase shifts to the light, and thus
we choose to keep it separate from ∆ (t) (instantaneous phase changes of the laser can be inserted as
Dirac Delta functions in ∆ (t), if desired [36]). Throughout this Chapter and treatment, we assume
that the atoms are spin-polarized, and we will only consider a single spin state (mF = +9/2).
Our theoretical approach will be as follows: for a given pulse sequence, the spectroscopy
laser may be on for a small segment of times (e.g., to produce the Ramsey pulses). As a result,
we can split up the evolution into two different stages in the limit of where the Rabi frequency ismuch larger than ∆ (t), which is almost always the case for multi-pulse sequences where the Rabi
frequency  2π × 10 Hz. The two phases of evolution are(1) Ω (t) = 0. Here the evolution is determined solely by the −∆ (t) σ̂z term of the Hamiltonian.
(2) Ω (t) = Ω0  ∆ (t). In this case, we can treat ∆ (t) using the theory of time-dependent
perturbations.
In the end, we will effectively stitch the solutions together to determine the evolution of the wavefunction.

219

a

b
ŷ

x̂
π/2

π

ẑ

−π

π

π/2

x̂

Θ

ŷ

Figure 6.1: The spin echo sequence being considered. (a) The spin echo pulse sequence we consider
here. First, a π/2 pulse rotates the quantum state to point along |yi [red arrow in (b)]. Then,a series of echo pulses, which alternate between π rotations about x̂ and −x̂, constitute the echo
sequence [blue arrow in (b)]. Finally, a final π/2 pulse is applied about ŷ (green). This final pulse
serves to rotate variations in the azimuthal angle that are due to local oscillator noise onto thepolar angle. (b) Bloch sphere depiction of the rotation operations. The colors in (a) correspond tothe operation depicted by the same color in (b). Measurements of total excited state population,
fexc reveal fluctuations in polar angle Θ with respect to the x-y plane.

Finally, we mention that although our treatment is completely general, and can be applied
to arbitrary pulse sequences, in this chapter we consider only periodic pulse sequences. This isbecause periodic sequences are the most experimentally relevant types for measuring noise at a
specific Fourier frequency. They also have clean closed-form analytic solutions. Specifically, we
consider a π/2 preparation pulse, followed by a series of even spaced π-pulses. This sequence is
depicted in Fig. 6.1a, and the corresponding rotations on the Bloch sphere in Fig. 6.1b. We notethat the sequence with evenly spaced pulses is different from the typical multi-pulse spin-echo
sequence, where the spacing between the first and last π pulses and the preparation and readout
π/2 pulses are half the spacing between π-pulses1 .
1

The timings for the normal echo sequence are so chosen so that there is no residual sensitivity to dc magnetic
fields/laser noise.

220

6.2

Limit of infinitely short pulses
In this section we consider the limit where the spin manipulation pulses are infinitely short,such that we do not need to consider dynamic noise processes occurring during the pulse. During
the time when Ω (t) = 0, the formal solution to the Hamiltonian of Eq. 6.1 is given by

 Z t

0
0
dt ∆ t ψ (t0 ) ,ψ (t) = R̂z −

(6.2)

t0

where R̂z (Φ) is simply the SU(2) rotation operation about the z-axis, given byiR̂z (Φ) = exp − σ̂z Φ .
2

(6.3)

Therefore, a general solution to a spin echo sequence as depicted in Fig. 6.1a is given byψ (t) = Ry [π/2] R̂z [Φn ] R̂x [π] R̂z [Φn−1 ] . . .
R̂z [Φ2 ] R̂x [π] R̂z [Φ1 ] R̂x [π/2] ψ (t0 )(6.4)

In this particular case we do a final rotation about y as the last pulse. The quantity Φi is given by
Z ti+1

Φi = −

dt0 ∆ t0



(6.5)

ti

Equation 6.4 can be simplified in the limit of perfect echo pulses toψ (t) = R̂y [π/2] R̂z [Θ] R̂x [π/2] ψ (t0 )(6.6)= R̂x [Θ] R̂x [π/2] ψ (t0 ) .

(6.7)

In the last equality, the accumulated rotations about z have been converted to a rotation about
x by the final π/2-pulse about y. Thus, the atomic population becomes a readout of laser noise.
Specificically, in Eq. 6.7, Θ is given byΘ = Φ1 +n−1
X(−1)j Φj + (−1)n Φn ,

(6.8)

j=1

where n is the total number of echo pulses. Equation 6.8 can be simplified to
Θ=−

Z t
t0



dt0 r t0 ∆ t0 .

(6.9)

221
1.0

r (t)

0.5
0.0
0.5
1.0
0.5

0.0

0.5

1.0

1.5

t/T0
Figure 6.2: The sensitivity function, r (t) of a four-pulse echo sequence in the limit of infinitely
short π-pulses. The x-axis has been normalized to T0 , the total duration of the echo sequence.

Here, r (t0 ) is a sensitivity function that in the limit of infinitely short pulses looks like a squarewave train. An example of a three-pulse echo sequence is shown in Fig. 6.2.
Ultimately then, in the limit of perfect pulses, our echo sequence under consideration has
the effect of integrating frequency fluctuations of the laser with the weighting function r (t) andthen converting these azimuthal fluctuations into polar angle fluctuations that are characterized
by the angle Θ. We have only considered a single experimental cycle in this section, it will be
our goal in the next section to treat the noise over many experimental runs and predict hΘ2i i
and h(Θi+1 − Θi )2 i/2 (Allan variance), where i represents the ith measurement of Θ via a direct
measurement of Sz (via measurements of atomic excitation fraction).

6.2.1

Computing variances from the sensitivity function

A large class of time-domain filtration operations can be described via a transfer function
approach [45], where, in contrast to performing operations on the time-domain data, a power
spectral density is weighted by the appropriate transfer function in order to calculate statistical222
quantities such as the variance. We consider as our experimental observable the accumulated phasedifference between a Ramsey atomic coherence and the probe laser that has been rotated into polar
angle by the final π/2 pulse. At the end of a given sequence, the polar angle on the Bloch sphere
for the ith measurement is (see Eq. 6.9)
Θm = −Z ∞∆ (t) rm (t) dt.

(6.10)−∞

Here, rm (t) is related to r0 (t), the sensitivity function for the first measurement and which isassumed to occur at t = 0 (e.g., the sensitivity function shown in Fig. 6.2) byrm (t) = r0 (t − mTc ) .

(6.11)

Here, Tc is the clock measurement cycle time, which includes the time necessary to prepare a new
atomic sample.

6.2.2

Computing the variance

As a first step, we might want to compute the quantity
Var [φ] = hΘ2i i − hΘi i2 = hΘ2i i.

(6.12)For the last equality, we assume that the that the noise has no mean value. We can rewrite hΘ2i i,
by making use of Eq. 6.10, ashΘ2i i =DZ ∞ Z ∞−∞E
dt1 dt2 ∆ (t1 ) ri (t1 ) ∆ (t2 ) ri (t2 ) .

(6.13)−∞

Equation 6.13 can be simplified in terms of autocorellation functions ashΘ2i i =Z ∞Z ∞
−∞
Z ∞

=

−∞

dt1 dt2 R∆ (t1 − t2 ) r0 (t1 ) r0 (t2 )dt (R∆ ? r0 ) r0 (t)(6.14)(6.15)−∞

where we have used the relationhξ (t1 ) ξ ∗ (t2 )i = Rξ (t1 − t2 ) ,(6.16)

223
where Rξ is the autocorrelation function. The star represents the convolution operation. We will
drop the “0” subscript on r0 (t) for the remainder of this section, although it is implied. In practice,
it will not matter which rm (t) we use, since rm (t) is just a time-shifted version of r0 (t).
Parseval’s theorem states that
Z ∞

Z ∞

∗

df X (f ) Y ∗ (f ) ,dt x (t) y (t) =−∞(6.17)−∞

where x (t) [y (t)] and X (f ) [Y (f )] are related by a Fourier transform. Thus Eq. 6.15 becomes,
Z ∞Z ∞dt (R∆ ? r) r (t) =−∞−∞df F [R∆ ? r] R∗ (f ) ,(6.18)

where R (f ) ≡ F [r (t)], and F represents the Fourier transform operation which we define, forcompleteness, asF [ξ (t)] =Z ∞dt e−i2πf t ξ (t) .

(6.19)−∞

Finally, we apply both the convolution theorem and the Wiener-Khinchin theorem to Eq. 6.18 toobtainhΘ2i i =Z ∞−∞df S∆ (f ) |R (f ) |2 .

(6.20)

Here, S∆ (f ) is the (two-sided) laser frequency noise power spectral density in units of angularfrequency. We can relate this to the more commonly reported laser frequency noise power spectral
density in units of Hz as S∆ (f ) = (2π)2 Sν (f ), leading to the final resulthΘ2i i = (2π)2

Z ∞
−∞

2

2

df Sν (f ) |R (f ) | = (2π)

Z ∞0df Gν (f ) |R (f ) |2 ,(6.21)

Where Gν (f ) is the single-sided power spectral density, as usual. At this point, the function r (t)is completely general, and this treatment applies to any valid r0 . We note that since the expression
depends only on the magnitude of R (f ), the expression is time-shift invariant.
In general, however, the variance is a bad way to characterize the noise. This is because it
will become infinite for noises that diverge as f → 0 for any function r (t) such thatZ ∞−∞dt r (t) 6= 0,(6.22)

224
or in other words if R (0) > 0. This is in fact one of the main reasons that the Allan varianceand similar methods have superseded the variance as a way to characterize time-domain stability.
So while it is helpful to go through the procedure to calculate the variance, for many sensitivity
functions r (t) we will get something that diverges due to the 1/f character of the laser frequency
noise. In the next section, we consider the connection between Allan Variance and the sensitivity
function.

6.2.3

Computing the Allan variance

As introduced in Chapter 1, the Allan Variance [44] is a useful tool for characterizing frequency noise. One of the reasons that it is so useful is that it does not diverge for many noise
processes that diverge as f → 0. We begin our calculation of the Allan variance by first considering
the two-sample case, and then generalize to the n sample case. The quantity we wish to computefor the two-sample Allan variance is2σΘ(Tc ) =

h(Θi+1 − Θi )2 i.
2

(6.23)

Here, Θi is defined as in Eq. 6.10. It is straightforward to see that we can perform the operation
Θi+1 − Θi by an extension of Eq. 6.10 if we make the replacement [317]ri (t) → r̃i (t) =ri+1 (t) − ri (t)√.
2(6.24)

We may thus write
Θi+1 − Θi
√
=
2

Z ∞
∆ (t) r̃i (t) dt.

