Enhancing optical clocks with ultrastable lasers andspin-squeezingbyJohn M. Robinson
B.S., University of Nevada, Las Vegas 2015

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 Science
2023

Committee Members:
Jun Ye, ChairJames K. Thompson
Ana Maria Rey
Konrad Lehnert
Joshua Combes

ii
Robinson, John M. (Ph.D., Physics)
Enhancing optical clocks with ultrastable lasers and spin-squeezing
Thesis directed by Prof. Jun Ye

Traditional optical atomic clocks are limited in their performance by laser frequency noise
and the intrinsic quantum noise of uncorrelated atoms. In this thesis, we advance the field of
optical clocks on both of these fronts. By developing the next generation of ultrastable laser technology, we enable clock comparisons that have approached the quantum projection noise limit. To
go beyond this limit, we build and operate an optical clock with the capability of spin-squeezing.
Employing conditional spin squeezing via quantum nondemolition measurements based on cavity
QED, we produce a spin squeezed state that yields a spectroscopic enhancement of 1.7 dB beyond the standard quantum limit. We then run a clock comparison between two spin squeezed
clock ensembles, making use of a movable optical lattice to individually squeeze and readout theensembles with cavity QED. This differential comparison between the two squeezed clocks directly
verifies enhanced clock stability of 1.9 dB beyond the quantum projection noise limit, and reaches
a measurement precision level of 10−17 . This constitutes the first direct demonstration of quantum
enhanced measurement in an operational optical atomic clock.

Dedication

To my parents.

Acknowledgements

First off, huge thanks to Jun for being the best research advisor imaginable. When I first
showed up, I was intimidated by Jun’s quick thinking and questions, he seemed to be moving
at 100 miles per hour. Over time, I realized that Jun’s tough questions were the embodimentof his dedication to his students, he was often thinking about our problems, both technical andfundamental. He simply wants to help - both in science and in life. I’ll always remember the lessons
I’ve learned working in the Sr lab, and I’m eternally grateful for the opportunities Jun has givenme.
When I first joined the lab, I spent the summer helping out Sr1, and I thank Shimon
Kolkowitz, Sarah Bromley, and Toby Bothwell for being supportive in those early days. Soonenough, Jun asked me if I was interested in stable lasers, and I began my journey over in S1B20.
That is when I started working with Wei Zhang. Wei is perhaps the most generous and kind person
I’ve ever met. His eagerness to help young students, sitting for hours on end until things were clear
for everyone, is truly admirable. Lindsay Sonderhouse was working on the comb and cavities at
the time, and I will always appreciate her careful thinking on all sorts of issues in the lab andher eternal kindness. Eric Oelker soon joined us, and brought with him some demystifying of thestable laser dark magic. Eric somehow also knew when I needed to go on with some issue/topic on
my own to grow, giving those little nudges in the right direction along the way. The more recent
members of Sr1 - Alex Aeppli and Kyungtae Kim were always eager to help and have been greatlabmates. The newly joined member Will Warfield will surely help that experiment continue toflourish.

v
One wonderful aspect to stable lasers is the collaboration with our PTB colleagues. When
Lindsay and I traveled to PTB to help work on the silicon cavity project at PTB, Dan Matei,
Thomas Legero, Fritz Riehle and Uwe Sterr were all so very kind and helpful. Their carefulattention to detail continues to be inspiring - always taking systematic steps, and somehow always
in the right direction.
Sr2 was always a very friendly bunch. Sara Campbell became a close friend, and time spentin the lab with her was always fun. She, along with other Sr2 folks: Ross Hutson, Will Milner, Ed
Marti, and Aki Goban, all became truly amazing colleagues over the years. Ross is one of the labs
wizards, somehow always either knowing the answer to your question, or knowing how to figure
it out. Will is incredibly giving with his time in fixing technical issues in the lab, and you canbe sure to have a good conversation if you ask him a physics question. Ed and Aki were amazing
post-docs in the Sr lab, each always being helpful to young graduate students. When Christian
Sanner showed up, he came over to stable lasers to help out on the Si4 upgrades. Christian’s
helpfulness and absolute technical wizardry made for some serious fun in the stable laser room, andthis carried on throughout his time at JILA. Working on Sr1 for a while was a great introductionof using stable lasers with the atoms. Colin and I worked closely on the Sr1 buildup cavity formany weeks, and during periods of setbacks, he always came into the lab with a positive attitude
nonetheless. Toby is a careful scientist, and was often extremely helpful during the building of Sr3,
using his experience on the Sr1 experiment to help guide us. Dhruv is a very skilled physicist, andhis work on Si6 took the next steps in stable laser research in exactly the right direction.
Some of the initial Sr3 effort involved all of us on Sr1, early cavity and chamber designs
were made by Colin and Toby. When Sr1 split into two experiments, I took on Sr3 and was firstjoined by Josie Meyer. Josie’s positive attitude was always infectious, and some of the Sr3 machine
would not be there without her. Shortly after, we were joined by Maya Miklos, an extremely sharp
physicist, and without her, many critical aspects of Sr3 would be vastly weaker. Yee Ming Tsojoined the project, and dove right into many critical aspects of the experiment. I’m certain the
future of Sr3 is in great hands.

vi
The work in this thesis would not be possible without support from many others, including
my Mom, Dad, Tom, Connor, Jake, Keene, Mark, Selina, Luna. Lastly, thank you to Raven Larson,for your unwavering support and love.

vii

Contents

Chapter
1 Introduction

1

1.1

Characterizing clocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Limits to clock performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

Overcoming the limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4

Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62 Ultrastable lasers based on optical cavities

8

2.1

Motivation and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2

Lasers stabilized to optical cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2.1
2.3

Fundamental limit - thermal noise . . . . . . . . . . . . . . . . . . . . . . . . 10

Technical noise budget of Si4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1

Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.2

Vibration noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.3

Residual amplitude modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.4

Photothermal effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.5

Path length noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.6

Electronics noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4

Stability measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5

Linear frequency drift : power dependence . . . . . . . . . . . . . . . . . . . . . . . . 30

viii
2.6

Inferring loss angle of SiO2 /Ta2 O5 coating . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7

Si3 : zero crossing of CTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Classical clock demonstrations and applications

46

3.1

Classical clock comparisons at the quantum limit . . . . . . . . . . . . . . . . . . . . 46

3.2

Dark matter limits from precision spectroscopy . . . . . . . . . . . . . . . . . . . . . 50

3.3

3.2.1

Sensitivity to fundamental constants . . . . . . . . . . . . . . . . . . . . . . . 50

3.2.2

Dark matter couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2.3

Dark matter Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2.4

Extra details on our analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Boulder Atomic Clock Optical Network . . . . . . . . . . . . . . . . . . . . . . . . . 58

4 Spin-squeezed clock : background and considerations

62

4.1

Basic theory and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2

Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3

How do we design a cQED system? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.4

How should we probe the atom-cavity system? . . . . . . . . . . . . . . . . . . . . . 68
4.4.1

Atomic motion and probing time scales . . . . . . . . . . . . . . . . . . . . . 72

5 Sr3 : a new spin-squeezed clock apparatus
5.1

77

UHV System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.1.1

Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.1.2

Assembly and bake-out procedure . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2

MOT bucket design

5.3

Accuracy considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3.1

Faraday shielding - DC Stark . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3.2

In-vacuum calibrated thermal sensors . . . . . . . . . . . . . . . . . . . . . . 84

Laser systems and cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

ix

5.5

5.4.1

461 nm laser system and blue MOT . . . . . . . . . . . . . . . . . . . . . . . 84

5.4.2

Repump lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.4.3

689 nm lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.4.4

Optical lattice and transport . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Clock laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.5.1

Fiber noise cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.6

Atomic state preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.7

Cavity-stabilization 813nm laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.8

cQED cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.9

5.8.1

Cavity design and construction . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.8.2

Cavity stabilization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Homodyne interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.9.1

Path length stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6 Generating and measuring entanglement on the Sr clock transition

113

6.1

Clock spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2

Entering the collective strong-coupling regime . . . . . . . . . . . . . . . . . . . . . . 119

6.3

QPN and the effective atom-cavity coupling . . . . . . . . . . . . . . . . . . . . . . . 122

6.4

6.3.1

QPN in slightly detuned regime . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.3.2

Measuring QPN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.3.3

Noise in QPN measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Conditional spin squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.4.1

Spectroscopic enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.4.2

Calibrating incident photon number . . . . . . . . . . . . . . . . . . . . . . . 128

6.4.3

Spin noise reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.4.4

Ramsey contrast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.4.5

Measured Wineland parameter . . . . . . . . . . . . . . . . . . . . . . . . . . 133

x
7 Squeezed clock comparisons

138

7.1

QPN and SQL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7.2

Self-synchronous clock comparison with cQED . . . . . . . . . . . . . . . . . . . . . 140

7.3

Optimal estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

7.4

Vertical extent of cloud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.5

Common-mode suppression of laser noise

7.6

Quantum efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.7

Spin state tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.8

Independence of the two ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152. . . . . . . . . . . . . . . . . . . . . . . . 149

8 Conclusion and outlook

156

Bibliography

158

Appendix
A Vacuum noise limits of cavity frequency determination

170

A.0.1 Heterodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
A.0.2 Homodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
B Calculation of effective atom-cavity coupling

173

C Calculation of free space scattering

175

xi

TablesTable
5.1

Key parameters of optical cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.1

Quantum efficiency contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

xii

FiguresFigure2.1

We illustrate the major contributions to the thermal noise for three optical cavities
at various temperatures and with length ranging from 6 cm to 40 cm. We use
the coating loss angle φc from our extraction [54] such that the noise values better
match our measured frequency noise. Fractional noise associated with coating andsubstrate scales inversely with the cavity length. Substrate thermo-elastic and spacer
Brownian noise are below 10−36 Hz−1 for the cryogenic systems. . . . . . . . . . . . 11

2.2Time domain step response of the Si4 thermal damping system. A step function isapplied to the active plate/shield, and we measure the response on the passive shield,
and the corresponding frequency change of the cavity measured via a heterodyne
beatnote against the MJM cavity.

2.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16(a) The mechanical layout of the Si4 system. The top view shows how the split plateis mechanically isolated from the main plate, and the only real physical connection
is a flexible thermal connection. (b) Measured coherence between the accelerometeron the AVI and the beat note of Si4 against a reference laser. (c) Vertical vibration
sensitivity in fractional frequency units per g of acceleration, versus the azimuthal
angle between the silicon crystal axis and the three-axis supports.

. . . . . . . . . . 18

xiii
2.4

(a) Vertical vibration sensitivity of Si4 measured versus the relative azimuthal angle
between the tripod support and the crystal axis of the silicon crystal. (b) Improved
high frequency noise performance of Si4 by sensitivity improvement, and optimization of AVI position.

2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18(a) Measured Si3-Si4 beat note. (b) Velocity waveform measured at the Briggsdale
seismic station. We can see the clear arrival of the different types of seismic waves,and the correlations with our optical frequency beat note measurement. . . . . . . . 21

2.6(A)Amplitude response of the photothermal effect in Si4. Dashed lines are phenomological fits, and the solid black line is a theoretical estimate. (B) Phase response of
the measured transfer functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.7

Technical noise contributions for the 4 K system, along with the sum of the technicalnoise (pink line). The sum of the technical noise sources is well below the coating
Brownian noise (red dashed line).

2.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Setup for measuring the beatnote between Si4 and Si3. Si4’s radiation shields areshown. The Si3 stability is checked using the Sr atomic clock. Comparison against
the UTC(NIST) timescale allows for calibration of the Si3 drift when the Sr clock is
not running.

2.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

(A) Modified Allan deviation for the Si4 cavity. (B) Fractional frequency noise
power spectral density (Sy ) of the Si4 cavity. In both panels, the shaded green
region indicates the spread of the estimated thermal noise floor given two previously
published values for the coating loss angle [65, 66]. . . . . . . . . . . . . . . . . . . . 31

2.10 (A) FFT of the beat measured at 1542 nm (black circles), fit to a Lorentzian lineshape(red line). (B) Histogram of the measured Si4 linewidths for 100 measurements. The
Si4 linewidths are obtained by multiplying the beat linewidth by 0.85. . . . . . . . . 32
2.11 Fractional frequency drift in units of 10−18 for Si4 at different operating temperature.
The Si3 drift versus intensity is shown as the open red squares, exhibiting a far
smaller sensitivity to optical power.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 34

xiv
2.12 Fractional frequency dift in units of 10−18 for Si4 at different operating temperature.
The Si3 drift versus intensity is shown as the open red squares, exhibiting a far
smaller sensitivity to optical power.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.13 The measured noise PSD at various temperatures (solid curves). The dashed lines
are the fits to Sy (f ) = a/f b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.14 Frequency-dependent mechanical loss angle for the SiO2 /Ta2 O5 coatings at different
operating temperature. The shaded bands indicate the 1σ uncertainty. . . . . . . . . 41
2.15 Mechanical loss of the individual coating constituents (SiO2 , Ta2 O5 ), and SiO2 /Ta2 O5
coatings measured by ringdown (orange diamonds [75], blue triangles [76], green
squares [77]). Other direct measurements are also included (black triangle [78], open
black square [25], and purple diamond [79]). Our results from direct CTN measurements are shown as the red circles and the red arrow. . . . . . . . . . . . . . . . . . 42
2.16 Si3 zero crossing measurement of the coefficient of thermal expansion. Measured
both against Si4 and the hydrogen maser. The fitted zero-crossings are consistent,with T0 = 123.345542(2) K from Si3-Si4, and T0 = 123.34594(7) K from the Si3-maser. 45
3.1(a) Schematic of the cross-correlation method. Each laser is fiber noise cancelled to
the frequency comb table, and we measure the beat between Si4 and Si3, and MJM
vs Si3. (b) The black data is the cross-correlation spectrum for the Si3 system (21cm Si). The red line is a laser model that we will use for Dick effect calculations anddark matter limit simulations.

3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47(a) Anti-synchronized clock comparison between Sr1 and Sr2. The solid gray lineis the callculated QPN, blue line is calculated Dick effect, orange dashed-dot is thequadrature sum of QPN and the Dick effect laser noise, and the dashed green line is√
the fitted stability, giving 5.5 × 10−17 / τ . (b) Synchronous comparison, showing a√single clock stability of 3.5 × 10−17 / τ . . . . . . . . . . . . . . . . . . . . . . . . . . 49

xv
3.3

(a) Fractional frequency deviation of the Si - Sr frequency comparison. The data
has been dedrifted by fitting the data to Eq. 3.13. (b) Power spectral density of the
Si-Sr data. The model for the Si cavity is shown as the dashed line. (c) Fractional
deviation of the Si - maser frequency difference. (d) PSD of the Si - maser data. The
maser model has a white noise term, and a broadened resonance to account for the
microwave link noise. The silicon model is the same as for (b). . . . . . . . . . . . . 53

3.4(a) 95% confidence limits placed on the dimensionless couplings to the electromagnetic field de , showing the improved limits set by fSr /fSi (red) and fH /fSi (blue)
ratios in the mass range above 1 × 10−19 eV. The maximum projected sensitivity fora search of the same 11-day duration without data gaps is included (light green),highlighting the potential of our technique. Limits derived from previous spectroscopic searches (black lines), with the network [95], Dy-Dy comparison [96], andRb/Cs [97]. Limits from equivalence principle tests (purple lines), with MICROSCOPE [98] and UW [99]. (b) Demonstration of a significantly improved limit on
the electron mass modulus (dme ) derived from the fH /fSi (blue) ratio. Limits fromequivalence principle tests (purple lines), including MICROSCOPE [98] and measurements from UW [99]. Shaded regions in both (a) and (b) are excluded at the
95% confidence level given the observed signal and noise models. . . . . . . . . . . . 55

3.5

Corrections to the dme limit from dedrifting and the unknown dark matter phase.
The vertical dashed line indicates the lowest Fourier frequency for which a dark
matter limit is computed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

xvi
3.6

(a) Schematic for the out-of-loop evaluation of the BRAN fiber link for the 2017/2018
JILA-NIST clock comparison. A stable laser (Si3) is fiber noise cancelled, with
the noise detected between the mirrors labelled as “ref #1” and “ref #2”. The
out-of-loop beat is measured between the local light and the “remote” light. (b)Overlapping Allan deviation of the inloop error for the fiber noise cancellation (red)and the out-of-loop beat (black points). (c) Histogram of the out of loop frequency
measurements, with the residuals to the Gaussian fit shown below. . . . . . . . . . . 60

3.7

Allan deviation of the three ratios on a given day. Also shown are noise traces forvarious components of the optical network, including the frequency combs, fiber link,
free-space link and a network loopback measurement. We can see good agreement
between the free space and fiber measurements of the Yb/Sr ratio. . . . . . . . . . . 61

4.1(a) Schematic of the detection scheme, where laser light is split into a probe pathand LO path, and these are combined on a beamsplitter. (b) By sweeping the probe
laser across resonance we can map out the I − Q response of the cavity response.
(c) Normalized I quadrature. (d) Normalized Q quadrature. . . . . . . . . . . . . . 70

4.2(a) The atom-cavity coupling g 2 (x, y), which follows the intensity of the cavity mode
standing wave is shown as the contour plot, normalized to a peak value of one.
The lines indicate the 1/e2 radius. Overlaid on top of g 2 is 100 different atomic
trajectories simulated for the particle motion in a single pancake of the optical lattice.
(b) Corresponding traces of g 2 (t)/δc for all of the different particles, showing thatmost atoms sample many standing waves well within 1 ms. (c) Fractional noiseof the cavity shift for a measurement time of 2 ms, versus the dwell time between
frequency measurements. We show the magnitude of the oscillations versus atomic
temperature (d) Same as (c), with a measurement time of 40 ms. We can see thatthe longer measurement time offers improved averaging of the radial motion. . . . . 74

xvii
4.3

Simulations of cavity frequency shift noise arising from themal atomic motion. Witha fixed seperation time between measurements of 10 ms, we predict fluctuations inthe shifted cavity frequency on the percent level. Both low atomic temperatures andlonger measurement time Tm are critical for minimizing this effect. . . . . . . . . . . 76

