A new record in atomic clock performanceby
Travis L. Nicholson
B.A., University of Colorado, 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
2015This thesis entitled:
A new record in atomic clock performance
written by Travis L. Nicholson
has been approved for the Department of Physics

Jun YeDr. Deborah Jin

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
Nicholson, Travis L. (Ph.D., Physics)
A new record in atomic clock performance
Thesis directed by Dr. Jun Ye

The pursuit of better atomic clocks has advanced many fields of research, providing better
quantum state control, new insights in quantum science, tighter limits on fundamental constantvariation, and improved tests of relativity. This thesis describes the construction and characterization of an 87 Sr optical lattice clock with a state-of-the-art stable laser. The performance of an atomic
clock is typically gauged by two figures of merit: stability and total systematic uncertainty. Stability is the statistical precision of a clock or frequency standard, and the total systematic uncertainty
is the combined uncertainty of all known systematic measurement biases. Several demonstrationsof clock stability are presented in this work, one of which was the first to significantly outperform
ion clocks. The most recent of these measurements resulted in fractional stability of 2.2 × 10−16 at
1 s, which is the best reported to date. These stability improvements are used for two systematic
evaluations of our clock. The first full evaluation at 6.4 × 10−18 total uncertainty took the record
for best clock performance. The second evaluation used improved strategies for systematic measurements, achieving a new best total systematic uncertainty of 2.1 × 10−18 . With a combinationof accurate radiation thermometry and temperature stabilization of the measurement environment,
we demonstrate the first lattice clock to achieve the longstanding goal of 10−18 level uncertainty
in the formidable blackbody radiation shift. Improvements in the density, lattice ac Stark, anddc Stark shifts were also a result of innovations that are described in this thesis. Due to the low
total uncertainty of the Sr clock, timekeeping based on this system would not lose a second in 15
billion years (longer than the age of the Universe), and it is sensitive to a gravitational redshiftcorresponding to a height change of 2 cm above the Earth’s surface.

Dedication

To my kind, brilliant, and loving wife, Dr. Huanqian Loh. And to my caring parents, Jeffand Johannah.

v

Acknowledgements

I would like to express my gratitude to those who have mentored me, worked with me,
educated me, and supported me during my time at JILA. Firstly, I want to thank my advisor, Jun
Ye. I have always admired Jun’s mastery of a wide range of physics, his intellectual rigor, andhis infectious enthusiasm. The time and energy he spends making sure his students meet his highstandards have greatly benefited those of us who were fortunate enough to be taken under his wing.
Under his mentorship and guiding hand, I have grown immensely as a scientist and a professional.
I am indebted to Jun for imparting his wisdom and technical expertise, and I thank him for his
dedication to my education and my success.
I worked on Jun’s second-generation strontium system (Sr2) from its construction until it
was retired, and during that time I interacted with a number of talented graduate students andpostdoctoral researchers. I am grateful to Martin Boyd, Gretchen Campbell, and Jan Thomsen for
their contributions to the Sr2 system in its early days. Marty was a clever physicist and one of
the funniest people I have worked with, and Gretchen impressed me with her ability to successfully
work full time on both Sr1 and Sr2. Professor Jan Thomsen is a frequent visitor to our group, andhe is a tremendously friendly person and a hardworking labmate. Sebastian Blatt and I were thetwo graduate students primarily assigned to Sr2, and we spent many long nights together building
this experiment. Sebastian is a hardworking and driven scientist who understands that, in the faceof long hours and the struggles that come with building a new experiment, people work better
together when they make it a point to get out of the lab and have fun with one another.
Shortly after we finished building the Sr2 system (first as a 88 Sr machine), Sebastian and I

vi
were joined by graduate student Ben Bloom. As we were wrapping up our first result on Sr2, which
was a study of optical Feshbach resonances, we were also joined by postdoctoral researcher Jason
Williams. After this first result was published and Sebastian graduated, taking a postdoctoral
position at Harvard, Ben, Jason, and I revamped Sr2 to be a 87 Sr clock. No description of our trio
could be better than to simply state that together the three of us achieved the best atomic clock,which is a feat I will always be proud of. As we were finishing our stability measurements, the lively
and talented graduate student Sara Campbell started on Sr2. I am particularly grateful for Sara’s
work on our Ti:saph laser system. Partway through the first Sr2 full systematic evaluation, Jason
accepted a permanent position at Jet Propulsion Laboratories. Around this time Ben decided tograduate early to take up an industry position at Intel.
During the second Sr2 systematic evaluation, graduate student Ross Hutson and undergraduate Rees McNally became part of our team. Ross is a bright student who is always willing tolend a helping hand. Rees is a mature, adept young scientist, and I wish him luck as he heads off
to Columbia University for graduate school. As Sara, Ross, Rees, and I were nearing the end of
the second evaluation, we were joined by postdoctoral researcher George (Ed) Marti, who quickly
proved to be a clever scientist.
With the second evaluation complete, I began writing my thesis, and Sr2 was torn down tomake space for Sara, Ross, and Ed’s new generation Sr system. Once, while they were clearing
away the Sr2 apparatus, they asked me to take a quick break from writing my dissertation and givethem a hand disassembling the main vacuum chamber. Although our last result was a fitting end
to the system I had worked on for so many years, I confess that it was painful to hear the chamber
lose vacuum as I turned the wrench. However, I am looking forward to hearing about the many
results that are sure to come from the new Sr apparatus.
My thanks are also due to my labmates on Sr1. Graduate student Michael Martin and I
overlapped for the majority of my graduate career, and I am grateful for his hard work on the Srclock laser. Mike is a thoughtful, gifted scientist, and I am certain he will continue to succeed at
CalTech and beyond. Matthew Swallows was a postdoctoral researcher on Sr1, and I thank him

vii
for his work on the first Sr clock comparisons. I am sure that Matt’s meticulous nature is serving
him well at AOSense. Michael Bishof, who has recently accepted a permanent position at Argonne
National Labs, is an effective scientist and a great guy to watch movies with. Visiting researcher
Yige Lin was a kind labmate and a thoughtful scientist, and I hope his Sr clock in Beijing is working
well. Postdoctoral researcher Xibo Zhang is a hardworking and thorough colleague, and I am sure
he will be a successful professor. Sarah Bromley is a friendly and dedicated young student who
is off to a great start in her career. Tobias Bothwell recently joined Sr1, and I am sure he willcontinue Sr1’s tradition of success.
I am grateful to postdoctoral researcher Wei Zhang for his work on the Sr clock laser, 689 nm
cooling laser, and frequency comb. Wei is one of the hardest working and most dedicated people I
have worked with. Lindsay Sonderhouse is a bright and friendly new graduate student, and I thankher for important contributions to the Sr laser systems. It has also been fun to interact with thegrandfather of our team, Jan Hall. Due to Jan’s encyclopedic technical knowledge and enthusiasm
for physics, conversations with him were always delightful and immensely educational.
The Sr2 team has been fortunate to collaborate with many great scientists over the years.
Paul Julienne provided invaluable theoretical work for the first Sr2 experiment, and since then heand I have collaborated on theoretical studies of intercombination line Feshbach resonances. It has
always been an honor and a pleasure to work with Paul. I also want to thank our collaborator
Weston Tew from the Sensor Science Division at NIST Gaithersburg. I am grateful for his masterfulwork on our thermometry project, which was one of the most important pieces of the second Sr2
systematic evaluation. I also wish to thank Gregory Strouse, also from NIST Sensor Science, for
his role in our thermometry work. JILA theorist Ana Maria Rey has been a good friend to the
Sr team, and I thank her for helping us understand the density shift, which was a central themeof the second publication from Sr2. I thank Murray Barrett for providing useful and enlightening
analytic theory of the Sr2 decay measurement. I am grateful to Marianna Safronova for her advice
and expert calculations.
I am thankful to my undergraduate advisor, Deborah Jin. Debbie was a great mentor who

viii
made it a point to provide me with a wide range of technical training. My work with her teamconfirmed my desire to pursue a career in physics, and I am grateful for the many opportunities
Debbie gave me to achieve that goal.
My thanks go out to the many skilled teachers who taught my graduate courses. The classesof Leo Radzihovsky, Paul Beale, Andrew Hamilton, and Tom DeGrand were enriching and fun.
Thanks also to Paul Beale for his continued friendliness and for always being happy to give me ahelping hand. Thanks to Andrew Hamilton for being a fun person to chat physics with.
I am continually impressed at how many people on the JILA technical staff are simultaneously
invaluable problem solvers and friends with everyone in the building. This includes instrument
shop staff Hans Green, Blaine Horner, Kim Hagen, Todd Asnicar, Kels Detra, and Tracy Keep;electronics shop staff Terry Brown and Carl Sauer; and computing group staff James Raith, Mike
Paige, and Jim McKown. Special thanks to Dave Alchenburger for assisting with some of our
toughest technical problems over the years. Thanks in particular to Hans Green for his first-ratework on so many of the instrument shop jobs that were critical to this thesis.
I am grateful to my parents for all they have done for me. Since childhood, my mother has
been a caring and generous parent, and I have always admired her cleverness and curiosity. My
father is a vibrant person, and some of my best memories from the past several years have beenof weekend ski days with him and Huanqian. My thanks go to my brother for his friendship andsupport. He is a smart guy with a strong conscience, and I am grateful to have him in my life. I
am thankful to my sister for her wonderful sense of humor and her friendship. I am proud of theperson she has become. Thanks to my brother-in-law for being a fun and friendly addition to ourfamily, and thanks to my little nephew for the joy he has brought to my life. My appreciation also
goes out to my parents-in-law. I feel lucky to have a kindhearted and reliable mother-in-law andan intelligent and friendly father-in-law. I am grateful to them for welcoming me into their family.
I also thank my brother-in-law, his wife, and their children for their kindness and friendship.
Lastly, and most importantly, my thanks go out to my wife, Huanqian. She is an excellent
partner for the same reason that she succeeds in so many aspects of life, namely that she approaches

ix
all matters large and small with thoughtfulness, care, dedication, and sincere humbleness. I am
grateful for her wonderful sense of humor, which was a welcome respite when the Sr2 lasers wereacting up. I hope that over the years, more of her many admirable traits rub off on me, and I lookforward to a lifetime of her company.

xContents

Chapter
1 Introduction

1

1.1

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

1

1.2

Metrological Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

Timekeeping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.4

Atomic Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.5

Optical Clocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.6State of the Art Frequency Standards . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.7

Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2 The Physics of the Strontium Lattice Clock9

2.1

The Sr Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Laser Cooling Sr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3

The Magic Wavelength Optical Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4

9

2.3.1

Optical Dipole Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.2

Retroreflected Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.3

Cavity Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.4

The Magic Wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Spectroscopy of Narrow Resonances in the Lamb-Dicke Regime . . . . . . . . . . . . 19
2.4.1

System Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

xi

2.5

2.4.2

Sideband Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.3

The Lamb-Dicke Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4.4

Anharmonicity and Sideband Broadening . . . . . . . . . . . . . . . . . . . . 26

2.4.5

The Lattice in the Presence of Gravity . . . . . . . . . . . . . . . . . . . . . . 28

Digital Servo to Atomic Spin Stretched States . . . . . . . . . . . . . . . . . . . . . . 31
2.5.1

Spin Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5.2

Locking to the Clock Transition . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5.3

Digital Lock to Alternate Stretched States . . . . . . . . . . . . . . . . . . . . 35

2.6

The Magic Wavelength Optical Lattice Revisited . . . . . . . . . . . . . . . . . . . . 37

2.7

Introduction to Stability and Systematic Effects . . . . . . . . . . . . . . . . . . . . . 37

3 The Sr2 Apparatus
3.1

3.2

3.3

3.4

3.5

40

The Sr2 Vacuum Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1.1

Sr Oven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1.2

Main Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Cooling on the 1 S0 →1P1 Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.1

461 nm Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.2

Zeeman Slower and Transverse Cooling . . . . . . . . . . . . . . . . . . . . . 45

3.2.3

Three-Dimensional MOT and Repump Lasers . . . . . . . . . . . . . . . . . . 47

Cooling on the 1 S0 →3P1 Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.1

689 nm Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3.2

The 689 nm MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

One-Dimensional Optical Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4.1

The Cavity Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.2

Tapered Amplifier Lattice Laser . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4.3

Titanium Sapphire Lattice Laser . . . . . . . . . . . . . . . . . . . . . . . . . 58

Spin Polarization into the Angular Momentum Stretched States . . . . . . . . . . . . 59

xii

3.6

3.7

3.5.1

Spin Polarization Bias Field Coils

3.5.2

Spin Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

The Clock Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.6.1

Cavity Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.6.2

Fiber Phase Noise Cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Digital Atomic Servo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.7.1

Clock Laser Frequency Modulator and Drift Cancellation . . . . . . . . . . . 64

3.7.2

Camera Imaging and Photomultiplier Tube Detection . . . . . . . . . . . . . 65

3.7.3

Clock Transition Scans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.7.4

Digital Servo Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4 Clock Stability
4.1

4.2

4.3

4.4

4.5

. . . . . . . . . . . . . . . . . . . . . . . . 59

70

The Allan Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.1.1

Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.1.2

Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Effects that Limit Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2.1

Quantum Projection Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2.2

The Dick Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2.3

Long-Term Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Stability in Optical Clocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3.1

The Stability of Lattice Clocks and Ion Clocks . . . . . . . . . . . . . . . . . 79

4.3.2

Using Longer Cavities to Reduce Thermal Noise . . . . . . . . . . . . . . . . 80

4.3.3

Self-Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Synchronous Interrogation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.4.1

Two Sr Clocks with a Shared Clock Laser Source . . . . . . . . . . . . . . . . 83

4.4.2

Excited State Fraction Correlation . . . . . . . . . . . . . . . . . . . . . . . . 85

Asynchronous Interrogation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

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4.6

4.5.1

Using Asynchronous Interrogation to Measure Independent Clock Stability . 88

4.5.2

Correlations and Allan Deviation . . . . . . . . . . . . . . . . . . . . . . . . . 90

High Stability Self-Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5 Systematic Uncertainty
5.1

5.2

5.3

5.4

5.5

97

Systematic Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.1.1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.1.2

Digital Lock-In Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.1.3

Lever Arm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.1.4

Model Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.1.5

Point-String Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Residual First-Order Zeeman Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2.1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.2.2

Background Field Servo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.2.3

Shift Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Second-Order Zeeman Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.3.1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.3.2

The Lattice Tensor Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.3.3

First and Second Evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Probe Ac Stark Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.4.1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.4.2

First and Second Evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Density Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.5.1

p-Wave Density Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.5.2

Density Shift in a Cavity Lattice . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.5.3

Large Lever Arm Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.5.4

Systematic Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

xiv
5.6

5.7

5.8

Lattice Stark Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.6.1

Lever Arm and Nonlinear Light Shifts . . . . . . . . . . . . . . . . . . . . . . 124

5.6.2

Technical Considerations

5.6.3

First and Second Evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.6.4

Hyperpolarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Dc Stark Shift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.7.1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.7.2

Electrodes and Lock-In Sequence . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.7.3

Observation of Dc Stark Shift and Removal of Background Field . . . . . . . 134

5.7.4

Active Control of the Dc Stark Shift . . . . . . . . . . . . . . . . . . . . . . . 136

Static Blackbody Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.8.1

Introduction to the Blackbody Radiation Shift . . . . . . . . . . . . . . . . . 139

5.8.2

Sources of Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.8.3

Radiation Thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.8.4

Kirchhoff’s Radiation Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.8.5

First Evaluation: Silicon Diode Radiation Thermometers . . . . . . . . . . . 148

5.8.6

First Evaluation: Non-Thermal Heat Bias . . . . . . . . . . . . . . . . . . . . 150

5.8.7

First Evaluation: Immersion Error . . . . . . . . . . . . . . . . . . . . . . . . 152

5.8.8

First Evaluation: Passive Temperature Control . . . . . . . . . . . . . . . . . 153

5.8.9

First Evaluation: Other Sources of Uncertainty . . . . . . . . . . . . . . . . . 154

5.8.10 Second Evaluation: Platinum Resistance Radiation Thermometers . . . . . . 156
5.8.11 Second Evaluation: Sensor Prequalification . . . . . . . . . . . . . . . . . . . 157
5.8.12 Second Evaluation: New Radiation Thermometry Hardware . . . . . . . . . . 158
5.8.13 Second Evaluation: Sensor Calibration at NIST . . . . . . . . . . . . . . . . . 159
5.8.14 Second Evaluation: Measurements in the Sr2 Chamber . . . . . . . . . . . . . 163
5.9

Dynamic Blackbody Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.9.1

A Scheme to Measure the Dynamic Coefficient . . . . . . . . . . . . . . . . . 167

xv
5.9.2

Decay Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

5.9.3

Fit Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

5.9.4

Systematic Bias Due to the Finite DFB Laser Pulse . . . . . . . . . . . . . . 174

5.9.5

Fit Bias Due to Stray Electromagnetic Radiation . . . . . . . . . . . . . . . . 176

5.9.6

Density-Dependent Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

5.10 Minor Systematic Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6 Outlook and Concluding Remarks

183

Bibliography

187

Appendix
A Mathematical Relations Not Derived in the Main Text

196

A.1 Two Time Signals Related by a Transfer Function . . . . . . . . . . . . . . . . . . . 196
A.2 Relating the Allan Deviation to the Power Spectral Density . . . . . . . . . . . . . . 197
A.3 Power Spectral Density of Discrete Time Signals . . . . . . . . . . . . . . . . . . . . 198
A.4 Detection Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

xvi

Tables

Table
2.1

Strontium Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

5.1

Uncertainty Budget for Both Systematic Evaluations . . . . . . . . . . . . . . . . . . 98

5.2

Diode Sensor Uncertainty Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.3

Long PRT Sensor Uncertainty Budget . . . . . . . . . . . . . . . . . . . . . . . . . . 167

5.4

Decay Measurement Uncertainty Budget . . . . . . . . . . . . . . . . . . . . . . . . . 180

xvii

FiguresFigure
1.1

The Target Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2.1

Strontium Level Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2

Repumping Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3

Hyperfine Levels of the 3 P1 State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4

A retroreflected lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5

A cavity lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.6

Cavity Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.7

Ac Stark shift hyperfine dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.8

Motional state quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.9

Sideband Rabi frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.10 Sideband spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.11 Sideband spectra with an improved Lamb-Dicke parameter . . . . . . . . . . . . . . 25
2.12 Sideband Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.13 The Lattice in the Presence of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.14 Transitions with Unpolarized and Spin Polarized Atoms . . . . . . . . . . . . . . . . 32
2.15 Locking to the Line Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.16 Spin Polarization Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1

Oven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

xviii
3.2

Vacuum Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3

Main Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4

461 nm Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.5461 nm Saturation Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.6

Zeeman Slower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.7

689 nm Saturation Spectroscopy and Cavity Stabilization . . . . . . . . . . . . . . . 49

3.8

689 nm Slave Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.9689 nm MOT Timing Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.10 689 nm Laser FM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.11 Bias Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.12 Cavity Lattice Mounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.13 Lattice Mount Cutaway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.14 Tapered-Amplifier-Based Lattice Laser . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.15 Titanium Sapphire Lattice Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.16 Spin Polarization Bias Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.17 Bias Coil Current Servo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.18 Clock Laser Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.19 Sideband Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.20 Resolved mF State Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.21 Stretched State Scan After Spin Polarization . . . . . . . . . . . . . . . . . . . . . . 69
4.1

Theoretical Allan Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2

Allan Deviation Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3

Total Deviation Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.4

Sensitivity Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.5

Atomic Clock Stability Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.6

7 cm and 40 cm Cavity Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

xix
4.7

Optical Clock Stability Comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.8

Comparisons with a Shared Clock Laser . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.9

Synchronized Raw Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.10 Simulated Correlation Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.11 Synchronous Interrogation Measurements . . . . . . . . . . . . . . . . . . . . . . . . 87
4.12 Correlation Data with the 40 cm Cavity . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.13 Allan Deviations with the 40 cm Cavity . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.14 4 hour Allan Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.15 Long-Term Drift Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.16 High Stability Self-Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.1

Lock-In Detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2

Probe Stark Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.3

Background Field Servo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.4

Minimizing σ Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.5

Second-Order Zeeman Shift Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.6First Probe Stark Shift Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.7

Second Probe Stark Shift Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.8

Density Shift Linearity in N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.9

Density Shift Lock-In Data Linearity in N . . . . . . . . . . . . . . . . . . . . . . . . 122
3/2

5.10 Density Shift Uatom Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.11 Density Shift Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.12 Lattice Stark Shift Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.13 Electrodes for Dc Stark Shift Measurement . . . . . . . . . . . . . . . . . . . . . . . 133
5.14 First Observation of a Dc Stark Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.15 Dc Stark Shift Cancellation Servo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.16 Dc Stark Shift Under Active Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

xx
5.17 Dc Stark Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.18 Simple Model of Radiation Thermometry . . . . . . . . . . . . . . . . . . . . . . . . 142
5.19 Simple Case of Kirchhoff’s Radiation Law . . . . . . . . . . . . . . . . . . . . . . . . 145
5.20 Silicon Diode Sensor Mounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.21 Non-Thermal Heat Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.22 Sr2 Chamber Passive Heat Shield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.23 PRT Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.24 Sensor Calibration Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.25 Sensor Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.26 Sensor Agreement After Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.27 Sensor Agreement After Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.28 Decay Measurement Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.29 Decay Measurement Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.30 Decay Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.31 Decay Data Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.32 Decay Data Simulated Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.33 Decay Fit for Finite Pulse Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.34 Decay Fit for 500 ns Pulse Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.35 3 D1 Lifetime as a Function of Atom Number . . . . . . . . . . . . . . . . . . . . . . 179

Chapter 1

Introduction

1.1

Precision, Accuracy, and Systematic Uncertainty
Throughout the 20th century, physics research pushed the frontiers of technology and trans-

formed civilization. The quest for better technology continues to motivate numerous physics research endeavors, which are driven by technological goals like renewable energy, quantum information, and biomedical devices. Good technology must be compatible and interoperable across the
Earth if it is to meet the demands of the global economy. This means that measurements—which
are foundational to the design, construction, characterization, and assessment of technology—must
be internationally and precisely consistent.
Two figures of merit that describe the quality of measurements are precision and accuracy,
which are exemplified in Figure 1.1. Here I use the famous target analogy for these two concepts.
Figure 1.1 depicts different distributions of 70 holes shot into a target. The shooter’s precision

a)

b)

c)

d)

Figure 1.1: a) The shooter is neither precise nor accurate. b) Accurate but not precise. c) Precise
but not accurate. d) Precise and accurate.

2is the width of the distribution, and the shooter’s accuracy is the distance between the center ofthe distribution and the bullseye. Similarly, measurement outcomes occur in distributions, where
the precision is the width of these distributions, and the accuracy is how much the centers of the
distributions differ from some true value.1
Systematic uncertainty can also be explained with the target analogy. To quantify theshooter’s performance, one must measure the accuracy of the distribution of holes. In Figures
1.1a or 1.1c, if one attempted to measure the difference between the center of the distribution
and the bullseye, this measurement would have uncertainty (known as the systematic uncertainty).
This is because of the inherent uncertainty in determining the center of the distribution. In Figure
1.1a, the distribution is wider than in Figure 1.1c, so there is more uncertainty in locating the distribution center and therefore more systematic uncertainty. This demonstrates a common featureof measurements: better precision leads to better systematic uncertainty.

1.2

Metrological Standards
Modern, global technology often requires precise and accurate measurement instruments.

Precision is needed to resolve small changes in measured quantities. Accuracy is necessary to ensurethat measurements are reproducible and agree worldwide. In this case, high accuracy typicallyentails that measurement instruments have been calibrated to high quality, internationally uniform
metrological standards.
Metrological standards are controlled systems that realize physical units. For example, atemperature standard could be a water bath that is pressure and temperature controlled such thatit maintains liquid, solid, and gaseous phases of water all at once (since the Kelvin is defined asexactly 1/273.16 the temperature of the water triple point). The worldwide uniformity of standards2
1

In principle, these quantities should be called the “imprecision” and the “inaccuracy”; however, I will be a bit
loose in making the distinction between “precision” versus “imprecision,” “stability” versus “instability,” etc. Muchof the clock literature is equally loose, and there is a good reason for this. These terms are used often in clock
publications, and it is a burden on careful readers to have to be concerned with things like whether “stability” makes
sense in its sentence instead of “instability.” In practice, it is often clearer to mostly use “stability,” “precision,” and“accuracy,” and rely on the context to shed light on whether “instability,” “imprecision,” or “inaccuracy” is actuallymeant.

3is ensured through the international Treaty of the Metre (or Metre Convention), which established
an international standards bureau as well as metrology conferences and committees of signatory
nations. Through these conventions, member states coordinate international metrology and, in
1960, formally established the International System of Units (SI). The SI is an evolving systemthat updates its definitions when superior standards based on new systems become available.3
Metrology researchers continue to develop more precise standards in anticipation that these
improved systems will be needed as technology becomes more precise and more global.
The most precise standards are those of frequency, which define the SI second. Currently, thedefinition of the SI second is “The duration of 9193631770 periods of the radiation corresponding
to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.”Many member nations of the Metre Convention maintain one frequency primary standard, which
is a microwave resonator slaved to the ground state hyperfine transition in a cesium fountain [48].
Primary standards are compared with those of other member nations, and they can also serve
as local standards against which other frequencies can be measured. In the United States, theCs primary standard is located at the National Institute of Standards and Technology (NIST) in
Boulder, CO.

1.3

Timekeeping
One of the most critical applications of metrological standards is globally coordinated time-

keeping, which provides precise timing and synchronization for modern technology. Some of thetechnologies that rely on precise timekeeping are GPS, secure communication, space navigation systems, power grid synchronization, and computer network timing. Timekeeping is based on clocks,
which are (at least) comprised of oscillators, counters, and references [56]. Examples of oscillators
found in clocks are pendulums (grandfather clocks), crystal resonators (quartz clocks), and elec2

Engineers often use the word “standards” differently. Here I use “standards” to mean only “metrologicalstandards.”
3
For instance, the SI meter was defined in units of wavelengths of a krypton transition. Later this definition was
updated to define length as “the distance traveled by light in a vacuum in 1/299792458 second” since the latter could
eventually be measured more precisely.

4
tromagnetic waves (atomic clocks). A clock counts oscillator cycles, converting them to quantities
with temporal dimensions by equating a unit of time with a given number of oscillator periods[56]. For instance, the resonators in many quartz clocks are laser trimmed until their fundamental
frequencies are 32768 Hz, and these clocks are designed to advance 1 second after 32768 oscillations
are counted.
Clock oscillator phase or frequency noise degrades timekeeping precision. For an oscillator
that can be expressed as A sin(ωt+φ), where A is a constant amplitude and ω is the (constant) basefrequency, allowing the phase offset φ to be a stochastic function of time describes both phase andfrequency noise [100]. If a clock considers 1 second to be exactly n oscillator periods, the oscillator
phase after n cycles is ωt + φ = 2nπ. This relation implies that, for a stationary phase noise
process, δt = δφ/ω (where δt is the root mean square (RMS) time signal noise, and δφ the RMS
phase noise). Therefore, good clock design means employing a phase-stable (small δφ) oscillator
with the largest possible base frequency.
To select the value of n correctly, one must measure the number of oscillator cycles in 1 s.
This is accomplished by referencing or calibrating the clock oscillator frequency to an accurate
frequency signal. Historically, this signal was often Earth’s rotational frequency (which could bemeasured with sundials) [56]. Now, for cutting edge clocks that use large base frequency oscillators,
microwave or optical atomic clock transitions are used as reference frequencies.4

1.4

Atomic Time
A frequency standard is typically a referenced oscillator [100]. Therefore, in principle, afrequency standard plus a counter yields a clock. The best clocks to date are atomic clocks, which
are based on frequency standards that use microwave or optical electromagnetic waves as oscillators
and atomic transitions as references. High performance atomic clocks exist mostly in academic andgovernment laboratories, such as NIST’s atomic clock that generates official US time (based on
their Cs primary standard) [48]. The Cs clock does not lose 1 s in 300 million years, in contrast to4

Any transition used as a frequency reference in a clock is called that system’s clock transition.

5
the high-quality quartz clocks found in marine chronometers, which lose & 10 s annually. Rather
than use better clocks for consumer timepieces, it is typically sufficient and cost effective to use
quartz clocks that are occasionally reset to official US time.
In practice, a Cs primary standard cannot run continuously, so many nations in the Metre
Convention utilize flywheels. In this case, a flywheel is an array of hydrogen masers, which provides the frequency that is counted to generate the official time of many countries, including theUnited States [67]. To maintain accuracy and long-term precision, the maser arrays are periodicallycalibrated to Cs primary standards.
Clocks around the world coordinate time by comparing with each other using clock signals
broadcast by GPS satellites [67]. Although GPS time signals are noisy, when two nations simultaneously compare their clocks with the same GPS clock, the difference between these comparisonscancels the GPS signal. This yields a comparison between the clocks of both nations that is free of
GPS time signal noise.

1.5

Optical Clocks
Cs clocks are based on phase-stable microwave oscillators (ω ≃ 2π × 9.2 GHz). However, withthe advent of phase-stable lasers [45], clock oscillators with frequencies of hundreds of THz are now
possible. Therefore, state-of-the-art atomic clocks are based on clock lasers, which are stable lasers
referenced to electronic clock transitions in atoms [93].
Advances in atom cooling and trapping also provide advantages for optical clocks and frequency standards. Cold neutral atoms and atomic ions can be confined in (approximately) harmonic
potentials, where kB T < ~ωtr , with T the atomic temperature and ~ωtr the harmonic oscillator
energy spacing of the trap. This condition ensures that atoms occupy few motional states of thepotential. In this regime, it is possible to interrogate clock transitions in a manner that is free of
Doppler and recoil shifts, which cause systematic uncertainties in Cs clocks (since Cs clocks are
based on atoms in free flight).
Optical frequency standards are now a mature technology, outperforming standards based

6
on microwaves. To realize an optical clock, one must also be able to count the cycles of opticalclock oscillators. Historically this was a difficult task, but in the 2000s this became possible with
self-referenced optical frequency combs [31]. The clock uncertainty introduced by the frequency
comb counter is currently negligible [97], so to make better clocks we seek to improve frequency
standards.
Two kinds of optical clocks have surpassed Cs: single-ion clocks and optical lattice clocks.
Ion clocks are less sensitive than lattice clocks to systematic effects that cause oscillator frequency
uncertainty. However, since lattice clocks use ∼ 103 atoms whereas ion clocks use only one, latticeclocks have the advantage in terms of precision. This is because the standard quantum limit to√clock precision scales as 1/ N, where N is the number of atoms in the clock. Until the work
described in this thesis, ion clocks held the record for lowest total systematic uncertainty for anumber of years.

1.6

State of the Art Frequency Standards
When we talk about one frequency standard outperforming another, it is important to beclear about how performance is gauged. To that end, it is useful to explain precision, systematic
uncertainty, and accuracy in the context of atomic frequency standards.
The precision of a frequency standard is also known as its stability, which in this case is theuncertainty in the mean frequency due to clock oscillator phase or frequency noise, δν (also called
the “statistical uncertainty”). Although precision gets worse with increasing δν, clock performance
improves with larger values of the oscillator base frequency ν = ω/2π; therefore, a good measureof a frequency standard’s performance is the stability in fractional units.
Systematic uncertainty arises because atomic frequency standards are typically based on baretransition frequencies of atoms. Bare transitions are useful because their frequencies are universal,
yielding international agreement between frequency standards based on the same atom. In practice,clock researchers must work perturbed transitions, which vary from laboratory to laboratory. Realmeasurement environments have stray electric and magnetic fields, heat, and atomic interactions

7
that introduce perturbations; however, it is possible to measure these systematic biases and thencorrect for them. The uncertainties in measurements of these biases (or systematic shifts) are thesystematic uncertainties of the frequency standard. Adding all of these uncertainties in quadrature yields the standard’s total systematic uncertainty, which is another important measure of afrequency standard’s performance. Systematic shifts are often measured by studying small changes
in the clock oscillator frequency, so these measurements are negatively affected by poor stability.
As explained in Section 1.1, good precision often lends itself to good systematic uncertainty.
The accuracy of an atomic frequency standard is the degree to which its quoted frequency
conforms to the true number of oscillations per SI second of the atom’s bare frequency.5

For astandard that is not based on Cs, accuracy can be achieved by comparing the standard’s frequency
with that of the Cs transition. This works fine for standards that have comparable or worse
performance than that of Cs systems, but for optical standards with total systematic uncertainties
that are orders of magnitude better than Cs, this is no longer a viable option. Any attempt tocompare optical and Cs standards would be severely limited by the Cs systematic uncertainty.
To improve upon the SI, timekeeping, and modern technology, there has been much discussion
in the atomic clock research community of redefining the SI second in terms of an optical frequency[39]. To further this eventual goal, this thesis describes the first lattice clock to surpass all ion clocks
and take the record for best clock performance, achieving stability among the best ever reported
and demonstrating the lowest total systematic uncertainty to date.

1.7

Thesis Overview
This thesis describes the stability characterization and systematic evaluation of an opticallattice clock based on a stable laser referenced to a transition in atomic strontium.6

This isthe second Sr lattice clock built by our team at JILA. I begin by discussing the physics of the5

A corollary to this is that the Cs primary frequency standard has perfect accuracy within its systematic uncertainty
6
This work will not discuss our frequency comb counter in detail, focusing instead on the strontium frequency
standard. This is because the counter does not limit clock performance, so any improvements to the clock mustcome from the frequency standard. For this reason, this thesis often uses the word “clock” to refer to the strontium
apparatus even though a discussion of the counter will be notably absent.

8strontium lattice clock, covering cooling, trapping, and spectroscopy of this atom. Next, I describe
the second-generation lattice clock apparatus (Sr2) that my labmates and I built during my timeon the JILA strontium team. Then, I discuss the stability of the Sr2 frequency standard, which
was improved upon over previous results from Sr1 using a more stable laser. Lastly, I detail the
results of two systematic evaluations of Sr2. The initial evaluation was the first instance of a latticeclock surpassing an ion clock, and the second evaluation reached a longstanding goal in the opticalclock community of low 10−18 level total systematic uncertainty.7

7

Fractional frequency units will be used throughout this thesis for both stability and total uncertainty. “Stability”
will refer to the statistical frequency uncertainty divided by the Sr clock transition frequency of 429.228 THz. “Total
systematic uncertainty” (or “total uncertainty” for short) will refer to the combined systematic frequency uncertainty
divided by the clock transition frequency.

Chapter 2

The Physics of the Strontium Lattice Clock

2.1

The Sr Atom
The strontium lattice clock is based on laser cooled Sr atoms trapped in an optical lattice,which is a periodic potential generated by laser light. The Sr atom has four stable isotopes (Table2.1), one of which is a fermion and the rest are bosons. Like other alkaline earth (Group II in the
Periodic Table) or alkaline-earth-like atoms, the bosonic isotopes all have zero nuclear spin.
Sr has two valence electrons, resulting in spin singlet and triplet angular momentum states.
This work often utilizes transitions between the 5s2 1 S0 ground state and the 5s5p 3 P0 or 5s5p 3 P1levels1 (Figure 2.1), both of which are forbidden by electric dipole selection rules. In the case of
the 1 S0 →3 P1 line, which violates the ∆S = 0 selection rule (where S is the total electron spinquantum number), angular momentum state mixing due to the spin-orbit interaction allows for aweak electric dipole intercombination transition [16].
In the bosonic isotopes of Sr, the 1 S0 →3 P0 transition is still electric dipole forbidden,
1

Here I use Russell-Saunders notation, in which angular momentum states are written as 2S+1 LJ , where S is thetotal electron spin, L is the orbital angular momentum, and J is the total angular momentum. Since there are twovalence electrons, S = 0 or 1, where the S = 0 case is referred to as a “singlet” state and S = 1 is the “triplet.”

Isotope
84 Sr
86 Sr
87 Sr
88 Sr

Abundance
0.56%
9.86%
7.0%
82.58%

Nuclear Spin
0
0
9/2
0

Table 2.1: Stable isotopes of strontium.

10(5s6s) 3S1(5s5p) 1P1
707 nm(5s4d) 1D2(5s4d) 3D1,2,3

679 nm461 nm689 nm(5s5p) 3P2(5s5p) 3P1(5s5p) 3P0

698 nm(5s2) 1S0
Figure 2.1: The Sr energy levels discussed in this thesis. All of the lasers used to access these levels
are represented by arrows.

and theory predicts that, for all practical purposes, it never decays [107]. In 87 Sr, however, thehyperfine interaction (that occurs because of the nuclear spin of 9/2) provides another state mixing
mechanism that allows the 1 S0 →3 P0 transition to take place [16]. The result is an ultranarrow∼ 1 mHz linewidth electric dipole resonance, which is used as the strontium clock transition.
The 1 S0 →3 P0 line is a good clock transition because it is naturally insensitive to straymagnetic fields [16], which can cause unwanted systematic shifts of the clock transition. For J = 0
states, the transition’s first-order Zeeman shift is given by the nuclear g-factor gI , which is smaller
than the Landé g-factor gJ (that determines the magnetic sensitivity of J 6= 0 states) by a factor
roughly equal to the proton-to-electron mass ratio. The 1 S0 →3 P0 second-order Zeeman shift isalso an order of magnitude smaller than that of many prominent ion clocks and several orders ofmagnitude smaller than that of the Cs clock transition [16].

2.2

Laser Cooling Sr
The source of the Sr atoms is an effusive oven that emits a hot atomic beam. To remove

enough kinetic energy from this fast moving beam to trap the atoms, laser cooling is employed.
Experiments that work with ultracold Sr typically use two laser cooling transitions: 1 S0 →1 P1

3

11

S1

P1

707 nm

3

P2
P1
3
P0
3

21

1

679 nm

ms

s
7m

0.7

5.3 ns

461 nm

21 ns

1.5

D2

36 ns

1

0.11 µs

s
6m
0.2

1

µs

S0

Figure 2.2: The repumping scheme. Lasers are represented as colored, solid arrows and decays aredashed arrows. The lifetimes of the decays are included next to the dashed arrows.

and 1 S0 →3 P1 . The 1 S0 →1 P1 461 nm transition is fully allowed by electric dipole selection rules
and has a γ = 30 MHz linewidth. This transition is suitable for Zeeman slowing and 3D magnetooptical trap (MOT) cooling to a Doppler-limited temperature [81] of hγ/2kB = 0.72 mK (where his Planck’s constant and kB is Boltzmann’s constant). The 1 S0 →1P1 transition is not completely
closed since one in 4.9 × 104 decay events populates the 1 D2 state (Figure 2.2) [124]. Most of thispopulation returns to the ground state through 1 D2 →3P1 →1S0 ; however, one in 1.4 × 105 decays
out of 1 P1 results in shelving population in the metastable 3 P2 state via 1 P1 →1D2 →3P2 .
To deal with this shelving, we use a 707 nm laser to drive the 3 P2 →3S1 transition. The 3 S1state decays into the 3 P manifold, so we also use a 679 nm laser to drive any population that decays
into the long-lived 3 P0 clock state back into 3 S1 . Eventually all the shelved population returns to
the ground state through 3 S1 →3P1 →1S0 .
After the atoms have been cooled to their Doppler temperature on the 1 S0 →1P1 transition,
we apply a second stage of 3D MOT cooling on the 1 S0 →3P1 intercombination line. This transition has wavelength of 689 nm and a linewidth of 7.5 kHz [32], which implies Doppler and recoil
temperatures of 180 nK and 230 nK (respectively) [70]. In the 87 Sr intercombination line MOT,

F = 7/2

908 MHz12

222 MHz(5s5p) 3P1

F = 9/2-1.24 GHz

Stirring
laser

F = 11/2

Trapping
laser(5s2) 1S0
Figure 2.3: The three hyperfine states of 3 P1 . The hyperfine levels are labeled by the difference
between the hyperfine transition frequency and the frequency of 3 P1 in absence of hyperfine structure. Using the language of Reference [86], the two lasers used in 1 S0 →3P1 cooling are the trappingand stirring lasers. These lasers are portrayed as red arrows.

complexities arise due to an interplay between the nuclear spin, the narrow linewidth, and the factthat the 1 S0 and 3 P1 levels have very dissimilar g-factors, leading to the lack of a trapping force forsome of the atomic spin magnetic quantum states mF [86]. This makes a traditional MOT basedon a single F → F + 1 hyperfine transition inefficient. A common solution to this problem is touse a stirring laser operating on the |1 S0 , F = 9/2i → |3 P1 , F = 9/2i transition to randomize mF
states along with the usual F → F + 1 cooling laser driving the |1 S0 , F = 9/2i → |3 P1 , F = 11/2i
transition [86]. After 500 ms of 461 nm MOT cooling and 260 ms of 689 nm MOT cooling, we
typically obtain samples of ultracold 87 Sr at temperatures of a few µK.

2.3

The Magic Wavelength Optical Lattice

2.3.1

Optical Dipole Traps

Throughout both stages of laser cooling, an optical lattice is superimposed on the atom cloud
trapped by the MOTs. The lattice potential collects the coldest atoms throughout the coolingprocess and retains them after the 689 nm MOT is shut off. Optical lattices utilize the ac Stark

13energy shift Eac to provide a conservative trapping potential. The ac Stark shift of quantum staten is given by [17]1Eac,n = − αn (ω)|E~ext (ω, t)|2 ,
2

(2.1)

where αn (ω) is the atomic polarizability, E~ext is an external electric field, and ω is the frequencyof the external field. As Equation 2.1 shows, for ω chosen such that αn > 0, Eac,n is negative andit is energetically favorable for atoms to be positioned in the most intense regions of the electricfield. Tightly focused laser beams can therefore confine atoms to small regions of space. Sincelasers oscillate at frequencies of order 1015 Hz, which is much too fast to have a measurable effecton the trap dynamics, |E~ext |2 in Equation 2.1 is replaced with its time average.
Lasers used for trapping are typically focused Gaussian beams [117],


 
2
2
−r
r
z
w
0
−1
~ z) = E~0exp− i(kz − ωt) − ik
+ i tan
,
(2.2)
E(r,
2
w(z)
w (z)
2Rc (z)zRpwhere |E~0 | is the peak electric field amplitude, w0 is the beam focal waist, w(z) = w0 1 + (z/zR )2 ,Rc (z) = z 1 + (zR /z)2 is the beam’s radius of curvature, zR = πw02 /λ is the Rayleigh range,k = 2π/λ, λ is the laser wavelength, and the unit vector E~0 /|E~0 | is the laser polarization. Thispexpression is written in cylindrical coordinates, with z the axial direction and r = x2 + y 2 theradial direction. The laser focus is located at r = z = 0.