(6.25)−∞

From this point, the same analysis of Eqs. 6.13–6.21 applies with r → r̃. The result for the two-point
Allan variance is1(Θi+1 − Θi )22Z ∞2= (2π)
df Sν (f ) |R̃ (f ) |2 .

σ 2 (Tc ) =−∞(6.26)(6.27)

225
Here, in analogy to Eq. 6.21, R̃ ≡ F [r̃ (t)]. For the two-point case, we can relate R̃ (f )2

to |R (f )|2

by an application of the Fourier shift theorem, such thatR̃ (f )

2

1 − ei2πf Tc=
2

2|R (f )|2 = 2 sin2 (πf Tc ) |R (f )|2 .

(6.28)

Here, Tc is the time between consecutive measurements of Θi .
If we wish to calculate the Allan deviation for longer averaging times, we make use of thedefinition of the Allan deviation
2 E
1D
Θ(i+1)n − Θin
2
!2
n−1
n−1
E
1D 1 X1X=
Θ(i+1)n+k −
Θin+k
.
2
n
n

2
σΘ
(nTc ) =k=0(6.29)(6.30)

k=0

It is then straightfoward to see that
r̃

(n)

n−1

n−1

k=0

k=0

1X
1X
(t) =
rn+k (t) −rk (t) .
nn(6.31)

Again, applying the Fourier shift theorem, we find thatR̃(n) (f )2=2

sin4 (πnf Tc )|R (f )|2 ,n sin2 (πf Tc )(6.32)

which reduces to Eq. 6.28 for n = 1. An interesting limit for Eq. 6.32 is where n → ∞. We obtainlim R̃(n) (f )

n→∞

2

=

comb (f TC )|R (f )|2 ,nTc(6.33)In this case we obtain∞2
σΘ(nTc ) →1 X|R (m/Tc )|2 Gν (m/Tc ) .
nTc(6.34)

m=0

Equation 6.34 has direct connections to the Dick effect, as seen in Chapter 2, specifically Eq. 2.62.

6.2.4

The sensitivity function R (f ) and experimental results

In the previous two sections, we showed that essentially any measurement of the varianceor Allan deviation of Θi is directly related to the Fourier transform of the time-domain sensitivity
function, r (t). Since the sensitivity function for the instantaneous pulse approximation is always a

226
10

a

0.200
0.150
0.100

σΘ (Tc )Gν (f )

1

b

0.1
0.10

0.01

0.1

1

10

100

1000

0.070
0.050

0.030
0.10 0.10
0.020
0.010
0.02

Fourier frequency (Hz)

*Dead-time free

*
0.05

0.10

0.15 0.20

Ramsey time (s)Figure 6.3: Diagnosing the laser spectrum with spectroscopy. (a) Laser noise spectrum assumed
for the Big ULE-stabilized laser (solid blue) compared to the measured frequency noise spectrum(red trace). The theoretical prediction of thermal noise-limited laser frequency noise for the footballcavity agrees with the measurement at low frequencies (dashed green), while the Big ULE thermal
noise floor is two orders of magnitude below (dashed black). (b) Prediction of Eq. 6.27 (solidblue) versus two-point Allan deviation of the optical clock polar angle Θ, which serves as a readout
for laser phase (black points). The horizontal solid black line is the measured/predicted quantum
projection noise for 2500 atoms. The dashed lines represent the purely thermal noise-limited casewith dead time (dashed black) and in the limit where the cycle time and the spectroscopy time are
the same (dashed grey).

periodic function of square waves as shown in Fig. 6.2, it is easy to find the Fourier transform forthe whole “train” by utilizing the Fourier shift Theorem. Then, all we need to know is the Fourier
power spectrum of a single square pulse, which is a sinc2 function. We find that the power spectrumof the time-domain sensitivity function for arbitrary numbers of infinitely short echo pulses is


h
i
1
πf T0
2 π
2
|R (f ; n)| = 2 2 sin(1 + n + 2f T0 ) tan
.
f π
2
1+n
2

(6.35)One important limiting case is where n = 0, which is the case of Ramsey spectroscopy. Here, we
obtain the Ramsey sensitivity function plotted given in Chapter 2 (and for r (t) corresponding to
Eq. 2.66)|R (f ; 0)|2 =sin2 (πf T0 )(πf )2(6.36)In late 2011, we utilized this sensitivity function to diagnose the laser noise spectrum. At
this time, the laser cavity was supported by long Teflon posts. We performed Ramsey spectroscopyon the clock transition with spin polarized 87 Sr using the sequence depicted in Fig. 6.1, but with

227
no echo pulses. A given measurement of Θi was obtained by measuring the population fluctuations
after the ithe spectroscopic sequence. For Θi ' π/2,
Θi ' 2pi .

(6.37)

Thus, by measuring a set of {pi } at a given T0 , we were able to measure σΘ (Tc ) for a variety ofspectroscopy times T0 . At the same time, we performed an optical beat with the “football cavity”[40]. While the low-frequency noise of the beat was dominated by the thermal noise of the footballcavity, we could see Fourier components at discrete frequencies as seen in Fig. 6.3a. We suspectedthat the first peak at 12 Hz was on the Big ULE cavity-stabilized laser and that it was caused
by a high-Q resonance in the support structure. By measuring σΘ (Tc ) as a function of Ramsey
spectroscopy time, T0 , we were able to confirm the existence of the 12 Hz peak from the characterof the oscillations in the noise as a function of spectroscopy time as seen in Fig. 6.3b. While themeasured data were completely consistent with the noise spectrum of the 12 Hz peak, we could
not specifically verify the presence of the other noise components at higher Fourier frequencies (welater verified these were due to acoustic noise and removed them by constructing the sound-proofbox). However, Fig. 6.3 was such strong confirmation of the problematic mechanical resonance that
we changed the laser suspension to the configuration described in Chapter 4.

6.3

Adding the effects of frequency noise during the pulses
As we saw in the previous section, the simple treatment of Ramsey pulse sequences yielded

insights into the laser noise spectrum, and confirmed the presence of an unwanted Fourier noise
component. In order to improve upon the technique, a natural extension is to include more echopulses in the sequence. However, experimentally we are limited in how much optical power wehave, making π-pulse times less that 1 ms difficult. As the number of echo pulses increases, theirfinite duration becomes an important modification to the transfer function derived in the previous
section. Also, it turns out that extending the time of the echo pulses effectively low-passes thefrequency sensitivity given by Eq. 6.35 to several or even tens of milliseconds, which can be useful.

228
Thus, we will now treat the case where the preparation and echo pulses are long enough that wemust treat laser noise during the pulse.
We again consider the Hamiltonian of Eq. 6.1. During the periods where Ω (t) = 0, theevolution corresponds to the rotation about the z-axis as before. Here, however, we consider thecase where Ω (t)  ∆, but not so much so that we ignore ∆ (t) during the laser pulse, as we did
before. In order to proceed, we split the Hamiltonian into two parts,
H = H0 + HI
H0 /~ =

Ω0
σ̂x ,
2

HI /~ =

(6.38)∆σ̂z2(6.39)

From the theory of time-dependent perturbations we can writeψ = e−iH0 t/~ ψI(6.40)dψI= eiH0 t/~ HI e−iH0 t/~ ψI .
dt

(6.41)

where

This becomes
i

dψI
∆ (t)
=−
[σ̂z cos (Ω0 t) + σ̂y sin (Ω0 t)] ψI
dt2Ĥeff (t)=ψI (t) .
~(6.42)

We can write the formal solution to Eq. 6.42 by using a Dyson series [131]. However, since thepulses are short and in general ∆ (t) × TPulse  1, it is sufficient to keep only the first order term in
the Dyson series, such that the time evolution operator for ψI , where ψI (t) = Û (t) ψI (t0 ) becomesσˆyÛ (t) ' 1 + i
2
+i

σˆz
2

Z t
t0
Z t
t0




dt0 sin Ω0 × t0 − t0 ∆ t0

(6.43)


dt0 cos Ω0 × t0 − t0 ∆ t0

(6.44)' R̂y (θQ ) R̂z (θI )(6.45)

229
Just to be explicit
θI = −Z t

θQ = −t0
dt0 cos Ω0 × t0 − t0 ∆ t0

Z t
t0




dt0 sin Ω0 × t0 − t0 ∆ t0

(6.46)(6.47)(6.48)

Thus, the total effect of the pulse plus noise isψ (tf ) = e−iH0 t/~ ψI = R̂x (A) R̂y (θQ ) R̂z (θI ) ψ (t0 ) .

(6.49)

Here A = Ω × (t − t0 ). We note that this is in agreement with the result obtained by Kotler et al.
[316]. Similarly, for the final rotation about y, we can write an expression for the error introducedin the rotation asψ (tf ) = R̂y (A) R̂x (−θQ ) R̂z (θI ) ψ (t0 ) .

(6.50)

This expression shows the importance of considering the laser noise during quantum state manipulations, as the laser noise couples to spurious rotations around both x and z axes when the rotation
is applied about y. This particular operation is important and relevant for the final rotation in thespin echo sequence shown in Fig. 6.1. It has also been used to measure both quadratures of theeffective magnetic field term, as parameterized by θI and θQ [316] (in this reference the “effective
magnetic field” was in fact a real magnetic field).
To illustrate the full model, we consider the echo sequence with only a single echo pulse. We
have taken a considerable amount of data using this sequence, as seen in Chapter 5. The time
evolution ishiψ (t) = Ry (A) Rx (−θQ ) R̂z (θI ) R̂z (Φ2 ) ×hiR̂x (π) R̂y (θQ ) R̂z (θI ) R̂z (Φ1 ) ×hiR̂x (π/2) R̂y (θQ ) R̂z (θI ) ψ (t0 ) .