5.1(a) Angle-view of the MOT bucket. The smaller MOT bucket pieces are welded
together on a 10 inch conflat flange that is bolted to the bottom of the chamber. (b)Side cut-view of the MOT bucket. The MOT coils and the cooling fixtures inside
the MOT bucket are labelled. In total, there are 5 in-vacuum welds. . . . . . . . . . 81

5.2

COMSOL simulations of the DC electric field at the location of the atoms. We
simulate the effect of having 1 nC of charge distributed uniformly on the surface of
one of the mirrors. Also shown is the effect of an applied voltage on that same PZT.
Even at the highest operating voltage, the overall shift should be on the order of
1 × 10−18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3(a) Top view of the optical cavity. Shown are the two thermal sensors that are
installed in the cavity spacer. (b) Calibration of Rgrey in a dry-block comparison
with three reference sensors. (c) Calibration uncertainty for the two sensors as afunction of operating temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4

Lattice phase noise measurements. The free running lattice phase noise is the red
trace, and the locked inloop error is the black trace. For comparison, the laser noisemodels for the two clock lasers MJM and Si3 are shown as the purple and cyandashed curves. The photodiode noise floor is the gray trace. . . . . . . . . . . . . . . 89

5.5

The left axis indicates the relative frequency detuning of the optical lattice beams
for transport, and the right axis is the corresponding velocity of the lattice. The
total transport distance is approximately 42 mm. . . . . . . . . . . . . . . . . . . . . 90

xviii
5.6

Differential fiber noise between the 0th order and 1st order beams. The black trace
indicates when we place the 1st order retroreflection near the 0th order retro, andred is placing the retro as close to the vacuum chamber as possible, just before theviewport. For reference, we plot the Si3 noise model, showing reasonably adequate
noise cancellation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.7

The top panel shows the experimental sequence roughly laid out in time. Below wegraphically indicate the location of the MOTs, and the subsequent loading of thelattice and transport into the cavity. We also note the magnitude and direction of
the relavant magnetic field for each step. . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.8

Measurement of the cavity finesse at 689 nm by monitoring the transmission after
abruptly shutting off the light. The black data and fit are before the bake out, giving
F = 20, 000. The red data and fit are after the chamber bakeout, giving F = 16, 000.
As a check, we temporarily move the photodetector to before the cavity (gray data),
to confirm our light shut-off time is sufficiently short. . . . . . . . . . . . . . . . . . . 97

5.9

Up close photo of the mirror assembly with the PZT stack. You can also see two of
the Ceramabond epoxy joints holding the mirror inside the mirror cup. The wholeepoxied assembly is bolted to the cavity spacer with four vented screws, of which
two are visible here. The other mirror, is identical apart from the overall height of
the PZT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.10 Full view of the cavity, as it would be seen from below after installation. In clear
view are the two mirrors. Also visible is the slot we cut in the cavity, to reduce Eddy
currents. The cavity is mounted onto an aluminum jig that was used to secure thecavity during installation of the various components. . . . . . . . . . . . . . . . . . . 101

xix
5.11 Response of the magnetic field when rapidly switching off the current through the
MOT coils. The magnetometer is placed inside a test cavity spacer having roughly
the same geometry as our actual spacer. We measure the response without the
spacer (orange), with the spacer (blue) and with the spacer but with a slot cut intoit (green). We show a dramatic effect of including the slot in the spacer, showing afaster damping time constant of induced Eddy currents in the spacer. . . . . . . . . 103
5.12 The optics layout and stabilization scheme of the cavity QED setup. The 813nm
laser is pre-stabilized, and the cavity is locked to that pre-stabilized laser via thecavity PDH lock. The 689nm laser is pre-stabilized to the red cavity system. The
689 ECDL is split into the LO and probe path. The leakage light from the cavity isdetected via homodyne detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.13 Frequency diagram of the cavity modes and of the different laser light tones. The
bare cavity frequency ωc is detuned from the bare atomic resonance ωa by δc /2π = 1
MHz. The cavity is shifted by the presence of atoms in the ground state |↓i. The
relavant frequencies are shown. The carrier is synthesized by frequency shift of
AOM-2, and the sideband that probes the cavity is generated by the EOM drive.
The LO is correspondingly frequency shifted to be on resonance with the sideband
for homodyne detection.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.14 Noise measurements of cavity QED stabilization scheme. The blue line is the inloop
error of the 813 ECDL lock to the comb. The gray line is the inloop error of the
PDH lock of the cavity to the 813 laser. The 689nm pre-stabilized laser then probes
the cavity - serving as the out of loop noise measurement. We can see the frequency
bands where we are still limited by the inloop lock of the cavity. . . . . . . . . . . . 106

xx
5.15 (a) Measured output of the homodyne voltage (blue) and the accelerometer (orange)
placed on the optical table near the homodyne interferometer. Both signals havebeen bandpass filtered in software to make the correlations clearly visible. (b) The
same signals in the frequency domain. After we subtract the accelerations from thehomodyne voltage, we can see strong cancellation of the noise peaks at 20 Hz and
30 Hz, without impacting the homodyne noise at other frequencies. We note herethat the photodiode voltage is amplified by a voltage factor of 25, on top of the 106
gain of the transimpedance amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.1

Sideband scans of the clock transition. Without any ramping of the lattice or additional cooling (green), with sideband cooling (red), and with the lattice ramp along(black). The lattice ramp gives a sample prepared primarily in the ground motionalband hnz i = 0, and a narrower blue sideband (radial temperature). The carrier
signal is heavily saturated in this clock scan. . . . . . . . . . . . . . . . . . . . . . . . 114

6.2

Radial Doppler scans of the clock transition. Red data points are without the latticeramp, black points are with the lattice ramp. The radial temperature extracted fromthe Gaussian fits (corresponding dashed lines) are T=290(10) nK for with latticeramp, and T = 447(20) nK without the lattice ramp.

6.3. . . . . . . . . . . . . . . . . 116

Rabi flops (a) Measured excitation versus the Rabi pulse time in seconds, with adrive Rabi frequency of ΩR /2π = 111 Hz. (b) Same as (a), with the drive Rabi
frequency set to ΩR /2π = 12 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.4

Ramsey contrast for increasingly complex pulse sequences. The pulses are labelledwith a sub-script, indicating the relative rotation angle. π/290 refers to an “align”
pi/2 pulse, required for squeezing. The introduction of the π0 is the π pulse we needto perform the pre-measurement in our squeezing experiments. . . . . . . . . . . . . 118

xxi
6.5

Center : we show the eigenvalue spectrum of the atom-cavity coupled system versus
the atom-cavity detuning δc /2π. Left : When we tune the cavity onto the bareatomic resonance, we observe the vacuum Rabi splitting, shown in the left plot.
This demonstrates entering the collective strong-coupling regime. Right : For the
QND measurements, we operate in the slightly detuned regime.

6.6. . . . . . . . . . . 120(a) Pulse sequence for measuring QPN. Black open pulses correspond to clock rotations, and the red pulses are cavity frequency shift measurements. (b) QPN measurement. The red datapoints have no rotation noise subtracted, and the model including
a linear term is fit (red line) giving g/2π = 5230(213) Hz. The open blue circles havean estimated clock rotation noise contribution subtracted, and is fit without a linear
term giving g/2π = 5127(192) Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.7(a) Pulse sequence used for measuring spin noise reduction. After an initial π/2, we
measure Jz,p , wait a dwell time τ , and then measure Jz,f . (b) The high-degree ofcorrelations are shown between Jz,p and Jz,f . . . . . . . . . . . . . . . . . . . . . . . 131

6.8

The left panel shows the pre-measurement (red points) and the final-measurement(open cyan points). The incident photon number per measurement is 4.2 × 104 .
The optimal subtraction Jz,f − βJz,p is shown in the right panel. The dashed lines
indicate the calculated QPN, and the solid lines indicate the calculated standard
deviation of the measured data. The spin noise reduction here is R = −6.3(4) dB. . 132

6.9(a) Pulse sequence used for measuring the initial contrast with no QND probing
applied. (b) Pulse sequence for measuring final contrast Cf with the QND probing.
(c) Open green squares is the Ramsey fringe with no probing, black points is with
2.3×104 photons per measurement, and red triangles is with 4.2 × 104 photons per
measurement. We do not observe any significant phase shift due to the probing. . . . 134

6.10 Measured fractional contrast Cf /Ci versus the incident photon number per measurement. The solid line indicates the calculated expected loss of contrast from free-spacescattering, described in Appendix C. . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

xxii
6.11 Relative spin noise reduction R (cyan circles), fractional contrast loss Ci /Cf2 (redcircles), and the corresponding Wineland parameter ξ (black circles). At the optimal
photon number, we directly measure a Wineland parameter of -1.8(7) dB. Expected
R given our estimated quantum efficiency of Q = 0.28 (cyan line), expected contrast
loss via free-space scattering (red line), and the corresponding expected Winelandparameter (black line, see Methods). . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.1(a) Clock pulses are the black pulses, measurements of ensemble A are the redpulses, and the transports are shown as the green and purple pulses. The Bloch
spheres depict the spin state distribution at various points during the sequence. (b)Ramsey fringe measured by varying the phase of the final π/2 pulse. (c) Pre andfinal measurements of ensemble A and B. (d) The final measurements of ensemble
A and B show strong correlations, allowing for the subtraction of the common-modelaser phase noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.2(a) Schematic showing the transported atomic cloud for squeezing/readout of eachensemble. The clock laser propagates vertically, addressing both ensembles in asynchronous fashion. (b) Fractional frequency stability given by the total Allan
deviation for the CSS - CSS comparison (green open squares) and the SSS - SSS
comparison (black circles). The green dashed line is the estimated QPN limit, andthe dotted blue line is the SQL. We note that we plot the observed comparison√
stability, so we have not divided by 2 to estimate single clock stability. . . . . . . . 143

7.3

The optimal estimators calculated for the spin-squeezed clock comparison data. (a)Optimal estimator βA for the pre-measurement of A. (b) Optimal estimator βB for
the pre-measurement of B. (c) Optimal estimator βD , which is the differential scale
factor between the two ensembles. The shaded region indicates the ± 1 standard
deviation for the datapoints beyond experimental shot 150, and the dashed line isthe corresponding mean value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

xxiii
7.4

Measured shift of the cavity-like mode versus vertical displacement of the atomic
cloud. The green dashed line is the Gaussian fit of the form of Eq. 7.8, giving afitted σz = 130µm.

7.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Measured cavity shift noise with artificial white frequency noise imposed on the clock
laser. Each ensemble (red and blue circles) becomes noisier as the magnitude of theadded noise is increased, but taking the difference (open green squares) gives a 10
dB common mode noise reduction factor. . . . . . . . . . . . . . . . . . . . . . . . . 150

7.6(a) Pulse sequence used for performing spin state tomography. (b) Relative spin noise
after a rotation angle of Ψ. We infer the spin noise just after the pre-measurementby assuming the noise in the pre and final measurement is uncorrelated. The black
dashed line is a unitary spin squeezed state, and the dashed-dot line indicates our
expected anti-squeezing for a quantum efficiency of 0.28.

7.7. . . . . . . . . . . . . . . 153

Independence of atomic ensembles (a) Measured correlation coefficient between Jz,Aand Jz,B versus the separation between the ensembles. (black circles). The blueline is a Monte Carlo simulation. (b) Corresponding change of the QPN due to thefinite overlap of the ensembles, with numerical Monte carlo simulation (blue) andanalytical calculation (orange). At our operating ensemble separation, the change
to QPN is 0.05 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Chapter 1

Introduction

One common thread in fundamental physics is the continual “zooming in” on the universe.
It also allows one to periodically “zoom out” and see the world as a whole. It is this simple pursuitthat has led us through developments of all branches of physics. The development and refinementof the tools used for investigating the universe aids this journey in a critical way. We will highlighta few aspects of this journey that leads to the current world of optical atomic clocks.
First, we need atoms. In 1909, Rutherford instructed his colleagues Hans Gegier and Ernest
Marsdeen to fire alpha particles (later known to be helium nuclei) at very thin sheet of metal foils
like gold and silver, and non-intuitively, look for diffuse reflections of the alpha particles at angles
greater than 90% [1]. To their astonishment, sitting in the dark and counting scintillations by eye,they found that these alpha particles were indeed being scattered at large angles. Rutherford sat
down for a few years to think about the ramifications of this experiment, and eventually developed
the picture of the modern atom [2]. This new model of the atom had a central nucleus with net
positive charge1 , and electrons orbiting outside that nucleus. We then have Niels Bohr to thankfor explaining that these electrons occupied quantized orbits around the atom [3]. It will be thetransition of atoms between these quantized energy levels that we will take advantage of in building
an atomic clock.
The question is - how can we probe these orbital levels inside the atom? I.I. Rabi showed theway forward by establishing the magnetic resonance method [4]. This involves tuning an oscillating
1

Neutrons were not known to exist at this time, but Rutherford was careful to simply state that the nucleus musthave a net positive charge.

2
magnetic field until it hits the same frequency as the Larmor precession frequency of the electron
inside the atom. When the oscillating field is on resonance, the radiation induces the nuclear
spins to transition to another spin state. One could use this technique to peer into the spectrumof different atoms and molecules, and measure the energy spectrum with a new level of precision.
Surely this method would lead to some fundamental discoveries about the nature of the atom. Willis
Lamb and Robert Retherford continued to zoom in when they employed microwave spectroscopy
and found a shift between the 2 S1/2 and 2 P1/2 levels in the hydrogen atom [5]. This discovery
spurred the exploration and further understanding of quantum electrodynamics (QED). Then, in
1950, Norman Ramsey introduced a new method - later named Ramsey spectroscopy - where one
uses two phase-coherent pulses seperated in time (or space) to probe the atomic resonance [6]. It
was clear that continued work in spectroscopy would yield further paths to explore physics.
I.I. Rabi first suggested using the methods of magnetic resonance techniques to build an
atomic clock in a 1945 lecture [7]. The idea is that one uses two energy levels of a particularatom as a frequency reference. The microwave radiation would be used as the pendulum, andone can count the waves of this radiation to provide the ticks of the clock. Sources of microwave
generation on their own are imperfect, possessing drift and random frequency deviations. However,by performing spectroscopy on the atoms, we can actively tune the microwave radiation to be on
resonance, thus fully stabilizing the light and giving us an atomic clock. The early developmentof atomic clocks moved very rapidly. The first atomic clock was an ammonia maser, with thecontruction led by Harold Lyons [8]. Further improvement in maser technology came in 1958 withthe development of the hydrogen maser in Ramsey’s group (led by Dan Kleppner) [9]. Aroundthe same time, scientists at the National Physical Laboratory developed the first cesium beamclock [10]. At what seems like a shockingly fast pace to us now, in 1967 the cesium hyperfine
transition frequency was chosen to redefine the second in the International System of Units (SI).
While the early development of microwave atomic clocks was revolutionary - we should continue to seek more refined tools to investigate nature. It soon became clear that these were waiting
for us in the optical domain. The invention of the laser and related optical technologies was another3
explosion of ingenuity and fast progress [11]. Venturing into the world of visible wavelength lasers
with femto-second timing resolution allows us to sub-divide the second into ≈ 106 more cycles ofradiation per second as compared to microwave clocks. This leads to an intrinsic improvement- missing one cycle of the radiation corresponds to a greatly reduced timing error. Shortly after
their first realization in the lab, optical clocks began to surpass their microwave counterparts inall aspects of their performance [12, 13]. Progress since then has been steady, with reported accuracies of ∆ν/nu = 10−18 [14, 15, 16]. State-of-the-art comparisons between independent clocks
are now reaching the 10−17 level at an averaging time of 1 second [17, 18]. Further, by comparing
sub-ensembles within a single chamber one can leverage the incredible precision and measure thegravitational redshift over a distance of just 1 mm [19].

1.1Characterizing clocks
It is important to establish metrics for clocks, to evaluate the performance of a given clockby asking the question - how well can a given clock tell time? Viewing this question from anotherperspective - how sensitive is a given clock to new physics and perturbations? The more sensitive
it is - the more we can learn.
There are two broad metrics for atomic clocks - accuracy and stability. Accuracy refers to
the fact that the atomic energy levels chosen for the clock transition are not completely immune
to systematic effects from the environment. For example, the largest systematic shift in a roomtemperature strontium atom clock is the dc Stark shift arising from black body radiation of thesurrounding environment. The fact that the thermal environment radiates microwave photons
means there is a finite electric field due to that radiation. Due to differential polarizability of theground and excited state, that electric field leads to a shift in the apparent clock frequency. In
other words, if you make a clock in the lab and someone across the world makes a similar clock,
but puts the atom in a very cold environment, the two clocks will disagree. It is the job of theclock scientist to carefully calibrate and understand the systematic shift. If we both write down
what we believe to be the shift, and our uncertainty in that shift, we can then compare clocks to

4
see if we still agree.
My thesis will focus more on aspects of stability, which is effectively a statement of how
noisy a clock (or laser) is. Put in other words, stability refers to how long it takes to reach agiven statistical uncertainty of the clock frequency. A clock that has more shot-to-shot noise will
have a worse stability - it will take longer to reach a given statistical uncertainty. We wish to
be quantitative about stability. Natural experience in measurement science is that when you take
datapoints close together in time, they end up more clustered together with some shot-to-shotnoise. This shot-to-shot noise could have several sources, such as technical electronic noise, or more
fundamental noise. But if we go home for the night, and come back the next day, we might find thatthe relative frequency had changed. There can easily be environmental conditions that influence
noise on longer time scales. We desire a statistical quantity that can capture this behavior - andby far the best for clocks is the Allan deviation,sσy (τ ) =(ȳi+1 − ȳi )2
2


(1.1)

where y = ν/ν0 is the fractional frequency, and the bar indicates that we have averaged over atime interval τ . This statistical operator allows us to separate and understand the influence ofnoise on different time scales. White frequency noise in the Fourier domain will result in an Allan√
deviation with a slope of 1/ τ , whereas flicker frequency noise (1/f ) will yield a plataeu in the
Allan deviation, indicating that more data is not giving a more precise measurement. We will use
the metric of the Allan deviation when looking at the stability of ultrastable lasers as well as the
optical clock. In fact, the fundamental noise floor of optical reference cavities comes in the formof a “flicker floor” in the Allan deviation. For clocks, the limit is usually arising from quantum
projection noise or aliased frequency noise, which we discuss in the following section.