The intensity of a harmonic complex electric field is given by [55]I(r, z) =where the laser power P =R 2π R ∞002P
ǫ0 c ~ ~ ∗(E · E ) =exp
2
πw2 (z)−2r 2,
w2 (z)



,

(2.3)

rI(r, z)drdθ, ǫ0 is the permittivity of free space, c is the speed oflight, and * denotes complex conjugation. Expressing Equation 2.1 in terms of this laser intensity
yields the potential for an optical dipole trap,
n
o
−2r 2
exp
2
1
αn P
w0 [1+(z/zR )2 ]UODT (r, z) = −αn I(r, z) = −
.
2
2ǫ0 c
[1 + (z/zR )2 ]πǫ0 cw0

(2.4)

14

k
Curvedmirror

Lens

Figure 2.4: A retroreflected lattice. A laser traveling in the direction of ~k is focused with a lens.
A curved mirror retroreflects the light upon itself, generating periodic intensity fringes. These areused as an array of atom microtraps.

2.3.2

Retroreflected Lattices

Retroreflecting a trap laser upon itself generates a one-dimensional standing wave known as
an optical lattice. A curved mirror is used to ensure that the beam foci of the incident and reflectedbeams overlap (Figure 2.4). This creates intensity fringes that are used as a periodic trap potential.
~ z) + E(r,~ −z), where E(r,~ z) is the expression
Therefore, the electric field of the lattice is E~lat = E(r,in Equation 2.2. This yields a trap potential Ulat of

4α P exp

1n∗=−Ulat (r, z) = − αn E~lat · E~lat
4πǫ0 cw02n−2r 2
w02 [1+(z/zR )2 ][1 + (z/zR )2 ]

o

cos

2




kr 2
−1
− tan (z/zR ) .
kz +2Rc (z)(2.5)

Each intensity fringe provides axial confinement within a length of λ/2. It is common to make the
approximations cos2 kz + kr 2 /2Rc (z) − tan−1 (z/zR ) ≃ cos2 (kz) and 1 + (z/zR )2 ≃ 1. For many
experiments based on ultracold atoms in optical lattices (including Sr2 and our first generation

Sr lattice clock, Sr1 [74]), these approximations are valid since, for typical beam parameters, theeffects of the neglected terms are negligible over the dimensions of trapped atom samples.
Using these approximations, the lattice potential is22Ulat (r, z) = −U0 e−2r /w0 cos2 (kz),(2.6)

where U0 = 4αn P/πǫ0 cw02 is the trap depth, defined as the difference between the asymptotic value

15

k
Input
mirrorOutputmirrorFigure 2.5: The cavity lattice. Incident light is coupled into the cavity with a lens, forming astanding wave in between the two mirrors.

of the potential and the potential minimum. Note that for the lattice, the trap depth is four times
greater than in the optical dipole trap case due to constructive interference.
To trap large numbers of atoms, it is desirable to choose the laser power P such that U0 is
several times greater than kB T , where T (again) is the sample temperature. In this regime, mostof the sample is located near the potential minimum, where the trap is approximately harmonic.
The trap frequencies are the radial and axial eigenfrequencies ωr and ωz at the minimum, which
can be derived using normal mode analysis [41],

ωr2 =

4U0
1 ∂ 2 Ulat
=
2
m ∂r
mw02
r=0

(2.7)

z=0

2k2 U01 ∂ 2 Ulat
=
ωz2 =
,
m ∂z 2 r=0
m
z=0

(2.8)
√

where m is the 87 Sr mass. Since w0 is typically tens of microns, ωz /ωr = kw0 /2 ≫ 1. The axial

trap frequency is often expressed in terms of the lattice recoil energy Erec = ~2 k2 /2m and thelattice recoil frequency ωrec = Erec /~,ωz = 2ωrec

2.3.3

r

U0
.
Erec

(2.9)

Cavity Lattices

The lattice employed by the Sr2 clock uses an optical cavity to enhance the lattice power.
In this case, atoms are trapped in the fundamental transverse mode of a standing wave cavity

16(Figure 2.5). The cavity is symmetric, meaning that the radii of curvature of the mirrors are equal,
and atoms are trapped approximately in the center of the cavity. Assuming the cavity mirrors are
separated by a distance L, we can derive the beam properties of the cavity lattice using the fact
that, for a stable cavity, at z = L/2 the beam’s radius of curvature Rc (z) is equal to R (the radiusof curvature of the mirror). Rc (L/2) = R implies that the beam’s Rayleigh range is

zR =

πw02
1p
L(2R − L),=
λ2(2.10)

which gives a cavity design equation for the beam waist w0 . It is worth noting that Equation 2.10
is not valid for L > 2R because in this case the cavity is unstable. It is straightforward to showwith cavity stability analysis [117] that for a symmetric resonator, the criterion for a stable cavity
is 0 < L < 2R.
The cavity intensity inside and outside the cavity is given by [87](1 − r1 )(1 − r2 )t,√(1 − t r1 r2 )2 + 4t r1 r2 sin2 (kL)√√√( r1 − t r2 )2 + 4t r1 r2 sin2 (kL)
Iref /Iin =,√√(1 − t r1 r2 )2 + 4t r1 r2 sin2 (kL)Itr /Iin =Ien /Iin =√(1 − r1 )t.
√√(1 − t r1 r2 )2 + 4t r1 r2 sin2 (kL)

(2.11)(2.12)(2.13)

Here Iin is the incident laser intensity (taken to be constant in these equations), r1 is the reflectivityof the input mirror, r2 is the reflectivity of the output mirror, and t is the cavity loss (fraction of
intensity lost after one pass through the cavity) from materials inside the resonator. The cavity isresonant when the round-trip phase accumulated by the electric field inside is equal to multiples of
2π, or equivalently kL is a multiple of π as in Figure 2.6. Itr is the intensity transmitted throughthe cavity, and Itr = Iin when the cavity is resonant, t = 1 (no cavity loss), and r1 = r2 . For t < 1,
Itr is always less than Iin . Iref is the reflected intensity, which comes from the field reflected by thelaser incident on the cavity as well as the field inside the cavity that leaks out of the input mirror.
In the impedance matched case, the input mirror reflectivity is chosen such that r1 = r2 t2 . This

17
b) 8

8

Intensity Fraction

Intensity Inside Cavity(fractional)

a) 10

6
4
2
0
0

π

2π

kL

3π

4π

6
4
2
0
0

π

2π

kL

3π

4π

Figure 2.6: a) The intensity enhancement of an optical cavity. Plotted is Ien /Iin for the r1 = r2 = rcase. The green curve depicts the enhancement for r = 0.5 and no losses in the cavity (t = 1).
For the red curve, t = 1 but r is increased to 0.9, which increases the enhancement and makes theintensity peaks narrower. In the blue curve, r remains at 0.9 but t = 0.95. Here the narrownessof the peaks remains but the intensity in the cavity is diminished. b) The cavity behavior for the
r = 0.9 and t = 0.99 case. The gray curve is the one-way intensity Ien /Iin inside the cavity. The
black curve is Iref /Iin , showing the reflected intensity decrease when the cavity enhancement ismaximal. The magenta curve is Itr /Iin .

condition causes Iref to vanish when the cavity is resonant, increasing the power inside the cavity.
Sr2 operates with mirrors from the same batch, so r1 = r2 .
The enhanced intensity Ien is the motivation for using a cavity lattice. The standing wave
inside the cavity is comprised of two counterpropagating electric fields, and the one-way intensity
Ien is the intensity associated with one of those fields. When trapping atoms in this standing wave,
both k and L are electronically controlled using Pound-Drever-Hall stabilization [10] to ensure thatthe cavity is resonant2 , which is necessary to achieve the maximum value of Ien (Figure 2.6).
The fact that light circulates several times in the cavity before leaking out means that the
optical power stored in the cavity can be significantly larger than the incident power. Trapping
atoms in a cavity-enhanced lattice allows us to leverage the extra power (compared to a retroreflected lattice) for a larger beam waist w0 while maintaining a good trap intensity and therefore a2

k is controlled by locking the laser to the cavity, and L is controlled by locking the cavity length with apiezoelectric actuator. See Section 3.4 for more technical details.

18
good value of the trap depth U0 . This leads to a larger spatial extent of the trap potential, which
proved to be critical for dealing with an important systematic frequency shift caused by atomic
interactions (as I will explain in Section 5.5). According to Equation 2.13, the enhancement factoris(1 − r1 )t.
√(1 − t r1 r2 )2(2.14)

To give a sense of the size of the enhancement, for t = 0.99 and r1 = r2 , this factor can be as large
as 25.

2.3.4

The Magic Wavelength

Although the ac Stark effect facilitates atom trapping, it also causes a frequency shift of theclock transition ∆νac

∆νac = (Eac,3 P0 − Eac,1 S0 )/h = −1∆α(ω)I(r, z),2ǫ0 hc(2.15)

where ∆α is the differential polarizability∆α(ω) = α3 P0 (ω) − α1 S0 (ω).

(2.16)

This shift is unwanted since the clock is based on the atom’s bare frequency (Section 1.6). Therefore,
we choose a lattice frequency ω at which α3 P0 = α1 S0 , causing the differential polarizability tovanish. This is the called the magic frequency, and the corresponding wavelength is the magic
wavelength [60, 57, 125].
The polarizability is given by [17]αn (ω) = 6πǫ0 c3

X
k

Ank
2
2 − ω2 ) ,ωnk (ωnk(2.17)

where the sum is performed over all hyperfine states except k = n. ωnk is the frequency of atransition between hyperfine states n and k, and Ank is the Einstein spontaneous decay coefficient19of this transition.
The polarizability is often expanded to make mF dependence explicit,

αn = αs,n + αv,n3m2F − F (F + 1)
mF
ξ k̂ · B̂ + αt,n(3|ǫ̂ · B̂|2 − 1),
2F
2F (2F − 1)

(2.18)

where αs,n , αv,n , and αt,n are the scalar, vector, and tensor polarizabilities. Also, ξ is the degreeof ellipticity of the lattice polarization, k̂ is the laser propagation direction, ǫ̂ is the direction of
the lattice polarization (assumed to be nearly linear), and B̂ is the quantization axis of the atomic
angular momentum. Using this form, ∆νac can be expressed as [121]on∆νac = − κs + κv mF ξ k̂ · B̂ + κt 3m2F − F (F + 1) (3|ǫ̂ · B̂|2 − 1) Ulat (r, z)(2.19)

where the scalar, vector, and tensor shift coefficients κs , κv , and κt are proportional to theircorresponding scalar, vector, and tensor polarizabilities. As explained in Section 2.6, the vector
term is strongly suppressed in the Sr2 clock, so the desired wavelength is that at which the scalar
and tensor terms cancel each other (Figure 2.7). The magic wavelength is approximately 813.4 nm[121].

2.4

Spectroscopy of Narrow Resonances in the Lamb-Dicke Regime

2.4.1

System Hamiltonian

To reference the clock oscillator to the Sr clock transition, this resonance will need to beinterrogated. The fact that the atoms are trapped in an optical lattice means that the dipole
approximation, which is often used for the spectroscopy of atoms, can no longer be made, leading
to distinct qualitative differences compared to the free atom case.
To analyze this case, I will treat the electric field of the clock laser as E~c = E~c0 cos(~kc ·~r − ωc t),where E~c0 is a constant electric field amplitude, ~kc is the clock laser k -vector, ~r is the position
vector, and ωc is the clock laser frequency. E~c is linearly polarized because only π transitions

Differential Shift (linear scale)20

κs U 0120 κ t U 00−9/2 −7/2 −5/2 −3/2 −1/2 1/2 3/2 5/2 7/2 9/2
Nuclear Spin State

Figure 2.7: The differential ac Stark shift corresponding to the peak lattice intensity. Here ǫ̂ · B̂ = 1(implying k̂ · B̂ = 0). The scalar and mF = ±1/2 tensor terms are included on the plot. The
shift vanishes for the angular momentum stretched states mF = ±9/2, which are used in clock
operation.

(transitions that do not change mF ) are used for clock spectroscopy. This expression ignores thespatial properties of the clock laser beam, which is valid because this beam is typically made several
times larger than the atom sample density to ensure uniform intensity across the sample.3

Also,

the trapping potential will be treated as a harmonic oscillator. The interaction Hamiltonian Hint
is −d~ · E~c , where d~ is the dipole operator. Transforming to a frame rotating with the laser frequency
and using the rotating wave approximation,

Hint =~Ω  i~kc ·~r~e|eihg| + e−ikc ·~r |gihe| .
2(2.20)~ · E~c0 /~ is theHere |gi and |ei are the ground and excited electronic state vectors, Ω = −he|d|gi
Rabi frequency, and d~ = e~r (where e is the electron charge).
As explained in Section 2.4.3, it is desirable to align the clock laser along the direction withthe largest trap frequency, which is the lattice axial direction ẑ (also referred to as the “latticeaxis”). In this case, ~kc · ~r = kc z. Since the atoms are trapped in a harmonic potential, z shouldpz0
be regarded as an operator z = √(â + â† ), where z0 = ~/mωz is the harmonic oscillator length,2
and â and â† are the annihilation and creation operators. These act on the harmonic oscillator
3This is done to ensure that the clock laser excites the atoms uniformly across the sample density.

21

-100

c)

-50

0

50

100

Cross Section

Cross Sectionb)

Cross Section

a)

-100

Detuning (kHz)

-50

0

50

Detuning (kHz)

100

-100

-50

0

50

100

Detuning (kHz)

Figure 2.8: The normalized clock laser absorption cross section (from [123]) for weak excitation
and T = 3 µK. The detuning is measured with respect to the free-atom resonance frequency in theatom’s rest frame. a) The black line depicts free-atom motional shifts. This is a Doppler-broadenedlineshape offset by the recoil shift νc,rec. The green bars are the trapped-atom cross section forη = 1. Under this modest confinement, the effect of motional quantization begins to appear. b)
For η = 0.6, motional transitions are well resolved, and each absorption event for nonzero detuning
involves gaining or losing motional quanta. The frequency spacing between the motional transitions
is νz , with the strongest transition now p
occurring at zero detuning instead of at νc,rec . The trap
frequency νz scales with 1/η 2 (since η = νc,rec /νz ). c) For η = 0.25, sidebands are weak, and thezero detuning transition (or carrier transition), corresponding to no change of motional quanta,
nearly dominates the cross section. Note that this carrier transition is not offset from zero detuning
as in the free-atom case. Here the momentum from the clock laser photon is taken up by the latticerather than the atom, resulting in recoil-shift-free spectroscopy [30].

state |nz i of the lattice potential with the usual ladder operator behavior â|nz i =√â† |nz i = nz + 1|nz + 1i.

√

nz |nz − 1i and

For many experiments, it is common to further simplify Hint with the dipole approximation,~in which eikc ·~r ≃ 1. This is valid when ~kc · ~r is much less than unity. For free atoms, the relevant

length scale of the atoms’ spatial extent is the Bohr radius a0 , so ~kc · ~r ∼ kc a0 ≪ 1, and the dipole
approximation holds. For atoms confined in the (approximately) harmonic potential of the lattice,
the length scale is now the harmonic oscillator length z0 , and kc z0 for an optical lattice is typicallycomparable to unity or greater. Therefore, the exponentials in Equation 2.20 must be kept.
The Hamiltonian of this system in the rotating frame (the same frame as Equation 2.20) is

1â â +2~Ω  iη(â+â† )†e|eihg| + e−iη(â+â ) |gihe| .
(2.21)
2√p
Here ∆ = ω − ωc is the detuning, η = kc z0 / 2 = νc,rec/νz is the Dicke parameter, νz = ωz /2π,
H = −~∆|eihe| + ~ωz

†

+

22
νc,rec = h/2mλ2c = 4.68 kHz is the recoil frequency of the clock laser, and λc = 698.4 nm is theclock laser wavelength. This system is typically analyzed using the basis |e, nz i = |ei ⊗ |nz i and|g, nz i = |gi ⊗ |nz i [123]. In this case |nz i is the z-directional motional quantum state. Although
the trapped atoms also have motional excitations in the x and y directions, H in Equation 2.21 does
not affect |nx i or |ny i. Therefore, according to this model, the x- and y- directional excitations stay
constant, so the trivial energy offset produced by these states is ignored. In this basis, an arbitrary
P
quantum state |ψ(t)i can be expressed as |ψ(t)i = nz [cg,nz (t)|g, nz i + ce,nz (t)|e, nz i], where thesum is taken from nz = 0 to ∞.

Spontaneous decay and laser decoherence are neglected in this treatment, implying that the
spectroscopy of the clock states is fully coherent.4 Spontaneous decay can be neglected when thepulse time of the clock laser interrogating the atoms is much shorter than the spontaneous lifetimeof the excited clock state. Also, laser decoherence can be neglected when the clock laser pulse time
is much shorter than the coherence time of the laser. These approximations are valid in this work
since the excited clock state is long lived and the clock laser is ultrastable.

2.4.2

Sideband Transitions

Analogous to the famous two-level Rabi solution, one can define a sideband Rabi frequencyΩeg in terms of the matrix elements of the interaction Hamiltonian,

~Ωeg
~Ω
†
= he, ne |Hint |g, ng i =hne |e−iη(â+â ) |ng i,22(2.22)

where ng and ne are the motional state quantum numbers associated with the ground and excitedelectronic states. This expression has been evaluated [123] as−η2 /2Ωeg = Ωesn< ! |ne −ng | |ne −ng | 2
(η ).
η
Ln<
n> !

(2.23)

Here n< is the lesser of ng and ne , n> is the greater of the two, and Lαj (x) is the jth generalized
4

Decay and decoherence could easily be treated with complex decay terms in the Hamiltonian of Equation 2.21
or with a Lindblad master equation and a Liouvillian superoperator to describe decay.

23

a)

3

P0
ne = ng + 1
ne = ng
ne = ng - 1

1

S0

ng

hνz

Figure 2.9: a) The carrier transition, for which ng = ne , is shown with a green arrow. It is also
possible for atoms in the state |g, ng i to be coupled to the excited electronic state when ne 6= ng ,allowing for motional state addressing. The ng → ng − 1 transition is the red sideband, andng → ng + 1 is the blue sideband. These resonances are separated in frequency from the carrierby νz . b) The sideband Rabi frequency for η = 0.5. Here ∆n = ne − ng . Ωeg is only definedat integer values of ∆n, but the curves are plotted as continuous functions to make the behavior
more apparent. At ng = 0, atoms are initialized in the lowest possible motional state, so losing a
motional excitation (∆n < 0) is impossible. Ωeg begins to vanish for ∆n = 0 as ng gets large.

Laguerre polynomial. This expression describes the optical coupling for the |g, ng i → |e, ne i transition, which is nonzero even for sideband transitions in which the motional state changes whendriving the electronic states (Figure 2.9). This is often interpreted as the Doppler shift when atomic
motion is quantized [30] (Figure 2.8).
Solving the Schrödinger equation with the Hamiltonian from Equation 2.21 is considerably
simpler in the resolved sideband regime. In this limit, sideband transitions are very well resolved
from one another, making it possible to drive one sideband transition without exciting others.5
This is a good approximation for spectroscopy of the clock transition, which is very spectrally
narrow. The system Hamiltonian in the resolved sideband approximation can be written in a 2 × 2
subspace −∆ + ωz (ne − ng )H/~ = 
1
2 Ωge

1
2 Ωeg

0




.

(2.24)

Equation 2.24 has an identical form to the familiar Rabi Hamiltonian, which describes the coherent
5

The resolved sideband regime occurs when γ ≪ νz , where γ is the full width at half maximum of the transitionlineshape.

24

Excited state population

1
n =0
g

0.8

n =1
g

n =2
g

0.6
0.4
0.2
0
-1000

-500

0
Detuning (Hz)

500

1000

Figure 2.10: Sideband spectra for η = 0.5, νz = 500 Hz, Ω = 100 s−1 , and t = π/Ω.

interrogation of a free atom at rest, −∆HRabi /~ = 
1 ∗
2Ω



1
2Ω 

0

.

(2.25)

Solving the Schrödinger equation with HRabi for the excited state population fraction ρee after asquare pulse of length t,

|Ω|2
sin2
ρee =
|Ω|2 + ∆2


 p
t
2
2
|Ω| + ∆ .
2

(2.26)

Therefore, the excited state population fraction ρne implied by Equation 2.24 is|Ωeg |2
ρne =sin2|Ωeg |2 + [∆ − ωz (ne − ng )]2

 q

t
2
2
|Ωeg | + [∆ − ωz (ne − ng )] .
2

(2.27)

Equation 2.27 describes transitions occurring for detunings ∆ = ωz (ne − ng ) (Figure 2.10).
2.4.3

The Lamb-Dicke Regime

The transition for which the motional state does not change (ng = ne ) is a carrier transition.
Carrier spectroscopy of lattice-trapped atoms has the advantage (compared to untrapped atoms)of being free of motion-related frequency shifts [30]. This is desirable since the clock is based on theunshifted bare frequency. However, at a sample temperature of a few µK, the Sr atoms may notExcited state population

25
1
n =0
g

0.8

ng = 1ng = 2

0.6
0.4
0.2
0
-600

-400

-200

0
200
Detuning (Hz)

400

600

Figure 2.11: Sideband spectra for η = 0.2, with νz , Ω, and t the same as in Figure 2.10. The carriertransition is robust for different values of ng . For η ≪ 1, the sidebands would not be noticeable on
this plot.

be entirely confined to the motional ground state of the lattice. As Figure 2.10 shows, the contrast
of carrier transitions can degrade if one interrogates a gas of atoms that occupy different motional
states.
To alleviate this problem, it is useful to work in the Lamb-Dicke regime. In this limit, theclock laser wavelength is much greater than the harmonic oscillator length along the lattice axis
(λc ≫ z0 , or equivalently η ≪ 1). Here spectroscopy of the carrier transition is like that of an atom
at rest, which is useful for an atomic clock based on a bare transition. The lineshape contrast is
maximal, and Doppler and recoil shifts are absent. For typical Sr2 lattice parameters, η = 0.2,
which is sufficient to reap these benefits (Figure 2.11).6

These advantages are not afforded toclocks based on free atoms (like Cs fountain clocks). Also, the magic wavelength discussed in
Section 2.3.4 alleviates the main drawback to introducing the trap, namely the systematic ac Stark
shift from the trap itself.
6

Probing the clock transition along the lattice radial direction, for which νr = ωr /2π = 120 Hz, would yieldη = 6. This is the reason for aligning the clock laser to the z direction.

26
2.4.4

Anharmonicity and Sideband Broadening

Since the Sr2 clock operates in the resolved sideband regime, measuring the frequency difference between the carrier and the red or blue sidebands (Figure 2.11) yields a value for νz . In fact,
in the harmonic approximation, the frequency difference between the carrier and the red or blue
sidebands is exactly νz . Since measurements of νz via the sideband frequencies will be important to
the lattice Stark shift systematic measurement (Section 5.6), it is important to consider how thesefrequencies are related to νz in the true potential, which is anharmonic.
The motional part of the Hamiltonian in Equation 2.21 was approximated as harmonic,
Hmotion = ~ωz (â† â + 1/2). Ignoring motion along the lattice radial direction, the full Hmotion isp2− U0 cos2 (kz),(2.28)2m√z0† ) as before, and p =√0 (â† − â), z = √(â+âwhere p = ipm~ωz . Expanding the potential in kz0 ,02
2
Hmotion =Hmotion = ~ωz

1â â +
2
†



−

1
1
~ωz ηL2 (â + â† )4 + ~ωz ηL4 (â + â† )6 + . . . ,1290(2.29)p√and ηL = kz0 / 2 = νrec /νz is the lattice Dicke parameter. The trivial potential offset −U0 has
been neglected. The blue sideband frequency νbsb (Figure 2.9) is thereforeνbsb = hnz + 1|Hmotion |nz + 1i/h − hnz |Hmotion |nz i/h≃ νz − (1 + nz )νrec .

(2.30)

Terms of order ηL4 and higher have been discarded since ηL is typically about 0.2. As Equation 2.30
shows, the blue sideband frequency is largest for atoms in the ng = 0 state.
The effect of radial motion can be analyzed by expanding Hmotion with the radial dependence
included [12],

27

aL

5
4
3
2
1

20
H´ 10-5 kHz-1 L

Lineshape function

H´ 10-3 kHz-1 L

Lineshape function

6

0

bL

15
10
5
0

80

85

90

95

100

80

Detuning from carrier HkHzL

85

90

95

100

Detuning from carrier HkHzL

Figure 2.12: a) The unbroadened (purple) and broadened (blue) lineshape functions describing
the blue sideband. These lineshape functions are area normalized. The carrier is located at zerodetuning. b) The broadened lineshape function alone, showing broadening toward the carrier.

22
p2− U0 e−2r /w0 cos2 (kz)
2m


p2
1 4 4 2r 2 2r 4 2k2 z 2 r 2
2 2
+
+
.
≃
− U0 1 − k z + k z − 2
2m
3
w0
w04
w02

Hmotion =(2.31)

Considering now the radial motional excitations nx and ny ,

νbsb = hnx , ny , nz + 1|Hmotion |nx , ny , nz + 1i/h − hnx , ny , nz |Hmotion |nx , ny , nz i/h≃ νz − (1 + nz )νrec − νrecνr(nx + ny + 1).
νz(2.32)

Since nx and ny do not change during spectroscopy, the r 2 and r 4 terms in Equation 2.31 do notcontribute to νbsb ; however, the r 2 z 2 term that couples the radial and axial states results in thedependence on radial excitations. To obtain the correct sideband lineshape, we must thermallyaverage over these states [12],Lblue (δ) =X e−nx hνr /kB Tr e−ny hνr /kB Tr e−nz hνr /kB Tz
2.
πΓZ n ,n ,n
1 + Γ42 (δ − νbsb )2xy(2.33)z

Here Lblue is the lineshape function, Γ is the power broadened linewidth of a zero-temperature lineshape (modeled here as Lorentzian), Tr is the temperature in the radial direction of the lattice, Tz

28
is the axial temperature, and Z =

Q

x,y,z

P

i exp(−ni hνi /kB Ti ) is the partition function. Although

there are enough radial motional states in the potential to approximate the nx and ny sums as
running from zero to ∞, the sum over nz should be taken from zero to the maximum number ofaxial excitations in the potential, which in this model is ⌊U0 /hνz ⌋ (where ⌊. . .⌋ denotes the floor
function).
This thermal averaging broadens the sideband asymmetrically toward the carrier (Figure2.12). Physically, this broadening can be viewed in terms of atoms experiencing quantized motionin the axial direction (hνz > kB Tz ) and classical motion (hνr ≪ kB Tr ) in the radial direction.
Atoms rolling back and forth in the radial direction will experience different axial trap frequencies
since the trap intensity they feel will vary depending on their radial position. The sharp sidebandedge at large detunings from the carrier corresponds to when atoms are in the most intense regionof the lattice laser (r = 0) and in the lowest axial motional state (Equation 2.30), resulting in thelargest axial trap frequency. Therefore, in light of Equation 2.30, the frequency difference νblue
between the blue sideband edge and the carrier is approximately

νblue ≃ νz − νrec .
2.4.5(2.34)The Lattice in the Presence of Gravity

As I will explain in Section 5.6, the lattice ac Stark shift is studied as a function of thepotential depth experienced by the atoms. This depth can be conveniently determined throughνz = νblue + νrec (Equation 2.34), which is measured with sideband scans of the clock transition(Figure 3.19). According to Equation 2.9, U0 = (νz /2νrec )2 Erec ; however, this expression is only
valid when gravity is ignored.
Gravity will change the sample’s equilibrium position, moving the atoms out of the mostintense region of the lattice. Since the position of the atoms is a function of U0 , the latticelaser intensity at the location of the ultracold sample will not scale linearly with the laser power.
Fortunately, the measured axial trap frequency can always determine the intensity at the location

29of the atoms even in the presence of gravity. To demonstrate this, I analyze this situation withclassical potential theory [41]. The lattice potential including gravity is

2

2

Ugrav = −U0 e−2x /w0 cos2 (kz) + mg(x cos θ + z sin θ),(2.35)

where θ is the angle the lattice axial direction subtends with respect to the horizontal (θ = 16◦ for
Sr2). Here I take x, y and z to be the same coordinates as in Equation 2.6: z is the lattice axis, x
makes an angle θ with respect to gravity, and y is taken to be perpendicular to both gravity andthe lattice axis.7
The equilibrium coordinates xeq and zeq of Ugrav are given by∂Ugrav=0eq∂x x=xz=z(2.36)∂Ugrav= 0.
Uz (xeq , zeq ) =eq∂z x=xz=z(2.37)Ux (xeq , zeq ) =eqeq

When the two axes are coupled, the angular trap frequencies are the square roots of the eigenvalues
of



U
(x
,
z
)
U
(x
,
z
)
xz eq eq 
1  xx eq eq,mUxz (xeq , zeq ) Uzz (xeq , zeq )(2.38)

where Uxx , Uxz , and Uzz are second derivatives [41]. The equilibrium position of the atoms isaffected enough by gravity to significantly diminish the potential depth at the location of theatoms (Figures 2.13a and 2.13b). However, as a function of νz , Ulat (Equation 2.6, which is the
optical part of Ugrav ) evaluated at xeq and zeq agrees with the simple expression (νz /2νrec )2 Erec at
the 0.1 Erec level (Figure 2.13c).
According to Equation 2.9, in the absence of gravity (νz /2νrec )2 Erec is equal to U0 =4 α1S0 P/πǫ0 cw02 (Equation 2.6), where P is the total trap power. As the above analysis shows,7

y is ignored in this analysis because its contribution is trivial. The equilibrium position along y is always y = 0.

30

aL
UlatHrÓeqL + U0 HErecL

Position HΜmL

-20

bL

20

-30
-40
-50
-60
-70

15

10

5
2

3

4

5

6

7

8

2

3

4

P HWL

5

6

7

8

P HWL

Potential depth HErecL

-50

cL

-100
-150
-200
-250

Simple Expression
Optical Potential

-300
50

60

70

80

90

100

110

120

Axial trap freq HkHzL
Figure 2.13: a) The lattice x direction as a function of the one-way lattice power. The z direction
is only affected at the 0.1 nm level for the Sr2 experimental parameters w0 = 160 µm and α1S0 =286 a.u. [105]. x = z = 0 is the laser focus where the atoms would be located without the influenceof gravity. 8 W is the maximum lattice power our system can achieve. b) The difference between
the potential depth at the equilibrium position of the atoms and the potential depth at the bottomof the trap (U0 ) in units of Erec as a function of one-way lattice power. The change is significant
even for large lattice powers. c) The potential depth at the equilibrium position and the expression−(νz /2νrec )2 Erec as a function of νz .

when gravity is considered, (νz /2νrec )2 Erec = 2 α1S0 Ieq /ǫ0 c, where Ieq is the laser intensity experienced by the atoms at their equilibrium position. Therefore, the potential depth at the location of
the atoms is related to the blue sideband edge frequency νblue as

Uatom =

νblue + νrec
2νrec

2

Erec .

(2.39)

31

2.5

Digital Servo to Atomic Spin Stretched States

2.5.1

Spin Polarization

The concerns about spectroscopy in a lattice raised in the previous section are addressed byworking with a small value of the Dicke parameter η (Section 2.4.1) and operating in the resolved
sideband regime. I now turn my attention to referencing the clock laser to the Sr clock transition.
The free running clock laser suffers from frequency drift that would preclude its use as a frequency
standard, which requires an accurate, stable frequency. Locking this laser to the clock transition
removes this drift and ensures that the laser frequency matches that of the atoms trapped in thelattice. With the aim of providing an overview of how this lock works, it is important to first discuss
magnetic effects on the clock transition due to interactions between external magnetic fields andthe atom’s spin.
Since F = 9/2 for both clock states, each state has the same 10 values of mF . As explainedin Section 2.1, the nuclear part of the clock transition g-factor determines its first-order Zeeman
shift. The differential first-order Zeeman shift of the clock transition is∆νZ1 = −µB δg~ = 9/2, mF i = − µB δg mF B,
hF = 9/2, mF |F~ · B|Fhh(2.40)~ is an external
where µB is the Bohr magneton, δg is the clock transition differential g-factor, B
dc magnetic field, and a π transition has been assumed. The g-factor has been measured [16] asµB δg/h = −108.4(4) Hz/G.
The clock transition Rabi frequency is also affected by the mF state. Using the Wigner-Eckart
theorem [17, 102] to make the mF dependence explicit,~ = 9/2, mF i · E~c0 /~ΩmF = −hF = 9/2, mF |d|F~
= −hF = 9/2, mF |F = 9/2, mF , 1, 0ihF = 9/2||d||F= 9/2i · E~c0 /~2mF= − √ Ωred ,3 11(2.41)

Excited state population

Excited state population

0.6 aL
0.5
0.4
0.3
0.2
0.1
0.0
-20

-10

0

10

20

1.0

32

bL

0.8
0.6
0.4
0.2
0.0
-150 -100 -50

Detuning HHzL

0

50

100

150

Detuning HHzL

Figure 2.14: a) ρmixture plotted t = 160 ms for B = 0 (purple) and B = 10 mG (blue). The latteris typical of stray laboratory magnetic fields. At zero field, the maximum achievable contrast is
0.61 (rather than unity, like in the case of the two-state population fraction ρee ). This is because
the Rabi frequency ΩmF is different for each mF state, each of which comprise 10% of the total
population. At small fields, the lineshape broadens and loses contrast since the states are slightly
split but unresolved. b) ρmixture for B = 250 mG (blue) and spin-polarized lines (red and orange).
At B = 250 mG, the ten π transitions are well resolved from each other. Polarizing the entire
population into one of the two angular momentum stretched states, shown in red and orange,
results in the Rabi lineshape of a two-state system.

where ΩmF is the Rabi frequency for the specific mF state, hF = 9/2, mF |F = 9/2, mF , 1, 0i is a~
Clebsch-Gordan coefficient, E~c0 is the clock laser electric field amplitude vector, hF = 9/2||d||F=
9/2i is the reduced dipole matrix element, and Ωred is the reduced Rabi frequency (Rabi frequency
corresponding to the reduced matrix element).
Since state randomization during MOT cooling (Section 2.2) prepares the atoms in an incoherent mixture of mF states with equal population in each state [86], the excited state population
fraction after this cooling is

1
ρmixture =2F + 1F
X

mF =−FΩ2mF
sin2Ω2mF + ∆2mF

 q

t
2
2
ΩmF + ∆mF .
2(2.42)In this expression, ρee from Equation 2.26 is evaluated for the appropriate Rabi frequency ΩmF
and detuning ∆mF = ∆ + µB δg mF B/~ and then summed over. Compared to the two-state
population fraction ρee , typical stray magnetic fields of a few mG result in line broadening andcontrast degradation of ρmixture (Figure 2.14). This is undesirable since broader lines and poorer

33
contrast tend to worsen clock stability.
To deal with this problem, the Sr clock operates with a bias magnetic field B of about 300
mG to fully resolve the ten π transitions (Figure 2.14). The bias field direction B̂, clock laser
polarization ǫ̂c , and lattice polarization ǫ̂ are parallel. To take advantage of the best possible
contrast, the atoms are spin polarized. Spin polarization is performed with an additional laser thatis resonant with the |1 S0 , F = 9/2i → |3 P1 , F = 9/2i transition. The laser has circular polarization(with switchable helicity) and propagates in the direction of the bias field, driving σ transitions8
until atoms are polarized into one of the angular momentum stretched states (mF = ±F ). The
resulting population fraction is ρee (Figure 2.14).

2.5.2

Locking to the Clock Transition

To reference the clock laser to the Sr clock transition, the laser is locked to the resonancecenter using a feedback loop (Figure 2.15). The excited state population fraction is measured on
one side of resonance, then the clock laser is stepped by the full width at half maximum (FWHM)
of the lineshape, and then the population fraction on the other side of resonance is measured. The
difference between these population fractions is the error signal ǫclock , which the servo loop nulls by
adjusting the laser frequency, locking the laser to the resonance center. This lock will be referred
to as an atomic servo.
The FWHM γ can be changed by adjusting the pulse duration of the clock laser. The relation
between γ and the probe time t is

γ≃

0.8
,
t

(2.43)

which is obtained by numerically finding the detuning at which ρee in Equation 2.26 is equal to
1/2, assuming a π-pulse.9 When the FWHM of the measured lineshape obeys this equation, it isoften referred to as the “Fourier-limited linewidth” for Rabi spectroscopy.
8

Transitions that change mF by one
A π-pulse occurs when the Rabi frequency of the transition times t is equal to π, yielding the maximum signal
contrast.
9

Excited state population

34
1.0

aL

b)

0.8

Correction
PMT

0.6
0.4
0.2
0.0
-20

Clock
Laser
-10

0

10

Modulator

20

Detuning HHzLCounting
Laser

Figure 2.15: a) The clock transition is probed on the left side of resonance, driving the excitedstate population fraction to ρL (indicated by the left blue dot). The laser frequency is then steppedby the full width at half maximum of the transition, which is set using the clock laser probe time.
The right side is probed, driving the population fraction to ρR (right blue dot). Measurements ofthese two populations fractions yields an error signal ǫclock = ρL − ρR . When the clock laser isdetuned from the clock transition, ǫclock 6= 0. The atomic servo adjusts the laser frequency untilǫclock = 0, locking the laser to the resonance center. b) A simplified block diagram of the servo loop.
The clock laser drives the atoms along the lattice axial direction. The counting laser fluoresces theground state atoms, and some of the fluorescence is collected by the PMT. A computer stores thesignal observed by the PMT, and once the population fraction has been measured on either sideof resonance, the computer processes the correction needed to center the laser on resonance. The
correction is made by varying an optical frequency modulator that changes the clock laser detuning
in attempt to null ǫclock .

The atomic servo measures the population fractions ρL and ρR (Figure 2.15) destructively,
meaning that each time either of these quantities are measured, the ultracold sample is lost. A new
sample needs to be cooled and loaded into the lattice again, resulting in servo dead time, during
which no information about the resonance center is added to the loop. For this reason, the atomic
servo is digital. One cycle of preparing atoms, interrogating them with the clock laser, and probing
the sample population takes about Tc = 1 s, where Tc is the cycle time. Corrections to the laser
frequency fcorr are computed (Figure 2.15b) using a digital proportional-integral-differential (PID)
filter,



Td
Tser
(ǫn + ǫn−1 ) +(ǫn − ǫn−1 ) ,fcorr,n = kp ǫn +
2Ti
Tser

(2.44)

where ǫn is the nth value of ǫclock = ρL − ρR (Figure 2.15) for the data record; Tser is the cycle

35
time10 of the servo; and kp , Ti , and Td are the proportional, integral, and differential gain constants,
which are found empirically.

2.5.3

Digital Lock to Alternate Stretched States

So far, I have explained techniques for dealing with the broadening and reduced contrast that
results from stray magnetic fields. Operating in a bias field ensures a nice Rabi lineshape rather
than something broader. Spin polarization improves the signal-to-noise ratio of clock transition
measurements. However, the bias field has the undesirable effect of frequency shifting the stretchedstates hundreds of Hz from their B = 0 values, which is an impediment for clocks based on a bareatomic resonance.
A stretched-states servo [112] is used to combat this problem. This technique involves fourinterrogations of the clock transition. Switching between the two stretched states, measurements
are taken on either side of both resonances (Figure 2.16). The modulator that controls the laser
frequency is an acousto-optic modulator (AOM), which shifts the frequency of the light passing
through it by the frequency of a microwave drive injected into the device. For each measurement,
the modulator drive frequency and the excited state population fraction are both recorded. With
this information, the value of the modulator frequency needed to set the laser frequency on theresonance center is computed for both states using Equation 2.44. For each cycle of these fourmeasurements, the ǫclock data is treated by the software as two independent atomic servo error
signals (one for each stretched state).11 This is accomplished by computing the correction for eachstretched state only from data taken for that resonance.
The modulator frequency required to detune the clock laser to a given mF state is

fmF = fSr +

µB δg
mF B,
h

(2.45)

where fSr is the modulator frequency that centers the laser on the zero-field resonance. With this10

The servo cycle time depends on how it is run. For a single atomic servo running by itself, Tser = 2Tc . For two
servos updating in an alternate fashion, like in Section 2.5.3, Tser = 4Tc .
11
Note that the phrase “stretched-states servo” implies two atomic servos.

36
Cooling andloadingσ+ polarization
Left side ofresonance
σ- polarization
Right side ofresonance

Time
Figure 2.16: Timing diagram (not to scale) for the spin polarization and locking sequence. Atomcooling and lattice loading takes up most of one measurement cycle. In the first cycle, after the
sample is prepared, atoms are spin polarized into the mF = F stretched state using σ+ -polarized
light. The clock transition is excited on the left side of resonance (Figure 2.15), and the excitedstate population fraction is recorded. In the next cycle, the helicity of the spin polarization laseris switched, polarizing atoms into the mF = −F stretched state. Again, the left side of resonanceis excited, and the population fraction is recorded again. The third and fourth cycles are the sameas the first two except that now the right side of resonance is measured.

stretched-state locking scheme, the data record of values for f9/2 and f−9/2 can be used to infer
fSr as the stretched state average frequency (f9/2 + f−9/2 )/2. This value of fSr can then be used
to produce clock laser light oscillating at the zero-field resonance frequency.12

These frequency

data can also be used to infer the magnetic field through the stretched state frequency difference
f9/2 − f−9/2 .
With the first-order Zeeman shift removed from the stretched state average frequency, thisalternate stretched state scheme has done much to eliminate the problems associated with locking
to an unpolarized resonance in the presence of stray magnetic fields. The one drawback is that thebias field creates a second-order Zeeman shift [16] that is not canceled upon averaging f9/2 andf−9/2 . This shift is regarded as a systematic effect that must be evaluated to understand how muchthe clock laser frequency differs from the bare Sr frequency. This is a worthy trade off since thesecond-order Zeeman shift is one of the more manageable systematic effects in the JILA Sr clock.
12This can be accomplished by splitting off some of the clock laser light before it enters the modulator in Figure
2.15b and then sending this light through a second modulator, which has a frequency that tracks the computed value
for fSr

37

2.6

The Magic Wavelength Optical Lattice Revisited
With an understanding of the spin states involved in clock operation, the need for a bias

field, and the bias field alignment, it is worthwhile to revisit the discussion of the magic wavelength
optical lattice. Recalling Equation 2.19,on∆νac = − κs + κv mF ξ k̂ · B̂ + κt 3m2F − F (F + 1) (3|ǫ̂ · B̂|2 − 1) Ulat (r, z),

it is clear that the vector shift term is triply suppressed. Since ǫ̂ and B̂ are aligned parallel to one
another (Section 2.5.1), the value of k̂ · B̂ is minuscule, arising out of small imperfections in thealignment of ǫ̂ and B̂. The lattice is also linearly polarized, so ξ is also very small but nonzero
due to imperfections in the polarization purity. Also, since the stretched-states servo cancels shifts
linear in mF , it diminishes the vector term as well. This strong suppression ensures that the vector
shift does not affect the Sr clock.
Furthermore, since mF = ±F for clock operation, the tensor term appears less ambiguousthan before (Section 2.3.4) when I claimed that the magic wavelength causes the scalar and tensor
terms to cancel (without elaborating on how this works given that the tensor term is different foreach mF state). The lattice Stark shift of the stretched-states servo is∆νac = − [κs + 2κt F (2F − 1)] Ulat (r, z).