(6.51)

Here, the composite operators in square brackets represent the first π/2 pulse, the middle π pulse,
and the final arbitrary rotation of angle A about y with noise included. The intermediate z-rotation

230

1.0

rt

0.5

T0

0.0
0.5
1.0

0.00

0.02

0.04

0.06 0.08
t ms

0.10

0.12

0.14

Figure 6.4: The time-domain sensitivity function r(t) for the echo sequence with 80 ms durationand 20 ms π-pulses.

operators are as before, where the dark-time evolution is described as a simple rotation about z asin Eq. 6.3.
If we assume that ψ (t0 ) = |gi, we can ignore R̂z ψ (t0 ). Similarly, if we ignore the R̂y operationof the second square bracketed term, which is justified if the state it is acting on is approximately
equal to an eigenstate of σ̂y (this approximation is valid when keeping only rotations to first order),then Eq. 6.51 reduces to 
ψ (t) = [Ry (A) Rx (−θQ )] R̂z Φ̃ × | − yi .
| {z }(6.52)|ψ(t0 )i

Here, Φ̃ is given by
Z t

0

0



dt sin θ t

Φ̃ =



0



Z t

∆ t =

t0



dt0 r t0 ∆ t0

(6.53)

t0

where
0



Z t0

θ t =


dt00 Ω t00 .

(6.54)

t0

This is exactly the result of [316] except that since we initialize our system in | − zi and they in |xi;
this difference is resolved by replacing sin → cos in Eq. 6.53. For a typical echo sequence, A = π/2

231
and thus the time-dependent Rabi frequency is given byΩ0 if 0 ≤ t < TRabi /2









0
if TRabi /2 ≤ t < (τ − TRabi ) /2








Ω0 if (τ − TRabi ) /2 ≤ t < (τ + TRabi ) /2
Ω (t) =
0if (τ + TRabi ) /2 ≤ t < τ − TRabi /2








Ω0 if τ − TRabi /2 ≤ t < τ








0
elsewhere(6.55)

This procedure is generalized to any number of echo pulses, where the Rabi frequency is apiecewise function corresponding to the periods where the laser is turned off or on, and r (t) is
given byr (t) = sin [θ (t)] .

(6.56)

Figure 6.4 shows the sensitivity function for a specific case of a single echo with a sequence durationof 80 ms and a π-pulse time of 20 ms. Qualitatively, this has the effect of rounding out the edges ofthe sensitivity function, which introduces a low-pass effect that “turns on” at the Rabi frequency.
Due to the periodic nature of the sensitivity function, we can apply the same trick that we did forthe case infinitely-short pulses: we use the Fourier shift theorem to derive the sensitivity function
for an arbitrary number of echo pulses as
2|R (f ; n)| = hΩ20
πf Ω20 − 4 (πf )3


i2


2
π
2πf cos (πf τ ) + Ω0 sin πf τ −
Ω0


×

sin2

π



2 (1 + n) (1 + 2f τ )cos2 (πf τ ).

(6.57)

Here, τ is the duration of one of the periodically repeated features of r (t) and is given by τ =T0 / (n + 1), where T0 is the total spectroscopy time and n is the number of echo pulses as before.
Figure 6.6 on page 233 shows a large array of sensitivity functions |R (f ; n)|2 for a sequence with
T0 85 ms, and a π-pulse time of 5 ms.

0.4

aGν (f )σΘ (Tc )

b

0.1

0.3
0.2
0.1
0.0
0.00

232

1

0.01
0.001
104
105

0.05

0.10

0.15

0.20

0.25

0.30

T0 (s)

106
0.01

0.1

1

10

100

Fourier frequency (Hz)Figure 6.5: Verification of improved laser noise spectrum with a spin echo sequence. (a) Measured
Allan deviation of polar angle Θ as a function of measurement time for a π-pulse time of 20 ms(markers). The three different solid lines correspond via color to the three noise spectra in (b). The
dashed black line is the quantum projection noise floor of the measurement. (b) The solid lines are
the three test laser noise spectra used to produce σΘ at left. The dashed line is Allan deviation2transfer function R̃ (f ) (See Eq. 6.28. Arbitrary units are used.) calculated for the case of finite
pulse time and for the measurement sequence with T0 = 120 ms.

We applied the calculated sensitivity function to test the new laser suspension and acoustic
isolation. Using a sequence with a 20 ms π-pulse and one echo, we measured σΘ (Tc ) as a functionof total echo sequence time T0 . In order to carefully account for fringe contrast decay, we were
careful to scale each noise measurement by the measured contrast for that given condition to makea proper estimate of set {Θi } for a given T0 . As also suggested by comparing the laser to thefootball cavity stabilized system, we found that the large spike at 12 Hz was gone. Also gone werethe large acoustic spikes at 50–100 Hz thanks to the sound-proof box (see Chapter 2, Fig. 4.21).
Figure 6.5 shows the measured noise and several test spectra. The old, noisy spectrum did noteven qualitatively fit the data. However, the laser was still above the noise floor predicted for a
flicker-floor stability of 1 × 10−16 . One way to account for this is to add a white noise floor. We
find that good agreement with the data is obtained for a laser frequency noise ofGν (f ) =

h−1
+ h0
f

(6.58)

where h−1 = 1.5 × 10−3 Hz2 and h0 = 3.3 × 10−3 Hz2 /Hz. This implies that the laser reaches athermal-noise dominated stability regime at Fourier frequencies below 450 mHz.

233
2

0.5
0.0

0.02

0.04

0.06

0.08

0.5

0.01

1.0

0.01

105

0.5

105

108

0.0

1011

0.5

|R (f )|

r (t)

1.0

1.0

0.1

Time (s)

1

10

100

1000

104

0.02

0.04

0.06

0.08

1.0

108
1011
0.1

0.01

1.0

0.01

0.5

105

0.5

105

0.02

0.04

0.06

0.08

0.5

108

0.0

1011

0.5

1.0

0.1

1

10

100

1000

104

0.02

0.04

0.06

0.08

1.0

0.1

0.01

1.0

0.01

105

0.5

105

108

0.0

10

0.5

0.04

0.06

0.08

11

0.5
1.0

0.1

1

10

100

1000

104

0.02

0.04

0.06

0.08

1.0

0.1

0.01

1.0

0.01

105

0.5

105

0.04

0.06

0.08

0.5
1.0

108

0.0

1011

0.5

0.1

1

10

100

1000

104

0.02

0.04

0.06

0.08

1.0

0.1

0.01

1.0

0.01

105

0.5

105

0.04

0.06

0.08

0.5
1.0

108

0.0

1011

0.5

0.1

1

10

100

1000

104

0.02

0.04

0.06

0.08

1.0

0.1

0.01

1.0

0.01

105

0.5

105

1.0

0.02

0.04

104

1

10

100

1000

104

1

10

100

1000

104

1

10

100

1000

104

1

10

100

1000

104

108

0.5
0.0

1000

1011

1.0

0.5

100

108

0.5
0.02

10

1011

1.0

0.0

1

1011

0.5
0.02

104

108

1.0

0.0

1000

108

0.5
0.02

100

1011

1.0

0.0

10

Fourier frequency (Hz)

1.0

0.0

1

0.06

0.08

108

0.0

1011

0.5

0.1

1

10

100

1000

104

1.0

0.02

0.04

0.06

0.08

108
1011
0.1

Figure 6.6: Overview of the sensitivity functions |R (f )|2 for increasing echo pulse number (increasing left to right, top to bottom). The upper left corner is the zero echo pulse condition (i.e.,
Ramsey spectroscopy). The bottom right-hand plot is the 11-echo pulse sequence. The horizontal
and vertical axes are all as labelled for the pair of plots in the upper left corner: the horizontal
axes of the plots are time in seconds and Fourier Frequency in Hz. The vertical plot axes are thetime-domain sensitivity function and its Fourier-space magnitude. Here, T0 =85 ms and the π-pulse
time is 5 ms.

234
6.3.1

Sensitivity function for Rabi spectroscopy

Finally, we briefly derive the Fourier-domain sensitivity function for Rabi spectroscopy. Rabi
spectroscopy is qualitatively different than Ramsey spectroscopy in that typically it is performed
with a static detuning ∆0 ' Ω0 such that
∆ (t) = ∆0 + δω (t) .

(6.59)

Thus, the perturbative approach used in the previous section is much more difficult to apply because
Eq. 6.1 becomes
1
1
1
Ĥ = ~Ω (r) σ̂φ − ~∆0 σ̂z − δω (t) σ̂z .
22
2(6.60)

Thus, the rotations are applied about non-orthogonal axes. While it is difficult to derive thegeneral rotation errors on the Bloch sphere, as we were able to do in the previous section, it
is straightforward to derive the population fluctuations caused by δω (t) for a given ∆0 . This isexactly the Rabi sensitivity function discussed in Chapter 2 and derived in Appendix A; it is definedby1δfexc =
2

Z T0

dtδω (t) rRabi (t; ∆0 ) .

(6.61)

0

Additionally, if the spectroscopy is performed such that the expected final population is 50%, thenmeasured population fluctuations correspond to polar angle fluctuations via Eq. 6.37.
The time-domain Rabi sensitivity function is (see Appendix A for a derivation)






∆0 Ω20ΩΩRabir(t; ∆0 ) = −sin (Ωt) − sin π+ sin(π − tΩ0 ) ,(6.62)Ω3Ω0Ω0pwhere the generalized Rabi frequency, Ω, is given by Ω = Ω20 + ∆20 . The magnitude of the Fouriertransform of rRabi (t) is given by
RRabi (f ; ∆0 )2=

∆2 Ω40Ω4 (πf Ω2 − 4π 3 f 3 )2




Ω sin

π2f
Ω0




sin

πΩ
Ω0




− 4πf cos

π2f
Ω0



2

sin



πΩ
2Ω0

2
(6.63)

When ∆0 ' ±0.80Ω0 , the final Bloch vector points in the x-y plane. In this limit, as alreadymentioned, we can calculate polar angle fluctuations on the Bloch sphere via measurement of
population fluctuations.

4
3

3

235

4

b

c

3

2

2

2

1

1

1

0

0

0

2

|R (f )| × 103

4

a

1

0

20

40

60

80

Fourier frequency (Hz)

100

120

1

0

20

40

60

80

100

120

1

0

20

Fourier frequency (Hz)

40

60

80

100

120

Fourier frequency (Hz)Figure 6.7: Linear scale plots of the Fourier-space sensitivity functions. As in Fig. 6.6, T0 =85 ms
and the π-pulse time is 5 ms. The number of echoes are one (a), five (b), and eleven (c). As seenin this figure, the sensitivity function is sinc2 -like and is sharply peaked. The width of the peakvaries inversely with the total sequence time.