1.2

Limits to clock performance
Now that we are acquainted with how we characterize clock performance, we ask what are

the limits to the performance? The finite signal to noise ratio of a given clock measurement will

5
ultimately set the limitations to the clock performance. There are two general sources of noise in
the clock - one coming from the interrogating clock laser and the other coming from the atoms
themselves.
The fundamental noise that arises from the atoms of the clock is quantum projection noise(QPN) [20]. This noise is associated with the projective measurement of a system placed into asuperposition of two states |↓i and |↑i. The standard deviation (denoted in this thesis by ∆) of the√atom number in |↓i and |↑i respectively is ∆N↓ = ∆N↑ = N /2. The fluctuations in the two clock√states are perfectly anti-correlated, so the standard deviation of the difference is ∆(N↓ −N↑ ) = N .
These population fluctuations are converted to fluctuations of the clock frequency by the slope of
the Ramsey fringe. The highest performing microwave clocks - the fountain clock - operate at thelimit of QPN [21]. Today, state of the part optical clocks approach the QPN limit [17, 18].
A primary roadblock to observing a clock operating at the noise limit given by QPN is laser
noise. Due to finite deadtime in the clock, we do not track the laser phase at all times. This meansthat the stroboscopic probing of the atoms leads to aliased high frequency noise, known in the clock
community as the Dick effect [22, 23]. The particular pulse sequence (Rabi or Ramsey) and thedead time of the clock dictates the magnitude of the laser noise impact on the clock [24].

1.3

Overcoming the limits
Viewed from a practical perspective, both the classical and quantum limitations to the noise

performance are critical to understand, optimize and improve and subsquently improve. For theclassical laser noise - we take a two-pronged approach. First, we push the boundary of stablelasers down to the 10 mHz laser linewidth by developing state-of-the-art lasers based on cryogenic
silicon cavities [25, 26, 27]. This laser stabilization pushed our clock performance up against the
QPN limit [17]. With the motivation of removing laser noise from clock comparisons, the Ye groupdeveloped the “self-synchronous” measurement technique, by comparing two parts of the sameensemble [28]. This was the technique employed in the measurement of the gravitational redshift
with a single atomic cloud [19].

6
The other fundamental limit - the quantum projection noise - can be a tricky noise source toimprove upon. Of course, one way forward is to make the QPN small by operating at large atom
numbers. However, there will always be some technical limit to atom number, so it is interesting
to wonder if we can obtain clock performance beyond. This can be done by introducing spinsqueezing. We can prepare states with reduced noise along a particular direction of the Blochvector, at the expense of increased noise along the orthogonal axis (anti-squeezed quadrature).
Since Ramsey spectroscopy is primarily sensitive to one quadrature of the spin state, we can alignthis spin-squeezed state along the phase-sensitive axis. There are a multitude of ways to generatesuch non-classical spin states [29]. One involves performing quantum non-demolition measurementsof the joint atom-cavity system [30, 31, 32, 33, 34]. With correlated quantum noise between themeasurements, taking the difference allows one to go beyond the QPN limit. This will be themethod we use when generating entanglement and using that in direct optical clock comparisons.
There has been much work on integrating spin squeezing in optical clocks [35, 36]. However, until
this work, there has not been an observation of an optical clock comparison operating below the
QPN limit. By combining the best clock laser technology with clock comparisons with cavity QED
squeezing and readout of sub-ensembles, we achieve this long-standing goal.

1.4

Outline of this thesis
This thesis discusses efforts to push the frontier of optical clocks on both the local oscillator

and atomic reference side.
Chapter 2 describes the details of the Si4 system, the first cavity to reach the thermal noise
floor below an operating temperature of 124 K, achieving this at both 4 K and 16 K. We present
a thorough noise budget of this system and stability measurements. Using the measured thermal
noise floor, we infer the loss angle of the high reflectivity dielectric coatings, providing valuable
information for both reference cavity and gravitational wave detector research.
In Chapter 3, we describe experiments involving classical atomic clocks that push the frontierof precision measurement science. These include demonstrating clock comparisons approaching the7
QPN limit, placing limits on dilaton dark matter, and optical frequency ratio measurements at the
10−18 level for the first time.
In Chapter 4, we discuss various aspects of a spin-squeezed clock, including design choices,probing atom-cavity system, and the influence of atomic motion on considerations of this probing.
In Chapter 5 we describe the construction of “Sr3”, the spin-squeezed clock apparatus.
Chapter 6 focuses on the first result coming out of Sr3, where we generate entanglement
directly on the Sr clock transition. We measure a Wineland parameter of -1.8 dB.
In Chapter 7, we apply spin-squeezing to a direct optical clock comparison. We directly
observe an enhancement in the clock performance of 2.0 dB beyond the QPN limit. This constitutes
the first direct observation of a spin-squeezed neutral atom optical clock operating below QPN.

Chapter 2

Ultrastable lasers based on optical cavities

2.1

Motivation and outline
Ultrastable lasers are at the core of most of the world’s best precision measurements, including

optical atomic clocks [37], tests of relativity [38, 39], and gravitational wave detectors [40]. Improved
optical coherence will open the door for more precise optical clocks [41, 42]. These lasers will further
studies in fundamental physics in several aspects, including the search for dark matter [43], atombased gravitational wave detectors [44] and many-body physics [45]. Furthermore, optical clocks willplay a defining role in the next generation of optical timescales [46, 47, 48]. In our lab, over the past
few years we have demonstrated numerous specific applications of ultrastable lasers. For example,
one involved comparing the silicon optical cavity against the Sr clock transition, setting a boundon various coupling of dark matter to the Standard Model [49]. This experiment leveraged thefact that this cavity-atom comparison is linearly sensitive to the fine-structure constant. A second
example was the first realization of an all-optical timescale [50]. In that work, the flywheel for thetimescale was the optical local oscillator instead of a microwave source. This all optical timescale
out-performed timekeeping that relies on microwave technology, showing the clear advantage of
optical clock technology.
All of these applications greatly benefit from improved short, mid, and long-term laser frequency stability. Looking forward to our spin-squeezed optical clock discussed in the later half of
this thesis, we show that possessing a low noise optical oscillator is a crucial ingredient to pushingthe quantum frontier of clocks. Without a pristine optical oscillator, it would become difficult or9impossible to debug various sources of noise, and achieve optical coherence at state-of-the-art levels.
We are thus motivated to explore the fundamental limits of ultrastable lasers, as progressin their stability directly advances the performance of optical atomic clock technology. Other
approaches for generating ultrastable laser light are being pursued with promising results [51, 52]
However, to this date, stabilization to optical reference cavities results in the lowest noise laser
light. In particular, the highest performing systems are based on cryogenic silicon cavities, taking
advantage of the reduced thermal noise floor at lower temperature, as we discuss in Section 2.2.1.
We begin with a short introduction to ultrastable lasers based on optical cavities, and thefundamental limit of thermal noise of these systems. We then describe in more detail a opticalcavity referred to as ”Si4”, which was a 6 cm cavity operating near 4 K cooled by a closed-cycle
cryocooler 1 . The noise budget, stability measurements, drift characterization, and inferring loss
angle will all be discussed. At the end we will also describe some of the details of the Si3 cavity,
which has been the work-horse optical cavity for Sr3 and all of the other Sr experiments at JILA,
including Sr1, Sr2 in Jun’s lab, and the Sr experiments in labs of Adam Kaufman and James
Thompson.

2.2

Lasers stabilized to optical cavities
All lasers start as free running, with their frequency dictated by the lasing conditions inside

the laser cavity. These free running lasers will have both short-term and long-term fluctuations, due
to a myriad of environmental perturbations through temperature, pressure, pump characteristics,
etc. Following the footsteps of our wiser predecessors, we know that one can make the frequency
of a laser more stable by locking to an optical cavity. From the outside a cavity looks like a simple
object - two mirrors facing one another, both fixed to a common cavity spacer. If the light enters thecavity exactly on resonance, it will experience a phase shift of modulo 2π, and thus exits the cavity
with no relative phase shift. The basic idea of using the optical cavity as a frequency discriminator
1

We use the past tense because since this work, Si4 has been upgraded to Si6 lead by Dhruv Kedar. This new
Si6 system implements crystalline optical coatings, with the motivation of further reducing the fundamental thermal
noise floor from the coatings.

10is that when the laser is detuned slightly from the cavity resonance, it will pick up a phase shift.
We can imagine that the magnitude of this phase shift for a given frequency detuning will dependon the quality factor of the cavity resonance . Cavities with a longer length or a higher finesse
have narrower cavity resonance linewidths. After undergoing many bounces in the cavity, someof the light will leak out of the input mirror - available to detect and measure. The leakage lightcoming out of the cavity can be described in I − Q space, which is just fancy language for saying
the leakage light has a real (I) and imaginary (Q) component. For example, if we can probe an
optical cavity and sense the corresponding Q-quadrature, or the dispersive response, then we canuse this as a feedback signal to the laser, to keep it on resonance.
This is the underlying idea of the Pound-Drever-Hall technique [53]. In the PDH scheme, one
applies sidebands to the laser well outside the linewidth of the optical resonator. When the laser
is incident on the cavity, the sidebands will simply bounce off of the front mirror, and they simple
reflect the present phase of the laser. Meanwhile, the carrier enters the cavity, and will “buildup”
in the cavity due to the high finesse resulting in many bounces per photon. This light experiences
some phase shift due to detuning between the laser and the cavity, and will then leak back out of
the cavity carrying that information with it. At low Fourier frequency below the cavity pole of thecavity, the PDH error signal is a frequency discriminator. At Fourier frequencies above the cavity
pole, the frequency detuning is integrated and becomes phase information. Sophisticated servo
loops will take this cavity pole in the transfer function into account, although we have found thisis not always necessary. With sufficient bandwidth, we can lock the laser to the cavity resonance
down to the mHz level.

2.2.1

Fundamental limit - thermal noiseState-of-the-art optical cavities are limited in their length stability by coupling to the thermal
environment. The presence of mechanical loss in the cavity constituents (spacer, substrates, andmirror coatings) leads to mechanical displacements which directly impact the frequency stabilityof the laser locked to the cavity [55]. A great overview of the different sources of thermal noise forFractional frequency noise at 1 Hz (Hz−1 )

11

10−31

Coating Brownian
Substrate Brownian
Substrate thermo-elastic
Spacer Brownian

10−32
10−33
10−34
10−35

4K
6 cm

16 K
6 cm

124 K
21 cm

288 K
40 cm

Figure 2.1: We illustrate the major contributions to the thermal noise for three optical cavities
at various temperatures and with length ranging from 6 cm to 40 cm. We use the coating loss
angle φc from our extraction [54] such that the noise values better match our measured frequency
noise. Fractional noise associated with coating and substrate scales inversely with the cavity length.
Substrate thermo-elastic and spacer Brownian noise are below 10−36 Hz−1 for the cryogenic systems.

12
rigid cavities can be found in both Chapter 3 of Michael J. Martin’s thesis and the book chapter
within the thermal noise book referenced here [24, 56]. We will briefly overview a few sources ofthermal noise.
When Einstein first considered Brownian motion, he thought about the random motion ofparticles suspended in a liquid. Einstein’s work is considered to be an antecedent to the fluctuationdissipation theorem. The fluctuation-dissipation theorem states that the thermally-driven fluctuations of a mechanical system can be directly tied to sources of friction or damping. The componentsof the optical cavity are bulk mechanical objects, and we can think of the system as responding
according to a modified Hooke’s law [57],F (f ) = − (1 + iφ) kX(f )(2.1)

where F (f ) is an applied force in the Fourier domain, k is the spring-constant, X(f ) is the displacement in Fourier domain, and φ is the loss angle. We refer to φ as the loss angle, because thefractional energy loss per cycle is 2πφ.
The task of calculating the Brownian motion for a macroscopic optical interferometer was
made much easier in 1997 when Yuri Levin published a direct approach [58]. Before Levin, themethod for calculating the thermal noise involved a computationally expensive normal-mode analysis. The conceptual idea of Levin’s approach is to apply an oscillating pressure (or force) tothe observed surface, and the displacement of the surface responds according to the fluctuationdissipation theorem. One very important takeaway from Levin’s approach is the role of a lossy
mirror coating. Levin’s analysis results in a thermal noise contribution due to localized dissipation on the surface (from the mirror coating) that scales as 1/r02 , where r0 is the beam radiuson the mirror. This means that that the loss angle of the coating can be critical for small beam
sizes (like in our optical cavities). Since the previously used normal-mode analysis assumes homogenous mechanical dissipation and thus ignored the possibility of localized dissipation from thecoating, Levin’s approach was pivotal for the development and understanding of precision opticalinterferometers.

13
Besides Brownian motion of all of the cavity constituents, there are several other types ofthermal noise. One that is important for ULE glass cavities that employ fused silica substrates
is substrate thermo-elastic noise. Thermo-elastic thermal noise refers to the fact that there arefundamental temperature fluctuations, and through the finite coefficient of thermal expansion, willlead to fundamental length fluctuations. A similar source of thermal noise is coating thermo-optic
noise, where the length fluctuations arise through the refractive index of the coating. These sourcesof noise have much longer correlation length, making it fundamentally different from Brownian
thermal noise.
Calculated thermal noise contributions for three different cavity systems in our lab are shownin Fig. 2.1. We note that for the cryogenic silicon systems, by far the dominate source is coating
Brownian noise. Due to the small mechanical loss of crystalline silicon, both the substrate andcavity spacer Brownian terms are negligible. For the room temperature ULE cavity with fusedsilica substrates, you can see the significance of the other sources of thermal noise.

2.3

Technical noise budget of Si4
To build a cavity-stabilized laser that is limited by the thermal noise, one has to characterizeand control all of technical sources of noise. In this section, we outline various sources of technicalnoise for the Si4 cavity system.

2.3.1

Temperature

Temperature fluctuations couple to the cavity length via the coefficient of thermal expansion(CTE). The expansion of the cavity and the corresponding change in fractional frequency can be
simply expressed in a Taylor expansion
∆L
∆ν
1
=
= α(T0 )∆T + α0 (T0 )∆T 2 + O(∆T 3 ),
L
ν
2

(2.2)

where ∆T is the temperature difference with respect to T0 , α(T0 ) is the coefficient of thermal
expansion evaluated at T0 , and α0 (T0 ) is the slope. This equation is what motivates some of the14more technical design decisions when considering what material to use and what temperature tooperate at. For silicon, there are two zero-crossing of α(T ), one near 123 K and another near 16 K.
As the temperature approaches zero, the CTE tends to zero, and so another strategy is to operate
at the lowest temperature possible. However, cooling down to such low temperature can introduce
detrimental vibrations induced by the cryogenics system used. This explains the natural decision
to explore the 124K operating point first, where an open-cycle cryogenics system using boiled liquid
nitrogen for cooling. Wei Zhang, the post-doc who was leading the initial Si4 effort, always said
that he would feel the cryocooler moving with his fingers and remark “that’s no good home for acavity”.
The Si4 system mainly operated near 4 K, and for a short time, near 16 K. The system iscooled by a commercial cryostat system from Montana Instruments. At the time, this system wascustom built, designed in collaboration between Montana Instruments and the stable lasers team,which at the time was Wei and Jun. The cavity operates at 4 K using a two-stage Gifford-McMahon
closed-cycle cryocooler [59, 60]. The temperature fluctuations of the cryocooler are typically 20 mKat an averaging time of 6 s. Given a CTE of ≈ 2 × 10−11 /K at 4 K, the corresponding frequency
instability is on the 10−13 level, which is several orders of magnitude above the thermal noise
floor. To suppress temperature fluctuations from the cryocooler, we use a custom, multi-stage
thermal damping system [26]. The shields are designed to maximize their heat capacity, while theconnections between the shields have a low thermal conductance.
The first shield is coupled to the 30 K stage of the cryocooler, and initially had no active
temperature control or monitoring. The cooling power for the next shield, called the active shield,
comes from the second stage of the cryocooler (2.2 K) through a mechanically flexible connection
called the cold finger. Between the cold finger and the bottom of the active shield, a cylindrical plate
made of holmium copper (HoCu2 ), which has a specific heat 2500 times larger than copper at 4 K,
is inserted to passively suppress temperature fluctuations from the cold finger. The temperature of
the active shield is controlled using a resistive heater. For additional thermal damping, we insert
a passive shield between the active shield and the cavity, which is mounted on a cylindrical ring

15
made of G10 fiber glass (not shown). The low thermal conductivity of G10 and the ring’s small
cross-sectional area (96 mm2 ) minimize the thermal conductance between the two shields. The
bottom of the passive shield is constructed out of HoCu2 . A cup-shaped structure made of G10
is attached to the base of the passive shield, upon which the cavity is supported. The support
structure has three nested layers to increase its effective thermal path, and has three 1 mm-long
G10 rods on top to support the cavity. The contact area of each rod is 0.75 mm2 to minimize thethermal conductivity.
In order to evaluate the impact of temperature fluctuations on the cavity we directly measured
the thermal response transfer function (Figure 2.2). We apply a step of the active shield temperatureof 0.5 K, and observe the low-passed response of the passive shield by plotting the temperature(dashed red). The optical cavity response is measured by a heterodyne beatnote against the MJM
cavity (blue). The thermal model is equivalent to an electrical second-order resistor-capacitor lowpass filter. The time constant is ≈ 1400 s between the active and passive shields, and is ≈ 600 s
between the passive shield and the cavity.
This was the state of the Si4 cryogenic system as of the end of 2017, after our first result.
In order to push this system to the fundamental thermal noise limit, we sought to improve various
aspects of the system. One important improvement was made regarding the first radiation shield.
Initially, the first radiation shield was a relatively thin aluminum radiation shield coupled to the
30 K stage of the cryocooler. On top of this, it was not actively controlled, only the 30 K stageof the cryocooler was controlled, which is far from the shield itself. We discovered a significant
coupling between the temperature of this supposed 30 K shield and the frequency of the cavity.
After discovering this coupling between the 30K stage and the cavity, we wanted to redesign andupgrade the radiation shield to a copper radiation shield with more thermal mass. Further, initially
we had no thermal sensors on the Al radiation shield. After opening up the chamber to begin our
upgrades, we attached a thermal sensor to this Al radiation shield and cooled back down (without
the cavity), and to our surprise we found that the shield was closer to 80-90K, with a massive 10
K gradient along the vertical extent of the shield; this made us happy that we already planned to16

Figure 2.2: Time domain step response of the Si4 thermal damping system. A step function
is applied to the active plate/shield, and we measure the response on the passive shield, and thecorresponding frequency change of the cavity measured via a heterodyne beatnote against the MJM
cavity.