(2.46)

Therefore, the desired magic wavelength is one for which the scalar and tensor shifts cancel whenmF = ±F and ǫ̂ · B̂ = 1.

2.7

Introduction to Stability and Systematic Effects
This chapter describes an ultracold gas of lattice-trapped Sr used as a frequency reference.

The technique for referencing a laser to the clock transition has also been described. To characterize the performance of this system, the stability and systematic uncertainty must be determined(Section 1.6).

38
Clock stability is limited by the clock laser’s spectral noise. Because the servo loop describedin Section 2.5.3 is digital, it has an inherent problem with aliasing, which degrades servo stability[28]. This problem is known as the “Dick Effect.”
Another important limitation to clock stability is the standard quantum limit, which, in thecontext of clocks, is called the “quantum projection noise limit” [54]. Since the clock is basedon locking a laser to the half-maximum points of a Rabi lineshape (Figure 2.15), the quantum
state of each atom is projected into either |gi or |ei with equal probability when the excited state
population fraction is measured. Therefore, quantum projection noise (or QPN) is given by cointoss (binomial) statistics. If several coins are tossed at once, the percentage that land on heads
will fluctuate about 50% each time another toss is made. If the number of coins is increased, thepercentage that lands on heads will fluctuate less. Similarly, more atoms leads to less QPN, which√scales like 1/ N , where N is the number of atoms.
As explained in Section 1.6, systematic uncertainty arises because the Sr lattice clock is basedon the bare clock transition. Perturbations that shift this transition must be well understood andtheir uncertainties must be quantified so that the clock oscillator frequency can be appropriately
corrected to the bare atom frequency. The uncertainty of the combined corrections is the total
systematic uncertainty, which is the uncertainty in a clock’s ability to provide the bare atom
frequency.
Most of the major systematic shifts that need to be considered for the Sr clock are electromagnetic perturbations. Despite operating the lattice at or near the magic wavelength, there canbe a residual ac Stark shift from the lattice. Also, as explained in Section 2.5.3, the bias field createsa second-order Zeeman shift. A residual first-order Zeeman shift might also exist even though thestretched-states servo removes most of this effect. Patch charges or faulty electronics can cause
dc electric fields that create clock transition Stark shifts. The clock laser itself causes an ac Stark
shift. The ambient room-temperature heat in our lab is an electromagnetic field that causes a Stark
shift. Furthermore, the clock transition is perturbed by atom-atom interactions.
There have been two evaluations of these systematic effects, as well as several other more39
minor shifts, in the JILA Sr2 clock. However, before I begin a discussion of these measurements, I
will describe the JILA Sr2 apparatus in the next chapter.

Chapter 3

The Sr2 Apparatus

3.1

The Sr2 Vacuum Chamber

3.1.1

Sr Oven

The different stages of the Sr2 clock measurement cycle were discussed in the previous chapter.
To summarize, first a hot atomic beam from the Sr oven is slowed (with a Zeeman slower) andthen trapped using a three-dimensional MOT. The first stage of MOT cooling operates on a dipole
allowed transition at 461 nm, reducing the temperature of the atoms to about 1 mK. Next, the461 nm MOT is shut off and a second-stage of three-dimensional MOT cooling begins. The second
MOT operates on a singly forbidden (intercombination) transition at 689 nm, cooling the atoms
to a few µK. These cold atoms are loaded into a one-dimensional magic wavelength optical lattice,
and a clock laser interrogates the 1.5 mHz linewidth clock transition. After interrogation by theclock laser, the change in the Sr excited clock state population is measured. This measurement isprocessed by a feedback loop that corrects the clock laser frequency, locking the laser to the clocktransition. This section describes the systems used to realize these measurement stages.
The hot atomic beam comes from a Sr oven that is based on a custom-made 2.75” vacuum
nipple with a bend (Figure 3.1). The nipple contains solid Sr and a microchannel array to collimate
the atomic beam. The oven is heated with two pairs of clamshell heaters. Homemade refractory
ceramic fiber spacers keep the clamshells in place. Several thermocouples are attached to thebent nipple for temperature monitoring. For insulation, the oven is wrapped with a ceramic fiber

41

Figure 3.1: The Sr oven after many years of use. The clamshell heaters and water cooling block
are not shown in this picture. One clamshell pair is wrapped around the conflat flanges closest to
the bend (labeled “reservoir”). The strontium is contained near these flanges. Another clamshellpair is used to heat the “nozzle” region. The water cooling block is used where the oven joins withthe rest of the chamber.

blanket, then a ceramic braid, and then aluminum foil. A water cooling block prevents heat fromconducting away from the oven to other parts of the chamber.
The heaters are driven with ac current provided by variacs that output 60 ac volts. For eachheater pair, both clamshells are wired in series, and their current can be gated with a solid state
relay. Temperature controllers from Omega Engineering sense on two of the thermocouples andactuate on the solid state relays. Typically, the Sr reservoir is heated to 575 ◦ C, and the nozzle(Figure 3.1) is heated to 625 ◦ C.

3.1.2

Main Chamber

The Sr2 main chamber is an 8 inch extended spherical octagon from Kimball Physics. The
atomic beam enters through one of the 16 1.33 inch conflats that makes a 16◦ angle with gravity.
A pneumatic gate valve separates the main chamber region from the oven region (Figure 3.2). Thisvalve is closed when the Sr2 clock is not running, protecting the main chamber region in case a leakwere to occur in the oven region. Opposite the atomic beam is a sapphire viewport that is heated
to 150 ◦ C to prevent Sr deposition.
The remaining 14 1.33 inch conflats, the two 8 inch conflats, and the eight 2.75 inch conflats

42

Figure 3.2: A CAD of the Sr2 vacuum chamber. A number of components—including pumps, coils,
and valves—have been omitted to show the fundamentals of the chamber. The Zeeman slower isshown here as a simple nipple, around which the slower coils are wrapped in the actual machine.
The Zeeman slower is joined with the main chamber using a flexible bellows. The side of the gate
valve containing the main chamber is referred to in this work as the “main chamber region,” andthe side of this valve with the oven is called the “oven region.” The oven region has two ion pumps
as well as an all-metal valve for turbo and roughing pump access. The main chamber also has an
all-metal valve for turbo and roughing pumps.

43

Figure 3.3: The main chamber region. The support structures and one ion pump are depicted.
Opposite the four-way cross (on which the ion pump is attached) is a nude ion gauge. The top
MOT coil can be seen as well. The two ion pumps in the oven region are not shown here.

are fitted with multiband-antireflection-coated fused silica viewports. The coating was performed
by Research Electro-Optics and provides reflectivity at the 0.1% level for 461 nm, 689 nm, 698 nm,
and 813 nm.
The main chamber has a 70 l/s ion pump (Figure 3.3). This configuration results in a pressure
at the mid 10−10 torr level. At this pressure, the number of atoms in the lattice exponentially decays
due to collisions with the chamber background gas with a time constant of a few seconds. Thisvacuum lifetime is sufficient for an atomic clock.

3.2

Cooling on the 1 S0 →1P1 Transition

3.2.1

461 nm Laser System

The first stage of cooling is based on the 461 nm |1 S0 , F = 9/2i → |1 P1 , F = 11/2i transition.
The three hyperfine states of the 1 P1 level are unresolved [61], so cooling on this transition is

44

Isolator

MasterTransverse
cooling

Fiber launchTransverse cooling laser
Slave
Saturated
spectroscopy

λ/4

AOM

-825 MHz

AOM

Slave

λ/2

-103.5 MHz

AOM-120 MHz

Block
λ/2

AOM

-103 MHz

AOM-114 MHz

λ/2

Zeeman slower laser

MOT

AOM

Slave

-72 MHz

SlowerProbe

MOT laser

Figure 3.4: The 461 nm laser system. The master laser is a New Focus ECDL that is frequency
stabilized using sub-Doppler saturation spectroscopy of a Sr discharge cell. A fiber is used toclean up the mode of the master laser. The AOMs here are labeled with the frequency shift theygenerate. All AOMs in this figure are aligned to provide light in their -1st order. The AOM before
the saturated spectroscopy lock can be adjusted to a -155 MHz shift for 88 Sr trapping. The AOMs
provide the detunings needed for the different cooling beams. The three slaves are injection locked
by the master laser through the rejected light port out of the laser isolators. Light for the Zeeman
slower and MOT is delivered to the Sr2 chamber by fibers. The slave laser for transverse cooling issent to the chamber through free space.

45
affected by the entire 1 P1 hyperfine manifold. Sr2 had two generations of the laser system used
for this cooling stage. In the first generation system, 922 nm light was frequency doubled to
461 nm using home-built linear doubling cavities based on periodically poled KTP. The 922 nm
laser originated from an Eagleyard diode that seeded an Eagleyard tapered amplifier. This systemsuffered appreciable power degradation after a few hours of operation and was difficult to workwith because of this. As we later learned, linear doubling cavities based on periodically poled KTP
suffer from thermally induced dephasing, which compromises frequency doubling efficiency [68].
Fortunately, over the past few years, single frequency diode lasers at 461 nm became available.
After encouragement from our research team, New Focus began selling external cavity diode lasers(ECDLs) at 461 nm, and Nichia also made an engineering sample of single frequency laser diodes
at this color. Using this new technology, a second generation 461 nm laser system was constructed(Figure 3.4). This system uses a New Focus 461 nm ECDL referenced to the Sr 1 S0 →1P1 transition
frequency with sub-Doppler saturation spectroscopy (Figure 3.5). Light from the New Focus ECDL
injection locks three Nichia slave diodes, which are dedicated to the Zeeman slower, 461 nm MOT,
and transverse cooling beams.
The Nichia diodes put out 100 mW of power. The New Focus laser originally emitted about
50 mW of power, but this number degraded to 35 mW over the course of a year and remained at
this value ever since. Nevertheless, 35 mW is sufficient for our application.

3.2.2

Zeeman Slower and Transverse Cooling

70 mW of light from the Zeeman slower slave laser is delivered by a fiber to the Sr2 mainchamber. Zeeman slowing is switched on and off with a mechanical shutter installed in the path of
this slave laser (Figure 3.4) before the fiber that delivers it to the atoms. The slower is wound in
spin flip or zero crossing configuration. It also has a compensation coil to cancel the slower fieldat the center of the main chamber where the atoms are trapped. After fiber delivery near the mainchamber, the Zeeman slower laser passes through a λ/4 waveplate and enters the chamber throughthe heated sapphire window.

46

Saturated spectroscopy

EOM

λ/2

AOM
-80 MHz

Dischargecell

PhotodiodeServo

Figure 3.5: Saturated spectroscopy for referencing the New Focus master laser to the Sr 1 S0 →1P1
transition [87]. The photodiode output is mixed with the 40 MHz EOM frequency, and the outputof this is sent to a lock-in detector referenced to the frequency at which the AOM intensity is
chopped (90 kHz). The New Focus master laser external cavity piezoelectric transducer is locked
to the resulting dispersive feature.

Figure 3.6: The Sr2 Zeeman slower. The right half of the slower has a zero crossing region. Coils
to the left of the zero crossing have current flowing in the opposite direction from the coils on theright. The zero crossing configuration reduces the amount of wire needed for the slower, which
reduces the resistive heating of the coils.

47
The output of the transverse cooling laser is split in two and sent into viewports mounted on
six-way cross (Figure 3.2). One beam is periscoped up and sent through the horizontal directionof the cross. This light is retroreflected through the atomic beam to better counter balance theforce in the horizontal direction. A similar retroreflected beam is present in the (nearly) verticaldirection. When these transverse cooling lasers are well aligned, they boost the number of atoms
trapped in the lattice by a factor of two.

3.2.3

Three-Dimensional MOT and Repump Lasers

A 3D MOT operating on the 461 nm transition is set up at the center of the main chamber.
The slave laser designated for MOT power is delivered to the main chamber with an evanescent wave
fiber-based one-to-three splitter from Evanescent Optics. The three output fibers are collimated to
2.5 cm diameter beams. The two horizontal MOT beams and one vertical beam are sent into themain chamber through two-color waveplates (3λ/4 for 461 nm and and λ/4 for 689 nm), creating
circular polarization. Each beam is retroreflected, double passing another two-color waveplate toprovide the usual three pairs of σ+ - and σ− -polarized beams for the MOT [81].
The MOT light passes through an AOM before being fiber coupled to the main chamber(Figure 3.4). This AOM is useful for fast intensity switching and servoing. The counting laser,used for atom number counting (Section 2.5.2), is taken from the zeroth order of this MOT AOM.
The counting laser has its own AOM to bring its detuning closer to zero.
The MOT coils are an anti-Helmholtz pair mounted on the top and bottom of our vacuum
chamber (Figure 3.3). The coils are made of square-shaped copper tubing1 that has an electricallyinsulating coating. Chilled water is flowed through the tubing to keep the coils from overheating,which would cause the electrical insulation to melt and short coil turns. The coils are driven by
a 60 A power supply, and the coil current is controlled with a servo that actuates on a MOSFET
bank. A Hall probe senses the coil current and feeds back through a PID filter to the gates of the
MOSFETs. The MOSFETs are mounted on a cooling block to prevent overheating. The coils are1

By “square shaped” I mean that the cross section of the tubing is square shaped.

48
designed to provide a gradient at the center of the chamber of 60 G/cm for 50 A of current.
As explained in Section 2.2, repump lasers are needed to deal with shelving in the 3 P2 state.
To target the many hyperfine states in the repumping process, the repump lasers are ECDLs
that are frequency modulated by sweeping both the laser current and piezoelectric transducer (or“piezo”) voltage, spectrally broadening these lasers. These lasers are left free running; however,frequency drift is not an issue since drift over several hours of operation is insignificant comparedto the large spectral widths of the modulated repump lasers.
With the Zeeman slower, transverse cooling, 461 nm MOT cooling, and repumping combined,
this first stage of trapping and cooling takes 500 ms and results in roughly ten million atoms trapped
at a temperature of about 1 mK.

3.3

Cooling on the 1 S0 →3P1 Transition

3.3.1

689 nm Laser System

As explained in Section 2.2, ultracold Sr is prepared using two stages of MOT cooling. The
natural linewidth of the second cooling transition, 1 S0 →3 P1 , is 7.5 kHz. This is significantly
narrower than the linewidth of ECDLs, which are typically of order 100 kHz. To ensure that thelasers used for the second stage of cooling provide sufficient power within the linewidth of thetransition, the 689 nm laser system makes use of cavity stabilization.
This system is based on home-built ECDLs with diodes from Hamamatsu. These diodes wereanti-reflection coated using the diode coating facilities at JILA. A master ECDL is stabilized to acavity, resulting in a 300 Hz laser linewidth. To remove the effect of slow cavity drift, the master isfrequency referenced to the 1 S0 →3P1 transition using saturated spectroscopy and a low-bandwidth
servo loop (Figure 3.7). The trapping laser, operating on the |1 S0 , F = 9/2i → |3 P1 , F = 11/2i
transition, and the stirring laser, operating on the |1 S0 , F = 9/2i → |3 P1 , F = 9/2i transition(Section 2.2), are slaved to the master with phase locks.
Each phase lock is based on one optical and one RF beat. The slave lasers are opticallyAOM-1.1 GHz

AOM-80 MHz

49

Rotator

Heat
pipe

λ/4

λ/2

EOM

CavityTo AOMfrequency lock

λ/2
λ/2

To PDHloop filter

To phase
locks

Saturated spectroscopy

Pound-Drever-Hall stabilizer

MasterFigure 3.7: A home-built master EDCL is double passed through a 1.1 GHz AOM and stabilized
to a cavity using the Pound-Drever-Hall technique [10]. The output of the Pound-Drever-Hall
photodiode is mixed with the EOM frequency to achieve an error signal, to which the master laser
current and external cavity piezo are locked. The cavity is under vacuum, temperature controlled,
and has passive vibration damping. The stabilized laser linewidth is 300 Hz. The saturation
spectroscopy setup is similar to Figure 3.5 aside from the fact that here the laser is locked to a heatpipe with Sr inside rather than a discharge cell. The photodetector output is mixed with the EOM
frequency and then measured with a lock-in detector referenced to the AOM chopping frequency.
A low-bandwidth loop filter takes the lock-in output as an error signal and feeds back onto the 1.1
GHz AOM frequency.

beat with the master, and the output of this beat on a photodetector is beat again with an RF
frequency (Figure 3.8). The output of this second beat provides a error signal to which the current
and external cavity piezos of the slave lasers are locked.2

In this case, the phase lock mixers areintentionally saturated, providing a steeper and more linear slope of the error signal about zero.
The frequency of each slave laser can be controlled using the synthesizer frequencies.

3.3.2

The 689 nm MOT

After the first stage of MOT cooling at 461 nm, the sample temperature is a few mK. At thistemperature, the Doppler width of the 1 S0 →3 P1 transition is roughly 1 MHz. To address all of
the atoms, the slave lasers are spectrally broadened by frequency modulating the phase lock syn2This technique is sometimes referred to as an “RF offset lock.”

50
RF

LO
Synthesizer 1

IF

PID

AOM

80 MHzCooling

To SrStabilized andreferenced master

AOM

80 MHzStirring

PID
Synthesizer 2

To Sr

IF
LO

RF

Figure 3.8: Two home-built ECDLs are phase locked to the stabilized and referenced master laser.
The lock is performed by beating each laser with the master on a photodetector and then mixing
the detector output with a synthesizer frequency. For both lasers, the stabilized output is send to
the Sr main chamber using fibers. AOMs are included for intensity stabilization and fast switchingof each beam.

thesizers. Each synthesizer has an external frequency modulation (FM) port and a programmablefrequency deviation. A triggerable two-channel arbitrary waveform generator controls the FM portsof the synthesizers.
The electric field amplitude of the frequency modulated laser Emod , assuming no phase noise,is [45]

Emod = E0,mod∞X

Jn (β)e−i(ω+nωmod )t ,

(3.1)

n=−∞

with the auto correlation Rmod

2
Rmod (τ ) = E0,mod∞X

Jn2 (β)ei(ω+nωmod )τ .

(3.2)

n=−∞

Here β = fdev /fmod is the modulation index, ωdev = 2πfdev is the frequency deviation, ωmod =2πfmod is the modulation frequency, and Jn is a Bessel function of the first kind. For MOT cooling,
we offset the broadened laser by the frequency deviation (Figure 3.10). Taking this into account,

51

Field Gradient(G/cm)

6040
20
0Fraction of Max
Intensity

1

0.5

0

FM voltage (V)

0

-0.5

-1
0

50

100

150

200

250

300

350

400

Time (ms)Figure 3.9: Timing diagram for intercombination line MOT cooling. The top curve (in red) is
the gradient from the anti-Helmholz field. A current servo (Section 3.2.3) controls the gradient.
The gradient ramps down from its 461 nm MOT value of 60 G/cm to 3 G/cm. After this ramp
completes, the 461 nm MOT light is extinguished. Broadband cooling on the intercombinationline, where the laser is significantly broadened from the FM, remains on. As the atoms cool, thefrequency deviation of the FM decreases to cool more efficiently, and the gradient ramps up tocompress the atoms. Once the gradient ramps up to 10 G/cm, the coil current is held for “single
frequency cooling” (beginning at the 305 ms mark). In this stage, the laser is modulated with asmall deviation until the 345 ms mark, at which point the modulation is shut off. The middle curve(blue) is the intensity of the trapping and stirring lasers as a fraction of their maximum value.
These intensities are servo controlled using the AOMs shown in Figure 3.8. The control voltage forthese intensity servos originates from a National Instruments DAC card. Intensity is ramped down
as the frequency deviation shrinks. This is to compensate for the fact that as the FM deviation
decreases and the sample cools, there is more power in the spectral components resonant with theatoms. The intensity is held constant during single frequency cooling. The bottom curve (cyan) is
the arbitrary waveform generator voltage sent to the FM ports of the frequency synthesizers usedin the trapping and stirring laser phase locks. The input voltage range of the FM ports is -1 V
to 1 V, where 1 V corresponds to the programmed frequency deviation (3 MHz in this case). The
oscillation frequency in this plot is 100 times smaller than the experimental value for the purposeof visualization.

52

3 MHz
-6

Smod Ptot HHz-1 L

10

1.5 MHz
None

10-7
10-8
10-9

-10

-8

-6

-4

-2

0

2

4

DSF HMHzL
Figure 3.10: Equation 3.3 plotted for different values of fdev (shown in the legend). This is the power
spectrum that results from the FM external control depicted in Figure 3.9. Here fmod = 36 kHz,as in the experiment. For this plot, Equation 3.3 is convolved with a Lorentzian to account for
the fact that the laser has phase noise. The FM deviation starts at 3 MHz, collapsing until themodulation is shut off for the latter part of single frequency cooling. The center of the spectrum
is shifted by the frequency deviation to ensure that the laser is always red detuned. At ∆SF = 0,
the laser is detuned by -200 kHz from resonance.

53
the power spectral density Smod (computed with the Wiener-Khinchin theorem) is∞
X
Smod (∆SF )
Jn2 (β)δ(∆SF − nωmod − ωdev ),
=
Ptot
n=−∞

(3.3)

where ∆SF is the difference between the laser detuning from resonance and the detuning during thesingle frequency cooling stage, Ptot is the total laser power, and δ(. . .) is the Dirac delta function.
This spectral density is plotted in Figure 3.10 for typical experimental parameters. For this second
stage of MOT cooling, the frequency deviation is dynamic, decreasing to allow for a laser linewidththat tracks the width of the atoms’ velocity distribution as they are cooling and the velocity
distribution narrows. Both the trapping and stirring lasers have this functionality.
After the trapping and stirring lasers are delivered to the optical table near the Sr2 mainchamber, they are combined and divided into three beams using beamsplitters. These 1 cm diameter beams are aligned along the 461 nm MOT beams, and the two-color waveplates provide thenecessary circular polarization. The intensities of these lasers are ramped on during the last 100
ms of 461 nm MOT cooling (Figure 3.9).
To ensure that the 689 nm MOT forms at the location of the optical lattice, Helmholtz coils
are used to adjust the 689 nm MOT center (Figure 3.11). Two pairs of large, square coils are placednear the main chamber to provide two dimensions of horizontal fields, and a vertical bias field iscreated by coils that are mounted to the chamber above (below) the top (bottom) MOT coil. The
coil currents are stabilized using servos that sense with Vishay Precision resistors and actuate with
MOSFETs (Figure 3.17). The servo locks the voltage across the sense resistors to a voltage basedon a stable LM399H source.

3.4

One-Dimensional Optical Lattice

3.4.1

The Cavity Lattice

Throughout both stages of laser cooling, the cavity lattice is left on. The cavity is comprisedof two mirrors, both with a radius of curvature of R = 20 cm. The mirrors are held in place using

54

Figure 3.11: The three pairs of Helmholtz bias coils. These coils are indicated with arrows. For
the remainder of this thesis, the square coils that form bias fields in the horizontal plane are called
the “xy bias coils” or “xy coils,” the circular coils that make a vertical bias field are the “z bias
coils” or “z coils,” and all three together will be referred to as the “xyz bias coils” or “xyz coils.”

custom mounts made by the JILA machine shop. The mounts clamp to two 1.33 inch conflat viewports on the main chamber. The mounts have adjustable tilt plates for cavity alignment (Figures
3.12 and 3.13). The lattice laser is locked to the cavity using Pound-Drever-Hall stabilization.
The mirrors are custom coated by Advanced Thin Films for reflection at 813 nm and transmission at 698 nm, allowing the clock laser to be aligned along the lattice axis (the need for whichis explained in Section 2.4.3). The mirror surfaces are separated by about 27 cm, resulting in a
160 µm beam waist at the focus and a 9 cm Rayleigh range. The cavity output mirror (Figure 2.5)
is adhered to a ring-shaped piezo stack. Scanning this piezo and measuring the cavity transmission
on a photodetector, the transmission fringe observed on an oscilloscope is similar to that in Figure
2.6. Cavities are often characterized by their finesse, which is the transmission peak separation
divided by the FWHM of the transmission fringes. Inferred from the cavity fringes, the Sr2 cavity
lattice finesse is 120.

55

Figure 3.12: The two cavity lattice mirror mounts. These mounts were home-built in the JILA
machine shop. The bodies of the mounts are anodized aluminum. The mounts clamp onto opposing
1.33 inch conflat viewports. Screws adjust the mirror position on a spring loaded tilt plate inside
the mount body. On one of the mounts, there is an adjustable tilt plate for mounting the fiber
phase noise cancellation mirror (see Section 3.6.2).

Figure 3.13: A cutaway of one lattice mount. The tilt plate inside the mount body changes thecavity alignment. The standoff is present because the other cavity mirror mount has a piezo stackactuator with similar dimensions, ensuring that the mirrors are equidistant from the center. Asidefrom tilt screws, ball bearings, and springs, the mounts are made out of anodized aluminum.

56
Aligning the cavity lattice makes temporary use of a laser resonant with the |1 S0 , F = 9/2i →|3 P1 , F = 9/2i transition. This laser is sent through the cavity (entering through the output mirror)
and aligned to extinguish the atoms in the 689 nm MOT.3 A laser frequency that effectively
extinguishes the atoms was found empirically and used for this purpose.4 The laser is also focusedon the 689 nm MOT to ensure that the alignment procedure is sensitive. Comparing the positionsof the lattice laser before the input mirror and the alignment laser after passing through the cavity,
the lattice laser position is adjusted to be counterpropagating with the alignment laser. With
this change in the lattice laser alignment, the cavity needs to be realigned by adjusting the cavity
mirrors until the transmission indicates a single cavity mode. This procedure tends to steer thealignment laser off the atoms (since the output mirror surface often needs to be aligned during thisprocess), so it needs to be iterated a few times until the cavity and 689 nm MOT are aligned. Aftera few iterations of this alignment procedure, the xyz coils (Figure 3.11) are varied to maximize theatom signal in the lattice.
The AR coating of the viewports has been observed to be birefringent. A cavity mode occurs
for lattice laser polarization at a few degrees angle with respect to gravity, and another mode
(resolved from the first) appears for lattice polarization nearly perpendicular to gravity. The cavity
lattice laser polarization is rotated to align with the polarization mode perpendicular to gravity.

3.4.2

Tapered Amplifier Lattice Laser

Tapered amplifiers (TAs) can be problematic sources of lattice light [114]. Amplified spontaneous emission from the TA can cause a large ac Stark shift of the clock transition that varies
in time. A better source of lattice light is a titanium sapphire (Ti:saph) laser, which has muchgreater spectral purity. However, a TA-based system is a more convenient source of 813 nm lightsince it requires little maintenance and start-up time. Therefore, when performing measurementsthat are insensitive to this amplified spontaneous emission shift, a TA system is preferable. For
3
4

The cavity mirrors transmit at this wavelength well enough for this procedure.
The detuning used here is -70 kHz from resonance. This could be a photoassociation resonance in 87 Sr.

AOM

30 MHzIsolator

ECDL

57

Mode
shapingTA
Clock laser

Cavity lattice
λ/2

Rotator

Dichroic mirror

For PDHlock

Figure 3.14: A free-running ECDL seeds a tapered amplifier (TA). Mode shaping optics (represented
here as a single lens) collimate the output in the horizontal and vertical directions and then telescope
it down to a smaller beam. An AOM is used for intensity control of the field in the cavity. This
light is delivered via fiber from the laser source to the Sr2 main chamber. A Pound-Drever-Hallservo locks the laser to the cavity resonance.

measurements of the lattice ac Stark shift or for running the system as a clock, a Ti:saph-basedlattice is necessary.
The TA for the lattice is seeded by an ECDL comprised of an Eagleyard diode laser in ahome-built enclosure (Figure 3.14). The laser is centered on the magic wavelength every few hours(using a Wavemeter) and left free running. Laser frequency drift about the magic wavelength is notproblematic since the drift rate is sufficiently small and this system is only used for measurementsthat are insensitive to lattice Stark shifts. Sidebands for a PDH lock to the lattice cavity are puton the laser by modulating the ECDL current. The laser diode current and the cavity lattice piezo
stack are locked to the PDH error signal. The cavity transition is separated from the clock laserusing a dichroic mirror and then measured on a photodetector. This detector is part of an intensity
servo loop that actuates on the AOM just before the lattice light is fiber coupled (Figure 3.14).
There is as much as 8 W of one-way power in the cavity.

58

A

OM
30 M
Hz

To cavity lattice

EOM

Lock ECDL
to bow tie

EOM

ECDL

Beat with
comb

Piezo

Verdi
Ti:saph crystal

Figure 3.15: The home-built Ti:saph-based lattice laser. The frequency of this system is stabilized
by frequency locking the Ti:saph to a Yb fiber comb (actuating on the ECDL external cavity piezo).
A Ti:saph crystal is situated inside a bow tie cavity and pumped with an 18 W Coherent Verdi
laser. The light reflected off of the bow tie cavity’s input mirror beats with the field leaking out of
the cavity for a Pound-Drever-Hall error signal. This error signal is used to lock the ECDL current
and bow tie cavity piezo for maximum cavity transmission. A double-passed AOM and the cavity
lattice piezo lock this laser to the cavity lattice. An EOM before the double-passed AOM puts the
sidebands on the light needed for a Pound-Drever-Hall lock to the cavity lattice.

3.4.3

Titanium Sapphire Lattice Laser

The lattice Ti:saph laser is a home-built system, injection locked [22] with an ECDL basedon an Eagleyard diode. When the Ti:saph laser is needed, frequency drift of the lattice wavelength
is unacceptable, so the diode is frequency locked (using the ECDL external cavity piezo) to a Yb
fiber comb that is referenced to the NIST Boulder maser array. This stabilization procedure allows
the absolute frequency5 of the lattice laser to be determined. In this case, a double-passed AOM
provides the high-frequency lock to the cavity lattice. Although this double-passed AOM lock has
a lower bandwidth (about 200 kHz, which is typical of AOM-based servos) than the current lockfor the TA-based lattice laser, the Ti:saph laser also has a purer spectrum (on the order of a 1 kHz
linewidth). Because of this, the AOM lock is sufficient for the Ti:saph laser. The maximum one-way
lattice power using the Ti:saph laser is about 8 W. This system also uses an intensity servo thatstabilizes the cavity transmission (actuating on the drive amplitude of the Ti:saph double-passed
5

The frequency in SI units

59AOM). For clock operation, the intensity servo is set such that the one-way power is 2.3 W.
Ramping the MOT field from 60 G/cm to 3 G/cm (Figure 3.9) causes the MOT coils to
shake, which shakes the cavity lattice as well. Although we checked to ensure that this shaking iswell dampened by the time clock spectroscopy is performed, the 813 nm source can come unlockedfrom the cavity lattice during the ramp. For the TA lattice, the 2 MHz lock bandwidth is large
enough for the laser to survive the shaking; however, the smaller-bandwidth AOM lock for theTi:saph lattice occasionally unlocks when the lattice cavity shakes. To combat this, the ramp is
made as gentle as can be, and the Ti:saph lattice also includes an auto-locking circuit.

3.5

Spin Polarization into the Angular Momentum Stretched States

3.5.1

Spin Polarization Bias Field Coils

With 87 Sr atoms cooled to ultracold temperatures and trapped in an optical lattice, spectroscopy of the clock transition is now possible. As explained in Section 2.5.1, clock spectroscopy is
performed in a bias field that is aligned parallel to the lattice and clock laser polarizations. To thisend, it is convenient to include another pair of Helmholtz coils mounted on the main chamber (Figure 3.16). These “spin polarization coils” (or just “polarization coils”) are also current stabilizedusing a sense resistor and MOSFET (Figure 3.17). Since the polarization coil field is not precisely
in the direction of the lattice polarization (as defined by the cavity polarization modes), additional
fields are provided by the xyz coils (Section 3.5.1) to align the bias field. For each of the xyz coils,
the field used to align the magnetic bias to the lattice polarization and the field that centers the689 nm MOT on the lattice are controlled with independent voltages, which are summed to create
a set point for the xyz coil servos (Figure 3.17). The fields created by these voltages can be tuned
and switched separately, acting as two independent coils.
The clock laser and lattice polarization are aligned by sending both these lasers through thesame broadband-AR-coated polarizer. The bias field is aligned to these two polarizations using
clock transition spectroscopy. Alignment using spectroscopy is possible because when the bias

60

Figure 3.16: A second pair of cavity mirror mounts are clamped to 1.33 inch viewports. These
were constructed with forms around which 125-turn bias coils are wrapped. The mirrors are notinstalled in this work.

MOT Centering
Error
in

V1
Align
bias
field

+

V2 + Σ Out+

61

Supply_PID

MOSFET

Out

Σ

+

Coil
pair

Current
Sense

Backgroundfield servo
Figure 3.17: Bias coil current servo. The loop stabilizes the magnetic field and allows for field
control. There is one of these servo loops for each pair of bias coils, allowing for independent
control of the x, y, and z coils. V1 and V2 are tunable dc voltages based on LM399H voltage
sources. These voltages are set by hand and do not typically need to be adjusted. They can also betuned and switched independently. A digital-to-analog converter (DAC) computer card provides athird voltage for the background field servo, which will be discussed in Section 5.2.2.

field is not aligned with the clock laser polarization, scans about the clock resonance will show σtransitions that do not have the same −108.4(4)mF Hz/G magnetic field sensitivity as π transitions(Section 2.5.1). Therefore, the bias field is aligned by nulling the amplitude of the σ transitions
using the xyz coil fields.

3.5.2

Spin Polarization

After the atoms are loaded into the lattice and the MOT fields are shut off, spin polarizationis performed. The zeroth order of the stirring laser AOM (Figure 3.8) is sent through another
AOM and then delivered by optical fiber to the main chamber. This acts as a spin polarization
beam. The beam passes through a liquid crystal half waveplate and then a quarter waveplate forcircular polarization. The liquid crystal waveplate allows for the resulting circular polarization to
be switched between σ+ and σ− helicity using a TTL signal (motivated in Section 2.5.1). This
beam is sent through clearance holes in the center of the polarization coil mounts (Figure 3.16).

62

Figure 3.18: A CAD of the cavity used to stabilize the clock laser. Also pictured is the Zerodur
cradle (in blue, false color). The cavity and cradle are separated by viton standoffs. The ultralow
expansion glass spacer is 40 cm long.

During spin polarization, a small 50 mG field is applied to the polarization coils, and the fields
from the xyz coils that align the bias field are shut off. The spin polarization beam is applied for
30 ms, driving σ transitions until the atoms are in one of the ground angular momentum stretchedstates (where the state is selected by the helicity of the liquid crystal waveplate). Like with thestirring laser, the |1 S0 , F = 9/2i → |3 P1 , F = 9/2i transition is used here. The field in this stage
is kept small to ensure that the spin polarization direction is defined by the bias field, yet not solarge as to shift some of the σ transitions far out of resonance with the spin polarization beam.
After this, the polarization coil current is increased and the xyz coil fields that align thebias field are turned on, resulting in a 300 mG field in the direction of the lattice polarization.
The atoms are spin polarized along the polarization coil direction (Figure 3.16), yet the clock
laser polarization is parallel to the lattice polarization; therefore, changing the bias field direction
after spin polarization causes the atoms to become depolarized with respect to the clock laser
polarization. This reduces the contrast of the π transitions used in the clock (Figure 2.14b). In
practice, this contrast reduction is small (at the few percent level), so no attempt is made to fix it.

63

3.6

The Clock Laser

3.6.1

Cavity Stabilization

The 698 nm stable clock laser was constructed out of an ultralow expansion glass spacer thatis 40 cm long [110, 89, 77, 8]. The mirrors were coated by Advanced Thin Films on fused silica
substrates. The cavity is supported against gravity by viton hemispherical standoffs placed at fourcontact points. The standoffs were positioned to make the cavity insensitive to vibration noise [91].
The cavity and its cradle (Figure 3.18) are placed inside a heat shield and two-stage vacuum
chamber. Active bipolar temperature control keeps the cavity temperature stable. The vacuum
chamber resides on a Herzan active vibration cancellation stage. Also, to prevent acoustic noise,
the vacuum chamber is surrounded by an acoustic shield.
The 689 nm light source is a home-built ECDL. First, this laser is current and piezo locked
to a prestabilization cavity. The laser then double passes an AOM, which is used to lock the laser
to the 40 cm cavity. A piezo on the prestabilization cavity also provides a lower bandwidth lock to
the 40 cm cavity. Measurements of this laser’s noise spectrum show that the fully stabilized clock
laser has a 26 mHz linewidth [8].

3.6.2

Fiber Phase Noise Cancellation

To prevent the ECDL from unlocking due to laboratory bustle, the clock laser cavity stabilization setup is kept in a separate room that remains unoccupied while the Sr2 clock is running.
The fully stabilized laser is sent via optical fiber to the Sr system. A fiber of the length required totransfer the clock laser between these rooms would cause broadening at the 1 kHz level, which would
be devastating for the Sr2 system, if it were not for fiber phase noise cancellation [75]. Fiber phasenoise cancellation reduces the decoherence from this fiber to levels that are currently negligible forSr2 [36].
For fiber phase noise cancellation, the source laser passes through an AOM before getting
launched into the fiber. A fraction of the fiber output is retroreflected back through both the fiber

64
and the AOM. The retroreflected electric field will acquire a phase factor e2i(ωAOM t+φnoise ) , whereωAOM is the frequency shift due to the AOM, φnoise is a stochastic function that describes thedecoherence (mostly due to the fiber) between the AOM and the retroreflecting mirror, and thefactor of two comes from double passing both the fiber and the AOM. Using a beat between the
retroreflected light and the source before the AOM, the retroreflected beam is phase locked to thesource laser using the AOM frequency, removing the effects of the fiber.6
After the clock laser is delivered to the Sr lab using a phase-noise-canceled fiber, its power
is not sufficient for both Sr1 and Sr2. Therefore, the light from the 40 cm cavity injection locks adiode, the output of which provides ample power for both systems. Light from the diode is divided
equally between Sr1 and Sr2 and sent to these two experiments using fibers, which are again phasenoise canceled. In this second stage of fiber noise cancellation, one can benefit from judiciously
positioning the retroreflecting mirror that sends light back through the fiber.
Any decoherence encountered by the laser light as it travels from the phase noise cancellation
AOM to the retroreflecting mirror is corrected by this cancellation technique. In other words, theretroreflecting mirror defines a plane at which the source laser coherence is reproduced (within the
bandwidth of the phase noise cancelation lock). Therefore, the phase noise cancellation retroreflecting mirror is mounted to the same structure that holds the cavity lattice mirrors (Figure 3.12).
In this case, the laser phase is adjusted to track any shaking of the lattice, removing the Doppler
shift that would result from atoms moving with respect to the clock laser.

3.7

Digital Atomic Servo

3.7.1

Clock Laser Frequency Modulator and Drift Cancellation

Another AOM, labeled as the “modulator” in Figure 2.15b, is used to control the frequencyof the clock laser for spectroscopy. The zeroth order of this AOM is retroreflected back through thefiber for phase noise cancellation, whereas the -1st order interacts with the atoms. The beam from6
The phase lock actuates on a voltage-controlled oscillator (VCO) centered at twice the AOM frequency. The
VCO frequency divided by two drives the AOM. The loop adjusts the VCO frequency to lock the beat to a stable
RF reference at twice the desired AOM frequency.

65
the 698 nm injection-locked diode is too powerful for normal clock operation, so a neutral density
filter with an optical depth of four is placed in the -1st order beam.
The frequency that drives this AOM comes from two synthesizers that are mixed together.
One “scanning synthesizer” is responsive to the experimental control software that steps the clock
laser frequency for scanning the clock transition or for the atomic servo. The other synthesizer
has a linear frequency drift with a fixed drift rate of about -30 mHz/s. This drift rate is adjusted
once or twice per day to cancel the drift of the 40 cm cavity. The cavity drift, which is attributed
to material creep, is linear and can be canceled effectively. The scanning synthesizer, which has 1µHz resolution,7 is connected to the Sr2 experimental control computer. The computer adjusts theclock laser frequency through a GPIB command that sets the frequency of the scanning synthesizer.
The modulator AOM that controls the clock transition frequency (Figure 2.15b) also has anintensity servo. Part of the -1st order is split off and sent to a photodiode. The photodiode output
goes into a PI-I controller (one proportional-integral gain stage and one pure integrator), which
actuates on the drive amplitude of the modulator AOM. Two versions of this servo have been used:
one with 50 kHz of bandwidth and one with 250 kHz. The setpoint for this servo is provided byan adjustable dc voltage (based on a stable LM399H-based voltage source) that can be switchedon and off using a 5 ns switch from Mini-Circuits. This allows for clean clock laser square pulses
with an easily adjustable pulse duration and amplitude.

3.7.2

Camera Imaging and Photomultiplier Tube Detection

As explained in Section 2.5.2, the error signal for the atomic servo is the difference between
excited state population fraction measurements. To measure the excited population fraction, threelasers are used. After driving the clock transition, the ground state is fluoresced with a resonant
counting laser on the 1 S0 →1 P1 transition (Figure 2.1). This strong transition scatters many
photons, which are detected with a photomultiplier tube (PMT), yielding a ground-state population
signal Vg that is proportional to the number of atoms in the ground state. The ground state
7This is a Stanford Research Systems DS345.

66
population scatters so many counting laser photons that it is quickly heated out of the trap andlost to the experiment. The excited state population is not resonant with the counting laser, so it
is unaffected. Next, the repump lasers (Figure 2.2) are applied, rapidly driving the excited state
population to the clock transition ground state. Atoms are then fluoresced again on 1 S0 →1P1 , andphotons are counted again with the PMT, providing the excited state population signal Ve . The
measured excited state population fraction is computed as8 ρmeas = Ve /(Ve + Vg ).
When the counting laser is applied to measure the excited state population fraction, fluorescence is collected through two viewports. A 3 inch diameter lens collects fluorescence through thelarge 8 inch conflat viewport on the top of the main chamber, focusing this signal down onto the
PMT with 2f imaging.9

The PMT output is sent to an oscilloscope computer card from GaGe

Applied Technologies. Each time the counting laser is applied to the sample to measure the ground(excited) state population, Vg (Ve )—given by the time integral of the fluorescence signal measured
by the PMT and the oscilloscope card—is added to the data record.
When Vg is measured, fluorescence is also collected by a 2 inch diameter lens outside one of
the 2.75 inch conflat viewports. This light is imaged on an Andor CCD camera with a wedged
window anti-reflection coated for 461 nm light. This camera is typically used for signal monitoring,but it has also been used for absorption imaging to calibrate the fluorescence counts in terms of
atom number.