6.4

Mapping the laser spectrum: multi-echo sequences
In this final section we describe continuing work to utilize the sensitivity functions developedin this chapter, and apply them to multi-pulse echo sequences. As this thesis was being written,
Sr1 teammates Mike Bishof and Xibo Zhang utilized a combination of Rabi, Ramsey, and echopulse sequences to map out the laser noise systematically [318]. As before, the spectrum wasconstrained with the help of the measured spectrum between the Big ULE system and the footballcavity. Several discrete resonances were added to the model of Eq. 6.58 to better match noise spikes
observed in the laser comparison frequency noise spectrum. As shown in Fig. 6.7, the Fourier-spacesensitivity functions are rather sharply peaked when viewed on a linear scale, and it is this feature,along with different regions of frequency sensitivity determined by the number of echoes, that makes
the transfer function a powerful diagnostic tool.
The final extrapolated spectrum is shown in Fig. 6.8a, and the corresponding noise plots
shown in Fig. 6.8b. For the single- and seven-pulse echo sequences, the computed deviation is
the standard deviation of the data sets and the theory corresponds to the same quantity. For
the Ramsey and Rabi measurements, the deviation used is the Allan deviation. The reason Allan
deviation was used for the Rabi and Ramsey cases is that the sensitivity function for the variancediverges with 1/f noise, but the sensitivity function for Allan deviation does not.

236

1

a

Deviation of Θ (rad)Gν (f )

0.1

0.25
0.2

b

Seven Echo Pulses
One Echo Pulse
Ramsey
Rabi
QPN

0.15

0.01

0.01

0.001
104

0.05

0.1

0.5 1.0

5.0 10.0

Fourier frequency (Hz)50.0 100.0

0

0.05

0.10

0.15

0.20

Total spectroscopy time (s)Figure 6.8: Multi-pulse extrapolation of laser noise spectrum. (a) Extrapolated laser spectrum
for the Big ULE-stabilized laser (solid black line). The 1/f flicker noise floor corresponding to astability of 1 × 10−16 is also shown (dashed red line). (b) Measured excitation noise (expressed
as noise in Θ) as a function of total spectroscopy time (T0 ). The pulse sequences are as indicatedin the legend. The bands indicate the range of uncertainty for the flicker and white noise floors
as indicated in the text. For the Rabi and Ramsey cases, the Allan deviation was computed. For
the echo sequences, the standard deviation of the measured {Θi } is reported. A measurement of
quantum-projection noise (QPN) is shown for reference.

Day-to-day fluctuations as well as discrepancies between different pulse sequences required
small changes in the fit parameters. Ultimately, the best fit parameters are h−1 = 1.5(5) × 103 Hz2and h0 = 3.3(3) × 10−3 Hz2 /Hz. The 1/f term agrees very well with the thermal noise floor
measured using the atomic clock correction signal as a frequency discriminator. This is gratifying
because those clock measurements were sensitive to the 1/f noise only at long time scales (seeChapter 4, Fig. 4.22), whereas these measurements are taken over a shorter timescale. As shownin Fig. 4.22, the flicker noise reaches 1.1(1) × 10−16 , which indicates h−1 = 1.6(2) × 10−3 Hz2 , ingood agreement with these measurements. We also note that this combination of flicker and white
noise indicates that the laser linewidth can be as narrow as 26 mHz [318].
The complete laser noise spectrum is directly relevant for the intercomparison reported in
Chapter 4. In Fig. 6.9 (page 238) we plot the Dick effect-limited clock stability using the newspectrum plotted in Fig. 6.8. As seen from Fig. 6.9, the Dick effect is dominated by the 1/f andwhite noise portions of the new spectrum. When we apply this spectrum to the conditions of thepclock intercomparison, we obtain that the Dick-effect-limited instability should be 5×10−16 / τ /1 s,237
which is close to the observed 4 × 10−16 /pτ /1 s, and approximately within the stated uncertainty

in the noise amplitudes. Thus, by characterizing the laser noise in such a detailed manner, future
clock experiments will have the benefit of precisely predicting the Dick effect or other spurious
Bloch sphere rotations.

6.5

Conclusion
In this chapter, we derived a general technique to treat errors arising from the local oscillator

frequency noise. We applied this formalism to a model laser spectrum and then used the measured
atomic population noise to adjust the model parameters. This was useful for demonstrating that thecavity support resonance was eliminated by changing the support strategy. We were additionallyable to demonstrate that the laser 1/f noise spectrum crosses a white noise floor at approximately
450 mHz and to verify and calibrate the presence of additional resonant features in the Fourier
spectrum.
These measurements show the promise of using spin echo techniques paired with opticalstandards to fully characterize the noise spectrum of ultrastable lasers. A natural next step will
be to develop a model-independent extension so that the noise can be mapped tomographically,
without need for a pre-existing model of the laser spectrum as obtained here by direct comparison
with a second system.
Finally, we note that these measurements give us confidence in something we have assumed allalong: that the atoms are “perfect” frequency references. While we know that there are perturbing
systematics, this is the first time that the clock atoms themselves have been put to such a stringent
test on short time scales. Since we only used a single spin state in these measurements, it is possiblethat some of the noise that was measured is due to the small ∼ 500 Hz/G magnetic sensitivity ofthe mF = +9/2. However, we note good agreement with the 1/f floor obtained from operating
the clock with both mF = ± spin states, as we did for the clock comparison measurements. Thus,
we do not foresee that technical perturbation to the atoms themselves will be an impediment topreaching QPN-limited instabilities of 1 × 10−16 / τ /1 s.

238

2  1015

Rabi and full noise
Ramsey and full noise
Rabi and flicker+white
Ramsey and flicker+white
Rabi and flicker

σy (τ = 1 s)

1  1015
5  1016

Ramsey and flicker2  1016
1  1016
QPN

5  1017

QPN

2  1017
0.01

0.02

0.05

0.10

0.20

Rab
i

Ram
sey

0.50

1.00

Spectroscopy time T0 (s)Figure 6.9: Dick effect-limited laser performance with the newly evaluated laser spectrum. Here,the loading time is 1 s, such that the cycle time Tc = 1 + T0 . We plot the quantum projection
noise (QPN) instability for Rabi and Ramsey spectroscopy (red and blue, respectively) and the
Dick-limited instability for a laser with flicker floor σy (τ ) = 10−16 for Rabi (solid black) andRamsey (grey) spectroscopy. As discussed in this chapter, the real laser spectrum has a white noise
component and discrete resonances in addition to the 1/f component. These specific cases are
p listedin the legend. In all cases the instability has been extrapolated to 1 s, so that σ (τ ) = σ (1) / τ /1 s.

Chapter 7

Conclusion

This final chapter summarizes the key results presented in this thesis, and we additionallydiscuss the outlook for the 87 Sr lattice clock in the context of both metrology and quantum manybody physics.

7.1

Summary of key results

Optical lattice clocks are powerful platforms for quantum measurement. As described in√
Chapter 2, the parallel interrogation of many atoms permits a N enhancement of measurement
precision, where N is the total number of particles in the clock, allowing fractional instabilities of√order 10−16 / τ to be within reach for even modest parameters. Yet until recently, only synchronousinterrogation of clocks had demonstrated the benefit of utilizing many atoms [234]. In this thesis
we described the development of an ultrastable laser system with fractional frequency instability of
1.1(1) × 10−16 . This system allowed the realization of independent clocks operating near this limit
set by quantum fluctuations. This represents an important milestone in the development of latticeclocks, and represents an order of magnitude improvement in measurement precision over that ofion clocks [27].
In addition to improving the local oscillator clock laser for the Sr experiments at JILA, we
reported the construction of a second laser in support of a joint PTB-JILA collaboration to construct
a laser cavity of monocrystalline silicon. The JILA support laser compared with the silicon cavity
demonstrated a record-low beat linewidth of 50 mHz [168]. While the silicon system operated

240
at 1 × 10−16 stability, showing the promise of cavities constructed from crystalline materials, it
was still limited by the amorphous mirror coating. We additionally reported the success of a new
mirror technology whereby the optical coatings are replaced by GaAs/AlGaAs heterostructures.
With these new mirrors, thermal noise limitations should be as low as 1 × 10−17 for cryogenic,
all-crystalline optical cavities.
An important systematic facing clocks with many particles is the potential for collisions toperturb the clock frequency. This is part of the motivation for the extensive study of collisional
effects that has taken place over the last several years [265, 135, 273, 293, 43]. However, the wellcharacterized environment and exquisite measurement precision of lattice clocks paired interacting
many-body systems represents a new opportunity for quantum science beyond timekeeping. In
Chapter 5, we described the discovery that p-wave effects dominate the density-dependent effects
within the 87 Sr lattice clock. In the many-particle trap sites, the p-wave interaction leads to strong,collective interactions. We uncovered and elucidated the quantum many-body nature of theseinteractions, and demonstrated the power and validity of a spin-model description of the manybody interactions. Furthermore, we reported the observation of spin-noise correlations as revealed
by noise tomography. Thus, the optical lattice clock becomes a platform for studying many-body
physics.
Finally, in Chapter 6, we described how atomic measurements employing periodic spin echo
sequences can act as a spectrum analyzer for the clock laser frequency fluctuations. Via this method,
we demonstrated a characterization of the laser noise spectrum. This knowledge will benefit future
clock comparisons as well as further experiments exploring the many-body nature of the 87 Sr latticeclock, and this general technique will support the development of the next generation of ultrastablelasers.