17
replace this shield! In hindsight - aluminum is not a good choice for the low temperature cryogenic
shields as compared to copper given the thermal conductivity, and the thin 1 mm thick walls both
gave a small thermal mass, and a small cross-section for propgation of heat to the top of the shield.
After replacing this Al shield for the Cu shield - we were able to cool that shield down to 40K withrelative ease, and achieved a sub 1K gradient along the entire extent of the shield.

2.3.2

Vibration noise

Vibrations are a significant source of short-term instability for the Si4 system because of theclosed-cycle cryostat. We minimize vibrations coming from the cryocooler by carefully designing
the mechanical layout of the system [26]. The mechanical design (Fig. 2.3a) isolates the cavity
from the vibrations of the cryocooler. The cryocooler sits on a main plate that is firmly fixed to amonolithic aluminum pillar that is bolted to the concrete foundation of the laboratory. There arethree mandatory mechanical links between the cryocooler and the chamber: a cold finger to cool
the active shield, a thermal link connecting the first stage of the cryocooler to the 30 K shield,
and a vacuum bellows to isolate them from the environment. All other mechanical connections
are eliminated by placing the vacuum chamber on a split plate, which is mechanically separated
from the main plate (see the top view in Fig. 2.3a). The gap between the main plate and the split
plate is 1.5 mm, and is exaggerated in the figure. The split plate is supported by three rods thatare bolted onto an active vibration isolation (AVI) table. The AVI rests on an optical table. The
optical table is supported by four aluminum blocks, roughly 1 foot tall, giving the setup an overall
low profile for minimizing tilting motion.
Our initial characterization of the vibration sensitivty of the Si4 system was obtained byusing ambient vibrations as the drive. We simultaneously measured the vibration noise and aheterodyne beat between the silicon cavity and a stable laser at 698 nm based on a 40 cm-longultra-low expansion glass (ULE) cavity (called the MJM cavity throughout this thesis) using a Yb
frequency comb. We found that the beat frequency against the MJM cavity was highly correlatedwith the vibrations measured by an accelerometer placed on the AVI table at Fourier frequencies

18

1.0

(b)

0.8
Coherence

(a)

0.6
0.4
0.2
0.0

Vibration sensitivity (g −1 )

(c)

10−9

H1
H2
Z

10−10
10−11 110102
Frequency (Hz)Figure 2.3: (a) The mechanical layout of the Si4 system. The top view shows how the split plateis mechanically isolated from the main plate, and the only real physical connection is a flexible
thermal connection. (b) Measured coherence between the accelerometer on the AVI and the beatnote of Si4 against a reference laser. (c) Vertical vibration sensitivity in fractional frequency units
per g of acceleration, versus the azimuthal angle between the silicon crystal axis and the three-axis
supports.

(a)(b)
Fractional frequency noise (1/Hz)

Vertical sensitivity (10−10 g−1 )

1.0
0.8
0.6
0.4
0.2
0.0
−0.2
−40

−30

−20

−10
0
10
Angle (degrees)

20

30

40

10−29

10−30

10−31

10−32

10−33

100

101
Frequency (Hz)102

Figure 2.4: (a) Vertical vibration sensitivity of Si4 measured versus the relative azimuthal angle
between the tripod support and the crystal axis of the silicon crystal. (b) Improved high frequency
noise performance of Si4 by sensitivity improvement, and optimization of AVI position.

19
above 10 Hz. To quantify the correlation between vibrations measured on the AVI table and thefrequency noise of the beat note we introduce the coherence : Cai ν (f ) = |Sai ν (f )|2 /[Sνν (f )Sai ai (f )].
Here, Sai ν (f ) is the cross-spectral density between frequency fluctuations of the beat signal andacceleration noise in the ith direction (i = x, y, z), and Sνν (f ) and Sai ai (f ) are the autospectral
densities for frequency noise and accelerations, respectively. Three accelerometers are mounted
to the AVI table, one for each direction labelled as H1 , H2 and z. The coherence between eachaccelerometer and the phase noise of the beatnote are shown in Figure 2.3b. A nonzero coherence
value at a given f indicates that the laser noise and vibrations are correlated at that spectral
frequency. This is an important prerequisite for measuring transfer functions from accelerations
to frequency noise, Hai (f ) = Sai ν (f )/Sai ai (f ), since phase estimates in the cross spectrum areonly trustworthy where significant frequency-domain correlation exists. From 10 Hz to 400 Hz, wemeasure appreciable coherence between frequency noise and vibrations along all three directions.
Within this frequency range, we compute transfer functions from each accelerometer on the AVI
table to the beat frequency spectrum by averaging Hai (f ) over an hour-long dataset. The measured
transfer functions along each direction are shown in Figure 2.3c. Si4 reached the level of 10−10 g −1
for 10 - 100 Hz, with some resonant features at higher frequencies. We have since made significant
improvement in lowering the vibration sensitivity by paying attention to the mechanical support
structure and mounting in the design of Si6 (an effort led by Dhruv Kedar).
After our initial result of [26], we sought to improve the short-term stability. Due to theanisotropic nature of the silicon crystal [61], the vertical vibration sensitivity is sinudoidally modulated, with a period of 120 degrees. When the cavity was initially installed, the vertical sensitivity
versus azimuthal mounting angle had not yet been experimentally checked, although Wei had placed
the cavity to the simulated zero-crossing of the sensitivity. We had reason to suspect how close wereally were to the zero-crossing, since the simulated zero-crossing can be modified by imperfectionsin the mechanical model of the cavity and in the finite-element-modeling. Since we were planning toopen the vacuum chamber in order to make the upgrades to the thermal system as discussed in Ch.
2.3.1, we planned to measure this. We directly measured the vertical vibration sensitivity versus the20
relative angle between the mounting points and the crystal axis orientation of the silicon. We wereable to reduce the vertical vibration sensitivity to (5 ± 2) × 10−12 /g at a driving frequency of 9.5
Hz. The horizontal vibration sensitivity in each direction was measured to be (2 ± 1) × 10−10 /g.
A reduction of vibrations at the cavity was obtained by fine tuning the relative position of thevacuum chamber and the cryostat. The combined improvements in sensitivity and noise provide
a tenfold reduction in the frequency noise power spectral density (PSD) for Fourier frequencies
of 10-50 Hz compared to previous work [26]. The horizontal sensitivity depends critically on thetransverse positions of the cavity mirrors with respect to the optical axis of the cavity, and thus amore precise centering of the mirrors with respect to the symmetry axis of the cavity will provide
a reduced horizontal sensitivity.
One other aspect related to vibration sensitivity is the impact of earthquakes. During a
JILA/NIST clock comparison data point, suddenly all of the clocks came unlock at the same time.
By chance, I went to the NYtimes website and saw that an earthquake had just occured in Mexico(February 16, 2018). We looked up the local seismometer station, and found that one can download
the ground velocity data, and this is plotted in Figure 2.5a. Looking at the beatnote between the
Si3 and Si4 cavity systems (Figure 2.5), we see clear correlation between the two signals. One cansee the arrival of the different components of the seismic waves produced by the earthquake. We
have a remarkably sensitive optical interferometer, that we have attempted to make as insensitive
to vibrations as possible. The fact that we can still see such a clear signal is indicative of therelatively large ground motion coming from earthquakes, even far away. It also reminds you thatlow frequency vibrations are not cancelled by our AVI table. This means that using a lower
frequency sensor of motion, like a tiltmeter, could give enhanced control over the low frequency
vibration environment.

2.3.3

Residual amplitude modulation

One of the most important technical noise sources for Si4 is residual amplitude modulation(RAM). RAM is a systematic effect in frequency modulation based stabilization schemes, which

21

Frequency (Hz)(a)

Velocity (10-4 m/s)

(b)

0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
3.0
2.0
1.0
0.0
1.0
7 minutes

2.0
3.0
0.005

0.010
0.015
0.020
+58165.98
Modified Julian date (MJD)

Figure 2.5: (a) Measured Si3-Si4 beat note. (b) Velocity waveform measured at the Briggsdale
seismic station. We can see the clear arrival of the different types of seismic waves, and thecorrelations with our optical frequency beat note measurement.

22arises when the two modulation sidebands are unequal in magnitude, not exactly 90◦ out of phase, ora mixture of the two effects. From an operational point of view, RAM that is completely stationary
is not a real problem, but in reality, the RAM always has some time-dependent component. This
time-dependent RAM manifests itself as time-varying baseline of the PDH error signal. Since the
PDH servo loop can not distinguish this change in the error signal from real frequency deviations
between the laser and the cavity, this RAM will be written as frequency drift of the locked laser.
There can be various sources of RAM, including time-varying RF pickup, changes in the EOMtemperature, and optical etalons. The most nefarious sources of RAM is time-varying opticalinterference, or etalons. If you have some sort of standing wave in your optical system, thatmeans the carrier and sideband can experience differential phase shifts or amplitude attenuations.
Then, as the lab temperature or pressure slowly varies, then this relative phase shift or amplitudeattenuation will appear as a slowly changing DC offset of the error signal. We point the interested
reader to [62] for an in-depth analysis of the role of etalons in different sections of the optics layout.
We employ active control of the RAM, following the strategy described in [63], with a few
minor differences. In our case, the RAM is detected by picking off the beam with a 90/10 nonpolarizing beamsplitter before the PDH PBS. We demodulate the RAM, with the phase set suchthat we are detecting the I quadrature. The relative phase for the I quadrature is typically set
by maximizing the RAM error signal. This demodulated signal is then sent through a servo loop
filter, and we apply the feedback as a DC voltage onto the EOM. The I quadrature dependence on
voltage is a sinusoidal function of the EOM temperature, and so to optimize the gain and dynamic
range, we tune the temperature so that this slope is maximized. For our setup, the Q quadrature
was always much too small to play a significant role.
The way to diagnose the out-of-loop RAM is by tuning the laser off of the cavity resonance,and observe the demodulated PDH error signal. If this baseline is fluctuating, then you can be
sure that this same fluctuation is impacting the PDH error signal when you tune on resonance. Bytilting optics, controlling temperature and pressure of the overall optics setup, and adding opticalisolation in the correct spots, one can reduce this RAM to the parts-per-million level or better.

23
The measured out-of-loop RAM for Si4 is shown as the red trace in Figure 2.7. We can see a broad
peak in the 10-40 mHz region, which we ascribe to fluctuating etalons in the system.
Of course, this off-resonance test is not perfect, and things could degrade further when you
are on resonance, since you are allowing the beam to enter the cavity. For example, the beam could
strike a piece of dust on a window in transmission of the cavity. This scattered light could make its
way back to the PDH error signal. It will appear as a fluctuating phase shift between the carrier
and the sidebands, and the PDH servo will apply feedback based on this signal. So while checking
the out-of-loop RAM as we have discussed is a first step, one must always check the stability of thestabilized laser by forming a three-cornered comparison with two other lasers.

2.3.4

Photothermal effects

Photothermal effects are those that change the frequency of the cavity depending on theamount of laser light circulating in the cavity. Photothermal effects can arise from different physicalmechanisms. At very low frequency, the absorbed power in the mirror coatings propagates andchanges the length of the spacer. At higher frequencies, the response tends to be dominated by thecoating, since the substrate contribution with silicon is negligible. Unfortunately there are not greatreferences for the different critical parameters. From the perspective of just trying to make thebest laser possible - it is sufficient to measure the transfer function, and then stabilize the intensity
to a high enough degree. In Figure 2.6a, we show the measured photothermal amplitude in units
of Hz/W, where W refers to the amplitude of the modulated power measured in transmission. We
measured the amplitude and phase response at different overall transmitted power, and found thatthe photothermal transfer function strongly depended on this transmitted power.
It is natural to ask what is limiting the transfer function in the different frequency ranges
in Figure 2.6. In order to attempt to understand the ferquency dependence of the photothermal
response, we calculate the expected transfer function based on the model developed in [64]. At
low frequency the dominate contribution is the thermal response of the cavity spacer. At Fourier
frequencies above 10−2 Hz, the photothermal response is dominated by the coating. There are

24

(a)

Amplitude (Hz/W)

109

108

300 nW
210 nW
120 nW
6 nW

107

106

Phase (degrees)

0

(b)

−50
−100
−150
−200 −5
10

10−4

10−3

10−2 10−1 100
Frequency (Hz)

101

102

103

Figure 2.6: (A)Amplitude response of the photothermal effect in Si4. Dashed lines are phenomological fits, and the solid black line is a theoretical estimate. (B) Phase response of the measured
transfer functions.

25
two contributions to the coating photothermal response - the thermo-elastic and photo-thermorefractive. The thermo-elastic effect refers to the fact that a temperature change results in achange in the coating thickness through the coefficient of thermal expansion. The photo thermorefractive effect arises due the the finite dependence of the refractive index on temperature dn/dT .
To calculate these effects, we need to plug in various material properties at cryogenic temperatures,
which can be difficult to assign. Nonetheless, we use the parameters from [64], and plot the resultof the calculation as the black line in Figure 2.6.
We find it difficult to assign the exact physical effect dominating the transfer function at
higher frequency. Photo thermo-refractive noise seems unlikely, since dn/dT tends to get small at
cryogenic temperatures. It is also interesting to note that the phase of the photothermal response
flips by 180◦ as we go from the low frequency cavity response to the higher frequency coating
contribution. Furthermore, the dependence on this photothermal transfer function on the opticalpower might suggest the presence of another light-matter effect.
From a practical perspective, we minimize the impact of the photothermal effect in Si4 byactively stabilizing the transmitted optical power, down to 40 nW. We can thus reduce the impact
of intensity fluctuations to be well below the Brownian thermal noise floor (see Figure 2.7). Further
efforts in our group on coating photo-thermal effect would reveal other interesting effects associatedwith different mirror coatings (in our case, crystalline coatings).

2.3.5

Path length noise

Another critical source of noise is path length fluctuation. When we are considering stabilizing
a laser to a cavity, the critical junction is when you split the light in front of the cavity, wheresome of the light heads toward the cavity and some heads toward either a beat note or toward aparticular optical clock experiment. Since any noise written onto the path length after the split
point is differential, one must be careful that the phase noise introduced in that path is sufficiently
small. For the Si4 cavity, we characterized this noise by performing a loop-back measurement.
We split the light in fiber before the output coupler, and send part through 5 m fiber across the26lab, and then loop the fiber back to the beamsplitter. We close the loop on the path length noiseof the fiber, and beat the cancelled light against the local light. The relative frequency noise iswell below the thermal noise limit of the the cavity. In actual operation, there is additional freespace path that is differential. In order to minimize the impact of noise in this free space path,
we built a brass enclosure that surrounds the entire optical setup. On top of this, we added an
additional layer of isolation around the optics using blackened aluminum foil. Around the optical
table we installed heavy-duty vinyl curtains to further supress both room temperature fluctuations
and pressure variations.

2.3.6

Electronics noise

There are various sources of electronics noise in a cavity-stabilized laser. One source of noise isfrom the photodetector (photodiode + transimpedance amplifier) used for PDH detection. In most
optical cavities, this noise source is dealt with by simply turning up the optical power such that thesignal to noise of the detected error signal is limited by photon shot noise. For Si4, due to significant
photothermal effects, both in noise and drift, we desire a low operating power. To this end, we
employ a resonant photodetector that has a noise equivalent power (NEP) of ≈ 60 nW, allowing
operation at the 100 nW level without significant contribution from the photodiodector. Anothersource of electronics noise can come from the servo used for PDH locking. We used a standard JILA
low delay servo, which is a circuit that has been around JILA for two decades, having undergone
several upgrades. For use in a stable laser system, we found that the low frequency noise had tobe improved. We achieved this by changing the input op-amp from a very high speed op-amp toa lowe bandwidth one, with a lower 1/f corner. The final PDH servo noise is shown in Figure 2.7
as the dotted curve. While this was low enough for Si4, Eric Oelker designed a further improved
servo for Si3 and Si6, to achieve an even lower noise floor.
The full technical noise budget of the 6 cm system is shown in Figure 2.7. We show fits
to the measured noise arising from temperature, Pound-Drever-Hall (PDH) servo, and the PDH
photodetector (PD). The dominant technical noise source is residual amplitude modulation (RAM),27
shown as the red data. The measured out of loop RAM has a prominent peak around 20 mHz, andsuch a broad peak can easily arise from time-dependent etalon effects. The sum of the technicalnoise is well below the predicted thermal noise floor (dashed red line).

2.4

Stability measurements
To determine the instability of the Si4 cavity, we measure a beat between Si4 and Si3 (seeFigure 2.8). Si4 and Si3 are both fiber-noise-cancelled to the comb table located within the sameroom as the cavities. Si3 consists of a 1.5 µm laser stabilized to a silicon cavity which operates at
124 K with a thermal noise floor of 4 × 10−17 [61, 25]. The short-term instability (averaging timesof 0.1 to 10 s) of Si3 is determined by a three-cornered comparison with Si4 and a ULE clock laserat 698 nm. The long-term instability (>10 s) is directly measured by a strontium optical latticeclock. These measurements show that Si3 is at its thermal noise floor for averaging times from 0.1
to 1000 s [17].
Measuring the noise of a beatnote at the 10−16 level and better is no easy task. We often
make use of frequency counters to measure beatnotes. However, while frequency counters are great
devices, they have broad bandwidth, and so they can be very sensitive to other spurious tones in
the RF neighborhood. Furthermore, the shortest gate time of the counter is 1 ms, which can leadto significant aliasing of noise above the Nyquist frequency of 500 Hz. Thus, we found that inorder to obtain reliable noise measurements with the counter, we needed to heavily band-pass thebeatnote signal. We employed an 8-pole active band-pass filter, with a 3 dB bandwidth of 1 kHz,and a center frequency of 10 kHz. At these lower frequencies, the counter can become sensitive toamplitude fluctuations, so our active filter is designed to give a square-wave output.
The modified Allan deviation of this beat, after subtracting the reference laser instability of
4 × 10−17 in quadrature, is displayed in Figure 2.9(A). The modified Allan deviation is calculated
from a 24,000 second long measurement record made with a dead-time free lambda-type counter.
We compute the instability after removing the linear drift of the beat with a magnitude of ∼3 × 10−18 /s. The Si4 instability reaches 6.5 × 10−17 for averaging times of 0.8 < τ < 80 s, which is

28

Fractional frequency noise (1/Hz)

10−28
10−29

RAM
Fiber noise
Temperature (fit)Intensity

PD noise (Fit)
PDH Servo (Fit)
Sum technical noise
Coating Brownian

10−30
10−31
10−32
10−33
10−34
10−35
10−3

10−2

10−1

100

Frequency (Hz)Figure 2.7: Technical noise contributions for the 4 K system, along with the sum of the technicalnoise (pink line). The sum of the technical noise sources is well below the coating Brownian noise(red dashed line).