3.7.3

Clock Transition Scans

The software for scanning and locking the clock laser to the clock transition was written in
MATLAB using the environment’s GUI building functionality. To begin the digital servo, firstthe clock transition is found by scanning over a 200 kHz window centered about the last known
scanning synthesizer frequency corresponding to the clock transition resonance. This broad scanis performed at zero bias field and with the neutral density filter removed from the clock laser
8

One detail not covered in this treatment is that once the atoms are all headed out of the lattice, a third
measurement is taken with the counting laser. This yields the background counts, which are subtracted off of theexcited and ground state population measurements (Section A.4).
9
When the lens is twice the focal length from the source

67

Excited State Fraction

0.15

0.1

0.05

0
-150

-100

-50

0

50

100

150

Detuning from Carrier (kHz)
Figure 3.19: A typical sideband scan for 1.5 kHz frequency steps, a 160 ms pulse, and the neutral
density filter (Section 3.7.1) removed from the clock laser. At this clock laser power, the clocktransition is saturated and dephases, achieving a final excited state fraction of about 0.5. Thisvalue is not portrayed in this data set due to the coarseness of the scan. The carrier transition
occurs at zero detuning, the blue sideband at positive detuning, and the red sideband at negative
detuning. The sidebands are asymmetrically broadened due to the effect described in Section 2.4.4.

beam (Section 3.7.1) to power broaden the transition. This power broadening helps to locate theresonance quickly using coarse frequency steps of 1.5 kHz. A scan over this range and at this large
power drives sideband transitions.
Data from this scan (Figure 3.19) resembles the calculation plotted in Figure 2.12. These
data can be used to infer the trap frequency. As explained in Section 2.4.4, the frequency difference
between the center of the carrier transition and the blue sideband edge, νblue , determines the lattice
axial trap frequency νz as νblue = νz − νrec .
The temperature in the lattice axial direction, Tz , can also be approximated using this scan.
For the case of a 1D harmonic oscillator potential, the entire population in the lattice can makea blue sideband transition whereas all population that is not in the motional ground state canmake a red sideband transition. Statistical mechanics gives the population in motional state n(where, again, n = 0 is the motional ground state) of [1 − exp(−hνz /kB Tz )] exp(−nhνz /kB Tz ).
Therefore, the population participating in a blue sideband transition divided by that participating
in a red sideband transition is exp(hνz /kB Tz ). To a rough approximation, this ratio is equal to the68

Excited State Fraction

0.08

0.06

0.04

0.02

0
-300

-200

-100

0

100

200

300

Detuning (Hz)Figure 3.20: With the bias field applied, all 10 mF states are resolved. Here the detuning ismeasured with respect to the zero field resonance. These data are mathematically described by theformalism presented in Section 2.5.1.

ratio of the maximum excited state fraction achieved for the blue and red sidebands. Therefore,ρmax,blue /ρmax,red ≃ exp(hνz /kB Tz ). This approximation typically agrees within 15% of a more
complex model that accounts for radial motion and trap anharmonicity [12].
With the clock transition located, the intensity is reduced with a neutral density filter andthe clock laser intensity servo, and the bias field is applied. Without spin polarization, a scan of
the carrier transition shows ten resonances corresponding to the different mF states (Figure 3.20).
This scan provides the clock laser AOM frequency corresponding to the two stretched states, andwith this information the stretched-states servo (Section 2.5.3) can be initialized.

3.7.4

Digital Servo Software

The software requires the user to input the initial modulator frequencies corresponding to thestretched state resonances (obtained from clock transition scans) and the linewidth of the feature
it is locking to. To lock to a lineshape, the software steps the scanning synthesizer half of the input
linewidth on one side of the resonance center and then measures the excited state fraction. The
scanning synthesizer is then stepped on the other side of the resonance by the input linewidth, andthe excited state fraction is measured again. The first iteration of the computed error signal ǫclock

69

Excited State Fraction

1
0.8
0.6
0.4
0.2
0
-15

-10

-5

0

5

10

15

Detuning (Hz)Figure 3.21: A scan of one stretched state in a bias field after the atoms have been spin polarized.
Here the pulse duration is 160 ms. The clock laser intensity is adjusted for maximum contrast.

will be large in magnitude if the initial modulator frequencies are off from the resonant values. Laser
frequency noise will also cause noise in ǫclock . A desktop computer records the fluorescence counts
and the line center, which is calculated from ǫclock with Equation 2.44. The next iteration of theservo uses the last computed value of the line center, and with this information the measurement
cycle begins again. The stretched-states servo locking procedure is performed for both stretchedstates using two independent servo loops, with the locking sequence described in Figure 2.16.
The lineshapes used in the stretched-states servo are Rabi π-pulses, which occur when ΩmF t =π (see Equation 2.42). Lineshapes very similar to those predicted by the Rabi solution are typicallyobserved (Figure 3.21).

Chapter 4

Clock Stability

4.1

The Allan Deviation

4.1.1

Definition

As explained earlier, total systematic uncertainty and stability are the two figures of meritused to gauge clock performance (Section 1.6). Stability is a measure of the noise of a frequency
standard, for it quantifies the uncertainty in the mean clock oscillator frequency due to random
processes such as frequency or phase fluctuations. In essence, better stability is useful because it
implies less measurement time. If an oscillator needs to be calibrated to a frequency standard,
this calibration measurement will typically involve averaging down the noise in a beat between
the oscillator and the standard. The noisier the standard, the longer the calibration will take toaverage. Also, stability affects total systematic uncertainty. Systematic effects are often studied
by modulating a parameter of the apparatus (like the amplitude of an electromagnetic field that
perturbs the clock states) and measuring the corresponding frequency shift of the clock transition.
Frequency noise in such a measurement needs to be averaged down to obtain precise uncertainties
of systematic biases. Better stability means that evaluating these uncertainties will take less dataacquisition time, which is usually the main practical limitation of systematic measurements.
According to the IEEE, the recommended statistic [50] for quantifying stability is the Allan
Deviation, σy (τ ), where

71
2

10

Drift
1/f
White

y

σ (τ) / σ

1

1

10

0

10

−1

10

0

1

10

2

10

τ (s)

10

Figure 4.1: The three processes found in Sr2 that degrade stability. Each are scaled to σ1 , which istheir value at τ = 1 s. For white noise, σy (τ ) averages down in the same manner as the standard
deviation of the mean, which decreases as the square root of the number of data points. There isno advantage to averaging pure 1/f noise since σy (τ ) does not improve with τ . In the case of drift,stability gets worse with averaging time.

" Z
2 #
Z
1 t+τ
1 t
′
′
′
′
y(t )dt −y(t )dt
.
2
τ t
τ t−τ

1
σy2 (τ ) = E(4.1)

Here τ is the averaging time, E [. . .] denotes the expectation value, andy(t) =νc (t) − νSr
.
νSr

(4.2)

νc (t) is the cavity stabilized clock laser frequency, which is a stochastic as a function of time, andνSr is the frequency to which the clock laser is referenced (that of the Sr atoms in this case).
It is straightforward to show (Section A.2) that

σy2 (τ ) = 2

Z ∞0

Sy (f )

sin4 (πτ f )df,(πτ f )2

(4.3)

where Sy (f ) is the one-sided power spectral density of y(t). The power spectrum of most atomic
Pclocks can be described with a simple model Sy (f ) = 2j=−2 hj f j , where hj is constant [50, 100].

y1(3)

y2(3)

y3(3)

72

y4(3)

yn
y1(3)
y2(3)
y3(3)
y4(3)
y5(3)
Figure 4.2: The Allan deviation estimator for overlapping and non-overlapping averages. The
frequency data yn (red points) is binned in two different ways. The binning scheme shown above
the points is for the non-overlapping Allan deviation. The points yn (m) are the average of the data
within a given bin, and m is the bin size (which is 3 in this figure). The binning scheme below thepoints is for the overlapping Allan deviation. Again, the points y n (m) are the average of the data
within a given bin; however in this case the bins overlap.

Typically only three noise processes govern the behavior of the Sr2 clock: white noise, flicker
frequency noise (or “1/f noise”), and linear drift.
For white noise, Sy (f ) = h0 , and using Equation 4.3, σy (τ ) =p

h0 /2τ . For 1/f noise,

Sy (f ) = h−1 /f , and σy (τ ) = 2 ln(2)h−1 . In the case of drift at a rate α, y(t) = αt, and Equation√
4.1 yields σy (τ ) = ατ / 2. Therefore, these three processes can be distinguished by the behaviorof the Allan deviation as a function of averaging time (Figure 4.1).

4.1.2

EstimatorsSuppose y(t) is sampled with a cycle time 2Tc and a total of Ny points.1 Let y1 , y2 , . . . , yNy

be equal to {y(2Tc ), y(4Tc ), . . . , y(2Ny Tc )}. The Allan deviation is often estimated as
σy2 (2mTc ) ≃

N −1

y
X
2

1
y n+1 (m) − y n (m) .
2(Ny − 1)

(4.4)

n=1

Here τ = 2mTc is the averaging time and m is an integer. The way this estimator is defined, mruns from a minimum of 2 to a maximum of Ny /2. yn is the average of data binned in a particular
1

Recall from Section 2.5.2 that the cycle time Tc is the time it takes to cool atoms, trap them in the lattice, andinterrogate the clock transition once. It takes two interrogations of the clock transition to get one resonance center
frequency, so 2Tc is needed to measure one point. If y(t) were the stretched state average frequency, 4Tc would beneeded to measure one point.

73

y#(t)

Original data

Time
Figure 4.3: Frequency points used to calculate the total deviation. A set of points y # is created byreflecting the original data set twice.

way. Overlapping and non-overlapping binning schemes are typical (Figure 4.2), but overlappingbins are much more utilized in the clock community [101]. When overlapping bins are used, thisestimator is typically referred to as the “overlapping Allan deviation.” Ny is the total numberof points after binning and averaging, and m is the bin size. Stability is typically analyzed by
calculating the Allan deviation estimator for different values of m to observe how noise diminishes
as a function of averaging time. The Allan deviation is usually plotted as a function of 2mTc . In
essence, the Allan deviation estimator quantifies fluctuations in the difference between averaged
subsets of data as a function of averaging time.
Another useful estimator is the total Allan deviation (or total deviation), σtotal (τ ) [43]. Thisis calculated using a set of points constructed by reflecting the frequency data twice (Figure 4.3).
###
Reflecting generates the points {y−N
, y−N
, . . . , yN}, which are defined as yn# = yn for 1 ≤ n ≤
y
y +1
y
#
#
Ny . For 1 ≤ j ≤ Ny − 1, y1−j
= yj and yN= yNy +1−j . Using this definition,
y +j2
σtotal(2mTc ) =

Ny h
i2
X
1
#
(m)
y#
(m)
−
y
,
n−m
n
Ny − 1(4.5)

n=2

1 Pm−1 #where y #n (m) = m
k=0 yn+k . This estimator is useful because it has less error at longer averaging

times [43].

74

4.2

Effects that Limit Stability

4.2.1

Quantum Projection Noise

For atoms in uncorrelated states, the fundamental limit to clock precision is quantum projection noise (QPN). QPN is the result of the binomial statistics (Section 2.7) associated withprojecting a coherent superposition of electronic states onto one of the levels of the clock transition. For a clock laser pulse chosen to drive the excited state population fraction ρee to a valuep (typically 1/2 during clock operation), the fluctuation in the excited state atom number σN isp
N p (1 − p). This is simply the standard deviation of a binomial distribution, which is approxi-

mately Gaussian for N & 50 and which averages down like white noise.

The fluctuation in the measured excited state population fraction σρ due to QPN affects thepclock signal. This is given by σρ = ∆N/N = p (1 − p)/N . The atomic servo cannot distinguishbetween these fluctuations and clock laser noise, so it will adjust the laser frequency in attempt tocorrect for the QPN. This negatively affects the stability of the locked clock laser.
When the clock laser is locked to a value of the excited state fraction, the slope s of the
Rabi lineshape at the lock point is the factor that determines the effect of the excited state fraction
fluctuation on the atomic servo.2

Differentiating ρee (Equation 2.26) with respect to detuningδ = ∆/2π and evaluating this derivative at the detuning for which ρee = 1/2 (since p is typicallyselected to be this value), s can be obtained numerically. Since QPN is a white noise process, thenoise amplitude averages down with the square root of the number of points averaged. For an
averaging time of τ and a cycle time of Tc , the number of points averaged is τ /2Tc , and the clockstability due to QPN is√2σρ
σQP N (τ ) =2s νSr

r

1
2Tc
=
τ
πQ

r

Tc
,
Nτ

(4.6)

where Q = νSr /γ, γ is given by Equation 2.43, and p = 1/2. Equation 4.6 is not exact for Rabi
2

Although there are two values of the slope (one on each side of resonance), which are equal in magnitude andopposite in sign, for convenience I take s to be the positive slope.

75
spectroscopy, 3 but it is good to 3%. The fact that Equation 4.6 has√

2σρ in the numerator insteadof just σρ is because the atomic servo requires two measurements of the excited state fraction toinfer the center frequency of a lineshape.
The factor of 2 appears in the denominator of Equation 4.6 because the atomic servo error
signal is twice as sensitive to clock laser frequency fluctuations as a single measurement of theexcited state fraction. To see this, suppose the atomic servo is locked to an excited state fractionof 1/2, and then the detuning due to laser frequency drift fluctuates by a small amount δǫ . The
two values of the excited state fraction measured by the atomic servo (one on either side of thelineshape) can be expanded about their lock points as ρee (±δ0 + δǫ ) ≃ 21 ± sδǫ , where δ0 is thedetuning corresponding to an excited state fraction of 1/2. The clock error signal is thereforeǫclock = 2sδǫ , so the sensitivity of the atomic servo to detuning fluctuations is 2s. Since the atomic
servo cannot discern between detuning fluctuations and QPN, it responds to QPN assuming thesame 2s sensitivity.

4.2.2

The Dick Effect

The main limitation to lattice clock stability is the Dick Effect [28]. The Allan deviation due
to the Dick Effect, σDick (τ ), is given by [42]2(τ ) =σDick∞

1X
Slocked (k/Tc ),
τ

(4.7)

k=1

where Slocked (f ) is the one-sided power spectral density of the clock laser after it is locked to theatoms.4

According to this equation, the Dick Effect is stability degradation caused by discretevalues of the clock laser’s noise spectrum. This is simply signal aliasing, whereby noise componentsof the locked clock laser with frequencies equal to harmonics of the sampling rate 1/Tc appear
to the atomic servo as lower-frequency signals. This causes high-frequency noise that is beyond
the atomic servo bandwidth to appear as low-frequency noise within the bandwidth of the servo.
3

It is exact for Ramsey spectroscopy with infinitesimal pulses
Like the spectral densities defined in previous sections, this is the spectral density of the clock laser frequency
fluctuation about the clock transition resonance in fractional units.
4

76
Since the servo cannot distinguish between aliased noise and actual low-frequency noise that it isdesigned to remove, stability is negatively affected as the servo attempts to correct for the aliased
frequencies. The Dick effect is simply the sum of these contributions.
The derivation of this expression makes use of a widely utilized relation for discrete time
signals (Section A.3), which is worth quoting because it provides intuition about the Dick Effect.
For a continuous signal φ(t) sampled every Tc ,Sdis (f ) =

∞
X
k=0

Scon



k
f+
Tc



,

(4.8)

where Scon (f ) is the one-sided power spectral density of the continuous signal, Sdis (f ) is the spectral
density of φ(nTc ) (the sampled signal φ(t)), and n is an integer. The terms for which the summationindex k 6= 0 arise from sampling the continuous signal and describe aliasing. Comparing this relation
to Equation 4.7, it is clear that the Allan deviation due to the Dick effect is just the aliased partof the locked clock laser power spectrum evaluated for low frequencies.
√
Note in Equation 4.7 that the Allan deviation averages like 1/ τ , which is characteristic ofwhite noise (Section 4.1.1). Dick’s original paper [28] explained this in the following way. Stability
degradation arises from the laser noise spectrum windowed about harmonics of 1/Tc . About sufficiently small windows, a complicated noise spectrum will have the characteristics of white noise;
therefore, the Dick Effect is the sum of white noise, which is itself white.
Before quoting the standard form of the Dick Effect, another signals relation is worth mentioning. Suppose for two continuous signals φ1 (t) and φ2 (t), with Fourier transforms Φ1 (f ) andΦ2 (f ), the transfer function G(f ) relates them as Φ2 (f ) = G(f )Φ1 (f ). It is straightforward to showthat S2 (f ) = |G(f )|2 S1 (f ) (Section A.1), where S1 (f ) and S2 (f ) are the power spectra of Φ1 (f )and Φ2 (f ). When these signals are sampled at a rate 1/Tc , Equation 4.8 implies that, for f = 0,5PP∞2
the spectrum of the sampled signal is ∞k=0 S2 (k/Tc ) =k=0 |G(k/Tc )| S1 (k/Tc ). Motivated bythis expression, it is advantageous to seek a transfer function that relates the locked spectral density
5

Here f is taken to be zero to give the same form as Equation 4.7.

77

1

1.0

bL

aL
Ègk g0 2

gHtL

0.8
0.6
0.4

0.01

10-4

0.2
0.0
0

1

2

3

10-6
1

4

2

5

10

20

50

k

t HsL

Figure 4.4: a) The sensitivity function for Rabi spectroscopy. Here Tc = 1 s and tp = 0.25 s. At
the end of the 1 s cycle, the clock laser pulse comes on and g(t) is no longer zero. b) The squared
magnitude of the normalized Fourier coefficients, |gk /g0 |2 , for the same Tc and tp .

to the spectrum of the free-running clock laser, which is easier to measure.6
With some work [42], it can be shown that the Dick Effect is2
σDick(τ ) =∞

1 X gk 2
Sclock (k/Tc ),
τ
g0

(4.9)

k=1

where Sclock (f ) is the one-sided power spectral density of the free-running clock laser, and

1
gk =
Tc

Z Tc

g(t)e−2πikt/Tc dt(4.10)

0is the kth Fourier coefficient of the sensitivity function g(t) [65]. The sensitivity function is thechange in the excited state fraction due to an instantaneous phase step of the clock laser. g0 inEquation 4.9 is simply gk for k = 0. The sensitivity function depends on the type of spectroscopy
used in the clock (i.e., Rabi, Ramsey), the measurement time, and the experimental cycle time.
For Rabi spectroscopy, when the clock laser pulse is on (at the end of a cycle),g(t) = −


 p




t − Tc + tp p 2
π2 κt − Tc p 2
2 + κ2 + sin
2 − sin
2
π
π
+
κ
π
+
κ
sin
.
tp
tp
(π 2 + κ2 )3/2
(4.11)

6
Here “free-running clock laser” means a laser that is cavity stabilized but not locked to the Sr atoms with theatomic servo.

78
In this expression, tp is the pulse time. g(t) is piecewise, and when the clock laser is off, g(t) = 0(Figure 4.4). κ is the product of detuning and pulse time for which ρee = 1/2 (where the atomic
 √
servo locks). Therefore, κ = 2.509, determined numerically by 1+κ12 /π2 sin2 21 π 2 + κ2 = 12 .
Equation 4.9 shows that the Dick Effect can be mitigated in two ways: reducing the sensitivity

to laser noise, and reducing the laser noise amplitude. The sensitivity function can be reduced bydecreasing the measurement dead time so that more of the experimental cycle is occupied by Rabi
spectroscopy [120]. Also, better laser stabilization improves laser noise.

4.2.3

Long-Term Drift

Over timescales of hundreds of seconds and longer, long-term drift can dominate over QPN
and the Dick Effect. The atomic servo removes frequency drift due to thermal variations in thelength of the cavity used to stabilize the clock laser; however, the clock transition frequency itself canchange. After all, the laser is locked to the perturbed clock transition, and laboratory perturbations
are subject to drift.
One example of this drift is a time varying laboratory temperature. As I will explain in
Section 5.8.1, the electric field associated with the ambient laboratory heat causes a blackbody
radiation (BBR) Stark shift that has a strong effect on the clock transition. Even a 1 ◦ C drift inlaboratory temperature causes a measurable change in the BBR shift. The experimental signatureof temperature drift would be an Allan deviation that averages down at first, then reverses direction
and begins increasing linearly with averaging time (Figure 4.1) when drift has accumulated enoughto become appreciable.
Long-term drift can be alleviated by stabilizing ambient temperatures and other sources ofclock transition perturbations.

79

Figure 4.5: Fractional clock stability per square root of averaging time (in seconds) for several
clocks. Stability quoted here is for a single clock rather than for a comparison. The record for best
Cs clock stability, from BNM-SYRTE, is shown on the left [118]. The NIST Al+ ion clock [23] andPTB Yb+ ion clock [52] have comparable stability to that reported by lattice clocks prior to thiswork. The JILA Sr1 lattice clock [74] and the NIST Yb lattice clock [63] are also shown here.

4.3

Stability in Optical Clocks

4.3.1

The Stability of Lattice Clocks and Ion Clocks

With QPN and the Dick Effect explained, the stability of the two prominent types of opticalclocks, ion clocks and lattice clocks (Section 1.5), can now be compared. Regarding QPN, ion
clocks tend to use shorter clock transition wavelengths [23, 93, 52] than lattice clocks [63, 74], andthis is beneficial for ion-based systems due to the factor of Q in σQP N (Equation 4.6). This is also
why optical clocks are better than Cs, which is based on a microwave transition. Furthermore,
ion clocks can operate with faster cycle times [23], which is a positive trait of these clocks due to√a superior Tc factor in the σQP N expression. However, ion clocks only have one atom whereas√
N is of order 103 for lattice clocks. Since σQP N (τ ) ∝ 1/ N , the larger atom number is a majoradvantage of lattice clocks. Due to larger atom numbers, the QPN of lattice clocks is typically atleast an order of magnitude better than that of ion clocks.

80
The stability limitation of ion clocks tends to be QPN [54]. For lattice clocks, QPN is small
and duty cycles are shorter (AKA more dead time), so the Dick Effect dominates. Prior to thework discussed in this thesis, the Dick-effect-limited stability of lattice clocks was about equal toreported ion clock stabilities (Figure 4.5). At the time, the laser causing the Dick Effect in the Sr1clock was based on a 7 cm long cavity [72], which was similar to those used by other lattice clock
teams.

4.3.2

Using Longer Cavities to Reduce Thermal Noise

When the Sr1 7 cm clock laser stabilization cavity was built, this new system made use of aclever cavity mounting scheme to remove ambient mechanical vibration noise [91]. Prior to theseadvances in cavity mounts, vibration noise was the dominant mechanism limiting cavity stability[126]. When vibration noise dominates, cavities should be made short since the amplitude ofthis noise scales with the size of the cavity. With vibration noise removed, the best stable lasers
became limited by thermal fluctuations of cavity mirror coatings [92]. This thermal noise degrades
cavity length stability, which negatively affects the frequency stability of clock lasers (since cavity
stabilization is the stabilization of the laser frequency to the length of the cavity).
The 7 cm cavity in Sr1 was designed to be short to reduce residual vibration noise notcanceled by the cavity mounting scheme. The tradeoff for using a short cavity is more thermal
noise. The stability of a laser locked to a thermal-noise-limited cavity scales as δL/L, where L isthe cavity length and δL is the fluctuation in L [90]. Although longer cavities could potentiallysuffer from worse residual vibration noise, if vibration noise cancellation could be made to work
well for a longer cavity, this would yield better clock laser stabilization and a reduced Dick Effect.
This possibility motivated the construction of a longer 40 cm cavity at JILA to improve thestability of our lattice clock (Section 3.6.1) [77]. This cavity proved to be vibration insensitive andthermal noise limited at stability an order of magnitude better than that of the 7 cm cavity (Figure4.6). Early characterizations of the clock laser based on the 40 cm cavity revealed a free-running

81

Allan Deviation

10 -1410 -15

10 -167 cm Cavity
40 cm Cavity

10

0

10

1

10

2

10

3

Averaging Time (s)Figure 4.6: Overlapping Allan deviations of diode lasers stabilized to 7 cm and 40 cm cavities.
Linear drift is removed to show the laser instabilities due to high frequency noise.

power spectrum7 with a flat Allan deviation, indicating a 1/f noise process [89]. This is consistent
with a thermal-noise-limited laser, which also has a 1/f noise spectrum [92].8
For a laser with a 1/f noise spectrum, Equation 4.3 can be used to write Sclock (f ) in termsof the Allan deviation of the free-running laser, σclock ,Sclock (f ) =2
σclock.
(2 ln 2)f

(4.12)

Using this expression, the Dick Effect is∞2
σclock
Tc X 1
2
σDick (τ ) =2 ln 2 τ
k
k=1

gk 2
.
g0

(4.13)

Therefore, since the free-running laser stability with the 40 cm cavity is 10 times better than withthe 7 cm cavity, and since the Sr1 stability with the 7 cm cavity was comparable to that of ionclocks, we expect this new cavity to make Sr2 clock stability an order of magnitude better than ion
7

The “free-running power spectrum” is the power spectrum of the clock laser stabilized to the cavity but not tothe atoms.
8
A later, more detailed study revealed 1/f noise, white noise, and resonances in the power spectrum [8].

82

Figure 4.7: Best reported Sr and Yb stability results compared to the ion clock stabilities in Figure4.5. Both the Sr and Yb advances were primarily due to better laser stabilization. The Yb result
is reported in Reference [49], and the Sr2 result is discussed in Section 4.6.

clock stability.
As I will explain in the rest of this chapter, this order-of-magnitude improvement over ion
clock stability is indeed what we have observed (Figure 4.7).

4.3.3

Self-Comparison

Self-comparison is a useful technique for characterizing clock stability. With self-comparison,a clock compares two independent atomic servos that are locked to the same stretched state andthat update in an alternating fashion. If processed correctly, the center frequencies measured by
the two servos yield the short-term stability of the clock.
Suppose that one atomic servo measures the center frequency ν1 , and immediately after thatthe other servo measures ν2 . Since these are scanning synthesizer frequencies, they contain thefree-running clock laser drift (due to stabilization cavity length variations) that is not removed bydrift cancellation (Section 3.7.1).9

To remove this drift, the center frequencies are processed as a

9
The atomic servo records the scanning synthesizer frequency required to set the free-running clock laser (after
drift cancellation) on the clock transition resonance. In the presence of uncanceled clock laser drift, the servo
will adjust the scanning synthesizer frequency to compensate for this drift, adding drift to the record of scanning

83
difference ν2 −ν1 . This works because the uncanceled laser drift is typically negligible over the 4Tc it
takes to make the two center frequency measurements. This difference also removes long-term driftof systematic shifts, so it does not determine long-term stability. However, (ν2 − ν1 )/2 determines
the short-term stability of the clock.
ν2 − ν1 is divided by 2 to extrapolate to the noise performance of the stretched state averagefrequency. Since the stretched state average frequency is an average of two atomic servo measurep
ments (Section 2.5.3), its noise (using the usual error propagation mathematics) is (σ/2)2 + (σ/2)2 =√√σ/ 2, where σ is the noise of a single center frequency measurement. The noise of ν2 − ν1 is 2σ,
so this difference must be divided by 2 to have the same frequency fluctuation as the stretched state
average frequency. Therefore, for frequency data {ν1 , ν2 , ν3 , . . .}, where the odd subscripts indicate
data taken by one atomic servo and the even subscripts are taken by the other, the self-comparison
data is { 12 (ν2 − ν1 ), 12 (ν4 − ν3 ), . . .}4.4

Synchronous Interrogation

4.4.1

Two Sr Clocks with a Shared Clock Laser Source

To study the stability of a frequency standard at different timescales, it is useful to beat thisstandard with a similar system. Therefore, we measure the Sr2 clock stability through a series ofcomparisons with the Sr1 system [89]. These comparisons were set up using a shared clock laser
source.
As explained in Section 3.6.2, light from the clock laser is divided between the two experiments
into two independent clock laser beams. Each system has its own AOM to realize independent
atomic servos (Figure 4.8a). As the frequencies of the each scanning synthesizer (one for the Sr2
AOM and one for Sr1) are adjusted to lock the two clock laser beams, these synthesizer frequencies
are recorded.10 One advantage to this setup is that a comparison between Sr1 and Sr211 can besynthesizer frequencies.
10
Both synthesizers share a stable RF reference.
11
Here “comparison” simply means the difference between the frequencies of the two clock laser beams after they
are locked to their respective Sr clock

84

a)

Sr1

Cavity Stabilized
Clock Laser

Sr2

b)

Sr1 Clock Pulse

Sr2 Clock Pulse

t

AOM1

RF1

AOM2
Sr1 Clock Pulse

RF2

Sr2 Clock Pulset

Figure 4.8: a) Comparison setup for the two JILA Sr clocks. b) Top panel: Stability with the
7 cm cavity (image on left) is studied using synchronous pulses (plot on right). Bottom panel:
Asynchronous pulses (plot on right) are used to study stability with the 40 cm cavity (image onleft). The Sr1 falling pulse edge and the Sr2 rising edge are separated by 10 ms for asynchronousinterrogation. The synchronous and asynchronous measurements use 160 ms Rabi π-pulses for both
systems.

performed in post processing. Since the two systems share a common clock laser source, taking
the difference between the recorded values of the two scanning synthesizer frequencies provides thesame information as beating the two clock laser beams (Figure 4.8a).
To rigorously study the stability with the new cavity (Figure 3.18), we use the setup shownin Figure 4.8a to perform comparisons between Sr2 and Sr1 with lasers stabilized to either the 7
cm or 40 cm cavities. Measurements with the 7 cm cavity made use of synchronous interrogation[9, 113]. In this case, the Rabi pulses of both Sr clocks were carefully synchronized such that they
always occur at the same time (Figure 4.8b top). The PID filters of both systems are also set tohave equivalent gain constants (Section 2.44), ensuring that both clocks respond to noise in thesame way. Pulse synchronization is accomplished by triggering Sr1 off of the Sr2 clock.
Since the synchronous clock comparison occurs in post processing, data analysis must correctly match each Sr2 frequency measurement with the simultaneous Sr1 measurement; therefore,sorting errors in the data record could destroy the comparison. To ensure that the data record isrobust against such errors, both systems used a shared counter (triggered by Sr2 to advance by 1

85
each measurement cycle) to stamp each measurement with a counter value. When comparing the
Sr1 and Sr2 data, counter values can be checked to ensure that they agree.
With synchronous interrogation, the two clocks always sample the same laser noise. Therefore, the difference between the clock frequencies benefits from common-mode laser noise cancellation, which will suppress the Dick Effect. Also, studying the effect that synchronous interrogation
has on stability is useful for quantifying the Dick Effect [89].
Synchronous interrogation is also useful for systematic evaluations. Systematic shifts can beevaluated using two synchronized clocks, where one clock modulates a parameter known to cause
systematic bias and measures the resulting change in frequency with respect to the other clock.
This would result in a systematic evaluation that is largely free of the Dick Effect [116]. However,a clock used for timekeeping is a stand alone system that has stability limited by the frequency
noise of the clock oscillator. To quantify the stability of a system for use as a clock, synchronouscomparisons will not work since they do not show the stability of independent systems. For this,we use asynchronous interrogation, which will be discussed in Section 4.5.1.

4.4.2

Excited State Fraction Correlation

Synchronous interrogation using the 7 cm cavity shows a clear effect on raw frequency data
(Figure 4.9). This measurement scheme should also affect the excited state fraction measurementsof these two systems [113]. When the two clocks are synchronized, clock laser noise will cause
measurements of ρee to fluctuate about 1/2, but synchronization means that these fluctuations will
be correlated between the two clocks.
To understand the effect of synchronous interrogation on the excited state fractions, it isworthwhile to simulate what the corresponding measurements would look like. Treating laser
fluctuations as a Gaussian white noise process, I simulate the excitation fraction data when both
Sr1 and Sr2 are locked to set points of 1/2 (Figure 4.10). When the two atomic servos use pulses thatare not synchronized, both laser noise and QPN (simulated for 1000 atoms) are uncorrelated. Thisis apparent because of the circle-shaped distribution of the simulated data in Figure 4.10a. Here an

86

Figure 4.9: Raw data measured by synchronized clocks. The scanning synthesizer frequencies forboth Sr1 and Sr2 during synchronized atomic servo operation are plotted with offsets between themremoved. The Rabi π-pulses are 160 ms long. The y axis has been offset by the scanning synthesizer
frequency at about 28 s. The linear laser drift removed by the atomic servos is apparent here. A
striking correlation between the frequency data is apparent. For uncorrelated data, the averagefrequencies would agree but the instantaneous frequencies would not.

Excited state fraction correlation (simulated)1a)

Sr1 excited state fraction

Sr1 excited state fraction

1
0.8
0.6
0.4
0.2
0

b)

0.8
0.6
0.4
0.2
0

0

0.2

0.4

0.6

0.8

Sr2 excited state fraction

1

0

0.2

0.4

0.6

0.8

1

Sr2 excited state fraction

Figure 4.10: Simulated correlation measurement data. Appreciable laser noise is assumed. a)
The two clocks are locked using pulses that are not synchronized. b) The clocks are locked withsynchronized pulses.

87
10-15

a)

b)

Unreferenced
Synchronized

0.8

Allan deviation

Sr1 excited state fraction

1

0.6
0.4
0.2

10

-16

10

-17

0
0

0.2

0.4

0.6

0.8

1

Sr2 excited state fraction

100

101

102

103

Averaging Time (s)Figure 4.11: a) Correlation data using a clock laser stabilized to the 7 cm cavity. The width is due
to uncorrelated noise, which can come from QPN, imperfect synchronization, or technical noise notcommon to both clocks. b) Overlapping Allan deviations of the Sr2 self-comparison data with andwithout the Sr1 reference. The 7 cm cavity is also used for this measurement.

excited state fraction measurement in one clock does not necessitate a value for the other system.
When pulses are synchronized, the data has a diagonal stripe distribution (Figure 4.10b). Herelaser noise causes the excited state fraction measurements to fluctuate, but the synchronizationof the two clocks ensures that measured values are similar for both systems. The width of thehorizontal stripe is given by quantum projection noise, which remains uncorrelated.
Using the noisier 7 cm cavity to test this situation, the 160 ms π-pulses from both clocks
are synchronized and the same clock laser source is used. A correlation is observed in the data,
showing that synchronous interrogation has succeeded in making much of the laser noise common
mode between the two clocks (Figure 4.11a) [89].
The effect of synchronous interrogation on clock stability can also be seen with Allan deviations. This is done by running Sr2 and Sr1 in a synchronized fashion while Sr2 takes self-comparison√
data. Self-comparison data fits to 1.2 × 10−15 / τ (Figure 4.11b, unreferenced data), where τ is inseconds, and the first few points shown on the unreferenced Allan deviation of Figure 4.11b havebeen ignored in the fit. If the Sr1 center frequency data is subtracted from the Sr2 measurements(where the difference is taken between points measured simultaneously), the boost in stability is

88√
appreciable (Figure 4.11b, synchronized data). The stability now fits to 4.6 × 10−16 / τ , which
is better than the unreferenced case by a factor of 2.6. This corresponds to a factor of 6.8 lessaveraging time.
The fact that the first few points of these Allan deviations are flatter is because for the first∼ 20 s, the Allan deviation is that of the free-running laser. After a few updates of the atomic
servo, the lock to the atoms takes over and the stability begins averaging down. Because of this,Allan deviations limited by the Dick Effect or QPN are typically fit by excluding the points before√
the servo becomes effective and then performing regression to a 1/ τ power law (and they will be
fit this way from here on). Clock stability is usually quoted in terms of this fit.12

4.5

Asynchronous Interrogation

4.5.1

Using Asynchronous Interrogation to Measure Independent Clock Stability

As explained in Section 4.3.2, given the Allan deviation of the free-running clock laser basedon the 40 cm cavity (Figure 4.6), we expect that upgrading to this cavity will provide a significant
boost in stability (over the laser based on the 7 cm cavity). As the previous section shows, one canuse synchronous interrogation to gauge the Dick Effect with this new laser. Also, after upgrading to
the 40 cm cavity, it is important to measure the independent clock stability, which characterizes theSr2 system as a frequency standard (Section 4.4.1). To do this, we use asynchronous interrogation,
which ensures that the Sr1 and Sr2 π-pulses never overlap in time. With this interrogation scheme,
the falling edge of the Sr1 pulse and the rising edge of the Sr2 pulse are separated by 10 ms (Figure
4.8b, bottom) so that the two clocks never sample the same laser noise.
It is important to note that two asynchronously interrogated systems are not 100% independent. Since the pulses have a fixed temporal relationship to one another, spectral components ofthe clock laser with frequencies that are integer multiples of 1/∆tp , where ∆tp is the separation
between the pulse centers, will affect asynchronous interrogation differently than they would an
12

For instance, “1 s stability” typically means the fit quoted at τ = 1.

89
independent clock. To analyze this issue,13 it is useful to express the sensitivity function g(t)(Section 4.2.2) in terms of the excited state fraction [106],1 1
ρee (t) = +2 2

Zg(t)ǫν (t)dt,(4.14)

where ǫν (t) is the stochastic frequency fluctuation of the laser about the clock transition frequency.
The integral is taken over the clock cycle under consideration. This expression describes the behavior of ρee when the clock laser is locked to the atoms. Here it is assumed that ρee is locked toa value of 1/2. This expression is a first-order approximation to the effect of laser noise on the
excited state fraction, treating ǫν (t) as a small-amplitude perturbation.
When ρee deviates from 1/2, the atomic servo error signal ǫclock ≃ 2(ρee −1/2) =R

g(t)ǫν (t)dt(using Equation 4.14). Since an asynchronous comparison is the difference between center frequencies computed by the Sr1 and Sr2 atomic servos—both of which operate with the same pulse
durations, gain constants, and cycle times—it follows from Equations 2.44 and 4.14 that the noiseRR
in the comparison is given by the integral [gSr2 (t) − gSr1 (t)] ǫν (t)dt = [g(t + ∆tp ) − g(t)] ǫν (t)dt.

Here, gSr2 (t) and gSr1 (t) are the Sr2 and Sr1 sensitivity functions. The expression g(t + ∆tp ) − g(t)can be viewed as the sensitivity function of the asynchronous comparison.

Using the fact that g(t) is periodic in Tc , the Fourier coefficients (Equation 4.10) of g(t + ∆tp )are gk e2πik∆tp /Tc (where gk is the kth Fourier coefficient of g(t)). This leads to the Dick Effect foran asynchronous comparison σasynch of∞

2 gk 2
1 X 2πik∆tp /Tc
Sclock (k/Tc )
−1
e
τ
g0
k=1


∞
X
πk∆tp
gk 2
4
=
Sclock (k/Tc ).
sin2
τ
Tc
g0

2
(τ ) =
σasynch(4.15)

k=1

Replacing the sin2 () factor with its average of 1/2 results in the Dick Effect for two identical but
independent clocks, for which one would simply add two identical single-clock Allan deviations
13

My thanks to Michael J. Martin for working this out first.

90
1

a)

0.8
0.6
0.4
0.2
7 cm cavity
40 cm cavity

0
0

0.2

0.4

0.6

0.8

1

Sr2 excited state fraction

Sr1 excited state fraction

Sr1 excited state fraction

1

b)

0.8
0.6
0.4
0.2
40 cm asynchronous
40 cm synchronous

0
0

0.2

0.4

0.6

0.8

1

Sr2 excited state fraction

Figure 4.12: a) Synchronous data with the 7 cm cavity (from Figure 4.11a) and asynchronous datawith the 40 cm cavity. The 40 cm cavity data shows a markedly narrower distribution, implying
less laser noise. b) Synchronous and asynchronous data using the 40 cm cavity. Synchronousinterrogation shows a bit of correlation.

in quadrature. Including this factor and using a good model for the clock laser power spectrum[8] changes the Dick Effect by 3% from its independent clock value. Therefore, asynchronousinterrogation reproduces independent clock stability at the 97% level.

4.5.2

Correlations and Allan Deviation

Measuring excited state fraction correlations with the clock laser based on the 40 cm cavity,
the distribution shows a dramatic difference. In Figure 4.12a, asynchronous data with the 40 cm
cavity is compared with synchronous measurements using the 7 cm cavity [89]. The distribution
is much narrower, implying less laser noise. Synchronous measurements with the 40 cm cavity
show a bit of correlation (Figure 4.12b), indicating that the Dick Effect has not been completely
suppressed below QPN [89].
Looking at Allan deviations of synchronous and asynchronous data taken with the 40 cm
cavity, a significant stability improvement over the 7 cm cavity is observed (Figure 4.13). Selfcomparison is also used to characterize the stability. Looking at self-comparison and synchronous
data on the same plot (Figure 4.13), it is apparent that the Dick Effect plays a much smaller roleAllan Deviation

91

Synchronous
Self Comparison
QPN

−16

10

−17

10

a)
0

1

10

2

10

10

3

10

Averaging Time (s)Allan Deviation

b)
−16

10

−17

10

Asychronous
Dick Effect + QPN
QPN
0

10

1

10

2

10

3

10

Averaging Time (s)Figure 4.13: Overlapping Allan deviations with the 40 cm cavity. a) Synchronous and selfcomparison data. b) Asynchronous data, which gives the stability of a comparison between two√
independent clocks. The Allan deviation fits to 4.4 × 10−16 / τ for the asynchronous comparison√
or, extrapolating to the performance of a single clock, 3.1 × 10−16 / τ . Using the known clock laser√
power spectral density [8], the theoretical one-clock Dick Effect is 3.5 × 10−16 / τ .

92

−16

Allan Deviation

10

−17

10

−18

10

1

10

2

10

3

10

4

10

Averaging Time (s)Figure 4.14: Overlapping Allan deviation for a 3.75 hour asynchronous comparison. 160 ms Rabi
π-pulses were used again here.

in the stability [89].14 Synchronous interrogation now results in stability that is only a bit below
the self-comparison (compared to the much larger effect shown in Figure 4.11b).
However, these measurements are only complementary to that shown in Figure 4.13b. This is
an asynchronous comparison between Sr1 and Sr2, which yields the full independent clock stability.
It is sensitive to long-term drift (that is not common to the two clocks), so it demonstrates thelong-term stability as well. Drift, which would be evident by the Allan deviation increasing linearlywith averaging time (Figure 4.1), is not present in the asynchronous data. This measurement√
achieves 3.1 × 10−16 / τ one-clock stability (the stability of a comparison between two identical√clocks divided by 2). When this stability was reported [89], it was the first instance of a latticeclock surpassing all ion clocks in an important measure of clock performance. The following year,√
the Yb lattice clock team at the Boulder campus of NIST also reported stability of 3.1 × 10−16 / τand averaged this down for 7 hours to 1.6 × 10−18 [49].
To study the Sr2 stability at longer averaging times, which ensures that drift is well controlled,
Sr1 and Sr2 are compared asynchronously for 3.75 hours [13]. No drifts are apparent; however, the√stability was a bit worse for this measurement, fitting to a value of 3.4 × 10−16 / τ (Figure 4.14).
14

Recall that self-comparison data is sensitive to the Dick Effect whereas synchronous measurements are not.