7.2

Outlook
The proposal for the lattice clock is barely over a decade old [82], yet in that time, remarkableprogress has been achieved. Only a few years after the first demonstration of magic wavelength

241
confinement and Lamb-Dicke spectroscopy [319], the 87 Sr lattice clock achieved uncertainty lower
than that of the Cs primary frequency standard [31]. In subsequent years, different groups havetackled different different systematic effects: at JILA we focused on understanding and controllingthe collisional effects [274, 125] culminating in an evaluation of the Sr2 density shift at 1 × 10−18[43]. Meanwhile, colleagues at SYRTE carried out a complete study of the vector and tensor
polarizabilities, in addition to the hyperpolarizability and M1/E2 effects, verifying that there are
no problems on the horizon above 1 × 10−17 due to the optical lattice [110]. Recently, colleagues
at PTB reported the first measurement of the DC polarizability and extrapolated the dynamic
polarizability, as is necessary for an accurate correction of the blackbody radiation (BBR) shift[115]. Finally, as we report in this thesis and in Ref. [43], the 87 Sr lattice clock has reachednear-quantum-limited performance due to advances in ultrastable laser technology.
As should therefore be evident, this progress owes its rapidity and breadth to the large international effort aimed at developing the Sr lattice clock. As of this writing three groups worldwide
have evaluated 87 Sr lattice clock systematics at the 1 × 10−16 level and the international agreementon the 87 Sr frequency is superb [31, 33, 34]. Thus, as even more groups around the world continue
to develop 87 Sr optical lattice clocks, we can expect further recognition of the 87 Sr lattice clock’s
potential role in redefining the SI second via an optical frequency standard [320]. At the very least,
it can continue to serve as a powerful secondary representation of the second along with otherhighly-successful optical clocks.
In terms of developing a working frequency standard that is competitive in systematic uncertainty with the best ion clocks, the next biggest hurdle for 87 Sr is obviously the BBR shift.
Unfortunately, Sr has the largest BBR shift coefficient of the alkine earth(-like) species being pursued as lattice clocks. However, there are several avenues forward. If one could maintain a known
and well-controlled blackbody environment at room temperature and 100 mK temperature uncertainty, then the BBR shift uncertainty would be below 1 × 10−17 . An (albeit complex) two-species
optical lattice clock (e.g., two separate but nearby lattices of 87 Sr and 171 Yb) could in principle
form a weighted clock transition, as proposed in [321], that is insensitive to temperature as long242
as the DC polarizabilities are known. Fortuitously, the DC polarizability for both Sr [115] and Yb[139] have been measured to high accuracy.
One final comment with respect to clock uncertainty is that the Sr lattice clock now finds
itself in an interesting situation: the 1 s stability of the 87 Sr lattice clock is very nearly equivalent
to its total systematic uncertainty, and the flicker noise of the clock laser presented in this thesis
is actually slightly below it. As we have mentioned in this thesis, this level of stability makes
measurement and control of many relevant systematic shifts a trivial undertaking; formerly monthlong measurement campaigns can be completed in one day. While this does not directly help tosolve the BBR problem, the improvement in precision will undoubtedly enable rapid progress in
reducing uncertainty through systematic measurement.
Sr and other alkaline earth atoms show great promise for quantum science beyond frequency
standards—with room for fruitful overlap at the boundaries. For example, many-particle latticeclock systems are highly decoherence-free testbeds for quantum measurement science, and thedemonstration of an entangled lattice clock would have profound implications for the ultimate
limit of clock stability [322]. Additionally, the confluence of quantum degeneracy and many-particle
clocks will yield rich scientific opportunity. In the case of 88 Sr, where the measured inelastic loss rate
is similar to that of 87 Sr while the elastic interaction strength is at least an order magnitude larger[290], degeneracy could yield an interaction-induced entanglement analogous to that obtained in
the microwave domain [284], allowing sub-standard-quantum-limit measurement precision. In thecontext of atomic interactions, dipolar couplings between clock atoms in three-dimensional lattices
have been studied as a potential systematic, and could also yield rich excitation dynamics [323].
Due to their SU(N ) symmetry, where N is the number of nuclear spin states, alkaline earth(like) atoms show promise for simulating complex condensed matter systems and realizing novel,
highly-frustrated quantum states [73, 74, 75, 76]. The presence of the metastable clock state
furthermore allows quantum simulation in the presence of an artificial gauge field [324]. The clocktransition can also serve as a readout and probe of the many-body spectrum [325]. However,
we note that while degenerate gases of alkaline earth(-like) atoms have been demonstrated [326,243
269, 270, 271, 272], including a recent demonstration obtaining quantum degeneracy by direct laser
cooling [94], they have not yet reached the levels of entropy required to observe these novel magnetic
phases, although the large numbers of spin-degrees of freedom in fermionic alkaline earth atoms
permits an effect similar to Pomeranchuk cooling to potentially reduce the entropy to the required
levels [327]. We additionally note that the large, electronically decoupled nuclear spin manifolds ofalkaline earth atoms represent a promising register for quantum information science [328, 329, 330].
Here, the electronic level structure of, e.g., 87 Sr could permit manipulation and readout of nuclear
spin-encoded qubits, and the ability to spectroscopically distinguish nuclear spin sublevels via clock
spectroscopy enables the implementation of phase gates between spatially separated qubits [329].
In the specific case of many-body physics with 87 Sr in a one-dimensional optical lattice,
Fermi degeneracy should more strictly enforce the approximations of the spin model, while at thesame time allowing the study of SU(N ) physics via clock spectroscopy and clean readout of manybody states. The operation of a Fermi-degenerate optical clock would be in itself an importantmilestone. In the case of the one-dimensional lattice, the attainment of single-site resolution would
yield enormous gains for the study of many-body effects, since each measurement cycle would yieldmany parallel measurements of the many-body dynamics and would not suffer from averaging.
Single-site resolution would be extremely challenging, but might be possible using a large magnetic
field or light-shift gradient to spectroscopically isolate the different spatial locations of each latticesite. Additionally, a non-retroreflected lattice laser could be employed, which would enhance thespacing between lattice sites.
Finally, we mention that alkaline earth atoms represent a novel regime for cavity QED where
both the critical photon and atom number can be far less than unity, leading to highly collective
quantum effects including steady-state supperadiance [331]. It has been proposed that such asupperadiant system of 87 Sr could operate at the heart of the ultimate laser, with linewidth of
1 mHz [252]. In that case, we could finally remove all reliance on length standards (optical cavities)
for probing the 87 Sr optical lattice clock for quantum metrology and explorations of many-body
physics.

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Appendix A

Coherent atom-light interactions

In this Appendix, we derive a time-evolution operator approach for finding the time evolutionof coherent spectroscopy. We then apply the formalism to derive the lineshapes and Dick sensitivity
functions used throughout this Thesis.

A.1

Derivation of the Hamiltonian
We consider a laser with electric field
1
1
E (r, t) = E0 (r) e−iφL (t) + E∗0 (r) eiφL (t) .
2
2

(A.1)

The Hamiltonian in the dipole approximation is given by [332]Ĥ (t) = ~ωa |eihe| + dge |gihe| + d∗ge |eihg| · E (r, t) ,(A.2)

where dge is the dipole between the ground and excited states being considered. Next we write
down |ψ (t)i as|ψ (t)i = c1 (t) |gi + e−iφL (t) c2 (t) |ei.

(A.3)

This represents a change of basis to the frame co-rotating with an angle equivalent to the laser
phase. Writing down Shrödingers equation yieldsi~d|ψ (t)i = Ĥ (t) |ψ (t)idt
= ~ωa c2 (t) e−iφL (t) |ei + dge · E (r, t) e−iφL (t) c2 (t) |gi+ d∗ge · E (r, t) c1 (t) |ei

(A.4)
(A.5)

265
We now project onto he| and hg|d|ψ (t)i = i~ċ1 (t) = dge · E (r, t) e−iφL (t) c2 (t)
dt

(A.6)

d
|ψ (t)i = i~ [ċ2 (t) − iωL (t) c2 (t)] e−iφL (t)
dt

(A.7)

hg|i~
Similarly
he|i~= d∗ge · E (r, t) c1 (t) + ~ωa c2 (t) e−iφL (t)

(A.8)

d
Here, ωL (t) ≡ dt
φL (t). This results in1 ∗1−i2φL (t)+ E0 (r) c2 (t)i~ċ1 (t) = dge · E0 (r) e
2
2


(A.9)

We now make the rotating wave approximation. The phase factor e−i2φL (t) is rapidly varying (it is
the laser’s total phase) so we drop this term. This leaves
11
i~ċ1 (t) = dge · E∗0 (r) c2 (t) = ~Ω∗ (r) c2 (t) ,2
2(A.10)

where we define the Rabi frequency asΩ (r) = d∗ge · E0 (r) /~ = e−iφ d∗ge · E0 (r) /~(A.11)

1i~ċ2 (t) = ~Ω (r) c1 (t) − ~∆ (t) c2 (t)2(A.12)

Similarly, we findwhere ∆ (t) = ωL (t) − ωa so that we can write d  c2 (t)   −∆ (t)
i~ 
=
dt
1
∗
c1 (t)
2 ~Ω (r)12 Ω (r) 0 c1 (t) ×c2 (t)(A.13)

Since we don’t care about global phase factors, we are free to add and subtract constantsfrom the diagonal. The time evolution finally becomes



 

−iφ |Ω (r)|
c
(t)
−∆
(t)
e
c
(t)
d  2
 1 
  2

i~ 
 = ~
×dt
2
c1 (t)
eiφ |Ω (r)|∆ (t)c1 (t)(A.14)For a real rabi frequency


d
1
1
i~ ~c =~Ω (r) σ̂φ − ~∆ (t) σ̂z ~c
dt
2
2
|
{z
}
Effective Hamiltonian

where σ̂i is a Pauli matrix with i ∈ {x, y, z} and σ̂φ = cos φσ̂x + sin φσ̂y .

(A.15)

266

A.2

Time evolution operator for lineshape calculation
We begin with the effective Hamiltonian of Eq. A.15 in the limit of constant detuning andRabi frequency,
11Ĥ = ~Ω0 σ̂φ − ~∆0 σ̂z .
22(A.16)

We can make use of the formal solution for the time evolution operator for independent Hamiltonians, such that
#


Ω
−iĤ
t = exp −i ~n · ~σ t .
Û (t; t0 ) = exp~
2"Here, Ω =(A.17)pΩ20 + ∆2 and ~n = {Ω0 cos (φ) , Ω0 sin (φ) , −∆0 }. The reader may recognize thisoperation as an SU(2) rotation about the vector ~n. We can use the properties of the Pauli matrices
to simplify Eq. A.17
Û (t; t0 ) = cos


 
 
Ω ˆ
Ω0
Ω
∆
Ω
t I −isint (cos φσ̂x + sin φσ̂y ) + i sint σ̂z .
2Ω
2Ω
2

(A.18)The usefulness Eq. A.18 comes from the simplicity with which it allows calculations of the time
evolution during spectroscopy sequences.