29

Si4

Optical power control

Si3
295K
40K

Sr

ift
y+dr
abilitSt

Er:comb

4K

Drift

Cryo

f
rep

Counter
UTC(NIST)
Beat
1.5 μm laserCounter

Figure 2.8: Setup for measuring the beatnote between Si4 and Si3. Si4’s radiation shields areshown. The Si3 stability is checked using the Sr atomic clock. Comparison against the UTC(NIST)
timescale allows for calibration of the Si3 drift when the Sr clock is not running.

30
consistent with the predicted thermal noise floor (green shaded region). At the time of this work, we
had limited knowledge of the actual loss angle of the dielectric coatings at cryogenic temperatures.
The uncertainty in the thermal noise floor is estimated from the two published values of the loss
angle at 4 K. [65, 66].
The corresponding frequency noise PSD for Si4 is shown in Figure 2.9(B). The PSD is calculated from the time-series of the beat obtained by the frequency counter. The thermal noise floor of
the Si3 cavity (Sy = 1.7×10−33 /f ) is subtracted from the beat PSD [25]. The Si4 laser is limited by
the thermal noise floor for Fourier frequencies over nearly three decades, from 5 mHz to 2 Hz. We
fit the measured PSD to a function Sy = af −1 and obtain the fit parameter a = 4.12(5) × 10−33 .
The thin noise spikes at 1 Hz and higher harmonics come from the cryocooler vibrations. The laser
deviates from thermal noise at Fourier frequencies below 0.5 mHz, potentially due to etalons ortemperature fluctuations.
We experimentally determine the laser linewidth from a Fast Fourier Transform (FFT) of the
Si3-Si4 beat. The beat is mixed down to 10 Hz and digitized with a analog-to-digital converter.
One example of such a measurement is shown in Figure 2.10(a). We use a measurement timeof 128 seconds and employ a Hanning window, corresponding to a Fourier limit of 10.9 mHz.
The expected linewidth for 1/f frequency noise is given by a statistical distribution [25]. Theq
2 + σ 2 ) = 0.85 in order to estimate the relative
distribution is multiplied by the ratio σSi4 /( σSi3
Si4
contribution of the Si4 laser to the beat linewidth. Here, σSi4(Si3) refers to the thermal noise floor of
Si4(Si3). We repeat this measurement 100 times and plot the histogram of results in Fig. 2.10(B).
The median laser linewidth for the distribution in Fig. 2.10(b) is 16 mHz, representing by far thelowest observed to date for a laser locked to an optical cavity placed inside a closed-cycle cryocooler.

2.5

Linear frequency drift : power dependence
We measure the drift of the Si4 system by counting the beat Si3-Si4 as shown in Fig. 2.8.

This requires careful calibration of the drift of the Si3 system. The Si3 laser is used as the clock
laser for a strontium optical lattice clock, giving a direct measurement of the drift [17]. As an

31

mod σy(A) 10−15

10−16

10−17
10−1

Fractional frequency noise (1/Hz)

(B)

100

101
102
Averaging Time (s)

103

104

10−2

10−1
100
Frequency (Hz)

101

102

10−28
10−29
10−30
10−31
10−32
10−33
10−3

Figure 2.9: (A) Modified Allan deviation for the Si4 cavity. (B) Fractional frequency noise power
spectral density (Sy ) of the Si4 cavity. In both panels, the shaded green region indicates the spreadof the estimated thermal noise floor given two previously published values for the coating loss angle[65, 66].

32

Power (arb. units)

1.0

(a)

0.8
0.6
16 mHz
0.4
0.2
0.0
−0.10

−0.05

0.00
Frequency (Hz)

0.05

20
25
30
35
Laser linewidth (mHz)]

40

0.10

0.20
(b)

Percentage

0.15

0.10

0.05

0.00

5

10

15

45

50

Figure 2.10: (A) FFT of the beat measured at 1542 nm (black circles), fit to a Lorentzian lineshape(red line). (B) Histogram of the measured Si4 linewidths for 100 measurements. The Si4 linewidths
are obtained by multiplying the beat linewidth by 0.85.

33
independent check, the drift of the Si3 system is continuously monitored against a hydrogen maser
from NIST via an optical frequency comb. This maser is then calibrated against UTC(NIST) as
depicted in Fig. 2.8. The long-term linear frequency drift of Si3 is −3 × 10−19 /s with 2.8 µW oftransmitted power. The measured linear drift of Si3 is removed from the Si3-Si4 beat, thus giving
the drift of Si4.
The linear frequency drift of the Si4 cavity is dependent on the transmitted optical power
as shown in Fig. 2.11. We vary the incident power and stabilize the cavity transmission at various
levels as shown in Fig. 2.8. With a cavity finesse of F = 500, 000, total transmission of T = 2 ppm,
and total loss of A = 4 ppm, a transmitted power of 40 nW corresponds to a circulating power of
2 mW. Each time the optical power in the cavity is changed, a frequency transient is observed. Inorder to extract the linear frequency drift, we wait several time constants for the transient to decay
away. Interestingly, the observed time-constant for the settling of the cavity to its asymptotic drift
rate depends on the final power setting, as shown in Fig. 2.12. This means that at the lowest
optical power (40 nW in transmission), we needed to take data for about 1 month to obtain areliable estimate of the frequency drift. We note that in order to achieve high performance at low
optical power by employing resonant photodetectors for both the PDH and the RAM detection,√
providing a shot-noise limited signal-to-noise ratio at 68 nW (input referred noise of 0.15 pA/ Hz).
The linear power dependence of the drift is striking evidence for a new mechanism of length
drift of an optical cavity at low temperatures. The sign of the frequency drift is always negative,
meaning the physical length of the cavity is getting longer over time. The slope of the power
dependence is roughly −7× 10−21 /s/nW. One potential explanation is thermal-induced mechanicalcreep of the mirror coating, where the mismatch in the coefficient of thermal expansion for thesubstrate and the coating gives a temperature-dependent creep. To reduce the impact of opticalpower on the long-term drift, Wiens et al. minimized the irradiation of their mirrors by periodicallyscanning the laser across the cavity resonance to measure the cavity frequency [39]. We present
the first rigorous characterization of a power-dependent frequency drift in an optical cavity. The
lowest operating power we have achieved is 40 nW in transmission, giving a fractional frequency

34

Fractional drift (10−18 /s)

0

−2
−4
−6

Si3, 124 K
Si4, 16 K
Si4, 4.9 K
Si4, 3.8 K

−8
−10

0

1

2
3
Transmitted intensity (W/m2 )4

5

Figure 2.11: Fractional frequency drift in units of 10−18 for Si4 at different operating temperature.
The Si3 drift versus intensity is shown as the open red squares, exhibiting a far smaller sensitivity
to optical power.

35

106

τ (s)

3.8 K
4.9 K
16 K

105

10−9

10−8

10−7

10−6

Transmitted power (W)Figure 2.12: Fractional frequency dift in units of 10−18 for Si4 at different operating temperature.
The Si3 drift versus intensity is shown as the open red squares, exhibiting a far smaller sensitivity
to optical power.

36
drift of −3 × 10−19 /s. This frequency drift is comparable to the previous state-of-the-art obtained
from a 124 K silicon cavity [67].
However, the implication of the current finding is tantalizing in that as we continue to reducethe incident power, we can access an extremely low value of cavity drift, making it possible thatsuch a cavity alone could be useful as a potential time scale. At this low power, the fractional noiseof the laser is higher than that showed in Fig. 2.9 by about a factor of two. The extra noise is dueto the photodetector, and will be mitigated with an improved design. We also note that since thiswork, crystalline mirrors have been implemented in the ”Si6” system. There, the overall magnitudeof the drift is much smaller [68]. This supports our understanding that the drift in the Si4 systemcomes primarily from some physical effect in the coating, or in some light-matter interaction in thecoating. It is interesting to note that a coating with more order (crystalline), yet with generally
higher loss (defects), still gives an overall smaller drift.
The advances presented here point to a clear direction for ultrastable lasers. To operatewith a minimal frequency drift, optical cavities at low temperature will have to operate at very
low optical power. Reduction of the thermal noise will be possible by replacing the conventional
SiO2 /Ta2 O5 mirrors with crystalline mirrors [69, 70]. Such crystalline mirrors have been shown
to exhibit a factor of 10 lower loss angle at room temperature [70]. Increasing the cavity length
further will also reduce the fractional frequency noise. We can now foresee a strong possibility ofachieving an ultrastable cavity with fractional instability < 1 × 10−17 using a continuously-running
closed-cycle cryocooler at 4 K.

2.6

Inferring loss angle of SiO2 /Ta2 O5 coating
As we discussed in Section 2.2.1, mechanical dissipation in the coating often dictates theamplitude of length fluctuations. These fluctuations are referred to as Brownian coating thermal
noise (CTN), and can limit the precision of interferometric measurements. Thus, there has been
considerable experimental effort to characterize the level of CTN present in current optical coatings.

37
There are two distinct methods for determining the Brownian CTN of an optical coating. The firstis referred to as the ”mechanical ringdown approach”. This involves measuring the mechanicalquality factor, Poisson ratio, and Young’s modulus for each coating material. These mechanicalproperties for the coating and substrate are then used to calculate the CTN for a given mirror.
This method of calculating the CTN may not account for all multilayer phenomena in the coating.
The second method, referred to as a ”direct measurement”, determines the CTN of the coating bymeasuring the frequency stability of optical cavities. This approach has the challenge of extracting a
coating loss angle from a single measurement over a broad frequency range, requiring input on othercoating and substrate properties. Due to the challenges in each approach, it is vital that both are
undertaken as independent and complementary research efforts. In this Section, we present direct
CTN measurements of SiO2 /Ta2 O5 using the most stable cavities in operation today at cryogenic
temperatures up to room temperature. To make calculations of the fundamental noise floor of agiven optical resonator, including both reference cavities or gravitational wave detectors, one must
know the ”loss angle” of the coating material. As of right now, all gravitational wave detectors
make use of amorphous coatings, mainly made up of SiO2/Ta2O5 dielectric stacks.
To extract a mechanical loss angle from the measured noise spectrum, we must performa characterization of all other noise sources in the reference cavity systems. The 124 K system
has been thoroughly characterized as described in [61]. Since that work we have made several
improvements to the setup, including active stabilization of the transmitted optical power, active
temperature control of the outermost vacuum chamber, and improved optical quality of the optics. All of these improvements are primarily focused on the lower frequency noise, allowing us tocharacterize the 1/f thermal noise over a wider frequency range.
The technical noise budget of the Si4 system is shown in Fig. 2.7. We show fits to the measured
noise arising from temperature, Pound-Drever-Hall (PDH) servo, and the PDH photodiode (PD).
The dominant technical noise source is residual amplitude modulation (RAM), shown as the reddata. The sum of the technical noise is well below the measured noise. Figure 2.7B shows theintrinsic thermal noise of the cavity, including Brownian noise from the spacer (dotted), substrate

38(dash-dot) and coating (dashed). The measured noise is shown (black) along with the sum oftechnical noise and thermal noise (green). The CTN is the dominant noise source from 5 mHz to
0.8 Hz.
The MJM cavity has routinely performed at a fractional frequency stability of 1 × 10−16 .
Since we have not developed a thorough noise budget for this system, we report only an upper
limit on the loss angle.
With the technical noise on the cryogenic cavities being sufficiently low, we can extract amechanical loss angle at each respective temperature. We assume that the mechanical loss in theparallel and perpendicular directions are identical (φk = φ⊥ ). Our measurements presented in thismanuscript do not distinguish these two loss components. Without loss of generality we lump them
together as a total loss angle. We also note a gradual shift away from the loss angle conventionof φk and φ⊥ in the LIGO community, and the loss angle can instead be defined in terms of bulk
loss φB and shear loss angles φS [71]. Approximately, the assumption of φB = φS is equivalent tothat of φk = φ⊥ , without detailed treatment of light penetration into the coating layers. However,
the reference [71] argue that using φk and φ⊥ can lead to negative strain energies for a non-zero
Poisson ratio σ 6= 0.
For the ith mirror, the expression for the fractional frequency PSD arising from coating
Brownian noise is then [72]Syi (f ) =2
 22kB T d 1 − σsubφc (f )
2
2
2
2
×
(1
+
σ
)
(1
−
2σ
)
+
Y
(1
+
σ
)
(1
−
2σ
)
Y
c
c
sub
sub
c
sub
2
2 )
π 2 f L2 ωi Ysub
Yc (1 − σc2 )(1 − σsub

(2.3)

Here, ”c” labels the coating and ”sub” labels the substrate. σc(sub) is the Poisson’s ratio, Yc(sub) is
Young’s modulus, d is the coating thickness, ωi is the 1/e2 beam radius at the ith mirror, φc(sub) is
the loss angle.
Note that we are allowing for the coating loss angle to have frequency dependence. Several
authors have reported a frequency-dependent loss angle for SiO2 /Ta2 O5 coatings [73, 74]. We fit
the measured spectra to the functional form of Sy (f ) = a/f b in order to allow for any potential

39

Fractional frequency noise (1/Hz)

10−29

10−30

10−31

10−32

10−33
10−4

4K PSD
16K PSD
124K PSD
124K CSD
288K PSD
10−3

4K fit16K fit
124K fit
288K fit
10−2
10−1
Frequency (Hz)

100

101

Figure 2.13: The measured noise PSD at various temperatures (solid curves). The dashed lines
are the fits to Sy (f ) = a/f b .

40
frequency dependence. A deviation from b = 1 for the CTN indicates a frequency-dependent loss
angle. The fits to the measured spectra are shown as dashed lines in Figure 2.13. We find weak
dependencies on frequency as summarized in Figure 2.14.
With the fits from Fig. 2.13 and Eq. 2.3, we derive φc (f ) and plot the results in Fig. 2.14.
The results are
φ4 K (f ) = (5.6 ± 0.9) × 10−4 × f −0.05±0.01(2.4)

φ16 K (f ) = (3.2 ± 0.3) × 10−4 × f −0.11±0.02(2.5)

φ124 K (f ) = (2.4 ± 0.3) × 10−4 × f 0.06±0.02(2.6)

The shaded band indicates the 1σ uncertainty. We find that the frequency dependence, althoughrather weak, has the opposite sign at 4 and 16 K compared with the 124 K result. The 4 and 16 K
results use mirrors from a different coating run than the 124 K cavity, and so this difference couldin principle be coating-dependent. However, the other measurements of the frequency dependence
to the loss angle all have positive coefficients (see [74, 73]). It might be that there is a real change
in the coating loss properties with the drastic change in temperature.
For comparison with other results published in the literature, we plot φc as a function oftemperature in Fig. 2.15. The orange diamonds [75], green squares [77], and blue triangles [76] areringdown measurements that do not explicitly consider φ⊥ and φk . The pink square [25], black
triangle [78] and purple diamond [79] are all from direct thermal noise measurements which assumeφ⊥ = φk . Our results are plotted at 1 Hz, and no frequency dependence has been assumed for thereferenced values, which are largely measured in the kHz frequency range. We measured CTN atseveral temperatures near 4 K, and average these values together. The x-axis error bar for thatdata point indicates the spread in temperature explored, while the y-axis error bar is the standard
deviation of the extracted φc . We do not see a strong temperature dependence between 4 to 16 K.
We have presented direct measurements of the thermal noise for SiO2 /Ta2 O5 HR coatings.
These measurements are complementary to those attained by ringdown spectroscopy. For thedesign of future gravitational wave detectors, independent measurements of coating thermal noise

41

288 K

Mechanical loss angle

10−3

4K

16 K

124 K

10−4 −3
10

10−2

10−1

100

Frequency (Hz)Figure 2.14: Frequency-dependent mechanical loss angle for the SiO2 /Ta2 O5 coatings at different
operating temperature. The shaded bands indicate the 1σ uncertainty.

42

14

Mechanical loss angle (×10−4 )12
10

SiO2 /Ta2 O5
SiO2
SiO2 /Ta2 O5
Ta2 O5
SiO2 /Ta2 O5
SiO2 /Ta2 O5 [36]
SiO2 /Ta2 O5 [This work]

8
6
4
2
0 0
10

101
Temperature (K)102

Figure 2.15: Mechanical loss of the individual coating constituents (SiO2 , Ta2 O5 ), and SiO2 /Ta2 O5
coatings measured by ringdown (orange diamonds [75], blue triangles [76], green squares [77]).
Other direct measurements are also included (black triangle [78], open black square [25], and purple
diamond [79]). Our results from direct CTN measurements are shown as the red circles and thered arrow.

43
are important. For ultrastable reference cavities, the use GaAs/AlGaAs crystalline coatings at
cryogenic temperatures is important to explore. Other potentially low thermal noise optical coatings
could also be tested using cryogenic ultrastable silicon cavities.