93

4.6

High Stability Self-Comparison
So far, all stability measurements discussed used 160 ms π-pulses with Tc ≃ 1 s. To infer thebest stability possible in Sr2, it is worthwhile to make the duty cycle larger, which reduces the Dick
Effect (Section 4.2.2). However, the best possible stability of Sr2 should, in principle, be better
than that of Sr1. An attempt to quantify the maximum Sr2 stability by comparing the two clocks
would be difficult once Sr2 has significantly better performance.
The reason why Sr2 should have higher stability is because of its cavity lattice. As explainedin Section 2.3.3, the cavity lattice is designed to have a large trap volume compared to the retroreflected lattice in Sr1. The advantage of this is twofold. First, the large volume means that thecavity lattice has a much better spatial overlap with the 689 nm MOT, allowing Sr2 to trap more
atoms than Sr1; therefore, Sr2 can obtain 2000 atoms with less MOT cooling time, resulting in lessdead time and a bigger duty cycle. Second, the larger volume implies that if Sr2 and Sr1 operatewith 2000 atoms, which is the number used for clock operation, then Sr2 will have lower density(by a factor of 30 for the actual trap parameters). This lower density allows Sr2 to use long pulses
while still operating at 2000 atoms. This is useful because Sr1 has observed resonance contrast decay, broadening, and lineshape distortion of the clock transition due to atomic interactions [6, 78].
These effects (which compromise stability) increase with density, and for typical operating atom
numbers, Sr1 was unable to achieve unbroadened lineshapes for pulse durations of 1 s and longer[78, 77].
Unfortunately, it is very inconvenient to upgrade the Sr1 lattice to a cavity-based trap because
the vacuum chamber viewports on that system do not have anti-reflection coating at 813 nm. Thiscoating on the Sr2 viewports (Section 3.1.2) is necessary to achieve good cavity enhancement, which
is negatively affected by power loss through the viewports (Equation 2.14). Therefore, we mustevaluate the Sr2 stability with self-comparison, which does not rely on another clock.
Since self-comparison is insensitive to long-term drift (Section 4.5.2), this technique alone
cannot determine the full independent clock stability. However, if we use previous comparisons to

94

a)

10 -16

Allan Deviation

Allan Deviation

10 -16

10 -17

10 -18b)

10 -17

10 -18
10 1

10 2

10 3

10 4

10 1

Averaging Time (s)

10 3

10 4

Averaging Time (s)c)

10 -16

Allan Deviation

Allan Deviation

10 -1610 2

10 -17

10 -18d)

10 -17

10 -18
10 1

10 2

10 3

Averaging Time (s)

10 4

10 1

10 2

10 3

10 4

Averaging Time (s)Figure 4.15: Simulations of stability data with drift. The noise is taken to be white, which isvalid after the few 10s of seconds it takes for the atomic servo to kick in. The blue data has√√
3.4 × 10−16 / τ stability, equivalent to that in Figure 4.14, and the red data fits to 2.2 × 10−16 / τ .
The cycle times of both data sets match those used in our experiments. a) A drift rate of 1.4×10−18
over the measurement time. Drift is not easy to discern from either Allan deviation. The red data
is turning up slightly, which could easily be mistaken for a different feature. b) 4.1 × 10−18 drift
over the measurement time is apparent for both curves. c) 1.4 × 10−17 drift over the measurement
time. d) 1.4 × 10−16 drift over the measurement time.

95
constrain the magnitude of long-term drift, self-comparison can determine the full stability at alevel given by this constraint (Figure 4.15). Figure 4.14 shows that the total long-term drift over
3.75 hours of averaging is below 3×10−18 , or a 2.2×10−22 s−1 drift rate. Therefore, self-comparison
stability is equal to the independent clock stability—so long as it is understood that the former
should not be extrapolated below 3 × 10−18 or longer than 3.75 hours of averaging time (whichevercomes first).
Using 1 s pulses and reducing MOT cooling time to achieve a 60% duty cycle15 , the self√
comparison (Figure 4.16) demonstrates one-clock stability of 2.2×10−16 / τ [88]. This value, which√is compared in Figure 4.7 with other prominent optical clocks, is the best stability per τ achieved
to date.

15
With two clocks operating at 50% duty cycle, one can realize a proposal by Dick, et al. [29], which wasdemonstrated in the microwave by Biedermann, et al. [5]. This involves using two clocks operating in an interleaved
fashion. When the first system measures the clock transition, the second is accruing dead time for sample preparation.
Then they switch roles, ensuring that the clock transition is always being measured and reducing dead time to zero.
This would affect the Dick-Effect-limited stability dramatically.

96

Excitation Frac.

(a)

0.8
0.6

0.9 Hz0.5 Hz

0.4
0.2
0.0
-2

-1

0
Detuning (Hz)1

2

Total Deviation(b) 10-16

10

-17

10

1

2

10
Averaging Time (s)103Figure 4.16: a) The effects of pulse duration on the clock transition. A typical line scan with a 1
s interrogation time is shown in open black circles. To explore the limit of coherence in Sr2, wescan the clock transition with a 4 s interrogation time and more atoms (solid green squares). Herethe linewidth and contrast are affected by the Fourier width and atomic interactions [6, 78]. b) A
new stability record (black circles) achieved by running with 1 s clock pulses and a 60% clock laser√
duty cycle for each preparation and measurement sequence. The stability fits to 2.2 × 10−16 / τ(red line). The previous result is indicated with the blue dashed line.

Chapter 5

Systematic Uncertainty

5.1

Systematic Measurements

5.1.1

Introduction

Improved stability, demonstrated in Chapter 4, is critical for better systematic shift measurements. This is because systematic uncertainty is often measured by the clock itself (Section
4.1.1). A parameter known to cause a systematic shift (such as the lattice intensity or bias field) is
modulated, and the frequency shift that results from changing this parameter is measured with ascheme similar to self-comparison. These measurements, which are typically limited by the stabilityof the clock, are extrapolated to operating conditions. Although most major systematic shifts aremeasured in this way, there is one important exception: the blackbody radiation (BBR) shift, forwhich a completely different approach must be taken.1
Systematic uncertainty arises because in practice the clock laser is locked to the perturbed
clock transition, but we want to extrapolate the locked clock laser frequency to the bare transition(Sections 1.6 and 2.7). With all systematic frequency shifts well known, real-time corrections could
be applied to Sr2 so that, when running this system as a clock (with the frequency comb counter
enabled), it ticks at a rate determined by the bare frequency.
1
Other teams have measured the BBR shift by looking at the effect of temperature modulation on their clocktransition frequency [2, 116]. This requires the ability to change the ambient temperature in a spatially uniform
manner, and the Sr2 clock is not equipped for this kind of temperature control. This approach to the BBR shift isdifferent than other systematic measurements performed in Sr2 (which are measured with a single clock) because it
requires the clock that is under evaluation to be referenced to a second, highly stable frequency standard. A referenceis needed because temperature cannot be modulated quickly, so drift of the clock laser cavity would ruin a BBR shiftmeasurement with only one clock. Referencing to a second system removes this drift from the evaluation.

98
Sr2 has performed two systematic evaluations. Prior to these, the record for the best atomic
clock2 was 8.6 × 10−18 total systematic uncertainty [23]. The first Sr2 systematic evaluation broke
this record with 6.4 × 10−18 total systematic uncertainty. The second evaluation, which began withthe goal of realizing low-10−18 -level performance, secured the standing of Sr2 as the best clock byachieving 2.1 × 10−18 total uncertainty.
Effect
1st-order Zeeman
2nd-order Zeeman
Probe Stark
Density
Lattice Stark
Dc Stark
Static BBR
Dynamic BBR
AOM phase chirp
Servo offset
Line pulling and tunneling
2nd-order Doppler
Background Gas Collisions
Total

Evaluation 1
Shift
Uncertainty-0.2
1.1-144.5
1.2
0.8
1.3
-4.7
0.6
-461.53.7-3.5
2.1-4962.9
1.8-345.7
3.7
0.6
0.4
0.4
0.6
0
< 0.1
0
< 0.1
0
< 0.6
-5921.2
6.4

Evaluation 2
Shift
Uncertainty
-0.2
0.2
-51.7
0.3
0.0
0.0
-3.5
0.4
-1.3
1.1
0.0
0.1
-4562.1
0.3-305.3
1.4
0.6
0.4
-0.5
0.4
0
< 0.1
0
< 0.1
0.0
< 0.6
-4924.0
2.1

Table 5.1: The uncertainty budget for both evaluations. The effects are listed in the order thatthey are discussed in this chapter. All uncertainties are quoted at the 1σ level.

5.1.2

Digital Lock-In Detection

The act of modulating a parameter to observe the change in the clock transition frequency
is similar to lock-in detection. The parameter under consideration is modulated at a certain rate,
and then the data record is demodulated at this rate in post processing, suppressing effects that
do not change with the modulation. From here on this procedure will be referred to as “lock-in
detection.”An example of how this works for systematic evaluations is the following. A chosen parameter(such as the lattice intensity or bias field magnitude) switches between two states. Each state of2

The word “best” is often used to describe the clock with the lowest total systematic uncertainty despite thatstability is also an important figure of merit for clock performance.

Experimental Parameter

99
Servo 1
State 2Servo 2

State 1

2Tc 4Tc 6Tc 8Tc 10Tc 12Tc
Figure 5.1: The lock-in measurement. A parameter is modulated by switching between two values(“State 1” and “State 2” in the Figure). If the cycle time is Tc , which involves one interrogationof the clock transition and an excited state fraction measurement, it requires 2Tc to measure thescanning synthesizer frequency corresponding to the resonance center (Section 2.5.2). When theparameter is in State 1, the clock transition is probed on either side of resonance, and the line
center is determined by the digital PID filter (Equation 2.44). The parameter is then switched toState 2, shifting the clock transition resonance center, which is determined by the second servo.
The red and blue points in the Figure depict the time and parameter value when the line center isdetermined.

this parameter corresponds to an independent atomic servo locked to the clock transition, and thelock-in alternates between these two servos (Figure 5.1). Aside from this modulation parameter,
the atomic servos operate in the same conditions.
Suppose the scanning synthesizer center frequency data taken in the lock in measurementis {ν1 , ν2 , . . . , νM }, where odd (even) subscripts mean that the data was taken with the lock-in
parameter in State 1 (2), and M is always an even number. For each of these points, the lockin parameter is known (either by measuring it each time a frequency point is acquired or bystabilizing the two lock-in states to be constant throughout the measurement). The lock-in data is∆νlock−in = {ν2 − ν1 , ν4 − ν3 , . . . , νM − νM −1 }. If clock laser drift is canceled well enough (Section
3.7.1) such that it is negligible over the 4Tc it takes to measure one lock-in point, then the lock-in
data is equal to the clock transition frequency difference between the two states of the modulated
experimental parameter.3
3

Many Sr lattice clock groups have performed systematic measurements while referenced to another clock; however,with good drift cancellation and point-string analysis (Section 5.1.5), the evaluations discussed in this thesis do not
require a second clock as a reference.

100
5.1.3

Lever Arm

Extracting the information needed from lock-in data requires a model of how these data varywith the lock-in parameter. As an example, take the probe ac Stark shift. This is the systematic
Stark shift caused by the clock laser. It is known that this shift is proportional to the clock
laser power P . The proportionality constant K is related to atomic structure and experimental
conditions such as the clock laser intensity distribution, so K is not known a priori. A lock-in pointν2 − ν1 , taken when the power is modulated between P1 and P2 , is equal to K(P2 − P1 ). Suppose P0
is the clock laser value during normal clock operation. The systematic shift of the clock transition
during normal operation is inferred from lock-in data as∆νprobe =h∆νlock−in iP0h∆νlock−in i,=
P2 − P1
L(5.1)

where h. . .i is the mean of the data set and, by analogy with simple machines, L = (P2 − P1 )/P0 is
known as the lever arm. Note that inferring ∆νprobe from h∆νlock−in i using Equation 5.1 relies on
the fact that the probe Stark shift is proportional to P . If this model were not accurate, Equation
5.1 would be an incorrect extrapolation of the data.
The uncertainty in ∆νprobe , δνprobe , isδνprobe =σlock−in
,
L

(5.2)

where σlock−in is the statistical uncertainty in the mean of the lock-in data, h∆νlock−in i. σlock−in
is given by the stability of the clock (Section 4.6). As Equation 4.6 shows, the lever arm provides
a means of measuring a systematic shift with precision much better than the clock stability. For
many systematic measurements, L > 10, allowing for a systematic shift to be measured at a given
precision more than 100 times faster than if clock stability were relied upon alone.

a)

1

x 10

−17

b)

0

1.5

x 10

101

−18

1

Extrapolated ShiftLock−in Data

−1
−2
−3
−4
−5

0.5
0
−0.5
−1

−6
−1.5

−7
−8
0

0.2

0.4

0.6

0.8

1

−2
0

0.2

2

∆I (mW/cm )

0.4

0.6

0.8

1

2

∆I (mW/cm )Figure 5.2: a) Simulated lock-in data. Here ∆I = I2 − I1 . The data scatter is such that χ2red takeson the ideal value of 1 (despite looking ugly, as sparse χ2red = 1 data often does). b) Simulated
shift extrapolated to the clock operation intensity. This is simply the lock-in data divided by themeasurement lever arm.

5.1.4

Model Testing

Lock-in data can also be used to test the model of the frequency shift as a function of thelock-in parameter. To demonstrate this, I have simulated a probe Stark shift measurement (Figure5.2). In this simulation, a clock laser intensity of I0 = 10 µW/cm2 is chosen as the clock operation
value. The lock-in modulates between I1 = I0 and I2 , and the frequency difference between thesetwo conditions is averaged until it reaches 1 × 10−17 statistical uncertainty. This measurement
is repeated for values of I2 ranging from 0.1 to 1 mW/cm2 .4

The mean of the lock-in datah∆νlock−in i for each value of I2 − I1 is shown in Figure 5.2a. The scatter is simulated as Gaussian
white noise with a reduced chi-square statistic χ2red = 1. The extrapolated shift, ∆νlock−in /L, whereL = (I2 − I1 )/I0 , is shown in Figure 5.2b. The simulation treats the shift as linear in intensity,and the mean and uncertainty of the mean of the extrapolated data is (−6.9 ± 0.5) × 10−19 . Thisagrees with the simulation’s true value of −6.5 × 10−19 .
In the context of a real measurement, since extrapolating data to the clock operation condition
4

As I will show in Section 5.4, in the actual probe Stark shift evaluation, we study this effect as a function ofclock laser pulse duration rather than intensity.

102
relies the shift being linear in intensity, it is tempting to worry if a next-order Stark shift term
(quadratic in intensity) might be present in a subtle but significant way. An F -test for an additionalterm [4] can be used to see if the data requires a quadratic model. This test compares the chisquare statistic for different models to see if an extra term improves chi-square. In the simulated∆νlock−in /L data, an F -test for an additional (quadratic) term results in the test statistic F = 0.37
(with 8 degrees of freedom). This corresponds to a test probability of 0.56.5

The F -test argues

for another term if the test probability is lower than a critical value that is conventionally taken
as 0.01 to 0.05 [4], so 0.56 is a strong indication that the data does not resolve quadratic behavior.
This is expected since ∆νlock−in values are simulated with a linear intensity dependence.
To study the effectiveness of the test, I add a small quadratic term (proportional to I22 − I12 )to the simulated lock-in data. In this case, the extrapolated shift has the form ∆νlock−in /L =α + β(I1 + I2 ), where α is the linear (in intensity) probe Stark shift at the clock operation conditionand β is a positive constant. When ∆νlock−in /L is studied as a function of intensity, the meanof the extrapolated data will be offset from α by the mean of the β(I1 + I2 ) term. Ideally, whenβ is small enough for the offset to be statistically unimportant, the F -test would conclude thatthe β(I1 + I2 ) term is not needed to model the data. It would also be desirable for the F -test
to conclude that β(I1 + I2 ) is required when the offset is significant. For most values of β, thesedesired properties of the F -test hold true.
However, there is a small range of β values for which the mean is significantly affected but the
F -test does not conclude that an additional term is needed. For example, β can be made as large as
5.8 × 10−19 (W/cm2 )−1 , yet the F -test will show a model linear in intensity with a comfortable test
probability of 0.15. For this β, the mean of ∆νlock−in /L is −2.3 × 10−19 , but β is still small enoughthat the simulation’s true value of the clock operation probe Stark shift, α + βI0 = −6.4 × 10−18 , isstatistically consistent with α. Meanwhile for β < 2.3 × 10−19 (W/cm2 )−1 , the offset of the meanof the extrapolated data is within statistical uncertainty, and the F -test reflects that another term
is not needed. Therefore, for this simulation, the linear shift model will introduce significant error
5This probability is computed using the F -test probability distribution given in Reference [4].

103
for the range of 2.3 × 10−19 (W/cm2 )−1 < β < 5.8 × 10−19 (W/cm2 )−1 , and the F -test will fail inthis regime.6
Since this blind spot results from neglecting higher-order terms, it may seem as if the clear
path forward is to always fit these terms even when the F -test argues that there is no statisticaljustification for doing so. However, for β in the blind spot range and below, fitting the simulated
data to a+b(I1 +I2 ) (where a and b are fit parameters) increases the uncertainty in the extrapolated
shift a + bI0 to four times above that when linear intensity dependence is assumed—even whenβ = 0. Therefore, a wide range of probe Stark measurements that would be well described by amodel linear in intensity are greatly affected by treating the quadratic term as unknown. Also, for
β in the blind spot range, the value for the b fit coefficient often differs from the simulation’s true
value by more than the fit uncertainty.
This analysis based on the probe Stark shift extends to the many systematic shifts that havehigher-order terms that are not known precisely, such as shifts from the lattice that are nonlinear
in intensity, the quadratic density shift, and the third-order Zeeman shift. Although the rangeof β values for which the statistical blind spot exists might seem insignificantly small, there has
been a great deal of concern in the clock community over higher-order effects that have not been
resolved but cause significant measurement bias [18, 121, 111]. If we choose to be concerned about
higher-order terms that might be significant but will not show up in an F -test, it is often difficult
to know how to proceed. Fits of higher-order terms will not be accurate, and they extract a big
toll, destroying the total uncertainty of the clock over a shift that may be negligible. Furthermore,
many of these shifts cannot be experimentally resolved even in the systems best equipped to do
so.7

Being confronted with the possibility of effects that cannot be statistically tested for, fit,or measured, yet there is still concern about their size seems like a contrived scenario, but it
6Note that this regime will vary with the parameters of the measurement set since the offset depends on thevariable that is modulated.
7
For instance, an M1-E2 lattice light shift was predicted to be very large [111]. Before this prediction, it had notbeen considered in Sr clocks, and theory suggested that clock uncertainty budgets need to be amended to account forthis effect. However, the weak dependence of this effect on trap intensity made it impossible to resolve in most lattice
Stark shift measurements. The Sr clock at LNE-SYRTE could achieve unusually large lattice intensity, which should
be conducive to observing an M1-E2 shift, yet they measured no shift within their precision [121]. They concluded
the shift is much smaller than predicted.

104
does happen [111, 121]. In the face of this, and because even seemingly sound calculations ofimportant shifts have turned out to be incorrect [94, 105, 111], we only accept higher-order termsin two situations: if our statistical tests demand them, or if another team has reported statisticallyresolved measurements of these effects that we have no cause to doubt. Although this approach
technically leaves open a small possibility of uncorrected measurement bias (should we be unlucky
enough for the coefficient of a higher-order effect to fall within the blind spot), to do otherwise would
leave our experimental methods and conclusions highly sensitive to experimentally unsubstantiatedconcerns of higher-order effects.

5.1.5

Point-String Analysis

During a lock-in measurement, the scanning synthesizer data can be affected by drift of the 40
cm clock laser cavity. This is because if the rate of the drift cancellation oscillator (Section 3.7.1) is
not optimized, the atomic servo removes the remaining drift by adjusting the scanning synthesizer
frequency to compensate for it. We have observed that when this rate is far from optimized, thelock-in data (which is comprised of scanning synthesizer frequency measurements) is contaminatedby this drift when it is analyzed as simple differences {ν2 − ν1 , ν4 − ν3 , . . . , νM − νM −1 }. To guard
against this problem, the residual drift is monitored by real-time plotting of the center frequency
computed by one of the atomic servos to look for a linear slope. If a slope becomes apparent, therate of the drift cancellation oscillator is reset to compensate. Also, the data is processed to remove
any residual linear drift. This drift-insensitive method of analyzing lock-in measurement data is
point-string analysis [109].
For residual linear drift at a rate rres , the probe Stark data will be

105



 

ν
ν
+
KP
1
of f set 1  


 


 

 ν2   νof f set + KP2 + 2Tc rres 

 


 

 ν = ν 3   of f set + KP1 + 2 × 2Tc rres  ,

 


 

 ν4   νof f set + KP2 + 3 × 2Tc rres   .  .
..
..

where νof f set is an unmeasured technical offset to the scanning synthesizer frequency. If these data
are used to create the vector










∆ν3pt = 









ν2 − 12 (ν1 + ν3 )
1
2 (ν2 + ν4 ) − ν3

ν4 − 12 (ν3 + ν5 )
1
2 (ν4 + ν6 ) − ν5..
.

1
2 (νM −2 + νM ) − νM −1











,









then the drift will be canceled, making each entry in the vector equal to K(P2 − P1 ) as desired.
Therefore,∆νprobe =h∆ν3pt i
,
L

(5.3)

and each entry in the vector ∆ν3pt is called a three-point string. This analysis is valid as long as theresidual drift is sufficiently linear over the 6Tc it takes to measure three center frequencies. Control
experiments with a known frequency offset added between the two states of the lock-in shows thatthree-point strings can effectively reproduce this added offset in the presence of uncanceled clock
laser drift.8
The three-point string vector ∆ν3pt has M −2 elements whereas the vector of simple difference
data {ν2 − ν1 , ν4 − ν3 , . . . , νM − νM −1 } has M/2 elements. Therefore, taking the uncertainty in the8

For the control experiments, the rate of the drift cancellation synthesizer is intentionally set off its optimum.

106
mean of ∆ν3pt to be the uncertainty in the lock-in measurement will underestimate the error bar(since ∆ν3pt has more elements than the M/2 statistically independent simple difference points).
The uncertainty in the mean of ∆ν3pt should be inflated to account for this issue.
Suppose a single center frequency measurement has uncertainty of σ. A single point in ameasurement of ∆νprobe , processed using a simple difference, is (ν2 − ν1 )/L. Using standard error√
propagation and assuming no uncertainty in L, the uncertainty in this measurement is 2σ/L, and√the uncertainty in the mean is 2σ/L M (for large M ). However, the first point of ∆νprobe processed
as a three-point string is ν2 − 21 (ν1 + ν3 ) /L. The single-point uncertainty and uncertainty in the√√√√mean are 3σ/ 2L and 3σ/ 2M L (again, for large M ). Therefore,δνprobe =r

8 σ3pt
,
3 L

(5.4)

where σ3pt is the statistical uncertainty in the mean of ∆ν3pt .
This approach can be generalized to higher orders of drift [109]. Point strings of length nremove polynomial drift of order n − 2. If νk is the kth point of {ν1 , ν2 , . . . , νM }, the n-point stringisnX(n − 1)!
(−1)k+m−1
νk+m−1 .
2n−2 m=1(n − m)!(m − 1)!
1

(5.5)

This expression is valid from k = 1 to k = M − n + 1, which is the total number of point strings.
When this formalism is used to remove drift, the uncertainty in the mean must be inflated by afactor

r

Pn

m=1

2n−1
h

i2
(n−1)!
(n−m)!(m−1)!

to extrapolate this uncertainty to that of statistically independent points.

(5.6)

107

5.2

Residual First-Order Zeeman Shift

5.2.1

Introduction

With an understanding of how many systematic measurements are performed and analyzed,
I will now discuss the Sr2 systematic measurements. Before I discuss the systematic shifts thatare measured with the lock-in technique, it is useful to first discuss the residual first-order Zeeman
systematic shift. This is because a discussion of this shift provides a good context for introducing
the background magnetic field servo, which is used for many other systematic measurements.
During normal clock operation, the stretched-states servo (Section 2.5.3) cancels most of thefirst-order Zeeman shift. However, if the background magnetic field (due to stray fields from lab
equipment or Earth’s field) drifts appreciably in between measurements of the two stretched-state
center frequencies, there can be a residual first-order Zeeman shift.
Let {ν1 , ν2 , ν3 , . . .} be a vector of center frequency data taken by the stretched-states servo.
Even (odd) subscripts correspond to data taken for the stretched state with a negative (positive)
first-order Zeeman shift. Also, let the stretched state first-order Zeeman shift per Gauss be ζ =9 µB δg/2h = 487.8 Hz G−1 (Equation 2.40). For an external field of B(t), which is comprised of theadded bias field and possible a drifting background field, ν1 = νSr − ζB(t1 ) and ν2 = νSr + ζB(t2 ),where t2 − t1 = 2Tc and νSr is the zero-field clock transition frequency.9 When the external field
is free of drift, the zero-field clock transition center frequency is inferred as (ν1 + ν2 )/2 (Equation
2.5.3).
When the external field does drift, (ν1 + ν2 )/2 = νSr + ζ [B(t2 ) − B(t1 )] /2. This stretchedstate average frequency, normally used to infer νSr , now has a residual first-order Zeeman shiftζ [B(t2 ) − B(t1 )] /2. The shift of the entire data set is∆νZ1 =91
ζhB(t + 2Tc ) − B(t)i,
2

(5.7)

Given that the two stretched state resonance centers are not measured one after another—rather, the data used
to determine these resonances is interleaved (Figure 2.16)—perhaps t2 −t1 should not be set equal to 2Tc . However, tosimplify the presentation and arguments in this Section, I consider the case of the resonance centers being measured
one after another. All of the conclusions of this Section can still be applied to normal clock operation.

108
where h. . .i is an average over the time values for which the data was taken.
5.2.2

Background Field Servo

The Sr2 system can experience background magnetic field drifts at the level of 10 mG over a
couple of hours. This is due to the magnet from an ion pump near the main chamber (Figure 3.3),
which produces a field that changes as the laboratory temperature fluctuates.10

To remove thisdrift, we implement a background magnetic field servo [13].
Measurements for this servo take place with the spin polarization beam extinguished. The
polarization coil current and V2 (Figure 3.17) are also shut off.11 In this state the clock transition is
unpolarized and the only remaining field is the residual background field. For a typical backgroundof about 10 mG, the unpolarized clock transition lineshape is broader and has worse contrast thanin the zero-field case (Figure 2.14a).
The background field servo adds a field to cancel this background. The servo accomplishes
this by adjusting the currents in the xyz coils to maximize the unpolarized clock transition contrast,
which is biggest for zero background field. One iteration of this servo requires three measurements
per xyz coil pair (nine total). The first three of these are taken holding the y and z coil currents
fixed while the x coil current is switched between the values Ix , Ix + ∆I, and Ix − ∆I (one for eachmeasurement). Here Ix is the initial coil current and ∆I is a small current step.
For each of these three currents, the clock transition interrogates the center of the unpolarized
line, and the excited state fraction is recorded. For a well-canceled background field, the excitedstate fraction would be the highest when the x coil current is Ix , and it would diminish equallyfor Ix + ∆I and Ix − ∆I. If the background field drifted off its optimum, the currents Ix + ∆I
and Ix − ∆I would yield different excited state fractions. Treating the on-resonance excited state
fraction as a parabolic function of external field, a digital PI filter computes a new value for Ix thatbalances the excited state fractions measured for currents Ix + ∆I and Ix − ∆I (where ∆I, which
10
Temperature stabilization of the space around the vacuum chamber, which will be discussed in Section 5.8.8,
helps with this problem.
11
V1 from Figure 3.17 is shut off after 689 nm MOT cooling, so it is off at this point as well.

109
is determined empirically to optimize the servo, remains fixed throughout servo operation). The x
coil is then held while this same procedure is performed on the y and then z coils.
These servo measurements cannot occur while the stretched-states lock is running; therefore,
normal clock operation involves switching between 81 cycles of the stretched-states lock and ninemeasurements for the background field servo.12

The background field servo adjusts the coilcurrents using a DAC card. Each of the three outputs of the card are summed into a current
stabilizer setpoint voltage for one of the xyz coils (Figure 3.17). After a few iterations of this servo,
the background field remains well canceled while this servo is engaged.
In a typical day, before the background field servo is initiated for the first time, we find xyz
coil servo voltages that roughly cancel the background field,13 allowing for the initial state of theservo to be nearly stabilized. However, to test the servo, we intentionally offset the xyz coils fromtheir optima and observe the servo effectively correcting the coil currents to cancel the backgroundfield (Figure 5.3).

5.2.3

Shift Analysis

With the background field well canceled, the residual first-order Zeeman shift ∆νZ1 must
be measured and assigned an uncertainty. This shift can be inferred with stretched-states servo
data. As explained in Section 5.2.1, for a time varying magnetic field, the first two stretched-states
servo points are ν1 = νSr − ζB(t1 ) and ν2 = νSr + ζB(t2 ). To analyze the shift, first I define astretched state frequency difference vector as {s1 , s2 , s3 , . . .} = {ν2 − ν1 , ν4 − ν3 , ν6 − ν5 , . . .}. Notethat s2 − s1 = ζ [B(t3 ) − B(t1 ) + B(t4 ) − B(t2 )].
Forming a new vector S = {s2 − s1 , s4 − s3 , s6 − s5 , . . .}, the mean of S is12
The background field servo operates with the scanning synthesizer at the frequency corresponding to the zerofield clock transition resonance. This frequency is known from the stretched-states servo. Just before pausing thestretched-states servo to allow for the background field servo measurements, the position of the zero-field resonancecenter is computed by averaging the most recently acquired stretched state center frequencies.
13This is done by setting the free running clock laser on the zero-field resonance and adjusting the coil currentsby hand (without feedback) to coarsely maximize the excited state fraction.

110

Excited State Fraction

0.5

0.4

0.3

0.2

0.1

0
1 3 5 7 9 11

−20

Servo Iteration

−10

0

10

20

Detuning (Hz)Figure 5.3: The zero-field-resonance excited state fraction as a function of servo iteration anddetuning. To test the servo, the xyz coil currents are intentionally offset from their optima. This
results in the broadened line of the right plot (red open circles). The servo is then turned on, andthe excited state fraction gets larger as the servo runs (left plot). After 11 iterations, a scan ofthe zero-field clock transition shows the familiar Rabi lineshape (right plot, blue circles), indicatingthat the 10 mF states are degenerate and the background field is well canceled.

hSi = 2ζhB(t + 4Tc ) − B(t)i(5.8)≥ 4ζhB(t + 2Tc ) − B(t)i.

(5.9)

where this second expression is equal to hSi for drift linear in time and less than hSi for higher-order
drift. Using Equation 5.7,1
∆νZ1 ≤ hSi.
8

(5.10)

This inequality puts an upper bound on ∆νZ1 . Since ∆νZ1 is usually consistent with zero, an upperbound is sufficient. Applying this analysis to the full record of stretched-states servo data, ∆νZ1 isless than (−1.6 ± 2.0) × 10−19 [88]. In practice, this value is quoted for ∆νZ1 .

111

5.3

Second-Order Zeeman Shift

5.3.1

Introduction

As explained in Section 2.5.3, using a bias field for clock operation introduces a second-order
Zeeman shift that cannot be removed with the stretched-states servo. This effect is predicted toshift the stretched states equally by (−0.233 Hz G−2 )B 2 , where B is the bias field magnitude [16].
To measure this shift, we use a lock-in measurement with B as the modulation parameter. Like inEquations 5.1 and 5.2, the shift of the clock transition for normal clock operation is h∆νlock−in i/L.
The lever arm for the measurement is L = (B22 − B12 )/B02 , where B1 and B2 are the lock-in states
and B0 is the field during normal clock operation.
It is convenient to measure this shift as a function of the frequency difference between thestretched states (which is proportional to the bias field magnitude) for normal clock operation [121].
As explained in Section 2.5.1, the first-order Zeeman shift per unit field is −108.4mF Hz G−1 , which
implies a stretched state frequency difference of 975.6 Hz G−1 . Using this number, the predictedvalue of the second-order Zeeman shift in units of fractional frequency per kHz of stretched state
difference is −5.71 × 10−18 kHz−2 . Working in these units, we choose a clock operation bias fieldcorresponding to a stretched state difference of S0 = 300 Hz. Therefore, the lever arm for themeasurement can also be written as L = (S22 − S12 )/S02 , where S1 and S2 are the stretched state
differences corresponding to B1 and B2 .
To measure the second-order Zeeman shift due to modulating the field, the lock-in data must
be free of the first-order Zeeman effect. This is accomplished using four atomic servos (rather than
two servos as discussed Section 5.1.2). For both lock-in states of the bias field, there is a stretchedstates lock (comprised of two atomic servos) generating a stretched state average frequency (Section
2.5.3) that is not affected by the first-order Zeeman shift. The difference between the two stretchedstate average frequencies after one cycle through the four atomic servos is one lock-in data point.
Also, since a drifting background field would introduce unwanted variation in the second-order
Zeeman shift measurement, the background field servo is enabled during this evaluation.

112
0.005

Fit Amplitude

0.004
0.003
0.002
0.001
0.000

-0.10

-0.05

0.00
0.05
z field (G)0.10

Figure 5.4: The data used to align the z coil field. The σ transition is scanned for different valuesof the z coil field. Measured lineshapes are fit, and the fit amplitudes are plotted here. These data
are then fit to a parabola, and the parabola’s offset is the desired field value.

5.3.2

The Lattice Tensor ShiftThis measurement could be negatively affected by the lattice tensor shift if care is not taken.
As shown in Equation 2.19, the lattice tensor shift ∆νt is∆νt = 36 κt (3|ǫ̂ · B̂|2 − 1)Ulat (r, z).

(5.11)

Due to the dependence on B̂, if the bias field is modulated in a manner that changes the field
direction, the measurement will be contaminated with a lattice tensor shift that differs between
the two lock-in states. Since the overall bias field is comprised of fields from the polarization andxyz coils, lattice tensor shift contamination can be prevented by adjusting all four coil currents sothat the bias field is well aligned with the lattice polarization ǫ̂ for both field states of the lock-in
measurement. As explained in Section 3.5.1, B̂ and ǫ̂ are aligned by first aligning the clock laser
polarization with ǫ̂ and then studying mF -changing σ transitions (Figure 5.4). σ transitions become
possible when B̂ is not aligned with the clock laser polarization.
For each state of the lock-in measurement, the polarization coil provides the bulk of the field

113
magnitude, and the xyz coils merely help to point the bias field in the correct direction.14

Twopolarization coil currents are selected as lock-in parameter states, and then σ transition scans areused to adjust the xyz coils such that the bias field is aligned with the clock laser polarization. This
procedure is repeated a few times to ensure that the bias field is well aligned.

5.3.3

First and Second Evaluations

The first evaluation of the second-order Zeeman shift is performed with lock-in parameters
S1 and S2 ranging from 300 Hz to 1.2 kHz [13]. This evaluation measures a second-order Zeeman
shift of (−5.78 ± 0.05) × 10−16 kHz−2 . The second evaluation uses S1 = 291 Hz and S2 = 1.66 kHz,measuring a second-order Zeeman shift of (−5.82 ± 0.07) × 10−16 kHz−2 . Extrapolating to theoperating stretched state difference of S0 = 300 Hz, the Sr2 second-order Zeeman shift implied by
the second evaluation is (−5.23 ± 0.07) × 10−17 (Figure 5.5a).
The shift per stretched state frequency difference squared is an atomic property, so evaluationsof this by different research teams can be treated as independent measurements of the same quantity(Figure 5.5b). Performing a weighted average of several reported measurements [74, 13, 88, 121,35], the shift and its root-reduced-chi-square-inflated uncertainty is (−5.75 ± 0.03) × 10−16 kHz−2 .
Extrapolating to the operating S0 , the final Sr2 second-order Zeeman shift is −5.17 × 10−17 withan uncertainty of 2.5 × 10−19 [88].

5.4Probe Ac Stark Shift

5.4.1

Introduction

As explained in Section 5.1.3, the probe Stark shift ∆νprobe is proportional to P , the clock
laser power. This shift is measured using the lock-in technique. For state 1 (2) of the lock-in, astretched-states lock uses clock laser π-pulses with a long (short) interrogation time, which meansthat a smaller (larger) Rabi frequency, and therefore less (more) laser power is needed to make a
Recall that the polarization coils point at an angle 16◦ with respect to the horizontal and the clock laser
polarization is nearly horizontal
14

114

a)
Allan Deviation

10 -17

10 -18

10 -19
10 1

10 2

10 3

Averaging Time (s)

b)

×10

-16

Shift (kHz -2)

-5.5

-6
Confidence Interval
Sr1 2008
SYRTE 2011
PTB 2011
Sr2 2013
Sr2 2014

-6.5

-7
1

2

3

4

5

Point Number
Figure 5.5: a) Overlapping Allan deviation of the second-order Zeeman shift extrapolated to theoperating value of S0 . This is the second Sr2 evaluation of this shift. b) Measurements of thefractional second-order Zeeman shift per stretched state frequency difference squared. These values
have been reported by four strontium clocks: Sr1 [74], Sr2 [13, 88], the Sr clock at LNE-SYRTE(Paris) [121], and the PTB (Braunschweig, Germany) Sr clock [35]. The uncertainty of the weighted
average is depicted as the gray band.

115
π-pulse. The difference between the two stretched state average frequencies is the lock-in data.
Two stretched-states servos are used in the lock-in measurement (rather than only two atomic
servos) because the background field servo is run during this evaluation (Section 5.2.2). The
background field needs to be canceled because the probe Stark shift has a tensor component thatwill vary with a drifting magnetic field. For the lock-in parameter values P1 and P2 , and an
operating laser power P0 , the lever arm for this measurement is L = (P2 − P1 )/P0 . It is easier toexpress L in terms of the pulse duration used for each of these laser powers, t0 , t1 , and t2 . Since
π-pulses are used, the Rabi frequency for the measurement Ω = π/t. Also, Ω ∝ Ec0 , where Ec0 is√
the clock laser electric field amplitude (Section 2.4.1). Since Ec0 ∝ P , ∆νprobe ∝ 1/t2 . Therefore, 2  2
the lever arm is L = tt20 − tt10 .
5.4.2

First and Second Evaluations

The first evaluation of the probe Stark shift uses t2 = 50 ms and t1 = 200 ms [13]. In thiscase, the pulse duration for clock operation is t0 = 160 ms, resulting in a lever arm of L = 9.6.
Modulating the clock laser interrogation time by this amount requires that the laser power be
modulated by a factor of (t1 /t2 )2 = 16 to maintain π-pulses. This is accomplished using the clock
laser intensity servo (Section 3.7.1). Dividing the mean and standard deviation of the mean of thelock-in data by L, the final probe Stark shift for the first evaluation is (0.5 ± 1.3) × 10−18 [13].
This treatment of the measurement lever arm assumes that both pulses perfectly achieve apulse area of π. The uncertainty in this assumption is assigned by studying the range of clock
laser powers for which the measured lineshape still resembles a π-pulse. For power slightly larger
or smaller than that corresponding to a π-pulse, the peak excited state fraction will be diminished.
Studying the change in clock laser power required to diminish the excited state fraction, treating
this range as uncertainty, and propagating it into the final processed signal, we calculate a negligible
correction for this effect.
For the second evaluation, we would like the probe Stark uncertainty to be characterizedat the mid-10−19 level or better. This is because our measurement goal for BBR dynamic shift

116

Allan Deviationa)

10 -17

10 -18

10 1

10 2

10 3

Averaging Time (s)b)

4×10 -17

Fractional shift

3
2
1
0
-1
-2
-3
0

10

20

30

40

50

60

Point number

Figure 5.6: a) Overlapping Allan deviation of the first probe Stark shift evaluation. This is themeasurement of the shift extrapolated to normal clock operation, meaning that it is the lock-in
data divided by L. b) Binned lock-in data. Each marker represents a mean of 25 points, and theerror bars are the standard deviation of this mean. The reduced chi-square χ2red is 0.92.

Allan Deviation

117

10 -1910 -20

10 1

10 2

10 3

Averaging Time (s)Figure 5.7: Overlapping Allan deviation of the second probe Stark shift evaluation. Here themeasured shift extrapolated to clock operation conditions reaches the low-10−20 -level in less thanan hour of measurement time.

coefficient (discussed in Section 5.9) would yield low 10−18 level uncertainty, and we want to reduce
as many other uncertainties as possible to negligible levels. To reduce the probe Stark uncertainty,
it is advantageous to improve the measurement lever arm. This is accomplished with a neutral
density filter attached to a stepper motor to move this filter in and out of the clock laser. This
allows the clock laser intensity to be modulated by a factor of 83.5.
t1 is fixed at 180 ms, and the clock laser intensity servo is adjusted for a π-pulse when the
motorized neutral density filter is removed from the beam. For the same intensity servo value, t2 isadjusted for a π-pulse when the filter is in the beam, resulting in t2 = 19.7 ms. If t0 = 160 ms as in
the first evaluation, we would have already achieved a lever arm of 65.2. However, to bolster the Sr2stability for our second evaluation [88], clock operation with 1 s pulses has now been demonstrated(Section 4.6). With t0 = 1 s as the new pulse duration for normal clock operation, L reaches a
staggering value of 2546. With this amazing lever arm, the second probe Stark evaluation measuresa shift of (−3.2 ± 1.7) × 10−20 .

118

5.5

Density Shift

5.5.1

p-Wave Density Shift

Although most major systematic shifts of the Sr2 clock transition come from electromagnetic
perturbations, there is one important exception: the density shift [66]. This is a frequency shift
due to interactions between particles in multiply occupied lattice sites. Since the 87 Sr lattice clock
is based on spin-polarized ultracold fermions, the s-wave part of these interactions is suppressed
unless there is a significant amount of excitation inhomogeneity [19, 99].
A substantial density shift was observed in the Sr1 clock, and early studies of this effectattributed the shift to inhomogeneous s-wave interactions [19]. Later, a study by the NIST latticeclock team identified p-wave interactions as the dominant partial wave interaction in ytterbium[64]. Further measurements in Sr1 revealed a p-wave contribution to the strontium density shift[7], meaning that s- and p-wave interactions most likely played a role in density shift measurements[109, 6]. More recent work in Sr1 showed improved excitation homogeneity, which suppressed swave interactions and left p-wave effects as the dominant density shift mechanism [78].
The p-wave density shift in the mean field approximation is given as [71]

∆νp =

π
n(~r)ET [vee ρee − vgg ρgg + (ρgg − ρee )veg ] ,h(5.12)

where n(~r) is the particle density and ET is the mean collision energy, which is proportional tokB T . ρgg and ρee are the ground and excited state population fractions, and vgg , veg , and vee are
the p-wave scattering volumes.
The density in a given lattice site is [110]



y2
z2
Nsite
x2
n(~r) = 3/2
exp − 2 − 2 − 2 ,Lx LyLz
π Lx Ly Lz
where Nsite is the number of particles in the site. The lengths Lx,y,z are given by(5.13)

119

Li =

r

~ p
2hni i + 1.
mωi

(5.14)

Here ωx,y,z is the trap frequency in the {x, y, z} direction (Equations 2.7 and 2.8) and the occupancy
factor hni i is

hni i =
q

1


.
~ωi
exp kB T − 1(5.15)

2kB T /mωi2 , which is the length scale of a thermal gas in apharmonic trap. For kB T ≪ ~ωi , Li becomes the harmonic oscillator length ~/mωi . Since thepradial trap frequency νr = 120 Hz, Lx and Ly are both equal to 2kB T /mωr2 to a very good

In the limit kB T ≫ ~ωi , Li →approximation (for typical lattice temperatures of a few µK).