A.2.1

Rabi lineshape

We now focus on the specific case of spectroscopy. In order to calculate the excited state
population, in general we will calculate2pe (t) = |he|ψ (t)i|2 = he|Û (t; t0 ) |ψ (t0 )i .

(A.19)

We can choose the initial phase such that φ → 0, which simplifies the time evolution operator of
Eq. A.18. Writing Eq. A.19 in vector representation with |ψ (t0 )i = |gi yields
 


 
Ω0
Ω
 0 
he|ψ (t)i = 1 0 · Û (t; t0 )   = −i
sin t
.
Ω
2
1

(A.20)

Thus,
 
Ω20
Ω
2
pe (t; ∆0 ) = 2 sin t,Ω2(A.21)

267

1.0

Excitation fraction

0.8

0.6

FWHM: 5 Hz

0.4

0.2

0.0
-40

-20

0

20

Frequency (Hz)Figure A.1: Rabi spectroscopy lineshape for a π-pulse time of 160 ms (line) fit to experimental
data (circles). An overall scaling factor has been applied to account for effects that reduce the peak
excitation probability.

where we recall that Ω =pΩ20 + ∆20 . This is the Rabi result. For the normal spectroscopic

sequence used for clock operation, t = π/Ω0 —the π-pulse condition—which yields unity excitation
when ∆0 = 0 and is given bype (∆0 ) =

Ω20
sin2
Ω2



πΩ
2Ω0


.

(A.22)An example of this function applied to real data is shown in Fig. A.1. An overall scaling factor isapplied to account for non-unity excitation due to two-body inelastic losses and slight excitation
inhomogeneity.

A.2.2

Ramsey lineshape

In this section, we again use the time evolution operator approach to calculate the lineshapefor Ramsey spectroscopy. Unlike Rabi spectroscopy, for Ramsey spectroscopy there are three
distinct periods of evolution:(1) Initial π/2 pulse, where Eq. A.18 applies (again we are free to choose φ → 0).

268

Excitation fraction

1.0
0.8
0.6
0.4
0.2
0.0
600

400

200

0

200

400

600

Frequency detuning (Hz)Figure A.2: Ramsey spectroscopy lineshape. The dashed black line shows the envelope of therapidly-oscillating excitation. The envelope is similar, but not identical, to the Rabi lineshape.
Here, the π-pulse time is 10 ms, and the free evolution time, τ is 50 ms; this results in a total
spectroscopy time of 60 ms.

(2) Free evolution time, where the dynamics are described by Eq. A.18 with Ω0 → 0 (and thusΩ → ∆0 ). We label the free evolution time τ .
(3) Final π/2 pulse. The time evolution operator is the same as in case (1).
Mathematically, we can write this as2

pe (t; ∆0 ) = he|Uˆ3 (t; t2 ) Uˆ2 (t2 ; t1 ) Uˆ1 (t1 ; t0 ) |ψ (t0 )i ,(A.23)

where the subscript indicates case 1–3 listed above. Here, the free evolution time τ is given byτ = t2 − t1 . We perform the matrix multiplication for the case where the first and last pulses areexactly π/2 pulses (i.e., t1 − t0 = π/ (2Ω0 ) = t − t2 ) and findpe (∆0 ) =









2
∆0 τ
πΩ
∆0 τ
Ω20
πΩ
2
Ω
cos
sin
−
2∆
sin
sin
.
Ω4
2
2Ω0
2
4Ω0

(A.24)

This function is plotted in Fig. A.2 for a π-pulse time of 10 ms and τ = 50 ms. If desired, theenvelope (i.e., the outline of the oscillatory part) can be found by summing in quadrature the
prefactors of the cos ∆20 τ and sin ∆20 τ terms.

269
A.2.3

Dick sensitivity function for Rabi spectroscopy

In Chapter 2, we discussed the Dick effect for optical frequency standards. The key element
for calculating the Dick effect is the sensitivity function for a given spectroscopic sequence. The
sensitivity function relates a given phase error at a specific time to a fluctuation in the final
population by [36]
1
δpe =
2

Z T0
dt∆ω (t) r (t; ∆0 ) ,(A.25)

0where ∆ω (t) describes fluctuations in the laser’s frequency about the nominal static detuning ∆0 .
Thus, we can calculate the sensitivity function as a response to an infinitesimal phase perturbation,
δφ at time t such that [36]r (t) = 2

∂δpe
∂δφt

(A.26)In order to proceed, we split the time evolution into two steps:(1) Non-perturbed evolution. We use the time evolution operator of Eq. A.18 with φ → 0.
(2) The phase perturbation occurs at time t1 . We use the time evolution operator of Eq. A.18
with φ → δφ.
This can be written as|ψ (t)i = Ûδφ (t; t1 ) Û0 (t1 ; t0 ) |ψ (t0 )i.

(A.27)

Here, the subscript indicates the phase, φ of the atom-light interaction. If we carry out the operator
multiplication for an infinitesimal phase step we find




Ω0
Ω
|ψ (t)i = Û0 (t; t0 ) − iδφsin(t − t1 ) σ̂y × Û0 (t1 ; t0 ) |ψ (t0 )iΩ2= |ψ0 (t)i + δφ|δψ (t; t1 )i,(A.28)(A.29)

where |ψ0 (t)i = Û0 (t; t0 ) |ψ (t0 )i, the unperturbed time evolution of wave function. We can thenfind δpe (t1 ) asδpe (t1 ) = |he|ψ (t)i|2 − |he|ψ0 (t)i|2= 2δφ< {he|ψ0 (t)ihδψ (t; t1 ) |ei} + O δφ2(A.30)
(A.31)

1.2
1.0

1.2

a

1.0

b

0.8

r (t)

r (t)

0.8

270

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0
0.00

0.02

0.04

0.06

0.08

0.00

0.02

Time (s)

0.04

0.06

0.08

Time (s)Figure A.3: Rabi Ramsey sensitivity functions. In both cases, the detuning ∆0 is negative andchosen such that pe = 0.5. (a) Sensitivity function for Rabi spectroscopy for an 80 ms spectroscopy
time. (b) Sensitivity function for Ramsey spectroscopy for an 80 ms spectroscopy sequence with a
10 ms π-pulse time and 70 ms free-evolution time τ .

For the usual case of a π-pulse excitation, we findr (t1 ) = 4< {he|ψ0 (tπ )ihδψ (tπ ; t1 ) |ei}

(A.32)

where tπ = π/Ω0 . When we carry out the operator multiplication we find that∆Ω20
r (t) =
Ω3

A.2.4




sin

πΩ
Ω0




 
π
.
−t
− sin (Ωt) − sin ΩΩ0(A.33)

Dick sensitivity function for Ramsey spectroscopy

In Chapter 6 we derived the perturbation of the Bloch vector angle Θ due to laser noise.
For Ramsey spectroscopy, δpe ' δΘ/2 for small fluctuations δΘ. Thus, the sensitivity function
we derived in Chapter 6 is exactly the Dick sensitivity function. We thus find, in the limit where∆0  1/τ ,r (t) = sin [θ (t)] .

(A.34)

Here, as in Chapter 6, θ (t) is the total pulse area applied to the atoms. The Fourier-domain
sensitivity |R (f )|2 is given by Eq. 6.57 with n = 0. In the limit that Ω0 → ∞, we recover theRamsey sensitivity function of Eq. 2.66. We note that in general, the effect of longer π/2 pulses inRamsey spectroscopy is to provide an additional pole of low-pass filtration at the Rabi frequency

271
corresponding to the π/2-pulse duration. The time-domain for both Rabi and Ramsey (finite
π-pulse case) are shown in Fig. A.3

Appendix B

Calculating the dipole moments from measured atomic lifetimes

In this Appendix, we summarize the method for relating transition dipole moments to experimentally measured lifetimes. The dipole moments are necessary for computing the AC polarizability
and the scattering rate, as calculated in Chapter 2.
We begin with the Wigner-Eckert theorem applied to the dipole operator. Simply stated,
the theorem relates a dipole moment (d̂ ≡ er̂) to a scalar quantity modified by a Clebsch-Gordan
coefficient. This can be stated mathematically as [111, 108]0hn, J, mJ |er̂q |n0 , J 0 , m0J i = hnJ||er̂||n0 J 0 i (−1)J −1−mJ

√

J0

1
2J + 1 
q m0J


J 
.
−mJ

(B.1)

Here, the quantum number n indicates an electronic configuration, J is the total angular momentum, and mJ is the angular momentum projection onto the quantization axis. The term inparenthesis is the Wigner 3-j symbol. This expression is only useful when the reduced matrix
element hnJ||er||n0 J 0 i is known. Luckily, it is related to the rate, ΓJJ 0 , at which an excited finestructure manifold decays to a lower fine structure manifold (n0 J 0 → nJ) by [108]Γn0 J 0 →nJ =ωn3 0 J 0 →nJ 2J + 1
2
hnJ||er̂||n0 J 0 i ,3π0 ~c3 2J 0 + 1(B.2)

where ωn0 J 0 →nJ is the transition angular frequency. However, even this expression is not so useful.
The most commonly reported quantities in the literature are the lifetimes (or “total” Einstein Acoefficients given by AT (n0 , J 0 ) = 1/τ , where τ is the lifetime) of an upper fine structure statewith electronic configuration labelled by n0 and angular momentum J 0 decaying to multiple finestructure states below.

273
In order to attack this problem, there is another fact we need to know. The reduced dipole
matrix element hnJ||er̂||n0 J 0 i for a given J state in an L manifold is related to the L reduced matrix
element, hnL||er̂||n0 L0 i, by [111, 108]



0 1


L
L
p
0
hnJ||er̂||n0 J 0 i = hnL||er̂||n0 L0 i (2J 0 + 1) (2L + 1) (−1)J +L+1+S,J 0 J S 


(B.3)

where the term in the curly braces is the Wigner 6-j symbol. Now we’re ready to use the publishedAT values to actually get somewhere. Using Eqns. B.2 and B.3, we can relate AT to a sum over
all the ground fine structure states by
 X
AT n 0 , J 0 =Γn0 J 0 →nJ
J

=



|hnL||er̂||n0 L0 i|2 2L + 1 X
2L0 + 13π0 ~c3

ωn3 0 J 0 →nJ (2J + 1) 2L0 + 1



 L

L0

2

1


J 0

J


S

J

.