2.7

Si3 : zero crossing of CTE
So far, we have focused on the Si4 cavity system, but for the optical clock experiments in

the Ye lab, we have only used Si3, the 124K cavity system. In this section, we will describe themeasurement of the zero crossing of the CTE of Si3. After the optical timescale data campaign [50],we were curious how much of the long-term instability was coming from temperature fluctuations.
When the Si3 cavity was commisioned, we performed a broad scan of the cavity temperature withthe help of PTB colleagues who had come to visit JILA. At that time, we beat the Si3 laser against
Si4, which did not have its power-dependent drift worked out yet. Thus, we decided to remeasure
the zero-crossing, since we weren’t exactly sure how far off we were. During the Christmas breakof 2017, a quiet lab allowed for taking the Si3 offline for some characterization work.
To measure the zero-crossing of the CTE, we want to sweep the cavity temperature over asmall range, say around 10 mK. We then record the cavity frequency (length) versus the temperatureof the cavity, and should observe a parabola when we cross through zero. Sweeping the cavity
temperature over a reasonable time scale can be tricky, since the time constant from the active
shield to the cavity is 12 days. Thus, one has to use “thermal inertia” in order to sweep thetemperature. We first make a large step to a lower temperature, wait some time, and then jump
up above the initial setpoint. By using a detailed thermal model of the cryogenic system, andcontinuously checking the parameters versus the measured temperatures, one can predict how thethermal system will respond, and fine-tuning of the step amplitudes and time can be performed.
The cavity temperature is read by a thermocouple, which can exhibit jumps in its reading. For
this reason, this sensor is only meant as a somewhat rough out-of-loop monitor, and the inloop
sensors are precision PT-100 thermistors installed on the radiation shields. So during the sweep, one
has to keep track of any inadvertant noise jumps in the data. This becomes especially important

44
when you have determined the zero-crossing point, but now need to steer the system to the correcttemperature reading.
With the cavity temperature now sweeping linearly, we measure the Si3 frequency against Si4
and the hydrogen maser reference from NIST (Figure 2.16). The measurement against the maser
has much more short-term frequency noise, but serves as a consistency check for the fact that wemeasured against Si4, another drifting system. We have measured the zero crossing to the µK
level, and can easily steer to this temperature to within 1 mK.

45

Si3 frequency (Hz)

0

−10
−20
from Si3-Maser
from Si3-Si4

−30
0.340

0.342

0.344
0.346
0.348
Cavity temperature - 123.0 (K)0.350

Figure 2.16: Si3 zero crossing measurement of the coefficient of thermal expansion. Measured
both against Si4 and the hydrogen maser. The fitted zero-crossings are consistent, with T0 =123.345542(2) K from Si3-Si4, and T0 = 123.34594(7) K from the Si3-maser.

Chapter 3Classical clock demonstrations and applications

In this chapter, we briefly overview various clock results using classical, uncorrelated atomic
ensembles. These experiments take direct advantage of the advances in ultrastable laser technology.
We first describe the clock comparisons between Sr1 and Sr2 that were undertaken during the second
half of 2018 [17]. Then, we discuss placing limits on ultralight bosonic dark matter, coming fromprecision comparisons between the Si cavity, Sr atoms and a hydrogen maser [49]. After this, we
switch gears toward time-keeping, discussing a measurement campaign made by the Bouder Area
Clock Optical Network (BACON), where in 2018 we measured three optical clock ratios at the 18th
digit. This lays the groundwork for the eventual redefinition of the SI second, and also provides
further opportunities for testing fundamental physics. These measurements and applications push
the frontier of what one can achieve state-of-the-art optical clock technology, without introducing
quantum entanglement. This chapter sets the stage for the rest of the thesis, where we report oureffort to push a further boundary by introducing quantum correlations and spin-squeezing.

3.1

Classical clock comparisons at the quantum limit
After many years of hard work spanning multiple PhD timescales [80, 24, 81], our groupdeveloped the first cryogenic silicon cavity to reach the thermal noise floor [25]. This new laser had
unprecedented frequency stability - reaching a laser linewidth below 10 mHz. Two of these systems
had been built at PTB in Germany, and it was time to transport one of the systems to JILA. Eric
Oelker and Lindsay Sonderhouse both travelled to Braunshweig Germany, assisted in packing up

Si3
21 cm Si

47

(b)
MJM
40 cm ULE

Si46 cm SiEr
Comb

Cross-correlation
measurement

Fractional frequency noise (1/Hz)

(a)

10−31

10−32

10−33
10−2

10−1
100
101
Frequency (Hz)Figure 3.1: (a) Schematic of the cross-correlation method. Each laser is fiber noise cancelled to
the frequency comb table, and we measure the beat between Si4 and Si3, and MJM vs Si3. (b)The black data is the cross-correlation spectrum for the Si3 system (21 cm Si). The red line is alaser model that we will use for Dick effect calculations and dark matter limit simulations.

the systems, and hand-carried the cavity to Boulder, Colorado. Eric led the effort to rebuild thesystem, and very quickly we were very happy to see that the Si3 system was dipping below theSi4 stability within just a few weeks of operation at 124 K. It seemed like a prime time to conduct
some new clock comparisons with this new clock laser.
As we mentioned in Chapter 1, neutral atom clocks make use of large ensembles in order tomake the influence of QPN negligible. However, it is quite difficult to operate a clock at the QPN
limit, due to frequency noise on the clock laser. Aliased high frequency noise, known as the Dickeffect, can contibute significant amounts of clock noise. Until this work, the laser noise had limited√
the direct observation of anti-synchronous clock comparisons to the low 10−16 / τ level, where τis the averaging time in seconds [41, 82, 83]. By employing the improved optical local oscillator,√
we sought to push single clock stability into the 10−17 / τ level for the first time. To this end, we
performed clock comparisons between the 1D Sr1 clock and the 3D Sr2 clock.
We start by characterizing the clock laser performance by using a cross-correlation measurement (Figure 3.1a). The measured cross correlation for Si3 is shown as the black data in Figure

48
3.6b. The noise model includes a thermal noise term of 1.5 × 10−33 /f . There are several peaks inthe spectrum. The peak near 5 Hz is a vibration peak that shows up on all three lasers, and at alarger magnitude on both the MJM and Si4 systems, making us suspect that the true magnitude
on the Si3 laser system could easily be smaller than observed and modeled here. Two peaks at 20
Hz and 30 Hz are likely to arise from HVAC motor vibrations. Also included are white frequency
and white phase noise terms.
For this comparison, the Sr1 1D clock (before the rebuild described in [84]) operated with1,500 - 2,000 atoms, probing the

1S , m = ± 9
0
F
2

→

3P , m = ± 90F2transitions. The ± label

refers to the averaging of the two transitions to become first order insensitive to magnetic field
fluctuations. The Sr2 system is a 3D optical lattice, and for this experiment was run without their
typical evaporation cooling stage [85]. The Sr2 system also operated on the magnetically insensitive
σ transition, 1 S0 , mF = ± 23 → 3 P0 , mF = ± 52 , which is 22 times less sensitive to the magnetic
field noise than the ±9/2 transition [86]. Since this work, several other groups/experiments havebegun using this particular clock transition [87, 18].
Furthermore, to achieve the best clock performance, we had to hunt for various sources ofnoise in the systems. We initially stabilized the 0th order of an AOM, and probed the atoms
with the 1st order. We found that differential noise between these two orders gave noise beyond
our desired level of precision. With some ingenuity (a combined effort of Christian Sanner, Ross
Hutson, Eric Oelker, Colin Kennedy and others on the Sr team), a simple solution was found for
Rabi spectroscopy. We retroreflect the 1st order beam for stabilization, but need to deal withthe turning on and off of the pulse accordingly, without introducing significant phase transients
when the laser is on. To work around this, we first detune the laser from the clock transitionby 1 MHz, turn on the light to a low level, and engage the fiber noise servo. The laser is thenswept onto resonance and turned to the clock drive power level. This solution works great for Rabi
spectroscopy because the time when the laser is swept onto resonance (ms level) is short comparedto the entire Rabi pulse duration of 550 ms. Furthermore, we characterized the overall frequency
shift uncertainty of this probing scheme to very high precision [14]. Ramsey spectroscopy is further

49

Anti-synchronized(a)Synchronous

(b)

Sr2
550 ms 570 ms

600 ms

570 ms

570 ms

600 ms

570 ms

550 msFit
Laser
Laser+QPN
QPN

10−17

10−18

101

102
103
Averaging time (s)

Fit
QPN

Total Allan deviation

Total Allan deviation

Sr1

10−17

10−18

101

102
103
Averaging time (s)Figure 3.2: (a) Anti-synchronized clock comparison between Sr1 and Sr2. The solid gray line isthe callculated QPN, blue line is calculated Dick effect, orange dashed-dot is the quadrature sumof QPN and the Dick effect laser noise, and the dashed green line is the fitted stability, giving√√
5.5 × 10−17 / τ . (b) Synchronous comparison, showing a single clock stability of 3.5 × 10−17 / τ .

complicated when you want to stabilize the 1st order, but has been implemented with clever tricks.
The comparison stabilities are shown in Figure 3.2. The anti-synchronized comparison is run
so that the two clocks have no temporal overlap of the clock pulses, with a pulse time of 550 ms,
and the dead time tuned so there is a buffer of 10 ms on either side of the pulse. The Dick effect is√
calculated to be 3.8 × 10−17 / τ using the laser noise model for the Si3 laser, plotted as the solid√
blue line. Shown as the gray line is he anticipated QPN level is 2.9 × 10−17 / τ , and we combine
the Dick effect and QPN for the orange dashed-dot line. The fitted stability of 5.5(3) × 10−17 is in√good agreement with the anticipated stability of 5.7 × 10−17 / τ .
To remove the common-mode laser noise, we perform synchronous comparisons (Figure 3.2b).

50√Here, the observed stability is 3.5(2) × 10−17 / τ , just 20% (1.6 dB) above the predicted QPN of√
2.9 × 10−17 / τ . This is the closest to the QPN level for any independent, neutral atom opticalclock comparison to date. It is this very fact that motivates the development of a spin-squeezed
clock in our lab. State-of-the-art clocks are running into the fundamental QPN limit.

3.2

Dark matter limits from precision spectroscopy
Overwhelming evidence from astrophysics points to the fact that dark matter makes up the

majority of the observable matter density in the universe [88]. However, direct detection of dark
matter has so far alluded the scientific community, despite significant efforts spanning generations ofexperimental astrophysicists [89]. Ultralight bosonic dark matter is one proposed type of dark matter out of a great soup of theoretical proposals [90]. Using our precision measurement techniques,
we can make measurements that are sensitive to fundamental constants.

3.2.1

Sensitivity to fundamental constants

We first consider the frequency ratio of our optical cavity to the Sr atom fSr /fSi , and show2ethat it is sensitive to the fine-structure constant, α = hc
. To see how this scaling comes about, we

make some simple considerations. The strontium optical transition frequency scales as
fSr ∼

Ry
h

(3.1)

where Ry is the usual Rydberg constant. Then, let’s suppose the silicon cavity frequency is given
by fSi ∼ λc , and that λ must be an integer multiple of the Bohr radius,
fSi ∼

c
N aB

(3.2)

where c is the speed of light, N is the number of silicon atoms making the length of the cavity,
and the aB is the Bohr radius. Then, in our measurements we look at the frequency ratio of thestrontium frequency and the silicon cavity frequency,
Ry N aB
fSr
∼
fSi
hc

(3.3)

51

fSr
∼
fSi

2
me e4
N m~e e2h2hc(2π)2 e2 N
fSr
∼
= bα
fSi
hc

(3.4)

(3.5)

where I’ve absorbed a constant factor b = (2π)2 N . We see that the linear dependence on alphanaturally comes out of this simple derivation. The next step is to take into account relativistic
corrections, which can be written quite clearly as an additive correction fSr ∼ Ry (1 + kα2 ).
Sensitivity to the electron mass is realized by comparing the silicon cavity against a hydrogenmaser. The hydrogen maser operates on the hyperfine ground state transition, which has the usual
scaling of

fH ∝

e2
aB ~





 2

me
α Frel (Zα) µp
mp

(3.6)

where α is the fine-structure constant, µp is the magnetic moment of the proton, me and mp are
the electron mass and proton mass respectively, and Frel (Zα) is a relativistic factor depending on
Z. The frequency of the silicon cavity is fSi = c/λ. Using the silicon cavity frequency from Eq. 3.2,
we take the ratio,


 3

fH
me
∝ N α Frel (Zα) µp
fSi
mp

(3.7)

Thus, the silicon cavity - hydrogen maser comparison is linearly dependent on the electron mass.
For an optical timescale to be established, we need to run the silicon cavity against strontium
over an extended (ideally forever) period of time. The comparison does not have to continuous, but
it is repeated many times nonetheless. We also naturally run the silicon cavity against hydrogenmasers in order to connect to the more conventional timescales. Establishing the optical timescalethus gives us “free” access to this search of ultralight dark matter.

3.2.2

Dark matter couplings

The standard way to search for dilaton dark matter is to consider phenomenological couplingsof the dark matter field φ(t) to various terms in the Standard Model Lagrangian. Following [91, 92],

52
we consider scalar couplings of the dark matter field φ to the Standard Model. If we consider
couplings to the electromagnetic term and the fermionic mass term, we get the following dilaton
Lagrangian (with ~ = c = 1),Lφ = κφde¯ e ψeFµν F µν − dme me psi
4e2


(3.8)

where dme and de are the dimensionless coupling constants, is the coupling of the dark matter toelectron mass me , and ψe is the fermion spinor [92]. The dark matter field oscillates with the form
φ(t) = φ0 cos (mφ t + δ)

(3.9)

The overall normalization is mass dependent
κφ0 =6.4 × 10−31 eV,
mφ(3.10)

where here (and only here), κ = m4πP l and mP l is the Planck mass. The 1/mφ scaling means that theamplitude of the expected dark matter increases for lower mass, such that the overall energy densityof the dark matter is held fixed. The fermionic term in the Lagrangian gives a time-dependence tothe mass of the electron (for details see Appendix A of [93],me (φ(t)) = me 1 + κφ0 dme cos (mφ t + δ) .

(3.11)and similarily for the electromagnetic field term, we obtain time dependence of α,α(φ(t)) = α 1 + κφ0 de cos (mφ t + δ) .

(3.12)

Thus, if one can find a measurement that is sensitive to the electron mass then a limit on themagnitude of dme may be set.
For the silicon-maser data, we use the Si3 cavity locked to stabilize an Er:fiber optical frequency comb (same setup described in [17]. A hydrogen maser (ST-14) signal at 1 GHz is transferredfrom NIST to JILA over a phase-stabilized optical fiber to the strontium lab [94]. The maser signal
is mixed with the repetition rate of the frequency comb to a frequency of 1 MHz and then countedwith a lambda-type frequency counter. This gives us the direct frequency comparison between the

53

darkmatter_data.pdf

Figure 3.3: (a) Fractional frequency deviation of the Si - Sr frequency comparison. The data
has been dedrifted by fitting the data to Eq. 3.13. (b) Power spectral density of the Si-Sr data.
The model for the Si cavity is shown as the dashed line. (c) Fractional deviation of the Si - maser
frequency difference. (d) PSD of the Si - maser data. The maser model has a white noise term,
and a broadened resonance to account for the microwave link noise. The silicon model is the same
as for (b).

silicon cavity and the hydrogen maser. For the optical data, silicon cavity is locked to the Sr1clock [14]. The control signal of the atomic clock lock yields the silicon cavity - strontium relative
frequency.
The data campaign spans from Modified Julian Date (MJD) 58443 - 58477. The raw data
for the two datasets are shown in Figure 3.3a and Figure 3.3c, in terms of fractional frequency
deviation versus the elapsed time. The data has been dedrifted by fitting to a model of the form,f (t) = a + bt + ce−t/d ,(3.13)

where (a,b,c,d) are the fit parameters. The power spectral density (PSD) of the residuals is computed using the Lomb-Scargle periodogram, shown as the red data points in Figure 3.3b for the
optical data and Figure 3.3d for the microwave data. The microwave PSD is calculated for Fourier
frequencies from 0.7µHz to 1 mHz. We leave out Fourier frequencies above 1 mHz for two reasons.
One is that due to the noise level, the bound is considerably higher than currently set limits. Another is to simplify the analysis, since the noise above 1 mHz is dominated by the microwave linknoise.
In order to set a dark matter bound, we require the 95% confidence limit of the PSD. Thisis achieved by establishing a noise model for the measurement and simulating many time-domain
traces that have the same noise as the model. In Figure 3.3d, we plot traces of the Si3 noisemodel (dashed green). Also plotted is the white noise level of the Hydrogen maser (dashed red) of54√
1.1 × 10−25 Hz−1 , which corresponds to an Allan deviation of 1.5 × 10−13 / τ . The measured noiseis largely dominated by the maser and some link noise at around 2 × 10−4 Hz. The link noise ismodeled as a sinusoidal signal at 2×10−4 Hz with phase noise introduced to approximate the shapeof the link noise. We then simulate 100 different time-domain traces, compute the Lomb-Scargle
for each generated trace and build a statistical distribution. The 95% confidence limit is thencalculated from the distribution.

3.2.3

Dark matter Limit

With the 95% confidence limit for the PSD in hand, one can calculate the dark matter limit.
We use the same normalization The experimental bound for the silicon-hydrogen maser is shown as
the black curve in Figure 3.4. This bound includes corrections coming from the impact of dedrifting
the frequency measurement and any potential degradation in sensitivity due to the unknown phaseof the dark matter. The discussion of these effects may be found in the following section.
In Figure 3.4b, we plot various other experimental bounds on the dme coupling. The torsion
pendulum bounds are obtained by searching for a fifth force, since the existence of a scalar field
φ would result in the exchange of virtual φ quanta [100]. The MICROSCOPE experiment is aspace-based torsion-pendulum [101, 98]). We note that in the 2018 PRL, the dme result is notdirectly plotted, but another paper shows the limit on dme [93].