It is interesting to look at how the density shift scales with the experimental parameters N ,T , and U0 . If one uses sideband cooling along the lattice axis, whereby the atoms are cooled to thepmotional ground state in this direction by driving a blue sideband transition, then Lz → ~/mωz ,

and

5/4

∆νdensity ∝ N U0 .
This expression uses the fact that ωr and ωz are proportional to(5.16)√

U0 (Equations 2.7 and 2.8).

The measured density shift ∆νdensity results from spatially averaging ∆νp . In this case, the p-wave
density shift is temperature independent.
For kB T ≫ ~ωz ,3/2∆νdensity ∝

N U0
√
.
T

(5.17)In Sr2, kB T is comparable to ~ωz ,15 , and treating the density as thermal is an approximationthat is good to 3%. Also, for consistency with Equations 2.7 and 2.8, this analysis has ignored
15
The biggest value of νz usually observed in Sr2 is 120 kHz. At this depth, the sample temperature is usually
7 µK.

120
the influence of gravity. Gravity can be included by replacing U0 in these scaling laws with Uatom(Section 2.4.5).

5.5.2

Density Shift in a Cavity Lattice

When operating a lattice clock, there is a tension between the density shift and QPN. A
small atom number is good for reducing the density shift, yet a large N is preferred for minimizing
QPN. Therefore, it is best to operate with N just large enough to make QPN a few times lower
than the Dick Effect. For the 160 ms clock pulses used in the first Sr2 evaluation, the Dick Effect√for typical experimental parameters is 3.6 × 10−16 / τ . Therefore, N = 2000 is a good choice since√
it results in QPN of 1.2 × 10−16 / τ . For the 1 s pulses used in the second evaluation, the Dick√√
Effect is predicted to be 1.8 × 10−16 / τ , and QPN for N = 2000 is 2.1 × 10−17 / τ . Although theclock operation atom number could stand to be reduced by an order of magnitude for 1 s pulses,
the Sr2 system is optimized for N = 2000,16 so it is convenient to use this value is used for both
160 ms and 1 s pulses. As I will show in Section 5.5.4, the density shift is manageable at N = 2000.
Although the density shift is large in Sr1 [74], it is significantly smaller in Sr2 due to thelarge-volume cavity lattice (Section 2.3.3). For both systems, the maximum lattice intensity islimited by the power available from the 813 nm light source. However, because of the intensity
enhancement from the lattice cavity (Equation 2.14), the Sr2 system can achieve a good lattice
depth for a trap beam waist of 160 µm (compared to the Sr1 waist of 40 µm). At this waist, Sr2
achieves a maximum axial trap frequency of νz = 120 kHz (corresponding to Uatom = 300 Erec or
50 µK), whereas Sr1 measures νz = 80 kHz (with Uatom = 130 Erec or 22 µK).
The two advantages of the cavity lattice that improve clock stability (Section 4.6) also improvethe density shift. First, the large volume means that Sr2 can operate with a smaller density at
N = 2000 than Sr1. For the same atom number, the Sr2 density is estimated to be 30 times lessthan that of Sr1 using a Poissonian lattice filling model [110].17 Second, the good spatial overlap
16
In particular, for the measured Sr2 detection noise of δN = 11.4 atoms, stability degradation due to detection
noise is worse than it is for QPN when N < 2 δN 2 = 260 (Section A.4).
17
This model assumes that the lattice overlaps a 1/e 689 nm MOT cloud width of 160 µm, compared to 30 µm

121with the 689 nm MOT means that Sr2 can trap an excess of atoms, which is useful for making
density shift measurements with a large lever arm.

5.5.3

Large Lever Arm Measurement

Due to the ability to trap many more atoms than are needed, N is a good choice for a lock-in
modulation parameter. Although Equation 5.17 suggests that Uatom might also be a good choice,it would result in a worse lever arm than if N were used. With a maximum Uatom of 300 Erec anda minimum reliable Uatom of 71 Erec , the best achievable lever arm is 8.4. Meanwhile, the lattice
can trap N = 1 × 105 . Extrapolating to an operating atom number of N = 2000, a lever arm of 50
is possible.
However, at some level, the density shift for Rabi spectroscopy becomes nonlinear in N [98].
To check the linearity of the density shift, this effect is measured as a function of ∆N , which is thedifference in lock-in parameters [89]. For this measurement, we use the lock-in scheme describedin Section 5.1.2, where two atomic servos (one for each value of the lock-in parameter) are locked
to the same stretched state. Because the atom number fluctuates, the total PMT counts (Section
3.7.2), which are proportional to the atom number, are recorded for every excited state fraction
measurement. Using the set of measurements of N , {N1 , N2 , N3 , . . .}, where Nk is controlled to belarge when k is even and small when k is odd, the lever arm for this measurement is treated as avector L = {(N2 − N1 )/N0 , (N4 − N3 )/N0 , . . .} (with N0 = 2000). Density shift data is processedby dividing each lock-in point by the instantaneous lever arm, {(ν2 − ν1 )N0 /(N2 − N1 ), (ν4 −ν3 )N0 /(N4 − N3 ), . . .}. Here νk is the center frequency taken when the atom number is Nk .18
We study the density shift as a function of ∆N = Nhigh − Nlow , which is the difference
between the large and small atom numbers used for each lock-in point. A plot of the difference
between νhigh − νlow (the frequencies corresponding to Nhigh and Nlow ) divided by ∆N should
appear constant as a function of ∆N if the density shift is linear, and this is the observed behavior
for the manner in which the Sr1 lattice is aligned.
18
Control of N is based on changing the 461 nm MOT laser intensity using the MOT light AOM (Section 3.2.3).
Fluctuations in N of 15% about its average value are typical, and N can also drift downward when one of the Sr2
cooling laser systems needs to be reoptimized.

122
×10 -19
Fractional Shift
at 2000 Atoms

5
Fractional Shift per Atom

4
3
2

10-16

10

-17

10-18
10

1

0

10

1

10

2

10

3

Averaging Time (s)

0
-1
-2
-3
1

2

3

4

5
×10 4

∆N

Figure 5.8: The measured Sr2 density shift as a function of ∆N (taken from Reference [89]). Eachpoint on this plot represents an average over a bin of 30 measurements of (νhigh − νlow )/(Nhigh −Nlow ). Our statistics show that these data extrapolated to N = 2000 do not support nonlinear terms
within an uncertainty of 8.2 × 10−19 . This measurement was performed with Uatom = 210 Erec , so
if the data is extrapolated to the clock operation Uatom of Uclock = 71 Erec , a nonlinear shift is notsupported within a 1.6 × 10−19 uncertainty. Inset: A single 2000 s long density shift measurementwith ∆N = 4.1 × 104 (average value). The shift per atom was measured and then scaled up to theclock operation atom number of N0 = 2000.

−16

x 10
2

Lock−in data

0
−2
−4
−6
−8
0

1

2

∆N

3

4

5
4

x 10

Figure 5.9: The linear trend is more apparent for the lock-in data. Here the data is binned morecoarsely, using bin sizes of 600, for clarity. The blue line is a linear fit that assumes the shift goesto zero for ∆N = 0.

123

−18

10

x 10

Residuals

5

0

−5
120

160

200

240

Trap Depth (E

280

)

rec

Figure 5.10: The density shift at N = 2000 studied as a function of Uatom . The residuals of
3/2
a fit to the expected Uatom power law (Equation 5.17) are plotted here to make the statisticalbehavior more apparent. The light blue band is the ±1σ confidence interval. The fit coefficient−3/2
is (−3.9 ± 0.3) × 10−21 Erec . The confidence interval has been inflated by the square root ofχ2red = 2.0.

(Figure 5.8). An F -test19 shows that measurement does not require a quadratic model, arguingfor a linear model within the measurement uncertainty of 1.6 × 10−19 . For large atom number
modulation, the shift averages down to 1 × 10−18 uncertainty in about 20 minutes (Figure 5.8) [89].
The data in Figure 5.8 is overscattered, resulting in a reduced chi-square statistic of χ2red =1.7. Although χ2red is typically very close to unity for most of the Sr2 systematic evaluations, it
is consistently around 2 for density shift measurements. An extra noise source could be present
that has not been identified, such as variations in optical trapping conditions. To ensure thaterror bars are sufficiently inflated to account for possible unidentified sources of noise, all final
uncertainties, including the 1.6 × 10−19 uncertainty from the data in Figure 5.8, are multiplied byqχ2red if χ2red > 1 [4].
We also study the density shift as a function of Uatom . For this measurement, we prefer to

use the Ti:sapphire lattice laser instead of the TA system (Section 3.4), which was employed in the
19With 305 degrees of freedom, the F -test results in a test statistic F = 3.36, corresponding to the probability
0.068.

124probe Stark and second-order Zeeman shift evaluations. This is because any study of clock behavior
determined by the lattice should use the light source that is part of normal clock operation. Uatom isdetermined using sideband spectroscopy to measure the trap frequency, which implies the potential
3/2

depth accord to Equation 2.39. The data fits to a Uatom model (Figure 5.10) with χ2red = 2.0.
5.5.4

Systematic Evaluation

Rather than use the results of the study of density shift linearity to assign a clock operation
value to this effect, we opted to re-evaluate the density shift (Figure 5.11). This is because shortly
after the linearity studies, changes to the Sr2 apparatus required the system to be re-optimized,
potentially changing experimental parameters that could affect the density shift. For the systematic
measurement, Uatom = 119 Erec , and the lock-in modulates between 2400 and 12000 atoms (averagevalues), leading to an average lever arm of 4.8. Extrapolating this to clock operation conditions
quoted in the first evaluation of N = 2000 and Uatom = 87 Erec , the final shift is (−4.7±0.6)×10−18[13].
This measurement was taken at values of Uatom lower than the best we could achieve (meaningthat the maximum atom number, and therefore the lever arm, was also lower). Despite that thelever arm is low for this measurement, it ultimately proved to be a good determination of the Sr2
density shift due to the length of the data record. This measurement is also good enough for thesecond evaluation, which quoted new clock operation conditions of N = 2000 and Uatom = 71 Erec ,implying a final shift of (−3.5 ± 0.4) × 10−18 [88].

5.6

Lattice Stark Shift

5.6.1

Lever Arm and Nonlinear Light Shifts

Using Equation 2.19, the lattice ac Stark shift for clock operation ishi∆νac = − κs (λ) + 36 κt (λ)(3|ǫ̂ · B̂|2 − 1) Uatom ,(5.18)

Allan Deviation125

10 -17

10 -18

10 -19
10 0

10 1

10 2

10 3

10 4

Averaging Time (s)Figure 5.11: The density shift extrapolated to operating conditions of N = 2000 and Uatom =71 Erec .

where the vector term has been ignored (Section 2.6). As explained in Section 2.6, the magic
wavelength for the optical lattice clock is the wavelength λ at which the scalar and tensor termsof ∆νac cancel one another for the ǫ̂ · B̂ = 1 case. Sr1 measured this wavelength, but at the timemeasurement precision was not high enough to resolve the tensor shift [74]. The Sr team at LNESYRTE achieved the precision needed to observe tensor shift effects, but they did not determine
the scalar-tensor cancellation wavelength, reporting instead the wavelength at which κs (λ) = 0[121]. Recently the Sr group at PTB measured this cancellation wavelength, obtaining a value ofc/368.554465(3) THz [33].
The two Sr2 lattice Stark shift evaluations are both performed with our Ti:saph lattice laser.
The laser wavelength is stabilized to a frequency comb that is referenced to the NIST hydrogenmaser array (Section 3.4.3), and the cavity lattice intensity is modulated for lock-in detection.
Modulation is performed with the lattice cavity transmission intensity stabilizer (Section 3.4.2),
which actuates on the drive amplitude of the Ti:saph double-passed AOM (Figure 3.15). Uatom is
modulated by about 180 Erec , resulting in a small measurement lever arm of about 2.
The calculation of this lever arm assumes that the shift is linear in lattice intensity. However,126
lattice hyperpolarizability, which is proportional to the square of the lattice intensity, has been
considered since the first proposals of the Sr lattice clock [58]. Also, a frequency shift from thelattice laser coupling to M1-E2 transitions, which goes as the square root of the lattice intensity,
has been discussed [111]. Attempts have been made to measure the size of these effects [121]. As
I will discuss in Sections 5.6.3 and 5.6.4, we have very carefully considered the treatment of theseshifts in the Sr2 systematic evaluations.

5.6.2

Technical Considerations

To perform a robust and rigorous evaluation of the lattice Stark shift, other systematic shiftsthat could couple to this measurement must be controlled. As with the probe Stark shift, thebackground field servo is used to prevent the tensor term (Equation 5.18) in the lattice shift fromdrifting. Also, we align the bias field to the lattice polarization to ensure that the tensor term is
first-order insensitive to any residual magnetic field drift that may exist.
Another potential problem is an uncontrolled density shift, which can mimic a lattice Stark
3/2

effect due to its Uatom dependence (Equation 5.17). Therefore, as the trap intensity is modulated for
3/2

lattice Stark measurements, we also modulate the atom number to ensure that N Uatom is the samefor both lock-in states. In this case, the density shift is common mode to both states, and therefore
3/2

it cancels in the lock-in data. To thoroughly enforce the constancy of N Uatom in the presence of a
3/2

fluctuating N , lock-in points for which the magnitude of N1 U1
3/2

in post processing until the data run average of N1 U1
precision.20

3/2− N2 U2

3/2− N2 U2is largest are removed

is well below our measurement
3/2

We calculate the maximum acceptable value of the magnitude of N1 U1

3/2− N2 U2

from the shift coefficients obtained from studying the density shift as a function of N and Uatom .
3/2

We then enforce that the data run average of N1 U1

3/2− N2 U2

is more than 10 times smaller than

this maximum value.21
20

For one lock-in point ν2 − ν1 , N1 (N2 ) and U1 (U2 ) are the atom number and observed Uatom for which ν1 (ν2 )is measured.
21
Great care must always be taken when removing data points. It is important to stress that no data pointis removed based on its frequency value; rather, measurements of the atom number and trap depth are used to
3/2
3/2
compute N1 U1 − N2 U2 , and the value of this difference determines whether frequency points should be removed.
Additionally, the cuts are performed in an automated fashion that is blind to the frequency values (looking only at127
Lastly, as explained in Section 3.4.2, a Ti:saph laser is needed for lattice Stark measurements.
This is because the amplified spontaneous emission from the TA-based lattice laser system can cause
severe frequency shifts. To get a rough idea of the magnitude of the TA amplified spontaneousemission shift, we measure the change in the clock transition frequency caused by putting a narrow
optical bandpass filter (centered on the magic wavelength) in the TA-based lattice laser, observing
a 10−15 -level effect. Furthermore, comparisons between Sr1 and the Yb lattice clock at NIST
suffered from 10−15 -level drift that has been attributed to the TA lattice laser Sr1 used at the time.
This implies that the Sr1 amplified spontaneous emission spectrum was time varying. In othermeasurements that used both an optical filter and a prestabilization cavity to remove amplified
spontaneous emission, Sr1 still observed significant time varying frequency shifts due to their TA
lattice spectrum, implying that filtration is not enough to remove this effect. The Sr group at
LNE-SYRTE observed similar issues with TA systems [114].
Fortunately this spectral drift will have no measurable effect on TA-based evaluations of theprobe Stark shift, second-order Zeeman shift, and other shifts that are not caused by the lattice.
Since a lock-in measurement is the difference between two data points taken within a few secondsof each other, the drift rate of the amplified spontaneous emission shift is slow enough that it iscommon mode between these two points. However, shifts caused by the lattice are measured overdays or weeks, and the TA spectrum drift is significant on these timescales. A spectrally pureTi:saph laser does not suffer from these problems.

5.6.3

First and Second Evaluations

The Ti:saph frequency of the first lattice Stark shift evaluation [13] is 368.5545965(3) THz.
If U1 and U2 represent the lock-in parameter values of Uatom , ∆U = U2 − U1 is varied from 34 to
184 Erec (Figure 5.12a). For each value of ∆U , lock-in data is averaged to 1 × 10−17 statisticaluncertainty. We also measured 87 Erec to be the lowest value of Uatom for which N = 2000 could
be reliably obtained, so 87 Erec is a good choice for Uclock , the clock operation value of Uatom .22
3/2

N1 U1

3/2

− N2 U2 ).

128

0.0

Differential Shift (10

−15

)

a)

−0.5

Evaluation 1
Evaluation 2

−1.0
0

20

40

60

80 100 120 140 160 180 200∆U (Erec )

−17

b) 4

x 10

272 E

rec

Differential Shift288 Erec
295 E

rec

2

306 E

rec

317 Erec

0

−2
40

60

80

100

∆U (E

120

140

160

)

rec

Figure 5.12: a) Lock-in data for the first evaluation (blue squares) and the second evaluation (redcircles). Data from the first evaluation and other data (not shown) is used to compute the latticefrequency for the second evaluation. For the second evaluation, the fact that the shift remains zerofor changes in the lattice intensity indicates that the magic wavelength has been identified. b) The
lock-in data for the second evaluation with the values of U2 specified (legend on right side).

To extrapolate this evaluation to the clock operation condition of Uclock = 87 Erec , we must
determine whether the data should be modeled as linear in Uatom or whether nonlinear terms like
hyperpolarizability should be included (Sections 5.1.4 and 5.6.1).
The LNE-SYRTE reports of the hyperpolarizability and M1-E2 coefficients do not specify
whether the density shift was under control during their measurements of these nonlinear effects(Section 5.6.1). Their determination of these coefficients involved modulating their lattice depth[119], and this could have been corrupted by an uncontrolled density shift. Also, measurements of
22

Using the analysis from Section 2.4.5, Uatom = 87 Erec corresponds to a one-way lattice power of 2.6 W.

129
nonlinear effects were performed by comparing two clocks with TA-based lattices [121]. Though theoutput of the TAs were filtered, one of the Sr systems used in these nonlinear shift measurements
was later reported to still suffer from the effects of spectral drift [114].
Therefore, we perform an F -test on our data to see if hyperpolarizability and M1-E2 shifts are
justified (Section 5.1.4). If hyperpolarizability were significant, the lock-in data ∆νlock−in would
need to be modeled as k1 (U2 − U1 ) + k2 (U22 − U12 ), where k1 and k2 are constants. The F -test
compares two fits of the extrapolated shift ∆νlock−inUclock /(U2 − U1 ): a constant fit (corresponding
to a Stark shift linear in Uatom ) and a fit to k1 + k2 x (linear shift plus hyperpolarizability), wherex = U1 + U2 . Comparing these two models, for 6 degrees of freedom the result is F = 0.44,
corresponding to a test probability of 0.53. This is a comfortable margin above the 0.01 to 0.05
range below which a new term is required.
√√If an M1-E2 shift were significant, ∆νlock−in should be fit to k1 (U2 − U1 ) + k2 ( U2 − U1 ).
A second F -test compares a constant fit of ∆νlock−in Uclock /(U2 − U1 ) to the fit k1 + k2 x, where√√x = 1/( U1 + U2 ). The result for 6 degrees of freedom is F = 0.40, resulting in the test probability
0.55. Therefore, the first evaluation does not support the addition of hyperpolarizability and M1-E2
terms.
Using reduced-chi-square-inflated least-squares fitting, the data from the first evaluation has−1 . For Ua shift per Erec of potential depth (at the atoms) of (−5.31±0.05)×10−18 Erecclock = 87 Erec ,the final value of the lattice Stark shift for the first evaluation is (−4.62 ± 0.04) × 10−16 .
The lattice Stark shift is known to also be linear in lattice frequency within tens of GHzof the magic wavelength [18]. Using this fact, data from the first evaluation and a handful ofmeasurements taken at different lattice frequencies imply that the magic wavelength is 368.554485
THz. Therefore, our second evaluation [88] of the lattice Stark shift is performed with a latticefrequency of 368.5544849(1) THz. For these data, the lattice Stark shift has largely vanished(Figure 5.12a and 5.12b). We also achieve a lower value of Uclock = 71 Erec by operating the Sroven reservoir at a hotter temperature of 600 ◦ C.23
23

Lowering the lattice depth means that theUsing the analysis from Section 2.4.5, this corresponds to a one-way lattice power of 2.3 W.

130trap will retain fewer atoms, so this Uclock is again determined as the lowest potential depth atwhich N = 2000 can be reliably obtained. Running the reservoir hotter increases the amount oftrapped atoms overall (for any lattice depth), implying that N = 2000 can be achieved for a lower
value of Uatom . We also find that the atom number is best when we load atoms into a deep lattice
and then ramp down the lattice intensity to the final value rather than load in a shallow lattice.
Performing an F -test on these data to look for hyperpolarizability and M1-E2 effects (inthe same manner as the first evaluation), the test yields F = 0.17 for hyperpolarizability and
F = 0.03 the M1-E2 shift for 22 degrees of freedom.24

These correspond to probabilities of 0.68

and 0.86, respectively. Therefore, we again model the data from the second evaluation as linear
in Uatom . Extrapolating the data to Uclock = 71 Erec , we obtain a final value of the lattice Stark
shift of (−1.3 ± 1.1) × 10−18 . The final shifts from the first and second evaluations imply a magic
wavelength of 368.5544845(3) THz, which is statistically consistent with the value used for thesecond evaluation. These evaluations also imply a lattice Stark shift per Uatom per detuning from−1 .
the magic wavelength of (−4.73 ± 0.02) × 10−17 GHz−1 Erec

5.6.4

Hyperpolarizability

Although we have chosen not to include previously reported values of the LNE-SYRTE−2 [114], the uncertainty in the correction at
hyperpolarizability coefficient (0.45 ± 0.10) µHz Erec

Uclock = 71 Erec would be only 1.2 × 10−18 . It is also possible to processes the data from thesecond evaluation (Figure 5.12) in a manner that is sensitive only to higher-order terms. In thepresence of hyperpolarizability, the extrapolated shift will have the form k1 + k2 (U1 + U2 ); therefore, differences between extrapolated shift data points that have different values of U1 + U2 will
depend only on hyperpolarizability (and other nonlinear terms). With this analysis, and treating
hyperpolarizability as the most significant nonlinear shift, we infer a hyperpolarizability coefficient−2 . Although this is consistent with the LNE-SYRTE value, it is also consistentof (0.3 ± 0.3) µHz Erec24
The degrees of freedom for this F -test is equal to the number of points minus 2. Here I quote 22 degrees of
freedom, yet in Figure 5.12, the two plots of the second evaluation data have 8 and 12 points. This is because theF -test is performed on unbinned data, whereas the points in Figures 5.12a and 5.12b are binned.

131with zero.
The magic wavelength of 368.554465(3) THz measured by the PTB Sr group [33] differsfrom our value by 19.5 MHz. One difference in their approach that can account for some of thisdiscrepancy is that they assume the LNE-SYRTE hyperpolarizability coefficient and subtract if off
of each of their lock-in measurements.

5.7

Dc Stark Shift

5.7.1

Introduction

A large systematic shift caused by dc electric fields was recently discovered by the LNESYRTE group [69]. Their in-vacuum lattice cavity mirror substrates had acquired patch charges
that caused clock transition shifts at the 10−13 level, and they were able to remove these charges
using UV treatment [69]. Motivated by this discovery, we looked for a dc Stark shift in Sr2, andan effect was indeed present [13]. The PTB group also measured a significant dc Stark shift thatthey attributed to an unwanted voltage across their MOT coils [33].
We measure the dc Stark shift using electrodes, with one placed above (below) our top(bottom) vacuum chamber viewports. Obtaining a value for the shift requires a measurement of
the clock transition frequency for each of three conditions: the top and bottom electrodes held atvoltages V and -V (respectively), the top and bottom electrodes held at voltages -V and V, andboth electrodes grounded. The chamber metal is grounded, and V is usually around 90 volts. Whenthe electrode field is applied, the clock transition frequency is

ν± = νSr −1∆α(0) |E~bg ± E~elec |2 ,2h(5.19)

where νSr is the unshifted clock transition frequency, ∆α(0) is the differential dc polarizability,
E~bg is the background electric field, and ±E~elec is the dc field from the electrodes. The ± sign
corresponds to the two directions of the electrode field. When the electrodes are grounded, theclock transition frequency is

132νgnd = νSr −12.
∆α(0) Ebg
2h(5.20)

These three frequency measurements can be combined to obtain the applied shift,νapp = −1
1
2= (ν+ + ν− ) − νgnd ,∆α(0) Eelec
2h2(5.21)(ν+ − ν− )21
2=∆α(0) Ebg.
2h
16 νapp(5.22)and the background shift,νbg = −
5.7.2

Electrodes and Lock-In Sequence

Two steel quadrant electrodes provide the dc electric field, one of which is mounted above
the top viewport and the other below the bottom viewport (Figure 5.13a). Both electrodes are
cut into four pieces, which can be hooked up in three different ways to create fields along threeorthogonal axes (Figures 5.13c and 5.13d). Also, a hole is drilled in the center of both electrodesfor MOT laser clearance. Finite element analysis of the chamber and electrodes shows an electricfield that is reasonably uniform over the ultracold sample (Figure 5.13b).25 The electrode voltage
is provided by two low-noise commercial power supplies, one for the positive voltage and one forthe negative. Reed relays allow us to choose which of these voltages are applied to which electrodeusing TTL logic.
The dc Stark shift sensitivity to the squared magnitude of this electric field is −7.2 ×
10−17 V−2 cm2 ; therefore, a small applied field of only a few V/cm2 (Figure 5.13b) will be sufficient to produce a large effect.26

To ensure that this measurement is stable and repeatable, it isimportant to use the background field servo, which prevents the tensor component of the dc Stark
shift from drifting. To facilitate the background field servo and the three clock transition mea25
High field uniformity is not a requirement for this measurement. All that is needed is for the field to have thesame magnitude when reversed and to be constant over the time it takes to measure one value of ν+ and one valueof ν− .
26
I calculated this sensitivity using the known value of the Sr clock transition’s dc polarizability [82]b)Electric Field Magnitude (V/cm)

133
10
8
6
4
2
0
-20

-10

0

10

20

Horizontal Position (cm)Figure 5.13: a) Electrodes are mounted above and below the vacuum chamber. They are separated
by 17.5 cm. b) Finite element model of the electrode field. The potential difference between
electrodes is 180 V. In this plot the field points in the vertical direction. Moving about the horizontal
plane, with the vertical position fixed at the center of the chamber, the field is reasonably uniform
over a 1 cm length scale. Along the vertical coordinate (not shown), the field is even more uniform.
c) The electrodes have holes for MOT laser clearance and are cut into quadrants. The quadrants
are colored cyan (pink) if they are held at a negative (positive) potential. If all four quadrantsof the top (bottom) electrode are shorted together, the electric field is vertical. d) Holding halvesof the electrodes at different potentials produces a horizontal field. The quadrants can also be
arranged to create a field in the other horizontal dimension.

134
×10

-15

0

DC Stark shift

-0.5
-1
-1.5
-2
-2.5

First Measurement
After UV and N 2

-3
-200

-100

0

100

200

Electrode Voltage (V)Figure 5.14: The first check for a dc Stark shift (red circles), and the shift after UV and nitrogengas treatment (blue squares). Both data sets are from measurements along the vertical direction.
The red data implies a background dc Stark shift of −1.3 × 10−16 . After removing the shift with
UV light and nitrogen, the shift is −1.6 × 10−18 .

surements needed to compute the background dc Stark shift (Equation 5.22), the lock-in sequence
uses four atomic servos. For two of these servos, the electrodes are grounded and a stretched state
lock is performed, yielding the zero-magnetic-field clock transition frequency that is needed for the
background field servo. For the other two atomic servos, there is one servo for each direction of theelectrode field, and they both operate on the mF = 9/2 stretched state. After one cycle through thefour servos, the three center frequencies measured for mF = 9/2 are used to infer the backgroundshift.27

The frequency measured for mF = −9/2 is used only for the background field servo anddoes not contribute to the dc Stark shift data.

5.7.3

Observation of Dc Stark Shift and Removal of Background Field

During the first check for a dc Stark shift in Sr2, we were surprised to find a large backgroundshift in the vertical direction. Fitting dc Stark shift data measured as a function of the potential
difference between the electrodes (Figure 5.14), a background shift of (−1.27 ± 0.05) × 10−16 is
27

These three frequencies are taken to be ν+ , ν− , and νgnd in Equation 5.22.

135
observed. This corresponds to a background field of (1.33 ± 0.03) V/cm.28
Simple checks for background fields caused by our MOT coils did not turn up anything;
therefore, we hypothesized that patch charges on the Sr2 vacuum viewports were causing the shift.
In attempt to remove this shift, we treated the vacuum viewports with radiation from a UV lamp[69]. The UV treatments did have a positive effect on the shift but progress was very slow, andwe projected from our measurements that it could take weeks or months to remove a −1.3 × 10−16
effect. After a couple of days of this procedure, the UV treatment was temporarily set aside sothat we could install temperature sensors in the vacuum chamber (discussed in Section 5.8.5). Thisinstallation required that the vacuum chamber be vented to high-purity nitrogen. After venting,the shift was orders of magnitude smaller.29
UV treatment followed by a nitrogen vent reduced the shift in the vertical direction to (−1.6±
1.0) × 10−18 (Figure 5.14). We also looked for a horizontally oriented background dc Stark shift
and obtained a value of (−1.9 ± 1.9) × 10−18 . The shift proved stable, so for the first evaluation we
combine the vertical and horizontal measurements for a final dc Stark shift of (−3.5 ± 2.1) × 10−18[13].
The horizontal shift was persistently below measurement resolution even when the verticalshift was quite large. If a patch charge on a Sr2 viewport were responsible for the shift, this patch
would have the field of an electric monopole [46]. For a monopolar field to cause a dc Stark shift in
only the vertical direction, it would need to be located directly above or below the atoms (dependingon the sign of the charge).30
28

Despite that the electrodes are separated by 17.5 cm, the fit tells us that the applied field behaves as if the plates
are separated by (41.4 ± 0.3) cm. We did not attempt to track down the cause of this discrepancy since it does notaffect our measurements.
29
Later, cleaning the top Sr2 viewport created a 10−15 -level dc Stark shift. We wiped the outer viewport surface
with material that should have removed static charge buildup, but the shift did not change. Therefore, we repeatedthe same procedure of UV treatment followed by venting the chamber to clean nitrogen. This removed the shift
again.
30
Reference [46] shows that laser light causes electron emission from anti-reflection coated viewports. The partsof the top and bottom viewports illuminated most brightly by the MOT lasers are directly above and below theatoms, so this is where positive patch charges resulting from electron emission would be located. Given the sign ofthe background field, the top viewport would need to be the bad actor. However, this explanation alone is insufficient
to explain the Sr2 dc Stark shift because it is unclear why electron emission would not be significant for the five otherviewports (all with the same coating) that MOT lasers are sent through.

In from
PMT136

DAC out
TTL
out

V

+
+

TTL in

Σ

10

Σ

10

In 1

-1

In 2
In 3

Multi- Out 1
plexer

-1

Out
2

+
+

Figure 5.15: The control loop that cancels the background shift. While the stretched-states servo
is running, two TTL lines from the computer set both multiplexer outputs to ground. In this case,the computer DAC card provides a voltage that, after amplification by a factor of 10, charges theelectrodes (shown here as a capacitor). The gain = 10 stages are composite amplifiers based onhigh-voltage op-amps controlled for noise reduction using low-noise, lower-voltage op-amps. The
DAC voltage is periodically adjusted to cancel the background dc Stark shift. Background shift
cancellation utilizes a fixed dc voltage source depicted as a circled V. To measure ν+ , the multiplexer
sets Out 1 = Vs and Out 2 = −Vs , where Vs is the voltage of the dc source (similarly, for ν− , Out
1 = −Vs and Out 2 = Vs ). The potential difference of ±20Vs across the electrodes is summed
with the amplified DAC voltage used to cancel the background Stark shift. If the background shift
is not well canceled by the DAC, ν+ − ν− will not be zero. Excited state fraction measurements
made by the Sr2 PMT are processed by the computer to create the error signal ν+ − ν− . A PI
filter (with gain constants optimized empirically) uses this error signal to compute the voltage thatthe DAC must apply to balance ν+ and ν− and cancel the background shift. After one of thesemeasurements, the multiplexer again grounds its outputs, and clock operation resumes with the
new DAC voltage.

5.7.4

Active Control of the Dc Stark Shift

The dc Stark shift was measured again for the second evaluation. This time around, thehorizontal shift remained unresolved, but the vertical shift seemed to be less stable. To remove
the (possibly time varying) vertical shift in a dynamic way, we use active control similar to thebackground field servo. This “electrode servo” is based on measurements using an applied electricfield (Figure 5.15). During clock operation, the stretched-states servo pauses after tens of cycles,
and the electrode servo takes two center frequency measurements. With the quadrant electrodes

137

a)

×10

-16

b)

5

-5
-10
-15
-20

-16

5
0

Fractional Shift

Fractional Shift

0

×10

ν -ν
+

ν -ν
-

gnd

-5
-10
-15
-20

gnd

ν -ν
+

gnd

ν -ν

gnd

-

-25

-25
20

40

60

80

100

Point Number

120

20

40

60

80

100

120

Point Number

Figure 5.16: Dc Stark shift frequency data. A background shift means that ν+ − νgnd and ν− − νgnd
will be different. a) Frequency data before the first UV treatment and N2 vent were performed. It
is only a coincidence that the background field nearly cancels the applied field in the + direction.
b) Frequency data with the shift under active control.

configured for a vertical field, the servo measures ν+ (ν− ), which is the center frequency when
the field is pointing up (down). The difference between these center frequencies is ν+ − ν− =2 ∆α(0) Ebg Eelec , which is proportional to the background field. This frequency difference is treatedas an error signal that uses a digital proportional-integral (PI) filter to compute the voltage at
which these two frequencies are equal.31

The electrodes are then held at this voltage, and thestretched-states servo resumes.
A key feature of this servo is the fact that the voltage that cancels the background field
and the voltage that creates the applied shift (used to measure ν+ − ν− ) come from independentsources (Figure 5.15). The DAC voltage that cancels the background field is always on, whereas
the voltage that creates the applied shift is fixed and is only used when ν+ − ν− is being measured.
This means that ν+ − ν− measures the residual field that is not canceled by the DAC voltage. In
principle, this technique can be scaled up to servo a background field that has components in all
three dimensions. There would need to be a DAC voltage and composite amplifier (Figure 5.15)
for each quadrant of both electrodes.32
31

The fact that the error signal ν+ − ν− is also proportional to the magnitude of the applied electrode field Eelec is
not an issue for this application. Eelec is fixed for this measurement, so it just acts like additional proportional gain.
32
We had these electronics ready, but in the end only vertical shift control proved important.

138

−16

x 10

Background Shift
Processed Pointwise

0

−2

−4

−6

−8
20

40

60

80

100

120

Point Number
Figure 5.17: Data from Figure 5.16b processed point by point using Equation 5.22. Processing
data in this way biases the mean toward negative values.

With the electrode servo operating, we observe a clear effect (Figure 5.16b). It is worth
noting that a well-canceled shift means that the dc Stark effect can be evaluated rapidly. To see
this, assume that each center frequency in Equation 5.22 has the same statistical uncertainty σ.
Propagating σ to νbg , the (squared) uncertainty in the background shift is

2
=
δνbg

1 νbg
2 νapp



1+3

νbg
νapp



σ2 .

(5.23)In this expression, νbg is treated as the background field plus the cancellation field from the servo.
Therefore, with the background field well canceled, the dc Stark shift averages down very rapidly.
About 20 minutes of averaging results in a total shift of (−0.1 ± 1.1) × 10−19 [88].
Care must be taken when processing measurements of ν+ , ν− , and νgnd to obtain values of νbg .
Since the applied electric fields for these measurements are small, νbg computed for each point (using
Equation 5.22) has unusual noise properties. The fact that the data scatters near or above zero
causes erratic fluctuations when it is processed point by point, making the usual statistical analysis
impossible (Figure 5.17). Simulating this measurement, the same fluctuations toward negative
numbers are reproduced when the simulated data is processed as in Figure 5.17, resulting in a data

139
mean that is much more negative than the simulation’s true value. However, the simulation does
show that the true value can be recovered if νbg is computed as h(ν+ − ν− )2 i/8hν+ + ν− − 2νgnd i.
Here h(ν+ − ν− )2 i is the mean of the numerator of Equation 5.22 processed point by point, and8hν+ + ν− − 2νgnd i is the mean of the point-by-point-processed denominator. The uncertainty inthis ratio is the propagated standard deviation of the mean of the numerator and denominator,
which is shown by the simulation to be a good assignment of uncertainty for the measurement.
Given the conclusions of this simulation, all dc Stark background shift data sets are averaged ash(ν+ − ν− )2 i/8hν+ + ν− − 2νgnd i.

5.8

Static Blackbody Shift

5.8.1

Introduction to the Blackbody Radiation Shift

The blackbody radiation (BBR) shift causes the largest systematic uncertainty in many
prominent lattice [114, 74] and ion [52, 76] clocks. Of all optical clocks currently in development,
Sr unfortunately has the largest BBR shift [104]. The effect is essentially the ac Stark shift causedby the electric field from the ambient heat in the laboratory. Since the electric field associated withheat is distributed over a range of frequencies, Equation 2.15 can be converted to an integral form
to express the BBR shift,−1∆νBBR =
2ǫ0 chZ ∞0∆α(ω) I(ω)dω,(5.24)

where I = dI/dω is the electric field intensity per unit angular frequency. For a blackbody, I is
the Planck distribution

I(ω) =~

ω3

π 2 c2 e~ω/kB T − 1.

(5.25)As Equation 2.17 shows, the polarizability ∆α(ω) varies over frequency scales given by atomic
transition energies. Since room-temperature BBR peaks at about a 10 µm wavelength, whereas therelevant atomic resonance wavelengths are orders of magnitude shorter, the atomic polarizability

140
is nearly constant over the room-temperature BBR spectrum. Therefore, Equation 5.24 is often
approximated by performing a dc expansion ∆α(ω) = ∆α(0) + 12 ∆α′′ (0)ω 2 + . . . on the differential
polarizability, which results inZZ∆α(0) ∞
∆α′′ (0) ∞ 2∆νBBR ≃ −I(ω)dω −ω I(ω)dω2ǫ0 ch 02ǫ0 ch 0
 6
 4
T
T
+ νdyn.
= νstatT0
T0(5.26)(5.27)The remaining terms in the polarizability expansion have been neglected since they result in corrections that are well below the current total uncertainty of the Sr clock. The first term in Equation
5.27 is called the static shift, which approximates the polarizability as its dc value, and the second
term is the dynamic shift. The reference temperature T0 = 300 K, andπkB T0 4∆α(0),
60c3 ǫ0~!
Agk
4π 4 kB T0 6 X Aek.
νdyn = −6 − ω621~
ωek
gk

νstat = −(5.28)(5.29)k

Here Aek and ωek (Equation 2.17) are the Einstein coefficients and atomic resonance frequencies
associated with coupling the excited clock state to the hyperfine state k, and Agk and ωgk correspondto the ground state coupling to k (where |ki 6= |gi and |ki 6= |ei).
5.8.2

Sources of Uncertainty

As Equation 5.27 shows, physically there are two sources of uncertainty in the BBR shift:uncertainty in the atomic response to BBR, which is described by the shift coefficients νstat andνdyn , and uncertainty in measurements of the ambient temperature T at the location of the atoms.
In 2008, uncertainty in the Sr1 clock from the shift coefficients was 7 × 10−17 [95, 74]. In 2012,
the PTB Sr group performed an important measurement of the differential dc polarizability of theclock transition, resulting in νstat = (−4.9629 ± 0.0001) × 10−15 [82]. With the static coefficient
uncertainty at 1×10−19 , it can be neglected for both Sr2 evaluations. The PTB group also calculated

141
the dynamic coefficient at 1.6% uncertainty [82]. Later, another team calculated νdyn with a bit
lower uncertainty [105], resulting in νdyn = (−3.48 ± 0.04) × 10−16 . Prior to these calculations, thedynamic coefficient had 7% uncertainty [105, 84].
Improving on the uncertainty in νdyn will not be necessary for our first systematic evaluation,
which has the goal of breaking the previous clock record of 8.6×10−18 total uncertainty; however, thesecond evaluation needs a more accurate determination of νdyn to reach low-10−18 total uncertainty.
Although uncertainty in νstat is negligible, the static shift is the dominant contributor to the portionof the BBR shift uncertainty caused by δT (the uncertainty in T ). Therefore, one can think of
the static shift and the dynamic shift as two separate effects that must be tackled with different
strategies. Reducing the uncertainty in the dynamic shift through a better determination of νdyn
requires measurements of Sr atomic properties [105]. Meanwhile, reducing the uncertainty in thestatic shift due to δT requires improvements to the Sr2 apparatus.
For many years, different Sr teams measured T by attaching several temperature sensors
to their clock vacuum chambers [74, 114, 33]. T was determined from this array of sensors as
the average of the hottest and coldest measurements, and δT was taken as half of the difference
between these two extreme values. With this scheme, δT was usually reported to be about 1 K,
which results in a BBR shift uncertainty of 7 × 10−17 . To improve the Sr BBR shift uncertainty
beyond this, a setup based on a cryogenic environment was proposed [83]. Cold environments,
which have successfully mitigated the blackbody shift in ion clocks [103] and Cs fountains [48],would reduce the shift immensely since ∆νBBR is a rapidly decreasing function of T .
Cold environments are unattractive to us because of their complexity and invasiveness. A
cold environment would at least require the Sr2 main vacuum chamber to be vented to install thenecessary hardware, meaning that much of the apparatus would need to be dismantled to bake thechamber and restore the vacuum. This feat could take the better part of a year. Also, it is possiblethat a good cryogenic system would demand an entirely new clock apparatus designed around it.
These technical issues motivated us to come up with a simpler, room-temperature strategy for
improving BBR shift uncertainty.

142

T1

r
R

T2

Figure 5.18: In this simple model, the chamber is treated as a sphere with a patch at temperature
T2 and the rest of the chamber at T1 . The atoms are located at the center.