(B.4)

One important fact about Eq. B.4 is that if we ignore the J dependence of ωJ 0 J , we can use thefact thatX(2J + 1) 2L0 + 1



 L

J 0

J

L0

2

1

J


S

=1

(B.5)

to recover an expression identical to Eq. B.2, only this time for a decay L0 → L. However, we
3 , suchchoose not to make this assumption, and instead treat the sum as a weighted average of ωJJ
0thatω̃n3 0 J 0 ≡Xωn3 0 J 0 →nJ (2J + 1) 2L0 + 1



 L

L0

2

1


J 0

J


S

J

.

(B.6)

This allows us to finally solve for |hnL||er̂||n0 L0 i|2 , namely
0 0hnL||er̂||n L i

20= AT n , J0 3π0 ~c3
ω̃n3 0 J 0

2L0 + 1
2L + 1

.

(B.7)

We note that it is also possible to write the branching ratio between different decay pathways as

2


0

 L L 1ωn3 0 J 0 →nJ
Γn0 J 0 →nJ
0=(2J + 1) 2L + 1
.
(B.8)

AT (n0 , J 0 )ω̃n3 0 J 0
J 0 J S 274
The term ζ (nJ; n0 J 0 ) ≡ ωn3 0 J 0 →nJ /ω̃n3 0 J 0 has sometimes been called the relativistic correction factor[112], because it corrects the effect of fine structure energy shifts on the dipole elements. For Sr,
this term is at most a 10% correction.
With these results, we are finally able to compute the desired matrix element given by Eq. B.1.
The final result iss

3π0 ~c3 AT p
(2J 0 + 1) (2L0 + 1) (2J + 1)×ω̃n3 0 J 0





0
0


L L 1 1 JJ 
0(−1)2J +L+S−mj ×.
J 0 J S  q m0 −mjj00hn, J, mj |er̂q |n , J , m0j i =(B.9)

For φi ∈ {3 P0 , 1 S0 }, the expression for the dipole is particularly simple. It is given byX
3π0 ~c3 AT (n0 , J 0 )2hφi |εq dq |n0 , 1, −qi =ω̃n3 0 J 0q
2L0 + 1 ×



L

J 0

L0

2

1

0


S

,

(B.10)

where the one remaining complication is the dependence on the relativistic correction factor inω̃n0 J 0 . This simplifies for 3 P0 toXq

h3 P0 |εq dˆq |n0 , 1, −qi2=3π0 ~c3 AT (n0 , J 0 ) (2L0 + 1).
9ω̃n3 0 J 0(B.11)

where the polarization can be arbitrary and the result still holds.
An even simpler expression is obtained for 1 S0 , and it is important to note that there is no
relativistic correction factor here due to the absence of fine structure in the 1 S manifold. In thiscase, the dipole depends directly on the lifetime and transition frequency asXq

h1 S0 |εq dˆq |n0 , 1, −qi2=3π0 ~c3 AT (n0 , J 0 ),ωn3 0 J 0

(B.12)For both 1 S0 and 3 P, as is expected, the is no εq dependence of the polarizability, which reflects
the isotropic nature of the J = 0 atomic states.
As discussed in Chapter 2, the hyperfine interaction breaks the symmetry and gives a mF
dependence to the polarizability. This effect is negligible due to the smallness of the hyperfine
energy splittings of 3 P0 . Although this is one mechanism by which the clock states can acquire

275
a vector and tensor polarizability in 87 Sr , the hyperfine-induced state mixing between levels [119]with the same principle quantum number F is the biggest contributor to orientation-dependent
light shifts [25, 112] .

Appendix C

Spherical tensor treatment of the light shift

In this Appendix, we present a detailed derivation of the scalar, vector, and tensor lightshifts. We follow the treatments of [108, 109]. It is my hope that the results of this section “takethe mystery” out of calculating vector and tensor shifts and to better elucidate the physics of lattice
trapping.
We begin our treatment with Eq. 2.12 of the main text (see, e.g., [109])
U (r) = −Eµ(−) Eν(+) αµν

(C.1)

where
αµν = 2X ωik hφi |dˆµ |φk ihφk |dˆν |φi i
.
2 − ω2
~
ω
ik
k

(C.2)

The electric field is now expressed with slightly different parameters such that E(+) ≡ E0 (r) /2 =
 (−) ∗E
. Next, we project the tensorial operator dˆµ dˆν onto states with quantum number n0 andangular momenta F 0 , along with the state for which we wish to calculate the polarizability, labeledby its angular momentum F such that we define an operator [333]Λ̂n0 F 0 =Xm,m0 ,m00|nF m00 ihnF m00 |d̂|n0 F 0 m0 ihn0 F 0 m0 |d̂|nF mihnF m|≡ d̂n0 F 0 d̂n0 F 0 ,

(C.3)
(C.4)

where the d̂n0 F 0 are the dipole operator projected between the ground1 state and n0 f 0 manifold.
1

By “ground” state we mean the state for which we intend to calculate the polarizability.

277
The polarizability projection between these same manifolds is thenα̂n0 F 0 = 2

ωnF →n0 F 0 Λ̂n0 F 0
.
2
2
~ ωnF
→n0 F 0 − ω(C.5)

and thus
U (r) = −E(−) ·X
n0 ,F 0hnF, m|α̂n0 F 0 |nF, mi ·E(+) .
{z

|

(C.6)

}

αµν

(0)

Since the tensorial operator α̂n0 F 0 is formed by a product of two vector operators, we can decompose
it into irreducible components
(1)

(0)

(2)

α̂n0 F 0 = α̂n0 F 0 + α̂n0 F 0 + α̂n0 F 0 .

(C.7)

The goal is to find the contributions of each of these terms to the dipole potential as written in
Eq. C.6.
We begin by first finding the jth representations of the operator Λn0 F 0 , as it directly relates
to α̂n0 F 0 . We begin by expressing the product of the vector operators dn0 F 0 in terms of a sphericaltensors [333]Λ̂n0 F 0 = d̂n0 F 0 d̂n0 F 0 =Xe∗q e∗q0 dˆq n0 F 0 dˆq0 n0 F 0

(C.8)

qq 0

XX

=

j,m q,q 0(j)e∗q e∗q0 T̂m
h1, q; 1, q 0 |j, mi.

(C.9)

Here, eq are the spherical unit vectors and are related to Cartesian coordinates by e±1 = ∓ (ex ± iey )and e0 = ez such thatd̂n0 F 0 =

X
q



e∗q dˆq n0 F 0 , dˆq n0 F 0 = eq · d̂n0 F 0 ,(C.10)

dˆq n0 F 0 dˆq0 n0 F 0 h1, q; 1, q 0 |j, mi.

(C.11)

and
(j)
T̂m
=

X
q,q 0

Thus, we find the jth irreducible component of ΛnF 0 is [333](j)Λn0 F 0 =

X
q,q 0

(j)
e∗q e∗q0 Tm
h1, q; 1, q 0 |j, mi.

(C.12)

278
Since the irreducible components of αnF 0 connect states with equal m, we can restrict ourselves to consideration of only the m = 0 components of the the spherical tensor. Thus
X
(j)
(j)
(j)
αn0 ,F 0 ≡ hnF, m|α̂n0 ,F 0 |nF, mi = hnF ||α̂n0 ,F 0 ||nF ihF, m|F, m; j, 0ie∗q e∗q0 h1, q; 1, q 0 |j, 0i. (C.13)|{z} q,q0(j)

αn0 F 0 (m)(j)

It can be shown that for the jth irreducible component of the projected operator, ΛnF 0 , the reduced
matrix element is given by [111, 108](j)hnF ||Λ̂n0 F 0 ||nF i = (−1)j+F +F

0p(2F + 1) (2j + 1)



1

1



j


F

F


F 0

2

hnF ||d̂||n0 F 0 i ,(C.14)(j)which we can use to compute αn0 F 0 (m) as(j)ωn0 F 0 hnF ||Λ̂n0 F 0 ||nF i(j) × hF, m|F, m; j, 0i.
αn0 F 0 (m) = 2
2
2
~ ωnF
→n0 F 0 − ω(C.15)

From Eqns. C.13 and C.15, we can solve for the contributions of the irreducible components of thepolarizability tensor as

1
(0)
(0)
E(−) · αn0 F 0 · E(+) = αn0 F 0 (m) √ E(−) · e∗−1 e∗1 − e∗0 e∗0 + e∗1 e∗−1 · E(+)3(0)
= −αn0 F 0 (m)

E(+)
√
3

2


1
(1)
(1)
E(−) · αn0 F 0 · E(+) = αn0 F 0 (m) √ E(−) · −e∗−1 e∗1 + e∗1 e∗−1 · E(+)
2
(1)


iα 0 F 0 (m)= n√
E(+) × E(−) · ez
2

1
(2)
(2)
E(−) · αn0 F 0 · E(+) = αn0 F 0 (m) √ E(−) · e∗−1 e∗1 + 2e∗0 e∗0 + e∗1 e∗−1 · E(+)
6


(2)
2
2
αn0 F 0 (m)
(+)
(+)
√
.
=
3 E · ez − E
6(C.16)(C.17)(C.18)The dependence on the light polarization comes directly from the sum over Clebsch-Gordan coef-

279
ficients in Eq. C.13. We also find22ωn0 F 0−1
 √ hnF ||d̂||n0 F 0 i ,(C.19)
2~ ωn0 F 0 − ω 2
3


s




1
1
1
2
2ωn0 F 0
3 (2F + 1)
0 +1
(1)
F
+F

αn0 F 0 (m) =(−1)
×
m
hnF ||d̂||n0 F 0 i , (C.20)
22
F (F + 1)~ ωn0 F 0 − ω
F F F 0 s
5 (2F + 1)
2ωn0 F 00(2) (−1)F +F
αn0 F 0 (m) =F (1 + F ) (2F − 1) (3 + 2F )~ ωn2 0 F 0 − ω 2





2
 2
 1 1 2× 3m − F (F + 1)
hnF ||d̂||n0 F 0 i .
F F F 0 (0)αn0 F 0 (m) =(C.21)
Now all that is left is to perform the sum as in Eq. C.6. The result is