3.2.4

Extra details on our analysis

In this sub-section, we expand on a few details regarding our data analaysis. Some otherdetails may also found in the supplemental materials of [49]. To check the normalization of theLomb-Scargle periodogram, we took two independent approaches. The first, most straightforward
check is by checking the amplitude of a sine wave. We generate a sine wave with a given amplitudeof 1 × 10−12 . In order to verify the amplitudes of the spectral densities, we simply integrate each
amplitude spectral density (ASD). The Scipy (a commonly used Python package) ASD gives us thecorrect rms value of 7 × 10−13 , whereas the Lomb-Scargle gives us an amplitude that is exactly a

55

Exclusion Limit (de )

(a)

10-2

10-3

10-4

10-5
H/Si Limit
Sr/Si Limit
Sr/Si Projected

Dy/Dy
Rb/Cs
Network
UW
MICROSCOPE

10-6
10-20
(b)

10-19
10-18
10-17
10-16
Dark matter candidate mass (eV)

Exclusion Limit (de )

10-2

10-3

10-4

10-5

H/Si
UW
MICROSCOPE
10-20
10-19
10-19
Dark matter candidate mass (eV)Figure 3.4: (a) 95% confidence limits placed on the dimensionless couplings to the electromagnetic
field de , showing the improved limits set by fSr /fSi (red) and fH /fSi (blue) ratios in the mass range
above 1 × 10−19 eV. The maximum projected sensitivity for a search of the same 11-day duration
without data gaps is included (light green), highlighting the potential of our technique. Limits
derived from previous spectroscopic searches (black lines), with the network [95], Dy-Dy comparison[96], and Rb/Cs [97]. Limits from equivalence principle tests (purple lines), with MICROSCOPE[98] and UW [99]. (b) Demonstration of a significantly improved limit on the electron mass modulus(dme ) derived from the fH /fSi (blue) ratio. Limits from equivalence principle tests (purple lines),
including MICROSCOPE [98] and measurements from UW [99]. Shaded regions in both (a) and(b) are excluded at the 95% confidence level given the observed signal and noise models.

56
factor of two smaller. One possibility is that the Scipy ASD is giving the one-sided spectral density
whereas the Lomb-Scargle is giving the two-sided. In any case, we will multiply the resulting LombScargle periodograms by a factor of two for amplitude, and by four for power spectral density (PSD).
The second check we did was to generate time-domain data of white noise, flicker frequency noiseand random walk frequency noise with known amplitude, and checking the Lomb-Scargle vs FFT.
Since our dark matter limit for dme extends down to a Fourier frequency of 0.7 µHz, we werecurious to check the effect of dedrifting the signal, as well as a random phase. In order to explore
this effect, we add a hypothetical sine-wave signal to the raw data (before dedrifting) with a known
amplitude of 1 × 10−13 . We then dedrift the data+sine by fitting the drift model given by Eq 3.13.
Then we can take a Lomb-Scargle periodogram of the data and check the RMS value of the signal
by integrating over the relevant frequency bins, and compare with the value we know we should
obtain for such a sine wave. The result of this simulation is plotted in Figure 3.5. We plot theratio of the obtained dark matter signal amplitude to the expected amplitude. Simulating the time
domain traces with a length of 30 days is computationally expensive, and so to reduce the timerequired for the simulations we only calculate the correction at the points plotted as the orange
circles. We then fit the correction to a 1 − e−x/f0 model so we can interpolate the correction to all
frequencies for the dark matter bound. We note that these corrections are more relavant for the Si- maser dataset since it is three times longer, allowing us to reach lower Fourier frequencies.
The scalar dark matter described by Equation 3.9 involves an arbitrary phase δ that couldin principle take on any value from 0 to 2π. Since the data is not perfectly sampled, we will havea non-uniform sensitivity function for dark matter fields possessing different phase δ. We have no
way of knowing what the correct phase of the signal could be, so we evaluate this effect by injecting
dark matter signals of known amplitude with varying phase δ. We then compare the obtained
amplitude from our least-squares fit of the data with the actual injected amplitude, and plot thisratio as the blue datapoints in Figure 3.5. Similar to the dedrift correction, we fit the ratio toobtain the corrections at our final Fourier frequencies. For the final corrections to the dark matter
limit, we multiply both the dedrift impact and the phase impact, and this is plotted as the solid

57

1.0

Correction

0.8

0.6
Dedrift correction
Dedrift fit
Phase correction
Phase fit
Total correction
Lowest frequency bin

0.4

10-7

10-6

10-4
10-5
Frequency (Hz)10-3

Figure 3.5: Corrections to the dme limit from dedrifting and the unknown dark matter phase.
The vertical dashed line indicates the lowest Fourier frequency for which a dark matter limit iscomputed.

58
black line in Figure 3.5.

3.3

Boulder Atomic Clock Optical Network
The Boulder Atomic Clock Optical Network (BACON) is a collaboration and corresponding

network between the Sr lab at JILA and the Yb and Al+ ion groups at NIST. The relative density ofhigh performing clocks in Boulder, Colorado is truly impressive. At the time of writing this Thesis,
the Yb clock has a systematic uncertainty of 1.4 × 10−18 [15], the Al+ ion clock is 9.4 × 10−19[16], and the Sr clock is at 2.0 × 10−18 [14]. In 2017, we embarked to measure the frequency ratios
between these three atomic clocks. This optical clock frequency ratio measurement serves as a
critical step toward the redefinition of the SI second. Optical frequency ratios are the best way tocheck consistency between optical clocks because they do not rely on cesium fountain clocks thatcurrently define the second, which would limit such a comparison to the 10−16 level. By using an
optical frequency comb as the flywheel, we can connect various optical standards across a wide
range of frequencies [102, 103]. Since the optical frequency combs introduce minimal noise, we canmeasure the ratio with the utmost precision [104].
One of the first steps in the comparison was establishing a link between the JILA Sr clock
and the clock/frequency comb labs at NIST. There exists a long network of optical fiber interconnecting various labs and institutes around the city of Boulder called the Boulder Research and
Administrative Network (BRAN). Before the clock comparison, we revived the BRAN fiberlink
between JILA and NIST. The servo loop had to be adjusted to prevent instability, since the finite
time-delay leads to phase shifts that can give π phase shifts and corresponding instabilities [105].
The bandwidth of the servo ended up being approximately 5 kHz. We set up the fiber link in a loopback configuration to evaluate the out-of-loop performance (Figure 3.6a). We install AOM-2 in the
retroreflection of the light from ref #2, so that the light is marked in frequency and is distinguishedfrom all of the other back reflections in the long fiber and interconnections. The inloop error ismeasured by out-coupling the error signal and counting it with a frequency counter (lambda-type,deadtime free), and is plotted as the red points in Figure 3.6b. We also measure the out-of-loop

59
noise, shown as the black circles. We see a two distinct bumps in the Allan deviation, likely due
to temperature/pressure fluctuations of the out-of-loop fiber. The Allan deviation averages down
to 5 × 10−19 , which we take to be the uncertainty of the fiber link. In Figure 3.6c, we plot ahistogram of the out-of-loop frequency measurements, and show the residuals to the Gaussian fit
in the bottom panel.
An example Allan deviation of a daily ratio measurement is shown in Figure 3.7. Within asingle day, the neutral atom clock ratio of Yb/Sr averages down to the low 10−18 level, while theratio with the single ion clock averages to 10−17 . The contributions to the instability are shown as
the dashed lines, showing that they are low enough noise as as to not contribute significant noise
to the ratio. The ratios were measured over several months, and in the end the Yb/Sr data wasover-scattered, partially due to this outstanding single-day precision. To treat the data from thiscampaign - we had to introduce the idea of “dark uncertainty”, which would account for unknown
fluctuating systematics (this data analysis was done in collaboration with NIST statistics expert
Amanda Koepke). The final ratios were measured to beνAl+ /νYb = 2.162887127516663703(13)νAl+ /νSr = 2.611701431781463025(21)(3.14)

νSr /νYb = 1.2075070393433378482(82),representing some of the most accurate measured quantities to date.
Using the same data from the clock comparison we can also measure the absolute frequencyof the strontium clock transition. A detailed analysis of the connection between the optical clock
and the international definition of the SI second determined the Sr absolute frequency to be [106]νSr = 429228004229873.19(0.15) Hz.

(3.15)

60

(a)

JILA

FM #1

Stable
laser

AOM-1

NIST
5 km

VCO

Er:doped
EDFA
out-of-loop
beat
BPF

5 km
AOM-2

Counter

FM #2

(b)

(c)

Probability density

Out of loop

10

−18

10−19

inloop10−20

Residuals

Overlapping Allan deviation

10−17

10−21
100

101
102
103
104
Averaging time (sec)

105

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0

0.060
0.030
0.000
−0.025−0.060−3 −2 −1 0 1 2 3-17Fractional offset
offset (×10(x10−17))
Fractional

Figure 3.6: (a) Schematic for the out-of-loop evaluation of the BRAN fiber link for the 2017/2018
JILA-NIST clock comparison. A stable laser (Si3) is fiber noise cancelled, with the noise detected
between the mirrors labelled as “ref #1” and “ref #2”. The out-of-loop beat is measured between
the local light and the “remote” light. (b) Overlapping Allan deviation of the inloop error for thefiber noise cancellation (red) and the out-of-loop beat (black points). (c) Histogram of the out ofloop frequency measurements, with the residuals to the Gaussian fit shown below.

61

Al+/YbAl+/SrYb/Sr: Free Space
Yb/Sr: Fiber

Fractional instability

10 16

Fr
ee
-S
10

17

pa
ce

Fibe

Li
nk

r Lin

Co

10 18

101

k

Ne
tw
or

k

Lo
o

pb

ac
k

m
bs

102
103
Averaging time (s)104

Figure 3.7: Allan deviation of the three ratios on a given day. Also shown are noise traces forvarious components of the optical network, including the frequency combs, fiber link, free-spacelink and a network loopback measurement. We can see good agreement between the free space andfiber measurements of the Yb/Sr ratio.

Chapter 4

Spin-squeezed clock : background and considerations

4.1

Basic theory and definitions
Neutral atom clocks that employ uncorrelated independent atoms are fundamentally limited

by QPN. We describe the state of this atomic ensemble using the collective spin operator Ĵ =Jˆx x̂ + Jˆy ŷ + Jˆz ẑ, which is formed by simply taking the sum over all of the individual spin vectors.
The x̂, ŷ, ẑ are unit vectors that form the basis for the collective state vector. The z-projection of
the collective spin operator can be expressed as population differences between the two spin statesasJz =N↓ − N↑,2(4.1)

where N↓ and N↑ are the atoms in the ground and excited clock states, respectively. We will concern
ourselves with Jz , because it contains our clock discriminator. Ramsey spectroscopy converts phaseaccumulation in the aximuth plane to a tipping of the Bloch vector along the z direction on the
Bloch sphere (Jz ).
Coherent spin states (CSS) describe the non-classical state formed by N identical spin-1/2
particles. We can express the CSS as a product state [24, 107]|θ, φi =

N
Y
i=1

cos

θ
θ
|↑i i + eiθ sin |↓i i
2
2

(4.2)

This state is represented by a simple arrow with the endpoint living on the Bloch sphere. It responds
to optical rotations in a classical fashion. We will consider a state pointing along the y direction
on the Bloch sphere after atoms are initially prepared in a coherent superposition of |↓i and |↑i.

63
The noise associated with this state is along the orthogonal directions z and x, and is obtained byapplying Heisenberg’s uncertainty principle,∆Jz ∆Jx ≥ |Jy |/2(4.3)

where ∆ denotes the standard deviation. We visualize the noise of this state as a Gaussian distribution on the surface of the Bloch sphere. This noise arises from quantum projection, which is thenoise associated with the projective measurement of a state placed into a superposition.
The ultimate phase resolution potentially achievable by a CSS is set by the QPN and is definedas the standard quantum limit. We express QPN in terms of the uncertainty of the z-projection of
the Bloch vector∆Jz,CSS =

√

N /2

(4.4)

QPN arises from the projective measurement of an ensemble in a superposition state, when theatoms “collapse” into either the |↓i or |↑i state. This QPN will lead to a limit in the phaseestimation of a Bloch vector with length N to∆θSQL =

√

1
N /2
=√
N/2
N

(4.5)

We note that we have assumed the contrast associated with the state is Ci = 1, and we treat thecase of Ci < 1 in Section 7.1. Equation 4.5 makes it very clear that operating with a higher atom
number is advantageous to push the SQL to the lowest phase resolution possible. It is difficult to run
a clock that operates at the SQL, and even the highest performing synchronous clock comparisonstend to operate a few dB above this level, often due to the presence of other technical noise in theclock [17, 19].
The idea of spin-squeezing is to engineer entanglement in the atomic ensemble in order tosurpass the SQL. Since the clock signal is captured in Jz , the simple idea is to shuffle the uncertainty to the anti-squeezed quadrature Jx , and improve the phase resolution while still satisfying
Equation 4.3. Experiments that involve spin-squeezing greatly increase the technical complexityof the optical clock. One has to engineer interactions between the atoms in order to obtain non-64
classical spin states. When introducing these interactions, one must preserve the coherence of theatomic ensemble.

4.2

Historical backgroundThe development of quantum entanglement has provided an exciting new direction for reduc-ing the impact of QPN in quantum sensors, offering the opportunity to greatly advance upon thisstate-of-the-art performance. Spin squeezing, in particular, was proposed early on to utilize quantum correlation to conceal noise from individual atoms and thus achieve improved measurement precision and bandwidth [108, 109]. The creation of entanglement for metrology has been explored in awide variety of atomic quantum sensors including microwave clocks [110, 111, 112, 30, 31, 34, 33, 29],
ion clocks[113, 114], magnetometers [115], and matterwave interferometers [116].
The first step toward implementing spin-squeezing in a cavity QED system is achieving
strong coupling between the atom (or atoms) and the electromagnetic cavity mode. This strong
coupling regime is also referred to the resolved vacuum Rabi splitting limit, where the vacuum Rabi
frequency,√Ω = 2g N ,(4.6)

is much larger than any other time scales in the system (Ω  κ, Γ). There are two general regimes
for strong coupling: microwave domain [117, 118], and optical domain [119]. In the earliest work,
both of these general approaches were pursued by coupling either single atoms, or beams of atoms
to the microwave/optical cavity. The approach in the Haroche and Walther group was to couple abeam of circular Rydberg atoms to a superconducting microwave cavity. In that case, the dipole
moment of the atomic transition was enhanced by using Rydberg atoms, and a very high quality
factor microwave cavity minimized κ. This enabled the realization of the strong-coupling regime
for the atomic beam passing through the microwave cavity. The second approach was pioneeredin the group of Jeff Kimble, where alkali atoms were coupled to a high finesse optical cavity via
dipole-allowed transitions. In this regime, strong-coupling can be attained for just single atoms

65[119].
In later work in the groups of Vladan Vuletic, James Thompson, Mark Kasevich, and others,
ensembles of atoms were trapped and coupled to optical cavities, enabling the inspiring work on
implementing spin-squeezing in microwave clocks [30, 31, 33, 29]. By coupling ensembles of many
atoms, one can enhance the collective coupling between the atoms and the cavity mode. We will
follow along this line, but instead run the clock in the optical domain, both for the cavity QED
coupling and the clock states.
Despite tremendous progress, spin-squeezing in atomic clocks has yet to yield enhancement
at state-of-the-art stability levels. A spin-squeezed microwave clock has observed 11 dB of enhancement [33] at the 10−10 stability level at 1 s, in contrast with microwave fountain clocks at 10−14 [120].
We note that the spin-squeezed clock in [33] operated at a considerably shorter Ramsey dark timethan the state-of-the-art atomic fountain clocks. For optical clocks, which operate at much higher
stability, generation of entanglement has been demonstrated by a measured Wineland parameterof -4.6 dB [35]. After subtraction of a laser noise model, an optical clock employing a SSS wasinferred to operate -4.4 dB below the SQL at a fractional frequency stability of 1.3 × 10−13 τ −1/2
(where τ is the averaging time in seconds) [35]. However, metrological applications require direct
observation of clock stability enhancement without post-processed removal of technical noise. It
remains to be seen whether spin-squeezing will enhance the operation of state-of-the-art atomic
clocks. The primary objection of the Sr3 machine is to demonstrate a competitive atomic clockwith cavity QED based squeezing, and directly observe metrological enhancement.

4.3

How do we design a cQED system?
What should we consider when designing a cavity QED system for an optical clock? To

answer this, we will examine some simple calculations of various physical quantities, and discuss
how different design choices impact various parameters through fundamental and technical scalings.
The Jaynes-Cummings Hamiltonian describes a single atom interacting with the quantized

66
electromagnetic field of the cavity mode,Ĥ = ~ωa σ̂+ σ̂− + ~ωâ† â + ~g σ̂− â† + σ̂+ â ,(4.7)

where the first term is the energy of the atom and ωa is the atomic transition frequency between
the energy levels 1 S0 →3 P1 = |ei. The second term is the energy of the photons, where â (â† ) is
the photon annihilation (creation) operator, so ââ† corresponds to the total photon number. ω isthe optical cavity resonance frequency. The last term describes the interaction between the photon
and the atom. σ̂− â† represents an atom going from the excited to ground state and emitting aphoton, while σ̂+ â describes and atom absorbing a photon and going from the ground to excitedstate.
As usual, we go into the rotating frame of the atom,Ĥ = ~δc â† â + ~g σ̂− â† + σ̂+ â ,(4.8)

where δc = ω −ωa is the relative detuning of the cavity from the atomic transition. A key parameter
here is the single atom cavity coupling [121],
r
g=µ

ωa
,
2~0 V

(4.9)

where µ is the atomic dipole matrix element, ωa is the atomic transition frequency, and V is thecavity mode volume. The atomic dipole moment µ is

µ=

1/2
30 ~λ3Γ8π 2(4.10)

We note that g only depends on properties of the atomic transition and the volume of the cavity
mode - which will depend on the cavity parameters.
To extend this Hamiltonian to describe N atoms interacting with the cavity mode, we replacePithe atomic operators with collective raising and lowering operators Jˆ± = 12 N
i=1 σ̂± , The TavisCummings Hamiltonian that describes the many-atom ensemble coupled to a quantized mode ofelectromagnetic radiation is thenĤ = ~δc â† â + ~g Jˆ− â† + Jˆ+ â .

(4.11)

67
Assuming we are in the weak-excitation limit, where the number of atoms excited to |ei, then wecan make an approximation to this Hamiltonian [32, 122],√ Ĥ = ~δc â† â + ~g N b̂â† + b̂† â .