5.8.3

Radiation Thermometry

Determining the BBR shift with arrays of sensors mounted to a vacuum chamber has large
uncertainty because it is difficult to characterize and model the chamber at the level required toaccurately infer the ambient temperature where the atoms are trapped. The vacuum chamberprovides a complex boundary that would need to be measured with a fine-mesh sensor array andwell modeled (accounting for both the realistic geometry and material properties of the chamber) topredict the temperature at the center with low uncertainty. Instead, we opt to put a well-calibrated
thermometer inside the vacuum chamber near the location of the atoms. This allows us to directly
measure the radiative temperature at the chamber center.
To see why an in-vacuum radiation thermometer is a dramatic improvement over a chamber
sensor array, it is constructive to consider a simple model. Treating the chamber as a sphericalshell at two temperatures (Figure 5.18) with the atoms at the center, one can gain intuition about
radiation thermometry even in a non-thermal (out of equilibrium) environment. Assuming steadystate, the electric field intensity per unit frequency at the atoms is I(ω) = Ω1 I1 (ω) + Ω2 I2 (ω).
Here I1 and I2 are the intensities per frequency at the two boundary surfaces (Figure 5.18). Ω1
and Ω2 are the area fractions, which are the ratios of the areas of the surfaces at T1 and T2 (Figure

1435.18) to the total surface area 4πR2 . Using this I(ω) in the BBR expression from Equation 5.26,∆νBBR =

νdynνstat(Ω1 T14 + Ω2 T24 ) + 6 (Ω1 T16 + Ω2 T26 ).
4T0
T0(5.30)First I will use this model to analyze the error inherent in using measurements of the chamber
temperature to infer the temperature at the atoms. One typically calculates the atom temperaturefrom measurements of the chamber surface temperatures as Tsurf = (T1 +T2 )/2, and the uncertainty
in Tsurf is computed as δTsurf = (T2 −T1 )/2. Since other Sr teams typically report 1 K temperature
variation across their chambers, I will take δTsurf = 1 K. The BBR shift is then calculated byevaluating Equation 5.27 at T = Tsurf . The uncertainty in this approach results from propagating
δTsurf to ∆νBBR , and the error is the difference between the true shift (Equation 5.30) and the
BBR shift calculated with Tsurf .
Writing the true shift (Equation 5.30) in terms of Tsurf and δTsurf and expanding it inδTsurf /Tsurf (since Tsurf is roughly 300 K), the error νerr between the calculated and true valuesisνerr ≃ −(Ω1 − Ω2 ) δνsurf .

(5.31)

δνsurf is the BBR shift uncertainty calculated by assuming that δTsurf is the measurement uncertainty,

δνsurf =∂δTsurf .
∆νBBR∂T
T =Tsurf(5.32)

Since in reality, the spatial behavior of the chamber temperatures is not well known, Ω1 and Ω2should be regarded as unknown quantities in Equation 5.31. Both Ω1 and Ω2 range from 0 to 1, so
it is reasonable to bound νerr within ±δνsurf ; therefore, δνsurf is a good choice for the uncertainty.
For δT = 1 K, δνsurf = 7 × 10−17 .
Now suppose a thermometer is measuring the radiative temperature inside the chamber very
near the atoms. The sensor absorbs thermal energy and provides a readout of its own temperature,

144
Tsens . Since the sensor equilibrates quickly, the heat energy it radiates per unit time can be treated4
as proportional to Tsens(via the Stefan-Boltzmann law). Assuming the sensor to be an idealR∞
4blackbody, Tsens∝ 0 I(ω)dω, orTsens = (Ω1 T14 + Ω2 T24 )1/4 .

(5.33)

Although the sensor exactly provides the static part of the true shift (Equation 5.30), which is also
proportional to the fourth power of temperature, it does not provide the exact dynamic part. The
error νerr in the radiation thermometer case is the difference between the true shift and the BBR
shift calculated with Tsens . To compare with the case of the chamber sensor array, it is useful towrite the radiation thermometry BBR shift error in terms of Tsurf and δTsurf . Expanding this,2νerr ≃ 6 1 − (Ω1 − Ω2 )



νdyn



Tsurf
T0

6 

δTsurf
Tsurf

2

.

(5.34)

Assuming Ω1 = Ω2 (which maximizes the error) and taking Tsurf ≃ T0 , νerr = −2.3 × 10−20 , which
is negligible for the total uncertainty goals in this work.
In summary, for a distribution of chamber temperatures that are within a degree of each other,
the radiation thermometer provides the static shift and a good approximation to the dynamic shift.
As I will explain later, a more complicated model of these processes (Section 5.21) also concludes
that radiation thermometry error is negligible.

5.8.4

Kirchhoff ’s Radiation Law

4
as proportional to the frequencySection 5.8.3 regarded the sensor as a device that treats Tsens

integrated spectral distribution of heat intensity incident on the sensor surface. For simplicity, thecase of perfect sensor surface absorption and emission (an ideal blackbody) was considered; however,even sensors that are coated with highly absorptive paint are not ideal blackbodies. If imperfectabsorption and emission significantly affect radiation thermometry in Sr2, we would need to perform

Tcav

145

Tobj

Figure 5.19: The cavity considered in the simple explanation of Kirchhoff’s radiation law given in
the text. The cavity temperature is fixed, and the temperature of the object inside can vary. The
temperature of the object can only change by heat emission or absorption.

the difficult task of characterizing the absorptive and emissive properties of the sensor surface.33
Fortunately, for a well-equilibrated environment, Kirchhoff’s radiation law [85] means thatimperfect absorption and emission are not concerns. To develop an understanding of Kirchhoff’slaw, it is instructive to consider a simple case. Suppose a cavity is carved out inside a mass ofthermally conductive material, which has a fixed temperature Tcav (Figure 5.19). Inside the cavity,
there is another thermally conductive object at temperature Tobj , which can vary. The innersurface of the cavity and outer surface of the object inside the cavity have emissive and absorptive
properties that are uniform over each surface; however, emission and absorption at these surfaces
is not necessarily perfect.
The electromagnetic power incident on the object is Pin = ǫcav Pcav + ρcav Pout . ρcav is thereflectivity of the cavity inner surface, which is defined as the reflected power divided by the incident
power [85]. ǫcav is known as the emissivity of the cavity inner surface [85]. For a mass at temperature
T , the emissivity is defined as the emitted power divided by the power of a perfect blackbody attemperature T . An emissivity of unity is considered perfect since heat radiation is maximal for a
perfect blackbody. Pcav is the power of a perfect blackbody at temperature Tcav . Pout is the power
directed away from the object in the cavity, due to either emission from that object or reflection
33
These surface properties are difficult to characterize because they generally depend on wavelength, angle ofincidence or emission, and temperature.

146
off of the object’s surface. This quantity can be expressed as Pout = ǫobj Pobj + ρobj Pin . Here ǫobjand ρobj are the emissivity and reflectivity of the object’s surface, and Pobj is the electromagnetic
power of a mass at temperature Tobj .
Solving for the incident and output power,

Pin =Pout =

ǫcav Pcav + ρcav ǫobj Pobj,
1 − ρcav ρobjǫobj Pobj + ρobj ǫcav Pcav.
1 − ρcav ρobj

(5.35)(5.36)In steady state, Pin = Pout , implying that

αobj ǫcav Pcav − αcav ǫobj Pobj
= 0.
1 − ρobj ρcav

(5.37)

Here I have made use of αobj (αcav ), the absorptivity of the object’s (cavity’s) surface. The absorptivity is defined as the absorbed intensity divided by the incident intensity, which is unity fora perfect absorber. This implies that αobj + ρobj = 1 = αcav + ρcav . In steady state, the second lawof thermodynamics requires that Tcav = Tobj , which implies that Pcav = Pobj (since the blackbody
power depends only on temperature). Under this condition, the expression in Equation 5.37 canonly be zero when

αobj
αcav
.
=
ǫcav
ǫobj

(5.38)

Since the absorptive properties of the two surfaces can vary independently with respect to one
another (same for the emissive properties), Equation 5.38 can only hold if both ratios are equal toa constant. Given that the absorptivity and emissivity are maximal at unity, it follows that

αobj
αcav
=
= 1.
ǫcav
ǫobj
This is Kirchhoff’s radiation law.

(5.39)

147
Now suppose that the object inside the cavity in Figure 5.19 is a temperature sensor. In
steady state, the power absorbed by the sensor is equal to the power emitted,

αobj Pin = ǫobj Pobj .

(5.40)

4 and P is proportional to the integral of the incident intensity per unit frequency,
Since Pobj ∝ Tobj
in

αobj
4
∝
Tobj
ǫobj

Z ∞
0

Iin dω.

(5.41)

Note that this expression resembles a relation that Section 5.8.3 relied upon, which assumed the
R∞
R∞
4 ∝
4
proportionality Tobj
0 Iin dω. As long as the proportionality factor in Tobj ∝ 0 Iin dω isa constant, the analysis of Section 5.8.3 is valid. Equation 5.41 shows why Section 5.8.3 hadR
4 ∝ ∞ I dω. Although theto assume the sensor to be a perfect blackbody in order to use Tobj
in0

proportionality factor in Equation 5.41 is constant, αobj and ǫobj are generally variable (as a functionof temperature, for instance); however, for a perfect blackbody, αobj = ǫobj = 1, and Equation 5.41

reduces to the desired relationship. As I have shown in this Section, the difficult-to-realize case ofa perfect blackbody is not required for accurate radiation thermometry since Equation 5.41 alsoR
4 ∝ ∞ I dω when Kirchhoff’s law holds.
reduces to Tobj
in0This simple development of Kirchhoff’s law is for equilibrium that is global to the entire

sensor-cavity system. In practice, the Sr2 vacuum chamber is near global equilibrium but has a 1
K temperature variation across it. In this situation, Kirchhoff’s law for global equilibrium holds
approximately; however, the error in this approximate treatment might not be small enough forthe high temperature accuracy needed for clock radiation thermometry. Fortunately local thermodynamic equilibrium ensures that the sensor surface properties do not effect the measurement [53].
If an object is in local thermodynamic equilibrium, each point on its surface absorbs and radiates
like a body at a well-defined thermodynamic equilibrium temperature; however, the temperatureacross this object is not necessarily uniform. Local thermodynamic equilibrium exists in the presence of small temperature variations in time and across the body of the object under consideration.

148
The criteria for local equilibrium are well met by our system. Temperature variations in time areslow and small (shown later in Figure 5.27), and variations in space occur over length scales muchlarger than the sensor dimensions. In local equilibrium, Kirchhoff’s law applies at each point on
the sensor surface [53], meaning that imperfect absorption and emission do not affect temperature
measurements.

5.8.5

First Evaluation: Silicon Diode Radiation Thermometers

For the first evaluation, the radiation thermometers are silicon bandgap temperature sensors
from Lake Shore Cryotronics (part DT-471-SD) [13]. These diode sensors are useful for their highsignal-to-noise ratio and low sensor self-heating properties. Two diode sensors, which have factoryspecified uncertainties of 16 mK, are used for this work. The sensors are coated black with vacuumcompatible paint (VacKote 48816, Ball Corp) to improve their absorptivities and emissivities. Eachsensor is held near the center of the chamber with a borosilicate glass tube (7 mm diameter, 1 mm
glass thickness) that has a mounting tip at the end (Figure 5.20a). The glass tube is useful because
it is a poor thermal conductor (conductivity of ∼ 1 W/m K) that is compatible with ultrahighvacuum. Electrical feedthroughs allow us to read the sensor from outside the vacuum. One sensor
is fixed at 2.5 cm from the center of the chamber to prevent it from being coated by the strontium
beam. The other sensor can move between the center of the chamber or 2.5 cm away from thecenter using an edge-welded bellows (Figure 5.20b). This allows the movable sensor to measure
the temperature at the center of the chamber where the atoms are located. Of course, the clock
cannot be running with the movable sensor in this position, so the sensor is retracted by 2.5 cm
during clock operation. The mounting structure for the fixed sensor is 8.1 cm from the sensor tothe conflat base, and for the movable sensor this dimension is 14.4 cm. The fixed sensor is also
referred to as the “short sensor,” and the movable sensor is also called the “long sensor.”
Prior to installation in the main chamber, the sensors were put in a test chamber. This test
chamber was baked at 200◦ C (limited by the melting temperature of silver solder) and pumped
down to the low 10−10 torr level to ensure that contaminants from the black coating and sensors

149

Figure 5.20: a) The shorter, fixed sensor. The black-coated sensor is at the tip of the glass tube. A
glass-to-metal seal joins the glass tube to a shorter steel tube, which is welded to a 1.33” vacuum
conflat. The conflat has a hole in the center (which cannot be seen in the photo) that allows
access to the inside of the hollow glass and metal tubes. Since the insides of the tubes are at
atmosphere, cartridge heaters for vacuum baking the sensors can be inserted in the tubes after thesensor mounts are installed in the main chamber. The leads of the sensor are connected to twoelectrical feedthroughs in the conflat for temperature measurements outside the vacuum chamber.
The hook-up wire is phosphor bronze for low thermal conductivity, and this wire is joined withthe sensor leads and electrical feedthroughs using silver solder. b) A CAD of both sensors in thechamber. The movable sensor is positioned near the center of the chamber. The mechanism used
to move this sensor in and out is depicted at its base. c) The sensors installed in the vacuum
chamber. The image is taken through one of the main chamber 2.75” viewports. The arrows point
to the two glass tubes that hold the sensors. The glowing spot in the center of the chamber is the
461 nm MOT.

150
were removed. Surprisingly, pumping the sensors to this pressure took several weeks. At thetime we attributed this to the black coating retaining water vapor. To transfer the sensors to
the Sr2 system, the main chamber was vented to clean nitrogen at a pressure a few PSI greaterthan that of the Boulder atmosphere. With this positive main chamber pressure, removing two
1.33” conflat viewports from the main chamber to make space for the sensor does not significantly
contaminate the chamber with water vapor. The sensors were transferred from the test chamber tothe main chamber in air. Once installed, the nitrogen was shut off and the chamber was pumped
out. The chamber did not pump down well at first until we added cartridge heaters (Figure 5.20a
caption) inside the sensor glass tubes to bake the mounting structures.34 Within a few days, themain chamber pressure had returned to normal. We refer to the technique of venting the vacuum
chamber to clean nitrogen to provide positive pressure so that the chamber can be opened safely
as a “nitrogen backflow.”
The diode is measured with electronics from Lake Shore. These electronics drive the diodes
with a constant 10 µA current and measure the resulting diode drop. Lake Shore provides thefit parameters (for a high-order fit function) used to calibrate the diode drop to a temperature
standard. The Lake Shore circuit is queried by the Sr2 control software, recording temperature
data each time the clock transition is measured. For clock operation, the BBR shift can be removed
from each frequency measurement in post processing using this temperature record and Equation
5.27.

5.8.6

First Evaluation: Non-Thermal Heat Bias

There are several sources of systematic uncertainty that must be mitigated to ensure accurate
radiation thermometry. The phenomenon with the greatest potential to bias temperature measurements is non-thermal heat.35

The sensors are factory calibrated to a thermal heat distribution(realized inside a highly isothermal cavity), so non-thermal heat will shift the clock transition in
These were baked at about 100◦ C.
“Non-thermal heat” is heat that occurs in an out-of-equilibrium environment; therefore, it does not have a
blackbody spectrum.
34

35

151

Bottom Window Temperature (K)
302

304

306

308

−1
Measured Temperature
Difference (K)Dynamic BBR Error(Frac. Freq × 10−19)

0

−2
−3
−4
−5

0.04
0.03
0.02
0.01
0

0

10

20

Distance To Atom cloud (mm)

0

0.1

0.2

0.3

0.4

0.5

Calculated
Temperature Difference (K)Figure 5.21: Simulating the non-thermal heat bias. The bottom 8” viewport temperature is raised
above the rest of the chamber, which is fixed at 301 K. The non-thermal heat correction is thencomputed. The inset is a measurement of the chamber gradient using the movable thermometer.

a manner that the sensors will not predict.36

Temperature gradients are the signature of non-thermal heat, so it is important to identify gradients and reduce temperature nonuniformity aroundthe Sr2 vacuum chamber.
Section 5.8.3 argued that variations in temperature of 1 K across the vacuum chamber causea negligible non-thermal heat bias. We also confirm this with a more elaborate analysis. We
simulate the heat spectrum from the Sr2 chamber and viewports, assuming local thermodynamic
equilibrium and allowing for temperature to vary across the chamber [13]. The emissivities of thechamber metal and viewports are included. To bound the non-thermal bias, we check an extreme
case in which the temperature of the entire 8” bottom viewport is raised above that of the rest of
the chamber. We find that for a 1 K variation across the chamber, the non-thermal bias is below
36

The BBR shift is caused by the small part of a heat distribution that interacts with atomic transitions out of theclock states. Much of a room temperature blackbody distribution is too far detuned from any resonance involving
one of the Sr clock states to have a significant effect on the BBR shift. On the other hand, the sensors measure theintegrated heat spectrum. Using the integrated spectrum to predict the shift relies on the heat distribution being
well known, which is the case in thermodynamic equilibrium. This is rarely the case for non-thermal heat; however,as I show in this Section and Section 5.8.3, the non-thermal heat bias in our system is well below the total systematic
uncertainty of Sr2.

152
the 10−19 level (Figure 5.21). This conclusion holds even if the material emissivities are changed
by 20%.37

We also measure the gradient inside the chamber using the movable sensor, and we

observe a 40 mK change over 2.5 cm (Figure 5.21 inset). Note that as long as imperfect sensor
absorptivity and emissivity do not affect the measurement (Section 5.8.4), non-thermal heat onlychanges the dynamic BBR shift. This is because when we can ignore imperfections in heat transfer
at the sensor surface, the sensor provides the total integrated heat, which is a direct measurementof the static shift (Equation 5.26) even in somewhat non-thermal conditions.
There are two sources of heat that require special attention: the Zeeman slower window andthe oven. The Zeeman slower window (Section 3.2.2) is heated to 150◦ C and has line of sight tothe atoms. Also, although the oven region is long, the oven is hot and has line of sight to theatoms. We have confirmed that the clock can operate with the Zeeman slower window cooled
to room temperature for several hours, so long as the window is heated when the experiment isoff; therefore, the Zeeman slower window heater is not used during clock operation. Also, theoven region has a rotary feedthrough that is used to block the atomic beam. The feedthrough is
motorized, which allows us to block the oven after atoms are cooled and ready for spectroscopy,
preventing oven heat from reaching the atoms.
Measuring the distribution of temperatures around the vacuum chamber, we observe that 1
K is indeed the temperature variation. This means that non-thermal heat bias can comfortably beignored.

5.8.7

First Evaluation: Immersion Error

Since we want the sensors to only measure the radiative temperature inside the chamber,
temperature measurements will be biased by any heat that conducts from the chamber along theglass mounting tubes (Section 5.8.5) and into one of the sensors. This is known as immersion error[122]. Immersion error is mitigated by the fact that the glass tubes emit and absorb radiation,
37
Since the metal emissivity is 4 orders of magnitude smaller than that of the viewports, the emissivities canactually change by a factor of 20 and the simulation will not be affected; however, 20% is a much likelier margin oferror for the emissivitiy values.

153
which attenuates heat that conducts along the length of the tubes.
To quantify the immersion error, we heat the conflat base of the fixed (shorter) sensor about
8 K above room temperature. The response of the sensor is minor and occurs after 2 to 3 minutes.
We also measure the radiative response of the sensor with a heat gun on its coolest setting. Pointing
the gun such that emitted heat passes through both the top vacuum chamber viewport and thecenter of the chamber, the sensor responds within seconds. This means that the sensor equilibrates
with the radiative environment orders of magnitude faster than with the vacuum chamber throughheat conduction, further reducing immersion error.
To bound uncertainty due to immersion error, we use the data from heating the sensor conflat
base. We extrapolate these data to the case of heating the flange by normal temperature variationsof about 1 K. Treating this error entirely as uncertainty, 0.7 mK uncertainty is added to that of
the sensor [13].

5.8.8First Evaluation: Passive Temperature Control

The Sr2 chamber is surrounded by a black enclosure (Figure 5.22) [13]. This enclosure shields
the atoms from radiation emitted by warm objects throughout the lab.38 The laser light required
for the experiment is delivered into the enclosure via optical fibers, meaning that there are no holes
in the enclosure to admit laser beams. Without holes, there is no way for stray thermal radiationfrom outside of the enclosure to enter the chamber.
To ensure that the system is robust against the usual laboratory temperature changes, temperature control of the space around the vacuum chamber is also attractive. This prevents temporaland spatial variations in the laboratory temperature from coupling too strongly to the chamber.
Since we only require the temperature to be stable at the 1 K level (out of few-K fluctuations),
passive temperature stabilization will be sufficient. The black enclosure around the chamber is also
insulating, providing the temperature control we need. For the first evaluation, measurements with
38

The chamber viewports have emissivities of 0.8-0.9 for IR wavelengths corresponding to thermal radiation, so
the absorptive properties of the viewports also help with stray heat.

154

Figure 5.22: A cartoon of the black enclosure around the vacuum chamber. Light is delivered into
this enclosure via optical fibers.

thermistors inside the enclosure show that the distribution of temperatures around the chamber isconsistently around 1 K. Also, records from the diode sensors show that temporal fluctuations of
the temperature inside the box are at the few hundred mK level even when the laboratory temperature changes by a few degrees K. This enclosure ensures that the temperature of the chamber
does not vary in a markedly inhomogenous way, which further keeps the non-thermal error undercontrol.

5.8.9First Evaluation: Other Sources of Uncertainty

Low sensor self-heating is one major advantage to working with diode sensors compared to
the widely used platinum resistance thermometers (PRTs). With PRTs, there is a competition
between signal-to-noise and sensor self-heating. PRTs are resistors, so more current will provide

Effect
Factory calibration
Immersion error
Sensor translation
Wire resistance
Lattice heating
Total

Correction (mK)
0
0
40
7.7
-15
32.7

Uncertainty (mK)
16
0.7
20
1.5
7.5
26.7

155

Table 5.2: Uncertainty budget for the diode sensors. All uncertainties are quoted at the 1σ level.
“Sensor translation” is the temperature difference between the center of the chamber and 2.5 cm
from the center.

more signal but also more self-heating, which compromises accuracy. For example, a 100 Ω PRT
with an excitation current of 1 mA will cause small-but-significant self-heating and produce a 100
mV signal; however, diode sensors are excited with only 10 µA of current, which leads to negligible
self-heating, yet the signal is a ∼ 0.5 V diode drop.
The mounting structures for the sensors (Figure 5.20) use 2-wire (rather than 4-wire) measurements. This means that the resistance of the phosphor bronze leads will affect the measurementof the diode voltage drop used to infer the temperature. Estimating this effect based on the lengthsof the leads and the diode calibration, the sensor measurement needs to have 7.7 mK added.39 A
generous uncertainty of 1.5 mK has been added to this correction [13].
As explained in Section 5.8.6, measurements with the movable sensor show that there is a 40
mK temperature difference between sensor readings at the center of the chamber (where the atoms
are located) and 2.5 cm away. Since the sensors are 2.5 cm from the chamber center during clock
operation, a 40 mK correction must added to the final temperature reading. A large uncertainty
of 50% is assigned to this correction [13].
Two other systematic biases were checked. When the lattice is turned on, the fixed sensor
heats up by 15 mK. We treat this as a correction and add another generous 50% uncertainty. We
also checked to see if the sensor was affected by any of the Sr2 magnetic fields and did not see achange.
39

The sensor diode drop decreases with increasing temperature, so extra voltage will cause the sensor to read atemperature that is too cool.

156
Combining all systematic uncertainties in the thermometry measurements, we obtain a final
temperature uncertainty of δT = 26.7 mK (Table 5.2) [13]. This corresponds to a static BBR shiftuncertainty of 1.8 × 10−18 and a total BBR shift uncertainty of 4.1 × 10−18 . Since this work, twoother lattice clock teams achieved BBR shift uncertainty below the 10−17 level. The Tokyo Sr grouprealized cryogenic lattice clocks, resulting in a 9 × 10−19 BBR shift uncertainty [116], and the NIST
Boulder Yb group achieved BBR shift uncertainty of 1.0×10−18 using a uniform, well-characterized
thermal environment [2].

5.8.10

Second Evaluation: Platinum Resistance Radiation Thermometers

To improve radiation thermometry for the second evaluation, we began a collaboration with
the Sensor Science Division of NIST (Gaithersburg campus). Our collaborators had their misgivings
about diode temperature sensors. In the 1980s, NIST tested diode thermometers for NASA andfound these devices to have poor reproducibility after thermal cycling (which is an issue for us since
we want to bake our sensors). The problem is that contact resistances at the junction between thesilicon and the metal leads can vary significantly with thermal cycling. In the 2000s, Lake Shore
published data showing more favorable thermal cycling properties [25], and new sensor models sincethen are likely to be even better. However, our collaborators did not feel that there was sufficient
data to verify good diode sensor performance after several thermal shocks.
Our collaborators recommended platinum resistance thermometers. PRTs have been wellstudied for long-term stability and performance during thermal cycling; therefore, we opted forthin-film PRTs [14], which have good vacuum properties and are insensitive to calibration shifts
caused by mechanical impacts. Motivated by a demonstration of a thin-film PRT that shifts only
1 mK after several significant thermal shocks [21], we chose 100 Ω model C416 PRTs from Heraeus
Sensor Technology.40

The Heraeus PRTs are not factory specified at uncertainties better thanwhat we achieved with diode sensors, but they can be calibrated at NIST at the 10 mK level or
40

The demonstration in Reference [21] was for a different Heraeus model that is no longer sold. After contacting
Heraeus, we were told that C416 is the replacement part for that used in Reference [21].

157
below.

5.8.11

Second Evaluation: Sensor Prequalification

Although Reference [21] demonstrated PRTs that shift by 1 mK after thermal cycling, thishigh performance was only observed in 20% of their sensors (all the same model). Motivated bythese observations, we perform prequalification tests on our Heraeus sensors to identify those withthe best temperature properties. These tests involve cycling the sensors between 0 ◦ C and 200 ◦ C.
Those that only change at the 1 mK level after several cycles are identified as the most robust.
The lower cycling temperature of 0 ◦ C is realized in an ice bath. The tests begin by measuring
the initial sensor resistances at 0 ◦ C. After the initial measurement, the sensors are heated in anoven at 200 ◦ C for an hour, cooled to room temperature for about 20 minutes, and then returnedto the ice bath for an hour. During this hour in the bath, the sensors’ resistances are checked every
30 seconds, and we record these resistances when the sensors have clearly equilibrated. Each sensor
is subject to 5 cycles of heating and cooling.
Since we are looking for 1 mK-level variations over several hours of measurement time, thetemperature at which the sensors’ resistances are checked must be stable at this level throughout
the measurement (otherwise poor temperature stability could be mistaken for calibration shifts).
This is why we need an ice bath, which can be stable to about 1 mK in the right conditions [47].
The bath is kept in a dewar. It is based on frozen distilled water broken up with a clean ice shaver.
The shaved ice is made into dense slush with more distilled water. The bath must be carefully
monitored, draining it or repacking it with ice as necessary, to prevent it from warming up due tothe ice melting. The sensors are mounted inside test tubes, plunged in the bath, and measured withphosphor bronze leads that feed through the lid of the dewar. After 5 rounds of thermal cycling,
one sensor maintained ±0.5 mK reproducibility and one was repeatable to ±2.0 mK. The ±0.5 mK
sensor is chosen for the fixed sensor mount and the ±2.0 mK sensor is picked for the movable mount.

158

Rsens

Current
Source

Rstd

Relay

Relay
+

-

Voltmeter
Figure 5.23: Electronics for measuring the PRT resistance. The sensor resistance Rsens is comparedwith a resistance standard Rstd . A stable current drives the two resistors, and both voltage drops
are recorded. Relays are used to switch between resistors. The measurement is repeated with thecurrent reversed so that data can be processed in a manner that removes thermoelectric effects.

5.8.12

Second Evaluation: New Radiation Thermometry Hardware

To prevent lead resistance from affecting the thermometry, we upgrade to four-wire sensor measurements [88]. In this case, two pieces of phosphor bronze hookup wire are soldered
to each sensor lead (one for sourcing/sinking current, one for measuring voltage). This requires
new electronics (Figure 5.23) and new sensor mounting structures that each have four electrical
feedthroughs. These electronics are based on comparing the PRT resistance to a 100 Ω commercial
resistance standard (Fluke 742A-100).
The standard and the PRTs are wired in series, and a commercial low-noise current source(Keithley 6220) drives the sensors. The voltage drop across the PRTs and the standard is measured
using an 8.5-digit voltmeter (Keithley 2002). The current direction is then reversed, and thesevoltage drops are measured again. For one PRT, the resistance inferred from these measurements
isRsens =

V+ − V−
Rstd ,
v+ − v−(5.42)

where V+ (V− ) and v+ (v− ) are the PRT and standard voltage drops measured for positive (neg-159
ative) current. Also, Rstd is the resistance of the standard (quoted at 1 ppm accuracy by the
manufacturer). Recording the voltage drops with two currents (one in each direction) makes themeasurement of Rsens insensitive to thermoelectric voltages, which do not change sign with the current. This method is also insensitive to any problems that would arise from the voltmeter having
an out-of-date calibration, relying only on the resistance standard to provide accuracy.

5.8.13

Second Evaluation: Sensor Calibration at NIST

Since the sensors will ultimately be used in ultrahigh vacuum, their calibration must be
performed under vacuum as well.41

We designed and built a vacuum chamber at JILA with thedimensions needed for the calibration facilities at NIST Gaithersburg (Figure 5.24). The sensors
were installed (on their glass mounts) in the calibration chamber, which was then pumped down
to a pressure similar to that of the Sr2 main vacuum. Like the diode sensors, the PRTs werepainted black with VacKote 48816 to increase radiative coupling; however, the chamber pumped
down in less than 24 hours this time, indicating that the difficulty obtaining good pressure withthe silicon sensors was not due to the black paint.42 The chamber was then transported to NIST
under vacuum. The safest and most cost effective option for transport was for me to hand carrythe chamber on a commercial flight. We obtained an official Department of Commerce request for
airport security to forgo x-ray scans of the chamber (which could damage the sensors) and performchemical testing instead.
The NIST calibration is performed in a water bath [115]. The calibration chamber (Figure5.24) is submerged in the bath, which is spatially uniform in temperature to 1 mK and has 1 mK
temperature stability. Two standard PRTs, calibrated to the NIST ITS-90 temperature scale to 1
mK accuracy, are also submerged in the bath. This allows the Sr2 sensors to be calibrated withrespect to the standard PRTs. PRT resistance as a function of temperature behaves according to41

One reason for this is that sensor self-heating would be markedly different if the sensors were calibrated at
atmospheric pressure. Convective heat transfer at the sensor-air interface mitigates self-heating, which will be worse
under vacuum.
42This may be because the Heraeus sensors are coated with a vacuum-compatible material, whereas the diodes
have more structure that could trap air.

160

Figure 5.24: a) CAD of the chamber used for shipping. The chamber is pumped through the 2.75”
conflat angle valve. The blanked-off arm of the chamber holds a nude ion gauge in the real setup.
The 1.33” conflat is the base of the long sensor, and the sensor’s four electrical feedthroughs can beseen here. Inside the black tube is the base of the shorter sensor. The tube, which is water tight,
is present so that the chamber can be submerged (in the calibration bath) up to the long sensor
base. b) A cross section of the chamber CAD (with false color for clarity). The 1.33” conflat sensor
bases are in pink. The glass tubes that hold the sensors can be seen. The chamber is designed suchthat the two PRTs are situated close to each other (in the center of the cyan-colored reducer). The
watertight tube is shown in cyan.

Short Sensor

Long Sensor

161

R vacuum − R He (Ω)

0.02

0.00

−0.02

−0.04
−2.0

−1.0
0.0
1.0
T flange −T primary (K)Figure 5.25: The calibration of the primary sensors. The primary sensors’ resistances are measured
with respect to the He calibration data as a function of the temperature difference across themounting structures. The longer sensor is less sensitive to immersion error, as expected.

the Callendar van Dusen equation,

R = R0 (1 + AT + BT 2 ),(5.43)

where T is in celsius (valid for T ≥ 0 ◦ C), and R0 , A, and B are fit parameters for the calibration.
For the Sr2 sensors, the manufacturer designed R0 to be 100 Ω. For platinum, A = 3.9 × 10−3 ◦ C−1 ,and B is on the order of −1 × 10−7 ◦ C−2 .
Immersion error is a bigger problem for the NIST calibration than it was with the diodesensors in the Sr2 main chamber [88, 115]. This is because at NIST the bases of the senors cannot
be submerged in the bath (for the sake of electrical measurements), so they are coupled to thelaboratory temperature. As the bath temperature is varied 5 to 10 ◦ C above and below room
temperature to calibrate the sensors, larger gradients than those present in the Sr2 main chamber
will cause increased heat conduction through the mounting structures.

162
To fix this, a two-part strategy is employed [115].43

First, a second pair of PRTs arecalibrated, and each of these PRTs are affixed to the conflat flange at the base of one of thesensor mounting structures. With these “flange sensors,” the in-vacuum resistances of the “primarysensors” (AKA the Sr2 PRTs that will be used as radiation thermometers) are studied as a functionof Tf lange − Tprimary , where Tf lange is the temperature of the relevant mounting structure conflat as
measured by one of the flange sensors [88]. Second, the primary sensors are calibrated in a mannerthat is free from immersion error. This is done using an exchange gas, which thermally couples thesensors and mounting structures more strongly to the walls of the chamber. High purity helium
was selected as the exchange gas, and after venting the chamber to He, immersion error could nolonger be detected.44

Combining the data from both parts of this strategy, the immersion error

for both sensor mounting structures can be understood (Figure 5.25).
The sensor resistances behave according to

Rprimary = RHe + CIE (Tf lange − Tprimary ) + Λ2= R0 (1 + ATprimary + BTprimary) + CIE (Tf lange − Tprimary ) + Λ,

(5.44)(5.45)

where the immersion error coefficient CIE and offset Λ are free parameters in a linear fit of the data
in Figure 5.25, and R0 , A, and B come from the calibration of the primary sensors in the presenceof the He exchange gas [88, 115]. To calculate the final temperature, this equation is solved for
Tprimary as a function of the measured quantities Rprimary and Tf lange . The flange sensors, which
will need to be installed in the Sr2 system for this equation to hold, are useful because they provide
real-time monitoring of the immersion error. It is worth noting the fit values of CIE for the two
sensors: (0.1358 ± 0.0075) Ω K−1 for the short sensor and (0.0183 ± 0.0006) Ω K−1 for the long43
My thanks goes out to Weston Tew for his excellent work on this problem and on the sensor calibration in
general.
44
Under vacuum, heat conduction along the glass tubes would cause the primary sensor readings to fluctuate withthe lab temperature. Argon was tried as an exchange gas, but under Ar the sensors still displayed small fluctuations
that tracked the laboratory temperature, indicating that immersion error was not suppressed. However, with ahelium exchange gas, the primary sensors read steady values and only varied with the bath temperature. In thiscase, measurement precision was high enough that if few-degree laboratory temperature fluctuations could not be
observed on the primary sensor with the short mounting structure, then immersion error is negligible (compared toother uncertainties) for the exchange gas measurements.

163
sensor. Despite that the long sensor mount is not even twice the length of the shorter mount, theimmersion error for the shorter sensor is an order of magnitude larger. This is because immersion
error decays exponentially with the tube length [122].
The final uncertainty of the sensors is 5.2 mK for the long sensor (Table 5.3) and 11 mK forthe short sensor45 [88]. Because propagated uncertainty in the solution to Equation 5.44 increases
with the value of CIE (Figure 5.27), the short sensor has larger total uncertainty than the longsensor; therefore, we use the longer sensor for determining the BBR shift. The short sensor will be
used primarily to ensure that the two sensors agree after installation in the Sr2 chamber.
Two other phenomena were checked during the NIST calibration. First, the sensor selfheating is studied in vacuum and under He. This is done by measuring the sensor readout as afunction of the driving current squared (which is proportional to the dissipated electrical power)
and then extrapolating the trend to zero current. Self-heating data plotted as a function of squared
current fits to a line, which can be used to extrapolate resistance measurements to zero self-heating[115]. For the data in Figure 5.25, all resistances are zero-heating extrapolated values. Second,the effect of vacuum pressure on the sensor readout is studied. In the presence of background gas,
the sensor will not be sensitive to radiation alone since convection will be present. However, weobserved that pressure effects are negligible below a vacuum pressure of 1 × 10−7 torr [115]. Since
the Sr2 main chamber is at the 10−10 torr-level, pressure effects can be comfortably neglected.

5.8.14

Second Evaluation: Measurements in the Sr2 Chamber

After the NIST measurements, the calibration chamber with the sensors inside was handcarried (this time by Jun) back to JILA. We pumped the chamber down again, and the sensors
were checked for agreement. They agreed within their combined uncertainties, indicating thattransportation did not significantly affect calibration (Figure 5.26).46 This is expected since thin45
These uncertainties are computed for negligible gradients across the mounting structures. The total uncertainties
of the sensors will be higher if significant gradients are present; however, significant gradients were not observed in
the Sr2 main chamber.
46
As our very knowledgeable collaborator Wes Tew (NIST Sensor Science Division) tells us, calibration shifts ofthin-film PRTs are random. If multiple sensors of the same model are subject to the same calibration-shift-causing
stress, the magnitudes and even the directions of their shifts will not be consistent. Therefore, agreement between

164
20

Temperature difference (mK)

15
10
5
0
−5
−10
−15
± 12 mK

−20
0

5

10

15

20

25

Measurement time (hr)Figure 5.26: Difference between the two sensors after transport. This measurement is performed
with the sensors in the calibration chamber. The data is represented in dark blue, and the light blue
region is the combined total uncertainty of the two sensors assuming zero gradient. Flange sensor
data has not been included. This data record is initialized right after shutting off the test chamber
ion gauge, which is a heat source that warms the sensors relative to their conflat bases. With thesensors at markedly different temperatures than the bases, there will be heat conduction along theglass mounting tubes, and the sensors will disagree since their immersion error constants are so
different. After an hour, the sensors have thermalized with the chamber, and sensor agreement iswithin the combined sensor uncertainty of 12 mK.

film PRTs are robust against calibration shifts caused by impacts, and for added security they werehandled carefully throughout shipping and calibration.
The sensors are again installed in the Sr2 main chamber using a nitrogen blackflow technique(Section 5.8.5). For the second evaluation, greater care is taken to ensure that the environment
inside the opaque chamber enclosure (Section 5.8.8) is uniform. In the first evaluation, one major
heat source in the enclosure was the Andor CCD camera used to image the atoms. This is unnecessary for clock operation, so it is removed from the enclosure. Also, the 8 ◦ C chilled water used
to cool the MOT coils (Section 3.2.3) was a major heat sink in the first evaluation.47

Replacing

the two sensors indicates that calibration shifts did not occur at a significant level.
47
Since water flows inside the MOT coils (Section 3.2.3), these coils are cooled very efficiently. Operating at fullcurrent (used in the blue MOT stage) for a few hours, the coils get hotter than room temperature. During clock
operation the field is ramped, and the coils settle at a temperature lower than that of the laboratory. Therefore,
before clock operation, we must turn the field ramps on and wait for an hour or two to let the inside of the enclosure
thermalize.

165
the chilled water with 15 ◦ C water balanced the heat from the coils much better, resulting in MOT
coils very near room temperature for clock operation.
With better uniformity of the temperature inside of the enclosure around the vacuum chamber, we check sensor agreement to make sure that installation and baking at 100 ◦ C did not affectthe sensor calibration. This bake was similar to that of the diode sensors, in which cartridge heaters
were used to bake the sensors while the main chamber was pumping down. Our checks confirm
that sensor agreement is still within uncertainty (Figure 5.27a) and quite similar to the agreement
observed in the calibration chamber after shipping (Figure 5.26) [88]. The temperatures measured
by the two primary sensors show that the temperature inside the enclosure is stable to 140 mK(Figure 5.27b). Since the insides of the sensor mounting structures’ glass tubes are exposed toatmosphere (Figure 5.20a caption), convection inside the tubes is suppressed by filling them withcotton insulation.
Although we use the flange sensors for thermometry measurements, we have found that they
are unnecessary because gradients across the temperature mounting structures are so small (Figure
5.27c). The observed temperature difference between the long primary sensor and its flange sensor is(on average) -0.45 mK, which indicates a well-thermalized environment inside the enclosure. Even
if the temperature difference across the long sensor mounting structure were as high as ±1 ◦ C,
the increase in uncertainty (Figure 5.27d) would not significantly contribute to the final sensor
accuracy; therefore, for the mounting structure temperature difference data shown in Figure 5.27c,
the correction to the sensor uncertainty is below 1 µK and can be comfortably ignored.
Lastly, the effect of translating the long sensor is studied. Since the translation mechanism
is a hand crank that can heat the mounting structure, one must wait for an hour or so after thesensor is translated to allow the structure to thermalize with the ambient temperature. Therefore,
the temperature in the enclosure will drift (Figure 5.27b) between data points measured at different
positions of the long sensor. Fortunately, each of these points can be measured with respect to theshort sensor temperature, which cancels ambient temperature drift. The temperature difference
between the chamber center and 2.5 cm away (where the long sensor is positioned during clock

25

a)

20

b)

± 12 mK

10
5
0
−5

20.8
20.7
20.6
20.5

−10
−15

0

5

10

15

20

25

20.4

30

0

5

c)

20
10
0
−10
−20
± St. Dev.

Long Sensor Uncertainty (mK)

Temperature difference (mK)

30

5

10

15

20

Measurement time (hr)

15

20

25

30

5.6
5.4
5.2
5
4.8

d)

4.6
19

0

10

Measurement time (hr)Measurement time (hr)−30Long sensor
Short sensor± St. Dev.

20.9

15

Temperature (°C)

Temperature difference (mK)

166

25

30

20

21

T

o

primary

22( C)

Figure 5.27: a) The difference between the long and short sensor after installation in the Sr2 mainchamber. The data is in dark blue, and the light blue region is the combined uncertainty of thesensors. b) The temperatures measured by the sensors. The standard deviation of the data (shownas the grey band) reveals that the temperature fluctuates by 140 mK. The average temperature ofthis data set is 20.7 ◦ C, but in longer data records (not shown) it averages to 20.6 ◦ C. c) Measured
temperature difference between the long primary sensor and its flange sensor. On average, thetemperature difference is -0.45 mK, and the standard deviation of these data is 12.8 mK (shown as
the gray band). d) Calculated total uncertainty of the long sensor, assuming Tf lange = 20.6 ◦ C. The
total uncertainty is determined as the propagated uncertainty in the solution to Equation 5.44. This
plot shows that temperature differences across the sensor mounting structures as large as ±1 ◦ Cwould not have a significant effect on the total clock uncertainty; therefore, the uncertainty accruedby the observed temperature difference of −0.45 ± 12.8 mK is orders of magnitude too small to beof any concern.