2
 2

3
|ε
·
e
|
−
1
z
3m − F (F + 1)m.
α (ε) = αs + iαv (ε × ε∗ ) · ez + αtF2
F (2F − 1)

(C.22)

Here, ε is the light polarization and we have introduced αs , αv , and αt , the “scalar”, “vector”,
and “tensor” polarizabilities, respectively. The normalizations of the vector and tensor shifts arechosen such that the coefficients are maximally unity for both αv and αt (e.g., the case of circularpolarization and m = F in the tensor case). We find the polarizabilities are (as in [108])2

X 2ωnF →n0 F 0 hnF ||d̂||n0 F 0 i

αs =
,
2
2
3~
ω
0F 0 − ω
0
0
nF
→n
nF

s

X
6F (2F + 1)  1
F +F 0 +1
αv =(−1)(F + 1) 
F
n0 F 0
2
1 1  ωnF →n0 F 0 hnF ||d̂||n0 F 0 i

,
2
2

~ ωnF
→n0 F 0 − ω
F F 02s
1 1 2  ωnF →n0 F 0 hnF ||d̂||n0 F 0 i
X
40F
(2F
−
1)
(2F
+
1)
0

αt =
(−1)F +F.
2
2

(1 + F ) (2F + 3)
~ ωnF
F F F 0 
→n0 F 0 − ω
n0 F 0(C.23)(C.24)(C.25)

We can relate the reduced matrix element hnF |kd̂||n0 F 0 i to hnJ||d̂||n0 J 0 i by [111, 108, 109]




0

J J 1
p
0 +J+I+1
F
0 0
0 0
hnF ||er̂||n F i = hnJ||d̂||n J i (2F 0 + 1) (2J + 1) (−1).
(C.26)

F 0 F I The procedure for calculating the reduced matrix elements hnJ||d̂||n0 J 0 i from the measured lifetimes
is presented in Appendix B.

Appendix D
The s- and p-wave interaction overlap integrals

In this appendix we derive the mode-dependent overlap integrals that appear in the manybody Hamiltonian for s- and p-wave interactions as presented in Chapter 5.
We begin by inserting the potentials of Eqns. 5.9 and 5.10 into Eq. 5.31, we find that
2 Z
4πa−
eg ~
eg
Uij =

d3 x |φi (x)|2 |φj (x)|2mZ
212πb3ss0 ~2
ss0~ j (x) − φj (x) ∇φ~ i (x)
Vij=
d3 x φi (x) ∇φ
m

(D.1)
(D.2)

We consider the specific case where the φ (x) are harmonic oscillator eigenstates such thatφi (x) = φnx,i (x) φny,i (y) φ0 (z) .

(D.3)

Here it is assumed that all the atoms are in the longitudinal ground state. We can then write
2
4πa−
eg ~
√
Ueg
=
S (nx,i , nx,j ) S (ny,i , ny,j )ij
2πmxz0

(D.4)

and
0

ss
Vij
=√

6πb3ss0 ~22πmxz0 (xr0 )2[P (nx,i , nx,j ) S (ny,i , ny,j ) + P (ny,i , ny,j ) S (nx,i , nx,j )] ,

(D.5)

where
Z
S (nξ,i , nξ,j ) =dξ φnξ,i (ξ)

2φnξ,j (ξ)

2

(D.6)

and
Z
P (nξ,i , nξ,j ) =dξ |φnξ,i (ξ)

p√nξ,j + 1φnξ,j +1 (ξ) − nξ,j φnξ,j −1 (ξ)− φnξ,j (ξ)

 2
p
√
nξ,i + 1φnξ,i +1 (ξ) − nξ,i φnξ,i −1 (ξ) | .

(D.7)

281
To derive Eq. D.7, we have used the relationship between the harmonic oscillator raising andlowering operators and a spatial derivative.
The quantities S (ni , nj ) and P (ni , nj ) are plotted in Fig. D.1. Due to the opposite scaling
with n for S (ni , nj ) and P (ni , nj ), which characterize the s- and p-wave interaction strengths,
respectively, the p-wave interaction energy is insensitive to radial temperature (see Chapter 5,
Fig. 5.15).

282

40

nj

30

a

20

10

x0r × S (ni , nj )

0

0.4
0.3
0.2
0.1
0

10

20

30

40

ni

nj 20 301040bx0r × P (ni , nj )

0

5

0

10

20

30

40

0

ni
Figure D.1: s- and p-wave mode-dependent interaction strength coefficients. (a) The modedependent s-wave interaction strength coefficient. The function S (ni , nj ) is scaled by the radial
harmonic oscillator length, xr0 , to remove the xr0 -dependence of S (ni , nj ). (b) The mode-dependents-wave interaction strength coefficient. As with the s-wave case, P (ni , nj ) is scaled with xr0 .

Appendix E

Site occupancy distribution

In this Appendix, we derive the site occupancy distribution for an optical lattice loaded by aGaussian-profile atomic cloud. In the computation of density-dependent experimental observables,such as the density shift, lineshapes, and spin noise, we have utilized the occupancy distribution
derived here to form a properly-weighted average of the desired quantity.

Sites with n

E.1

We begin by considering the situation depicted in Fig. E.1. A Gaussian-profile MOT is loaded
into a 1-D optical lattice during the red MOT stage of the experiment (see Chapter 2). Thus, if a
total of Ntot atoms are loaded into the lattice, the probability of a given site labelled i having n
atoms is given by a Poissonian distributionpi (n) =λni e−λi
,
n!

(E.1)

where the mean values λi are given byx2i

λi ∝ Ntot e− 2σ2 .

(E.2)

Here xi is the position of the ith site with respect to the MOT center and σ is the MOT width.
For the 87 Sr lattice clock, we estimate that σ ' 30 µm. The loading is depicted schematically in
Fig. E.1. The constant of proportionality should be chosen such that the sum of λi will be equal
to Ntot .

284
Sr MOT

λ1

w

λ0
λ−1

σ

Figure E.1: Lattice loading from the red MOT and longitudinal distribution. The red MOT has
longitudinal extent σ, which causes each trap site to have a position-dependent loading, with sitedependent averages {λi }. The lattice sites are spaced a distance w = 813/2 nm. The figure is not
drawn to scale. In the experiment, σ ' 30 µm.

The most important quantity we need to calculate is the distribution of site occupancies (i.e.,
the average number of sites with n atoms), which we call S (n). This quantity is given by
S (N ) =Xpi (n) =i

X λn e−λi
i

i

n!

.

(E.3)

We note that this sum can be carried out numerically. We approximate the sum by an integral
such that
X
i

→

Z ∞

dx
−∞ w

(E.4)

where w is the spacing between lattice sites, and in this case w = λlattice /2. It is then obvious thatthe proper continuous distribution for λi , λ (x), is given byx2
wNtot − x22(E.5)
λ (x) = √e 2σ = Ntot e− 2σ2 ,2πσ 2√where we have made the substitution  ≡ w/ 2πσ 2 . We plug this continuous distribution into

Eq. E.4 and obtaindx [λ (x)]n e−λ(x)
.
n!
−∞ w

Z ∞
S (n) =

(E.6)

285

100

a

200

n × S (n)

S (n)

80
60
40

b

5000
1500
1000

150 500
100
50

20
0
0

10

20

30

Occupancy n

40

50

0
0

10

20

30

Occupancy n

40

50

Figure E.2: Distribution of site occupancies. (a) The average number of sites containing n atoms.
The color scheme is as labeled at right. (b) The atom-number-weighted site occupancy distribution.
As the total atom number increases, the majority of atoms reside in highly-occupied sites. The
numbers label Ntot for a given color. The color scheme is the same at left.

Inserting the continuous distribution of Eq. E.5 into Eq. E.6 and performing a Taylor expansion on
the exponential yields an infinite sum of Gaussian integrals, each with its own analytical solution.
We replace each Gaussian integral in the sum with its analytic solution to finally obtain∞

S (n) =

1 X (−1)j (Ntot )j+n
√
,
n!
j!
j+n

(E.7)

j=0

√
where as before  ≡ w/ 2πσ 2 . The reader is cautioned that for large Ntot and large n, the sum
converges very slowly, so high numerical precision should be used in evaluating the sum (double
precision is woefully insufficient). Finally, we note that the sum can be truncated to a total numberof terms given by ∼ 2eNtot . For typical conditions of σ = 30 µm and Ntot = 5000, this meansthat around 100 terms of the sum must be evaluated. We plot the function S (n) in Fig. E.2a. A
more useful quantity is the atom-weighted S (n) given by n × S (n). This quantity gives the total
number of atoms that share, on average, a site with n atoms. This quantity is plotted in Fig. E.2b.

E.2

Linearity of the density shift and average occupancy
In this section we show that the Poissonian site occupancy statistics enforce that the total

density-dependent shift should be linear in atom number under simplifying assumptions.

Average occupancy

286
20
15
10
5
0

1000

2000 3000 4000
Total atom number

5000

Figure E.3: Average site occupancy as a function of total atom number.

We consider the case where the density shift scales as n (n − 1), which will generally be thecase for a many-body interaction potential of the form of Eq. 5.45 and in the absence of two-bodylosses, which are nonlinear. Thus, the average interaction energy per particle when averaged overtrap occupancy is proportional to the S (n)-weighted sum of n (n − 1) as
u∝

1

X

Ntotalnn (n − 1) S (n) =1
NtotalXXnin (n − 1)[λi ]n e−λi
n!

(E.8)

From the properties of the Poisson distribution, we find that this reduces to
u∝

1 X 2
λi .
Ntot

(E.9)

i

If we use the approximation of Eq. E.4 combined with the probability distribution function of
Eq. E.5 we find
1 X
Ntot
n (n − 1) S (n) = √ ,Ntot n2(E.10)

1 X 2
Ntot
n S (n) = √ + 1,
Ntot n2(E.11)√
where  ≡ w/ 2πσ 2 . It is then also seen then that atom-number weighted average occupancy isgiven byN=

thus proving its linearity with respect to Ntot . The average site occupancy as a function of Ntot isplotted in Fig. E.3. Thus, as seen from Eq. E.10, the average interaction energy per particle (andthus the frequency shift) is proportional to Ntot .