Effect
Bath non-uniformity
Bath SPRT calibration
Bath temp. stability
Sensor self-heating
Electrical errors
Sensor translation
Thermal cycling
Calibration coefficients
Total

Uncertainty (mK)
1.0
1.0
1.0
0.5
0.07
0.03
2.0
4.5
5.2

167

Table 5.3: Uncertainty budget for the primary sensor with the longer glass tube. All uncertainties
are quoted at the 1σ level. Here “SPRT” is the standard PRT used in the calibration.

operation) is (1.45 ± 0.03) mK [88]. Again, this small gradient indicates that the chamber is wellthermalized.
Combining all of the PRT sensor measurements at JILA and NIST, the total uncertainty of
the long sensor is 5.2 mK (Table 5.3).48

For the Sr2 operating temperature of 20.6 ◦ C, the static

BBR shift is −4.5621 × 10−15 with an uncertainty of 3.48 × 10−19 [88].

5.9

Dynamic Blackbody Shift

5.9.1

A Scheme to Measure the Dynamic Coefficient

As explained in Section 5.8.2, dynamic BBR shift uncertainty is dominated by uncertainty inthe coefficient νdyn . Furthermore, uncertainty in νdyn is dominated by insufficient accuracy in the
Einstein A coefficients for transitions out of the clock states. The relationship between νdyn andthe Einstein coefficients is given by Equation 5.29,

4π 4νdyn = −
21



kB T0
~

6 X
k

Agk
Aek
− 6
6
ωek
ωgk

!

.

This expression is proportional to the curvature of ∆α(ω) (the differential polarizability of the clock48
Lattice light heating was not observed this time around. With the lattice at its operating power, no effect could
be discerned from background temperature drift. This may be due to the reduction of scattered lattice light after
realigning the lattice and cleaning up dust around the chamber. Scattered light must have been the cause of latticelight heating in the first evaluation because, at the position of the sensors, the Gaussian beam equation predicts thatthe trap light intensity is smaller than its maximum value by a factor of e48828 .

168transition) at ω = 0. Curvature in ∆α(ω) is caused by resonances, so νdyn is determined by thelowest-frequency transitions out of the clock states (as long as the corresponding resonance widths
are substantial).
98.2% of the dynamic shift is caused by the 3 P0 →3D1 transition at 2.603 µm, and uncertainty
in the Einstein A coefficient for this resonance dominates the uncertainty in νdyn [105]. In Yb, thissame transition dominates νdyn (which is 6.6 times smaller than in Sr), and a scheme to measure
the 3 P0 →3 D1 Einstein coefficient was demonstrated by the NIST Yb lattice clock team [3]. The
NIST approach involves measuring the total decay rate of the 3 D1 state and then inferring theneeded Einstein coefficient through the 3 D1 →3 P0 branching ratio. We use this same scheme tomeasure νdyn in Sr.
We did not need to measure this decay rate in our first evaluation. The 3 D1 →3P0 Einsteincoefficient had already been determined at the 1% level [82, 105], which results in 3.7 × 10−18
uncertainty in the dynamic BBR shift. This uncertainty is low enough for the first evaluation’s
goal of becoming the best clock, but for the second evaluation to reach low-10−18 -level total clock
uncertainty, the dynamic coefficient must be measured more accurately; therefore, we aim for a
0.5% measurement of νdyn . Since the 3 D1 →3 P0 branching ratio is known to 0.1%, and since a
combined uncertainty of well below 0.1% arises from all terms in the νdyn expression aside fromthat due to the 3 D1 →3P0 line, νdyn can be determined at our accuracy goal by measuring the 3 D1
decay rate to 0.5% [105].
The decay measurement begins by preparing atoms in the 1 S0 ground state, driving population into the 3 P0 state using a clock laser π-pulse, and then driving the 3 P0 →3D1 transition witha 2.6 µm laser (Figure 5.28). The 3 D1 state decays into the 3 P manifold. Of the 3 P states, only
3 P has any measurable decay on experimental timescales. Photons emitted from 3 P →1S decay110

are collected on a PMT, and the number of collected photons is recorded as a function of time.
Treating the five-level system depicted in Figure 5.28 with rate equations,

169
3

3

D1
3

1

S0

Clock
laser

D1

3

P2
3
P1
3
P0

2.6 µm
DFB
laser

3

D1

3

P2
3
P1
3
P0
1

PMT

P2
3
P1
3
P0

1

S0

S0

Figure 5.28: The decay measurement scheme. (Left) Population is prepared in the electronic groundstate and then driven to 3 D1 using two laser pulses. (Center) The 3 D1 state decays into the 3 P
manifold. The branching ratios to 3 P2 , 3 P1 , and 3 P0 are 1.9%, 38.7%, and 59.5% (respectively).
(Right) The population in the 3 P1 state then decays to the ground state. Photons emitted during
this decay are collected with a photomultiplier tube.

N2
dN2
=−
dt
τtot
N2 N1
dN1
=
−
,
dt
τ2
τ1

(5.46)(5.47)

where N1 (N2 ) is the population in the 3 P1 (3 D1 ) state, τtot is the lifetime for 3 D1 to decay into
the entire 3 P manifold, τ1 is the lifetime of the 3 P1 state, and τ2 is the inverse of the decay rate forthe 3 D1 →3P1 pathway. Solving this,N1 (t) =τtot τ1
N0 e−t/τ1 − e−t/τtot ,(τ1 − τtot )τ2(5.48)

where the initial conditions are N1 (0) = 0 and N2 (0) = N0 (assuming that the 2.6 µm laser pulse
is infinitesimal). This is proportional to the number of photons emitted by the 3 P1 →1 S0 decay,
which is proportional to the experimental signal (Figure 5.29).

5.9.2

Decay Measurement

One cycle of the 3 D1 decay measurement begins by laser cooling 87 Sr atoms and trappingthem in the TA-based lattice. The atoms are spin polarized and driven into the 3 P0 state using a 3

170
300
250

1

N (t)

200
150
100
50
0

0

20

40

60

80

100

Time (µs)
Figure 5.29: Equation 5.48 plotted as a function of time. Correct values have been used for the
lifetimes τtot = 2.16 µs, τ1 = 21.28 µs, and τ2 = 5.58 µs. An initial atom number of N0 = 1000 has
been assumed.

ms clock laser π-pulse. At this pulse time, the Fourier-limited clock transition linewidth is 270 Hz.
This comparatively broad linewidth is useful because it makes the decay measurement less sensitive
to clock laser drift, which is a greater concern (compared to other systematic measurements) since
atomic servos are not used here. We also spin polarize the atoms because this allows us to drive
population into the 3 P0 state with near unity efficiency.49
With nearly all population in the 3 P0 state, the atoms are driven into the |F = 11/2,3 D1 i
state with a 2.6 µm DFB laser.50

The DFB laser is from NanoPlus, and it is factory specified

to have a 2603.1 nm center frequency, a 3 MHz linewidth, ±1.5 nm wavelength tunability, and 2
mW of output power. We observed output power closer to 4 mW. The frequency drift of the DFB
laser is very low, and we can use it free running for an entire evening without having to adjust thefrequency.51 The DFB laser is pulsed with a 90 MHz AOM.
49
The peak excited state fraction for short π-pulses like these and a spin-polarized sample is typically 0.95. Recall
that for unpolarized lines, the peak excited state fraction is 0.61 (Figure 2.14a caption).
50
Prior to the decay measurement, we performed survey spectroscopy with an OPO laser and found the |F =9/2,3 P0 i → |F = 11/2,3 D1 i line at 2603.128 nm. This is the Stark shifted resonance wavelength observed in the
TA-based lattice.
51
The resonance frequency of the |F = 9/2,3 P0 i → |F = 11/2,3 D1 i transition does drift if the lattice intensity is

171
Data
Fit

PMT Counts ( ×103 )

10

8

6

4

2
0

50

100

150

Time (µs)
Figure 5.30: Decay data. Decay measurements are taken in 40 ns time bins. This figure is generated
by summing all data for each time bin. The magenta dots are the fit of data taken after the pulse isoff. The y-axis offset is due to dark counts on the PMT, and the x-axis offset is primarily a timing
delay added to ensure that all the early data is captured.

3 P →1 S fluorescence resulting from the 3 D →3 P →1 S cascade is collected with a 3”
1
0
1
1
0

diameter lens and focused onto a Hamamatsu PMT.52 To get rid of dark counts, the PMT iscovered with two optical bandpass filters centered on 690 nm (the fluorescence is at 689.4 nm).53
The output of the PMT is sent directly to an SR430 photon counter.
Since atoms decay back to the ground state, more decay measurements can be performed
without having to prepare another ultracold sample; therefore, 50 decay measurements are madeeach time we cool and trap atoms. Decay back into the ground state depolarizes the atoms into
other mF states, so for each cooling and trapping cycle the spin polarization laser is applied for 5 ms
in between decay measurements. Also, a bias field is turned on when the clock laser is pulsed, andthen it is shut off before the DFB laser is applied. Keeping the field off during the 3 D1 →3P1 →1S0
decay prevents systematic bias due to Zeeman quantum beats [3].
The photon counter records the number of detected photons in time bins of 40 ns. Data is
not stabilized, resulting in a time varying ac Stark shift.
52This a second PMT in addition to the one used to measure the atom number and excited state fraction.
53
We found that one optical filter was not enough to eliminate dark counts.

300

200

a)

172

b)

Standard Deviation

200

Counts

100
0
−100

100

50

Binned residuals
Pure Poisson process
0.515 power law

−200
25
−300

50

100

Time (µs)

150

3

4

5

6

7

8 9 103

Mean ( ×10 counts)

Figure 5.31: a) Fit residuals of the decay data. b) Comparing statistical moments of the residuals.
For this plot, the residuals are binned (bin size of 15 points), and the mean and standard deviationis computed for each bin. Plotted on a log-log scale, a pure Poisson process would have a slope of
0.5. The trend is somewhere between a Poisson process and a power law with a 0.515 exponent.

stored in the SR430 local memory for speed and offloaded after several minutes. The decay data
(Figure 5.30) was taken over one week in two shifts, one in the daytime and one overnight, for roundthe-clock acquisition. The data set corresponds to about 8 million single decay measurements, andthe trend in the data strongly resembles the rate equation solution shown in Figure 5.29 [88].
Motivated by the solution to the rate equations (Equation 5.48), the signal is fit with the functionhi−(t−t0 )/τ3 P−(t−t0 )/τ3D
1 − e
1 ,
y(t) = y0 + A e(5.49)

where y0 , A, t0 , τ3 P1 , and τ3D1 are fit parameters. y0 and t0 are added due to the presence oftechnical offsets (Figure 5.30).

5.9.3

Fit Uncertainty

Binning performed by the photon counter introduces concerns about fit uncertainty. For eachtime bin, an offload of photon counter data provides the sum of all counts acquired since the previous
offload. These sums have an averaging-like effect on the bins, smoothing statistical fluctuations.
Therefore, in principle each bin should also have an error bar representing the statistical uncertainty

173
−1

Fractional Uncertainty

10

Simulated, unweighted
Simulated, weighted
Fit, unweighted
Fit, weighted

−2

10

−1

0

10

10

1

10

Measurement Time (days)
Figure 5.32: Simulated fit uncertainty in τ3D1 . Here the experiment is simulated assuming Poissionian noise. For each simulation, a weighted and unweighted fit to Equation 5.49 are performed.
The fit weights treat the statistical uncertainty of each simulated time bin as the square root of thebin mean. The blue and red lines represent the simulated uncertainties. To compute the simulated
uncertainties, simulated data is generated and fit repeatedly, resulting in a distribution of fit valuesof τ3D1 . For the blue (red) line, fit values are computed by an unweighted (weighted) fit. The
simulated uncertainty is taken to be the standard deviation of this distribution, which describes
the true scatter in the fit. In practice there is only one data set, so uncertainty in the fitted valueof τ3D1 must come from the uncertainty generated by the fit routine. If the fit routine provides
the correct uncertainty, it will agree with the simulated uncertainty. The green and cyan lines are
uncertainties generated by the fit routine, showing that an unweighted fit significantly disagrees
with the simulated uncertainty whereas a weighted fit (cyan) gets the uncertainty correct.

in each sum, and a fit of the data to Equation 5.49 should be weighted by these error bars. The
photon counter does not provide uncertainties, but it is possible to understand what they shouldbe by studying the measurement noise process.
The noise is characterized through the fit residuals of the decay data (Figure 5.31a). Plotting
the standard deviation of the residuals against the mean, we observe a trend consistent with a(nearly) Poisson process (Figure 5.31b). This implies that the measurement is dominated by photon
counting noise, with no other noise sources significantly contributing. Since Poissonian noise has
a standard deviation equal to the square root of its mean, the data fits should be weighted by
the square root of the counts for each bin (Figure 5.32). Without weighting, the fit routine will

No Cuts

174

With Cuts

1.5

0.1

b)

a)
Fit Error (%)

Fit Error (%)

0.08
1

0.5

0.06
0.04
0.02

0

200

300

400

Pulse Duration (ns)

500

0

200

300

400

500

Pulse Duration (ns)Figure 5.33: a) Theoretical fit bias due to finite pulses. This is based on numerically solving the
Lindblad master equation for different pulse durations. The situation is treated as a four-statesystem of 1 S0 , 3 P0 , 3 P1 , and 3 D1 . 3 P2 is ignored because its branching ratio is small, and it
does nothing except shelve population (including this state would merely increase the 3 D1 lifetime
a bit, which would not affect the conclusion). A realistic Louivillian is used in the calculation,
assuming accurate values of state lifetimes and an estimate of the DFB laser linewidth based on
factory specifications. The master equation solution provides numerical data for the system density
matrix, which yields the 3 P1 population (proportional to the experimental signal) as a function oftime. This population is fit to Equation 5.49, and the error is computed as |τf it − τtrue |/τtrue , whereτf it is the fitted value of τ3D1 and τtrue is the true value used in the master equation. The error isnearly equal to the measurement accuracy goal of 0.5% for pulses of 250 ns. b) The same masterequation data with points cut from the fit. Here the fit only considers data after the DFB pulse isoff, which reduces error to a very comfortable level.

underestimate τ3D1 uncertainty.
Fitting the data in this way, the measured 3 D1 lifetime is τ3D1 = (2.18 ± 0.01) µs, which isslightly better than the 0.5% measurement goal [88]. The fit also yields a new best value for the
3 P lifetime (which is of interest to other research teams) of τ
3 P = (21.28 ± 0.03) µs
1
1

5.9.4

Systematic Bias Due to the Finite DFB Laser Pulse

The function used to fit the data (Equation 5.49) is motivated by the solution to the rateequations (Equation 5.48). The initial conditions for the rate equations assume an instantaneous
DFB laser pulse.54
54

To check if the length of the pulse has any effect, I numerically solve theEquation 5.48 resulted from assuming the initial condition of all population in the 3 D1 state. This implies an
instantaneous DFB laser pulse because for a finite pulse, population would leak into 3 P1 while the DFB pulse is on.

175
Numerical Data
Fit
−3

0.2

x 10

P1 population

0.16

5

3

0.12

10

0.08

0
0

3

1

P population

15

50

0.04

100
150
Time (ns)

200

0
0

20

40

60

80

100

120

140

Time (µs)

Figure 5.34: The master equation solution for a 500 ns pulse fit with Equation 5.49. Inset: Although
the instantaneous pulse model appears to fit the data well, closer inspection at early times reveals
disagreement. The non-instantaneous system response at time = 0 (due to dynamics while thepulse is on) causes the master equation solution to be qualitatively different than the fit function
at short times. This causes a fit bias of nearly 1.5% (Figure 5.33), which is unacceptable for our
accuracy goal of 0.5%.

Lindblad master equation for a variety of pulse durations (Figure 5.33a). This calculation uses
realistic values for state lifetimes and the DFB laser linewidth. The numerical master equation
data is fit using the function from Equation 5.49 to extract the 3 D1 lifetime, τ3D1 . The error, whichis the disagreement between the true lifetime value used in the master equation and the value
obtained by the fit, is higher than the measurement accuracy goal of 0.5% for pulses longer than
250 ns (Figure 5.33a). The error is caused because the signal edge when the DFB laser is turnedon gets smoothed (with respect to the instantaneous DFB pulse model) due to dynamics occurring
during the DFB pulse (Figure 5.34).
Treating the decay data while the pulse is on would require a good model of the real DFB
pulse shape and linewidth, and we would also need confirmation that these quantities do not varyin time. However, the system still admits an easy solution after the DFB pulse switches off.55 The
55

My thanks to Murray Barrett for being the first in our team to work out the analytic treatment of the population
decay for finite pulses.

176
rate Equations 5.46 and 5.47 still hold in this case, but the initial conditions are now N1 (0) = N01
and N2 (0) = N02 , where N01 (N02 ) is the atom number in 3 P1 (3 D1 ) when the pulse is extinguished.
The difference between this case and that used to derive Equation 5.48 is the number of atoms N01
that accumulate in 3 P1 when the DFB laser is on. Here t = 0 is taken to be the instant when the
DFB pulse reaches zero intensity. The solution isN1 (t) =τtot τ1
N01 + N02
τ2 (τ1 − τtot )e−t/τ1 − N02τtot τ1e−t/τtot .
τ2 (τ1 − τtot )(5.50)

τ1
and Aet0 /τtot =Defining constants A and t0 through the equations Aet0 /τ1 = N01 + N02 τ2 (ττtot
1 −τtot )τ1
N02 τ2 (ττtot, it is apparent that Equation 5.50 is faithfully modeled by the fit function that assumes
1 −τtot )an instantaneous pulse (Equation 5.49). Therefore, as long as we only fit data taken after the pulse
switches off, the fit function will provide the correct 3 D1 lifetime. This conclusion is independently
verified by fitting master equation data for times after the pulse is off (Figure 5.33b).
As the master equation analysis shows, two strategies can be used to reduce fit bias due tothe finite pulse duration: keep the DFB pulses at 200 ns or shorter, and only fit data after the
DFB pulse is off. To doubly suppress this bias, we take both approaches. We confirm the amountof data that must be cut from the fit by comparing (on an oscilloscope) the 200 ns DFB pulses ona high-speed photodetector to the electronic pulse used to trigger the photon counter.56

Basedon the master equation analysis, we assign a conservative upper bound of 0.1 ns on the systematic
uncertainty in τ3D1 due to the remaining finite pulse bias.
5.9.5

Fit Bias Due to Stray Electromagnetic Radiation

Stray electromagnetic radiation resonant with the 3 P0 →3 D1 transition will affect decay
dynamics and bias the τ3D1 measurement. The two sources of stray radiation at a 2.6 µm wavelength
are the heat in the Sr2 laboratory and stray beams from the DFB laser. The bias due to laboratory
heat is extremely small, and it can be estimated using the Einstein approach to radiative processes
56

The fit of the data in Figure 5.30 is plotted only for the range of data included in the fit. The fact that thefitted curve is absent right when the decay process begins is because early data is not included in the fit due to theissues described in this subsection.

177[117]. Laboratory heat will cause unaccounted for absorption and stimulated emission on 3 P0 →3D1 ,meaning that the 3 D1 decay rate will be

1
τ3D1

+B

Z ∞

ρ(f )g(f )df,

(5.51)

0for spontaneous decay and stimulated emission. Here B is the stimulated emission Einstein coefficient, ρ(f ) is the energy density of blackbody radiation from the lab, and g(f ) is the lineshapefunction of the transition. Since g(f ) is orders of magnitude narrower than ρ(f ), g(f ) can be treatedas a delta function δ(f − f0 ) in the integral, where f0 is the 3 P0 →3D1 resonance frequency. Using
the relations between Einstein coefficients, the fractional systematic uncertainty in τ3D1 is simply
1/(ehf0 /kB T − 1), which is the average number of blackbody photons at frequency f0 . Although
this calculation neglects state degeneracy (which modifies the answer by a factor of order unity), it
also results in 10 ppb uncertainty, so this analysis could be off by a factor of 100,000 and the uncertainty would still be negligible. A similar treatment for absorption results in another very small
correction. Therefore, we assign a conservative upper bound of 10 fs on the systematic uncertainty
in τ3D1 due to contamination from laboratory heat.
To study the uncertainty due to stray light, we set the AOM (used to pulse the DFB laser)
in its off state and scan the DFB frequency across the 3 P0 →3 D1 line. We expose the atoms toresidual light that may still be in the AOM diffracted order for three orders of magnitude moretime than the experimental pulse duration. No excitation of the 3 P0 →3D1 resonance is observed.57
Treating the background signal fluctuation in this measurement as the upper bound on 3 D1 state
excitation, another master equation calculation is used set a bound on the effect of residual lightfrom the AOM diffracted order used in the decay data. The results show a negligible effect.
Although residual light in the AOM diffracted order is not significant, we do observe a straylight effect when the DFB laser frequency is such that the AOM zeroth order is resonant with the
3 P →3D transition. This effect is observed for an artificially long exposure time of 200 ms, and it
0
1
57

Excitation is measured by driving the clock transition, applying the DFB laser, and then measuring the clocktransition excited state fraction in the usual way. Excitation into 3 D1 will diminish the excited state fraction.

178
appears to be coming from the AOM zeroth order beam (rather than from the DFB beam before it
enters the AOM) because the effect disappears when we close a mechanical shutter installed after
the AOM. After measuring the population we can excite for 200 ms of exposure to the stray AOM
zeroth order light, the master equation is used to put an upper bound on the Rabi frequency fromthis light. From this information, the bias due to stray light in normal experimental conditions isinferred using the master equation.58

Again, the effect is negligible. The small size of this effectis due to both the low intensity of the stray light and also because during the decay measurement,
the DFB frequency is such that the AOM zeroth order is detuned by 90 MHz from the 3 P0 →3D1
resonance.
These studies of the effects of stray DFB light allow us to put an upper bound of 10 fs on fit
bias due to stray DFB beams.

5.9.6

Density-Dependent Effects

During their measurement of the 3 D1 lifetime in Yb, the NIST Boulder lattice clock teamobserved a density dependence to fitted values of τ3D1 [3]. Effects like superradiance and radiation
trapping could cause the 3 D1 lifetime to vary with density. Uncertainty in extrapolating the measured lifetime to zero density was the dominant limitation to the precision of the NIST measurement[3].
The Sr2 experimental sequence used to measure decay data also records the atom counts(with the 461 nm counting laser and the PMT described in Section 3.7.2). Each cycle of thesequence involves preparing the atoms in the lattice, applying the clock and DFB lasers, measuring
fluorescence from the 3 D1 →3 P1 →1 S0 cascade 50 times, and then fluorescing the 1 S0 →1 P1transition to record how many atoms remain after each set of 50 decay measurements. Every 30
cycles, an interrupt shuts off the clock laser and photon counter trigger so that the atom number canbe recorded in a manner that is decoupled from the effects of the clock and DFB lasers. Treating58

To calculate this bias, I account for the fact that the stray light interacts with the atoms whenever the shutter
is open during the decay measurement, which is on a tens of ms timescale. The calculation is performed by running
the simulation with the stray light included and then fitting the result. The disagreement between the fitted and true
values of 3 D1 is the fit bias.

179

3

Fitted D1 decay rate (µs)

2.5
± 1σ
2.4
2.3
2.2
2.1
2
1.9

1

2

3

4

Atom counts

5

6

7
4

x 10

Figure 5.35: Fitted 3 D1 lifetime as a function of atom counts, which is proportional to the atom
number. Here the data from Figure 5.30 is reanalyzed to look for density dependence. No trend isobserved. The blue confidence interval represents the 1σ uncertainty in theqweighted mean assuming

the decay is constant in density. This uncertainty has been inflated byfor these data.

χ2red , where χ2red = 1.12

this one-in-thirty measurement as the atom number for each of the previous 29 cycles, sets of decay
data (as a function of time) are binned according to atom number, then summed and fitted as in
Figure 5.30.
The fitted 3 D1 lifetime as a function of atom counts on the PMT does not show the same
striking trend observed by the Yb group [3]; in fact, it does not seem to be affected by density at
all [88]. To test the data for density dependence, a constant model is compared to a linear trend,
which would be the first-order description of density-related effects. The F -test comparing thesemodels results in F = 0.045 for 11 degrees of freedom. This corresponds to a test probability of
0.84, which concludes that the data does not resolve a density dependence; therefore, density effects
are excluded from the uncertainty budget for the decay measurement.
Two more systematic uncertainties are considered. First, 0.4 ns timing uncertainty is assigned
to the photon counting setup. Second, the decay measurement assumes that the 3 D1 lifetime does
not depend on the large number of F and mF states that are populated during this experiment.

Effect
Fit uncertainty
Photon counter timing
Finite pulse duration
Hyperfine correction
Stray laser lightLaboratory heat contamination
Total

Uncertainty (ns)
10
0.4
< 0.1
< 0.1
< 0.01< 0.01
10

180

Table 5.4: Decay measurement uncertainty budget. All uncertainties are quoted at 1σ or as upperbounds on a 1σ uncertainty. Fitting uncertainty dominates this budget, with all other effects being
negligible at the 1 ns level.

One typically sums the decay rate over all product hyperfine states, resulting in decays that do
not depend on either the initial or final hyperfine quantum numbers [44]. This approximation
breaks down when hyperfine quenching becomes important. The hyperfine quenching correction is
calculated to be below 0.1 ns.59
Combining all of these uncertainties, the final value of the 3 D1 lifetime is τ3D1 = (2.18±0.01) µs(Table 5.4) [88]. This results in a dynamic coefficient of νdyn = (−148.7±0.7) mHz, or −3.464×10−16
with uncertainty of 1.6 × 10−18 [88]. For the Sr2 operating temperature of 20.6 ◦ C, the dynamic
BBR shift is −3.053 × 10−16 with uncertainty of 1.4 × 10−18 .

5.10

Minor Systematic Shifts
With all major systematic uncertainties now discussed, I will briefly mention a few minor

uncertainties that were checked. First, phase transients can occur when the clock laser AOM(Section 3.7.1) is pulsed [26]. We study the AOM phase transients by looking at a beat of the -1st
AOM order with the zeroth-order on a digital phase detector. The phase detector was found tohave transients of its own that were measured, fit to a high-order polynomial, and removed fromthe final measurement. We calculate the effect of clock laser phase transients on the atoms usinga model developed in Reference [34]. This results in systematic uncertainty due to an AOM phase
chirp of (6 ± 4) × 10−19 . This value has been used in both evaluations.
59

My appreciation goes out to our theory collaborator, Marianna Safronova, for performing this calculation.

181
Evaluations of several different optical clocks have considered a servo offset [63, 33, 52], whichis steady state error in the atomic servo resulting in a frequency bias of the clock laser. This offset
is computed as the mean of the entire history of atomic servo error signal data converted fromexcited state fraction units (Section 2.5.2) to frequency. This is done by dividing error signal datataken with a given clock laser pulse duration by 2 times the calculated slope of a Rabi lineshape(corresponding to the correct pulse time) at half the maximum excitation. The factor of 2 comesfrom the fact that the atomic servo error signal is the difference between excited state fractions
measured on each side of a resonance, so the error signal change per unit frequency excursion is 2
times the Rabi lineshape slope. For the first evaluation, the servo offset is (4 ± 6) × 10−19 , and itis (−5 ± 4) × 10−19 for the second evaluation.
A phenomenon called “line pulling” has the potential to cause small clock transition shifts.
This occurs when the tails of off-resonant spectroscopic features make the clock transition resonanceappear slightly off its true center. These features can be present due to imperfect spin polarization
resulting in population in mF states aside from ±9/2, residual clock laser ellipticity driving mF changing σ transitions, or clock transition sidebands that result from tunneling between latticesites. Population in the mF = ±7/2 states is most likely to cause off-resonant excitation, butsome common-mode cancellation of this effect will be present in the stretched-states servo. This isbecause equal population in the ±7/2 states will pull the ±9/2 levels toward the clock transition barefrequency equally, and averaging the ±9/2 stretched state data will remove this effect. Calculating
line pulling for these cases and bounding the calculations using the fact that, with our measurement
precision, we do not observe any off-resonant features near the clock transition (where they would
be the most harmful), we put a conservative upper bound on this effect of 1 × 10−19 . This is valid
for both evaluations.
In the ideal case of a perfectly stable lattice, probing the Sr atoms with the clock laser aligned
along the lattice axis will be free of a first-order Doppler shift. If mechanical noise causes the lattice
to shake with respect to the clock laser, a first-order Doppler shift is possible. Fiber phase noise
cancellation (Section 3.6.2) removes this problem, but a second-order Doppler shift (the second-

182
order term in a low-velocity expansion of the relativistic Doppler expression) can still exist. We
estimate this effect to be at the 10−21 level. To buffer this calculation against errors, we give it acomfortable 1 × 10−19 upper bound for both evaluations.
The background gas in the Sr2 main chamber that limits the vacuum lifetime will collide withthe trapped Sr sample and cause frequency shifts. For the Sr2 vacuum pressure, the backgroundgas is mostly hydrogen. Using this fact, we employ the model of Reference [38] to put an upperbound of 6 × 10−19 on frequency shifts due to collisions between Sr and the background gas. Thisbound is used for both evaluations.
Finally, there is added uncertainty associated with treating the BBR static and dynamic shifts
as two separate effects. When the static and dynamic uncertainties are added in quadrature, thisignores the fact that they are not statistically independent. With a bit of uncertainty propagation
mathematics applied to Equation 5.27, it is straightforward to show that the static and dynamic
 4pshift uncertainties must be added in quadrature with a covariance-like term 48νstat νdyn TT0 δT
T0 ,which is equal 8.2 × 10−19 for the first evaluation and 1.4 × 10−19 for the second. The value forthe first evaluation is just barely negligible (a slightly larger value would affect the second digitof the total clock uncertainty), and for the second evaluation this correction can comfortably beignored.

Chapter 6

Outlook and Concluding Remarks

I started as a first-year graduate student on the JILA Sr team shortly before Sr1 achieved
1.4× 10−16 total uncertainty, which was the first time a lattice clock surpassed the total uncertaintyof the Cs primary standard at NIST Boulder [74]. 1.4 × 10−16 total uncertainty [20] was a great
accomplishment at the time, but despite our optimism about the future of lattice clocks, it is safe to
say that it was unclear whether an optical lattice clock would ever overtake ion clocks. Three major
issues needed to be addressed for lattice clocks to realize the best performance: clock stability, thedensity shift, and the BBR shift.
Although lattice clocks seemed promising because of their potential for good stability, thiswould require better laser stabilization. Unfortunately, for over a decade the best laser stabilization
efforts had not managed to surpass laser linewidth records set in the late 90s [126]. Then cavity
stabilization took a leap forward [89, 49], resulting in laser linewidths an order of magnitude narrower than the previous best [8]. Clock stability resulting from these laser improvements enabled
the record total uncertainty we later achieved.
The density shift is not present in single-ion clocks. When Sr1 achieved 1.4 × 10−16 total
uncertainty, a density shift was observed in Sr1 evaluations [74, 20]. This shift was a major contributor to Sr1 uncertainty budgets, yet its physical cause was confusing. It implied collisions between
ultracold spin-polarized fermions, which naively seemed to be forbidden by the Pauli Exclusion
Principle. After many experiments and theoretical studies aimed at understanding and mitigating
this shift [19, 99, 37, 64, 73, 6, 89, 78], it is now a small contributor to the Sr2 clock uncertainty.

184
The BBR shift has limited many of the best optical clocks [114, 74, 52, 76]. Ion-based
systems have an advantage on this front due to naturally smaller sensitivity of ion clocks to roomtemperature BBR. When Sr1 first outperformed Cs, BBR shift uncertainty was at 1 × 10−16 . Given
that the sensor arrays used to measure the BBR shift in many lattice clocks [74, 33, 114] were merely
workarounds for systems not originally designed with this shift in mind, there was plenty of room
for progress. However, beating the 3×10−18 uncertainty in the miniscule BBR shift of the Al+ clock
seemed a formidable task. When the time came for the lattice clock community to confront the
BBR shift head on, the first three teams to make serious efforts on the BBR shift all succeeded using
distinctly different techniques [13, 2, 116]. It is now clear that the BBR shift is not a limitation at
the 1 × 10−18 level, and it will no doubt improve as the lattice clock community continues to workon this effect.
Despite the atomic clock performance records presented in this dissertation of 2.1 × 10−18√
total uncertainty and 2.2 × 10−16 / τ stability, lattice clocks still have tremendous potential. For
the clock probe times, duty cycles, and atom numbers of 1 × 105 achievable in the Sr2 system, I√
calculate the QPN-limited stability to be 2 × 10−18 / τ . This would allow a clock to average to agiven precision 10,000 times faster than Sr2 can currently achieve. Although the density shift would
be a problem for large atom numbers in a 1D cavity lattice, multidimensional cavity lattices could
fix this [109]. The Sr2 blackbody shift due to temperature uncertainty is already at the low 10−19level, and the uncertainty of the full BBR shift could match this with more precise measurements ofνdyn . The NIST Yb clock would also see improvement with a more precise measurement of νdyn [2],reducing their total BBR shift uncertainty to the 10−19 level. Even cryogenic environments haveroom for improvement [116] despite that advances in this technology would be more technicallychallenging. For example, colder cryogenic environments can be used to further reduce the BBR
shift. Also, to confirm or reject previous determinations of higher-order lattice light shifts, it will
be important for another Sr team to measure these effects.
Better clock lasers are also on the horizon. Novel cavity spacers based on silicon [59] andnew mirrors with crystalline coatings [24] have been demonstrated to improve cavity stabilization.

185
Our team is in the process of combining this technology to build a new clock laser, which will
surpass the previous best stable laser by a factor of 10 according to our simulations. Also, proof-ofprinciple tests of proposals for novel stable lasers that do not use cavity stabilization [80, 79] havebeen demonstrated [15]. If these proposed schemes become practical laboratory technology, clockstability could take another giant leap forward.
Timekeeping at the level realized by Sr2 would not lose a second in 15 billion years (greaterthan the age of the universe), yet there are good scientific reasons to push performance further.
Sr2 can now resolve 2 cm height differences near the surface of Earth due to the gravitational
redshift. With this sensitivity, optical comparisons of two remote clocks [96] would be sensitive tovariations of Earth’s surface due to tidal forces. Better clock precision combined with efforts torealize portable clocks (currently underway at NIST and PTB) could make these systems useful for
relativistic geodesy. Clocks are also useful for studying fundamental physics like time variation of
the fine structure constant or the electron-to-proton mass ratio [40, 51]. Although ion clocks are
more sensitive to fundamental constant variation, with better precision lattice clocks may be in aposition to make competitive measurements.
Sr clocks have also been used to put limits on the coupling of fundamental constants togravity [11], which is predicted by some unification theories. Improvements to clock performance
could serve to improve these limits. Furthermore, upgrading to the 40 cm cavity enabled Sr1 tospectroscopically observe many-body physics [78] and novel particle interactions [127, 108]. It is
likely that many more interesting quantum effects would be uncovered with better measurement
precision.
Theoretical proposals have argued for replacing the current system of global timekeeping witha worldwide network of optical clocks [62]. Although such a network is far from being realized,
it exemplifies the potential of linking up precise optical clocks on an international scale. Finally,
it was argued that atomic clocks are useful systems for dark matter searches [1, 27]. We may beentering a regime in which atomic clocks can answer some of the most pressing physical questionsof our time.

186
In summary, the outlook for lattice clocks is bright, and I expect that the lattice clock
community will continue with the same impressive pace it has maintained for the past 10 years. It
has been a pleasure to participate in this wonderful field.

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Appendix A

Mathematical Relations Not Derived in the Main Text

A.1

Two Time Signals Related by a Transfer Function
Two time signals x1 (t) and x2 (t) have corresponding Fourier transforms X 1 (f ) and X 2 (f ),respectively. The frequency functions are related by a transfer function H(f ), such that X2 (f ) =H(f )X1 (f ). Equivalently, according to the convolution theorem, x2 (t) = h(t) ∗ x1 (t), where ∗denotes a convolution, and h(t) is the inverse Fourier transform of H(f ). The cross correlation
R12 (τ ) between X 1 (f ) and X 2 (f ) isZ ∞h(t′ )hx1 (t)x1 (t + τ − t′ )i dt′
R12 (τ ) = hx1 (t)x2 (τ + t)i =−∞Z ∞h(t′ )R1 (τ − t′ ) dt′ = h(τ ) ∗ R1 (τ ),
=

(A.1)

−∞

where R1 (τ ) is the autocorrelation function of x1 (t). Furthermore,Z ∞R2 (τ ) = hx2 (t)x2 (t − τ )i =h(t′ )hx1 (t − τ − t′ )x2 (t)i dt′−∞
Z ∞h(t′ )R21 (−τ − t′ ) dt′ = h(−τ ) ∗ R21 (−τ ) = h(−τ ) ∗ R12 (τ ),
=

(A.2)

−∞

where the cross correlation identity R21 (−τ ) = R12 (τ ) was used. Putting equations A.1 and A.2
together, the relationship between the autocorrelation functions of x1 (t) and x2 (t) is

R2 (τ ) = h(−τ ) ∗ h(τ ) ∗ R1 (τ ).

(A.3)

197
According to the Wiener-Khinchin theorem, the power spectral density (PSD) is the Fouriertransform of the autocorrelation function,

S2 (f ) = F[R2 (τ )] = F[h(−τ )] · F[h(τ )] · F[R1 (τ )]= |H(f )|2 S1 (f ).

(A.4)

where F denotes a Fourier transform. This is the desired relation.

A.2

Relating the Allan Deviation to the Power Spectral Density
The Allan deviation is given by
" Z
2 #
Z
1
1 t−τ
1 t
2
′
′
′
′
σ (τ ) = E,
y(t ) dt −y(t ) dt
2
τ t
τ t−τ

(A.5)

where y(t) is the fractional frequency fluctuation of the local oscillator and E [. . .] is an expectation
value. The integral terms can be combined in impulse-response form,
"Z
2 #
∞
hτ (t′ )y(t′ − t) dt′
σ 2 (τ ) =
,

(A.6)

−∞

where hτ (t) is an asymmetric window filter:− √1 , −τ < t < 02τhτ (t) = √1
,0<t<τ

.

(A.7)

2τ

The transfer function of this filter is:
Z τ
Z 0
Z ∞
1
1
e−2πif t dt − √e−2πif t dt =hτ (t)e−2πif t dt = √Hτ (f ) =
2τ 0
2τ −τ
−∞
√ sin2 (πf τ )
2
=
.
iπf τ
R∞
Let f (t) = −∞ hτ (t′ )y(t′ − t) dt′ ,(A.8)

σ 2 (τ ) =f 2 (t) = hf (t)f (t − τ )i |τ →0 = Rf (0) (autocorrelation)
Z ∞
i
h
=
Sf (f ) e2πif τ
df
τ →0

−∞

=

Z ∞

−∞

Sf (f ) df =

Z ∞

−∞

2

|Hτ (f )| Sy (f ) df = 2

Z ∞sin4 (πf τ )Sy (f ) df ,2−∞ (πf τ )

(A.9)

198
where the relation developed in Appendix A.1 has been utilized. Here Sy (f ) is the PSD of y(t).
Note that Sy (f ) is the two-sided PSD. It is convenient to define a one-sided PSD,
S 1−sided (f ) = 2S 2−sided (f ) ,for 0 ≤ f < ∞ .

(A.10)

This factor of 2 is chosen so that both spectral densities give the total power when integrated
over all frequencies for which these functions are defined,

Total power

=
=

Z ∞

S

Z−∞
∞

2−sided(f ) df = 2

Z ∞S 2−sided (f ) df

(since PSD is even)0

S 1−sided (f ) df .

(A.11)

0

Thereforeσ 2 (τ ) = 2

Z ∞
0

A.3

sin4 (πf τ ) 1−sided
S(f ) df .
(πf τ )2 y

(A.12)

Power Spectral Density of Discrete Time Signals
Suppose that y(t) is a stationary time signal. This function is sampled with a period of Tc .

For the nth cycle, the discrete time signal autocorrelation R(nTc ) is defined as

R(nTc ) =

∞
X

m=−∞

y(mTc ) y[(m − n)Tc ].

(A.13)The PSD of the discrete signal Sdis (f ) is given by the discrete time Fourier transform of R(nTc ),defined as

Sdis (f ) = Tc∞
X

R(nTc )e−2πif Tc n .

(A.14)n=−∞

Let R(τ ) be the autocorrelation of the continuous signal y(t). Then let Scon (f ) be the PSD
of y(t), obtained via a continuous Fourier transform F of R(τ ). It follows that

199

Sdis (f ) = Tc∞
X

−2πif Tc nR(nTc )e

n=−∞
"
∞
X

= F Tc
=

∞
X

k=−∞

= Tc

∞
X

n=−∞

n=−∞

#

R(nTc )F[δ(t − nTc )]"R(nTc )δ(τ − nTc ) = F R(τ )ih

F R(τ )e−2πikτ /Tc =

∞
X



∞
X

k=−∞

Scon f +k=−∞e−2πikτ /Tc

k
Tc



.

#
(A.15)

where δ(t) is the Dirac delta function. Here I have used the well-known relation between a sum
over delta functions and a sum over complex exponentials. This is the desired relation between
Sdis (f ) and Scon (f ).

A.4

Detection Noise
Suppose that the counts measured on the PMT after fluorescing the counting laser (Section

3.7.2) are Ce , Cg , and Cbg , which are the excited, ground, and background counts. Also, Ve =Ce − Cbg and Vg = Cg − Cbg (Section 3.7.2), and each are proportional to their corresponding atom
numbers by a factor of β. The measured excited state fraction isρmeas =

Ce − Cbg,
Cg + Ce − 2Cbg

(A.16)

where the total atom number N is proportional to Cg + Ce − 2Cbg .
The detection noise is the number of counts measured when the atomic beam and lasersthat cause background counts are blocked. The observed detection noise is δN = 11.4 atoms per
measurement of Ce , Cg , and Cbg . Propagating this error into ρee , the uncertainty due to detection
noise is
s

∂ρee
β δN
∂Ce

2

+



∂ρee
β δN
∂Cg

2

+

This assumes that the clock is locked to ρmeas = 1/2.



∂ρee
β δN
∂Cbg

2

δN
=√
.
2N

(A.17)

200
Just like in the case of QPN, this fluctuation in ρmeas causes frequency instability in the clock
laser. Using arguments similar to those in Section 4.2.1, the frequency stability due to detection
noise isδNσdet = √2N s νSr

r

Tc
.
τ

(A.18)√
Therefore, detection noise scales like 1/N , QPN scales like 1/ N , and the two stabilities cross one
another when N = 2 δN 2 . For atom numbers less than 2 δN 2 , detection noise dominates over QPN.