(4.12)

where the collective operators Jˆ± are replaced with new creation and annihilation operators b̂ =√√J+ / N and b̂ = J+ / N . We have finally arrived at the enhancement of atom-cavity coupling by√using an ensemble of N atoms, increasing the frequency by a factor of N .
One might think that choosing a transition with the largest dipole moment µ leading to alarger g will give the maximum achievable squeezing. This is not true, however, and instead the
key parameter that dictates the maximum achievable squeezing through QND measurements is the
single-atom cooperativity C, expressed as
C=

(2g)2
.
κΓ

(4.13)

Examining this expression, we can see that the cooperativity C does not depend on Γ (or thestrength of the atomic transition dipole moment). We can understand this intuitively: a transition
with a larger dipole moment will increase the numerator of Equation 4.13, but that increased dipole
moment simultaneously increases the rate of incoherent scattering of photons into free space. The
spirit of Equation 4.13 is that what matters is the tradeoff between photons leaving through thecavity mirror as useful information and the leakage into the universe via free space scattering. Thismeans that one can either use a broad transition (1 S0 →1 P1 , Γ = 2π × 35 MHz), or a narrowertransition (1 S0 →3 P1 , Γ = 2π × 7.5 kHz). Reasons for choosing one transition over another are
thus more technical rather than fundamental. For example, working high finesse cavities at shorter
wavelengths can be problematic in terms of mirror coating damage [123].
The spectroscopic enhancement using QND measurements scale with the spin noise reduction
and the shortening of the Bloch vector (loss of Ramsey fringe contrast). We assume that the contrast
loss is limited by free space scattering, so we can write the spectroscopic enhancement as

ξ=

∆Jz0
∆Jz,QP N

2e−ms(4.14)

68
where ms is the number of scattered photons and ∆Jz0 is noise of a differential Jz measurement.
In the far-detuned limit where the atom-cavity detuning is much greater than the vacuum Rabi
splitting δc  Ω and assuming negligible readout noise of the final measurement, and assuming theloss of coherence is dominated by free-space scattering, the potential spectroscopic enhancement
scales as [32]
ξ −1 =

QN C.
e(4.15)

This means that the maximum achievable squeezing only depends on the collective cooperativity N C
and the total quantum efficiency Q. This result is independent of the atomic transition linewidthΓ.
One important assumption of the above equation is that we are working with a cyclingtransition. This is more complicated when working with the fermionic isotope of Sr, where weneed to consider the complicated hyperfine levels. Transitions that take the atom out of the initial
ground-state will lead to loss of coherence and an increase of noise via diffusion [32]. For thesereasons, we choose to operate on the 1 S0 , F = 9/2, mF = 9/2 →|3 P1 , F = 11/2, mF = 11/2i. We
apply a bias magnetic field along the direction of the cavity, and probe the cavity with circularly
polarized light. Barring any impure polarization or relative misalignment of the magnetic field andthe k-vector, we should realize a cycling transition with relatively small probability of a spin flip.

4.4

How should we probe the atom-cavity system?
We want to probe the atom-cavity system such that we optimize the information gained

about the collective atomic state versus the incoherent collapse of the single-particle wave function.
This means we need the most precise information about the state of light leaving the optical cavity.
In QND measurements, we aim to utilize the phase of the light leaking from the cavity as
the witness for the ground state population of the atomic ensemble in the cavity. The leakage lightthat comes out of the cavity through the input mirror can be written relative to the incident field69
Ei , [32]Er= I + iQ,
Ei

(4.16)

where the I quadrature response is
I = 1 −


1+

βr


δ
κp /2


2 

(4.17)and the Q quadrature is
Q=

βr



δ
κp /2



δ
κp /2


2

(4.18)

1+p
where βr = κ1 /κ is the radius of the I − Q circle for the light coming back through the input
mirror. Here, κ1 is the rate of coupling out of the input mirror, κ2 is the rate of coupling out of thesecond mirror, κ = κ1 + κ2 + κL is the total rate of power decay for the cavity mode. The detuning
between the probe laser and the cavity mode is δ. To show a direct example of this, we measure
the I−Q response of our cavity with the setup in Figure 4.1. We scan our probe laser across thecavity resonance and record a heterodyne beat between the probe and the LO beam. By directly
digitizing this signal, we can demodulate the RF signal in software. The normalized I−Q responseis plotted parametrically in Figure 4.1b. The individual I and Q responses are plotted versus theprobe laser frequency in Figure 4.1b and c. The red line in each panel is the result of a simultaneous
fit to Eq. 4.17 and Eq. 4.18. We can see some slight deviation from the theoretical response, as
seen in the baseline drift in the I quadrature. This can come from a slight miscalibration of thedemodulation waveform, and does not significantly impact the extraction of the cavity frequency,
as we can restrict the fitting window to the most sensitive parts of the frequency sweep (closer toresonance).
The QND measurement can be distilled down to a measurement of the leakage light to extract
the phase. It is this light that carries information about the atomic state inside the cavity. However,in cavity QED experiments, this probe power must be kept at an extradorinarily small level, in ourcase around 10−15 W. Since it is such a small amount of probe power, we opt to measure the phaseshift against a stronger local oscillator beam, in order to make the photon shot noise dominant

70

(b)

(a)

0.50

probe
laser

0.25

LO

Q/I0

frequency
sweep δ(t)

-

0.00

−0.25

−0.50

(c)
1.0

0.2

0.4

0.6

0.8

1.0

0
δ (kHz)

100

200

I/I0

0.50

0.8

0.25

I/I0

Q/I0

0.6
0.4

0.00

−0.25

0.2
0.0
−200

0.0

(d)

−100

0
δ (kHz)

100

200

−0.50
−200

−100

Figure 4.1: (a) Schematic of the detection scheme, where laser light is split into a probe pathand LO path, and these are combined on a beamsplitter. (b) By sweeping the probe laser acrossresonance we can map out the I − Q response of the cavity response. (c) Normalized I quadrature.
(d) Normalized Q quadrature.

71
over the detector noise, giving a shot noise limited signal-to-noise. The amplitude of the resulting
interference term is proportional to the product of the electric fields, or equivalently the square rootof the optical powers. For sufficient LO power, the noise of the photodetector and measurement
system is negligible compared to the LO photon shot noise. In this case, the signal-to-noise ratio
is simply proportional to the signal beam,√p
Ps PLOSN R ∝ √= Ps
PLO(4.19)To achieve the photon shot noise limit, we also have to be careful about classical intensity noiseof the LO, which is more severe at greater LO powers. We combat this by employing balanced
detection in addition to active control of the LO intensity.
While this simple scaling gives good intuition on quantum optical measurements, there is the
choice between “heterodyne” and “homodyne” detection schemes. This refers to the fact that one
can tune the LO beam to be at a different frequency from the cavity probe and look at the ACsignal, or tune to the same frequency and extract the information at DC. The heterodyne method
can be simpler to implement, but suffers from a loss of 3 dB of signal to noise ratio. This is seen
by considering the influence of the “image frequencies”. When you form a heterodyne beatnote,
you are looking at the interference of two distinct optical frequencies, which produce a beamnote
at the difference frequency fhet = νLO − νS , where νLO is the optical frequency of the LO andsignal respectively. However, on the other side of the LO is not a tone, but shotnoise, that gets
mixed to the same beat frequency fhet . Hence the shot noise contribution gets doubled, resulting
in the loss of 3 dB in signal to noise. The most straightforward way around the 3 dB loss is byperforming homodyne detection. There, the LO is tuned to the same frequency as the signal beam.
By adjusting the relative phase of the LO beam with respect to the signal beam, one can choose
the particular quadrature of detection of I or Q. Making the signal beam and the LO beam to be
the same frequency, there is no image shot noise, and hence no extra loss of 3 dB of signal to noise.
While homodyne detection optimizes the signal-to-noise, it also allows for further optimization by performing “fixed tone measurements”, which refers to fixing a probe tone on the resonance72
of the cavity mode as opposed to sweeping the laser frequency across the resonance. For example,
if you sit exactly on resonance with heterodyne, the signal will spend half of the time in the Q
quadrature, and thus half of the probe will not carry useful information. However, if we perform
homodyne detection, then the entire probe field will carry the most sensitive phase information.
It’s also clear that sweeping will not provide the optimal measurement. This is because the optimalmeasurement quadrature varies from the I to Q and back the I again during the sweep. So unless
one varies the detection phase during the sweep, one loses the 3 dB gain.
With all of these considerations - it is clear that the best QND measurement will be madeby performing homodyne detection and probing with a fixed frequency probe on resonance. This is
the primary motivation for operating our system in a detuned regime. In this regime, we can parka probe tone near the shifted cavity-like mode, and the atom number fluctuations are small enoughsuch that we stay in the linear response of the homodyne signal. The one con of this approach isthat we lose the common mode suppression of laser noise that one enjoys by probing the vacuum
Rabi splitting. We combat this by building a low noise cavity stabilization setup, discussed in
Chapter 5.8.1.

4.4.1

Atomic motion and probing time scales

In the experiments described in this Thesis, we keep all intra-cavity light to a minimum,
and thus do not confine the atoms along the cavity axis. The atomic motion is dictated by theirtemperature and the trapping potential of the vertical trapping optical lattice. At higher order, theoptomechanical effects from the probing light may play a role [34], but for now we will ignore thiseffect. We treat each pancake of the vertical optical lattice as a 2D harmonic oscillator. The atoms
occupy the radial modes of the 2D harmonic oscillator according to the corresponding Boltzmann
distribution. To estimate the order of magnitude for the spatial spread, the thermal radius for ourcloud can be estimated from 21 kB T = 12 m87 (2πνR )2 σy2 , where m87 is the mass of 87 Sr, νR is theradial trap frequency. From this, we get a thermal cloud radius of σy = 26 µm. We note that thisdefinition is consistent with our density distribution defined in Appendix B. The cavity mode gives

73
rise to the spatial dependence of the atom-cavity coupling222

gi2 (X, Y, Z) = g02 e−(Y +Z )/w0 sin2 (2πX/λ689 )(4.20)

So as the atoms move, they will experience a time-dependent atom-cavity coupling, g(X(t), Y (t), Z(t)).
In the far-detuned regime (δc  Ω)), the time-dependent frequency shift for the total atomic
ensemble can be decomposed into the dispersive shifts of each of the individual atoms,δνi (t) =gi2 (t).
δc(4.21)We seek to simulate the atomic motion of atoms in the 2D harmonic oscillator, and use thosetrajectories to calculate gi2 (t) for each ith atom. We can then calculate the mean cavity shift due
to each atom, and simulate how the cavity shift depends on probing time scales or the atomic
temperature.
Simulating the atomic trajectories is carried out as follows. We randomly sample initial X
and Y positions for each atoms according to the normal distribution with σY = 26 µm. With thespatial coordinate, we then pull from Boltzmann distributions to assign the initial momentum forpx and py . The distributions along the two directions are taken to be independent, each having a√standard deviation of σp = 2m87 kB Tr . We then iteratively solve Hamilton’s equation of motion
to obtain new values of position and momentum. These trajectories are plotted in Figure 4.2a as
the white traces. They are plotted on top of the intensity distribution of the standing wave inside
the cavity. The solid lines indicated the 1/e2 radius of the cavity mode, w0 = 71 µm.
For each atom, we track the motion and plot the quantity gi2 (t)/((2π)2 )δc ), giving units of
Hz/atom. We can see that after a very short amount of time, most of the atoms traverse several
standing waves of the probing light. This is consistent with a rough estimate made by considering a
1D Boltzmann distribution and calculating the fraction of atoms that traverse 5 standing waves to
be around 0.1 ms. Thus, we can be rest assured that sufficient thermal averaging along the cavity
mode is achieved after a relatively short amount of probing time.
However, the story for the radial motion is not as straightforward. This is because the thermal
cloud radius σY = 26 µm is not that much smaller than the cavity mode waist of w0 = 71 µm.

74

150

(b)

100

0:8

50
0:6

y (m)

0
0:4

50

g 2 (t)/δc (Hz/atom)

(a)

60
40
20

0:2

100

0
150

150

100

50

0
50
x (m)

100

0.0

150

0.05

0.8

1.0

−0.05
−0.10

0

20

40

td (ms)0.10
Tr = 2 µK
Tr = 1 µK
Tr = 0.3 µK

P

0.00

60

Fractional change of

Tr = 2 µK
Tr = 1 µK
Tr = 0.3 µK

νi

0.10

P

νi

0.4
0.6
Time (ms)

(d)

(c)

Fractional change of

0.2

0.05
0.00
−0.05
−0.10

0

20

40

60

td (ms)Figure 4.2: (a) The atom-cavity coupling g 2 (x, y), which follows the intensity of the cavity mode
standing wave is shown as the contour plot, normalized to a peak value of one. The lines indicate
the 1/e2 radius. Overlaid on top of g 2 is 100 different atomic trajectories simulated for the particle
motion in a single pancake of the optical lattice. (b) Corresponding traces of g 2 (t)/δc for all of thedifferent particles, showing that most atoms sample many standing waves well within 1 ms. (c)Fractional noise of the cavity shift for a measurement time of 2 ms, versus the dwell time between
frequency measurements. We show the magnitude of the oscillations versus atomic temperature(d) Same as (c), with a measurement time of 40 ms. We can see that the longer measurement timeoffers improved averaging of the radial motion.

75This means that the radial motion will lead to significant changes in the atom-cavity coupling. We
consider the impact of radial motion on a two window measurement of the cavity frequency shift,similar to what we implement for our QND probing of Jz in Chapter 6 and 7. In Figure 4.2c, weplot the total fractional shift versus the dwell time between the two measurements td for a shortmeasurement window of Tm = 2 ms. We can see the radial oscillations are damped out as the radial
motion is not perfectly coherent and there exists a spread of radial trapping frequencies. If we set
the measurement time to Tm = 40 ms (Figure 4.2d), we can see that there is further averaging of
the radial motion giving a drastically smaller effect.
Now we want to simulate the magnitude of noise in dressed atom-cavity frequency measurements that can occur due to the atomic motion. We fix the dwell time td = 10 ms, chosen to
represent the time it takes to perform a π-pulse. We then vary the measurement time, and look at
the standard deviation of the fractional cavity shift difference (νp − νf )/νp (Figure 4.3). Further, byvarying the atomic temperature, we can see that lower temperature atoms will give smaller noisedue to the atomic motion - matching intuition. We can also see that a measurement time of 20 ms
or longer is a safe choice for mitigating motional effects. As we will discuss further, experimentally
we use a measurement time of Tm = 40 ms. This choice is primarily influenced by considering two
effects: the atomic motion discussed in this Section, and the influence of cavity - laser technicalnoise, discussed in Chapter 5.

76

0.10
Std deviation of (νp − νf )/νp

2 µK
1 µK
0.3 µK

0.08
0.06
0.04
0.02
0.00

0

10

20

30
Tm (ms)

40

50

60

Figure 4.3: Simulations of cavity frequency shift noise arising from themal atomic motion. With afixed seperation time between measurements of 10 ms, we predict fluctuations in the shifted cavity
frequency on the percent level. Both low atomic temperatures and longer measurement time Tm
are critical for minimizing this effect.

Chapter 5

Sr3 : a new spin-squeezed clock apparatus

In this Chapter, we describe the design and construction of a new experimental apparatus
that integrates a high finesse cavity with a state-of-the-art optical lattice clock. We have named
this system as “Sr3”, since it is the third atomic clock system being operated in the Ye lab. The
first goal of this experiment was to directly observe improvement in clock performance due to spinsqueezing. To achieve this, we require the preparation of 1,000 - 10,000 strontium atoms inside
a high finesse cavity and in a well controlled environment. The high finesse optical cavity andsupporting structure is designed for low noise frequency measurements, allowing us to make nondestructive measurements of the atomic population with resolution below the quantum projection
noise. We also implement a movable optical lattice, in order to apply squeezing and read out
independent atomic ensembles with the cavity probe. With all of this in mind, we took one stepfurther and designed a system that would enable future performance at the state-of-the-art level
for accuracy, including black-body radiation, DC Stark, ac Stark, and other systematic shifts.
We split the experiment into two main regions inside our vacuum chamber, referred to as the“MOT bucket” and the “science cavity”. The MOT bucket sits below the cavity, and serves as thelocation where we prepare the initial blue and red MOTs. These atoms are loaded into the movable
optical lattice, which transports the atoms ≈ 40 mm vertically into the science cavity. The cavity
enables us to engineer squeezed states of several thousand atoms, while simultaneously providing
a clean environment in terms of clock systematics. The following will describe the design of theultrahigh vacuum (UHV) system, the MOT bucket, the science cavity, and some of the surrounding

78
hardware.

5.1

UHV System
One of the requirements for a cavity-QED optical clock system is achieving UHV. A poor

vacuum would impact our experiment in a number of different ways. Broadly speaking, a higher
background pressure means more collisions of these gas molecules with Sr, leading to loss, or in thecase of a grazing collision, leading to a systematic clock frequency shift. Thinking practically about
the impact of a finite vacuum lifetime for Sr3, we consider a potential future experiment where
we will use the QND nature of the probing to recycle the atomic ensemble instead of reloading
atoms each cycle of the experiment. This type of experiment would have the advantage of reducing
the dead time of the clock, leading to improved stability due to mitigation of the Dick effect. Of
course, a short vacuum lifetime would be a limitation to such an experiment, due to the eventual
loss of atoms and the corresponding rise of the QPN. When designing Sr3, we knew we would beplacing a lot of different materials in vacuum, including: the copper cavity spacer and supporting
structure, mirrors, epoxy, temperature sensors, kapton wires, viton. Thus, we could not expect
the same vacuum quality as the Sr2 machine described in [124]. We ended up setting the goal of< 10−10 Torr, which should lead to a > 10 s vacuum-limited lifetime.
Each component that was placed in vacuum have different levels of handling, and different
cleaning procedures altogether. In the following we describe the UHV system, the pre-bake andcleaning procedures we implemented, and finally the bakeout of the entire system.

5.1.1

DesignWe employ an AOSense strontium cold beam source, which includes the oven crucible, Zeeman
slower and two 2D MOTS. After transporting the chamber from the auxillary lab to the main Srroom, we found that one of the heater pins for the oven was shorted to the vacuum chamber. Aftersome moments of anxiety, we realized we could actually run the heater in this configuration. The
one worry is that if we flow current directly into the vacuum chamber, it could find its way back to

79
Earth ground through an undesireable path (for example through the breadboard → photodiode).
We thus attached a thick tin-coated copper braid (the type typically used for electronic cableshielding) to serve as the return for the current, and by checking with a clamp current reader, we
confirmed the current was making its way back to the power supply through this attached braid.
The oven is attached to the chamber via a long skinny vacuum tube. The differential pumping
speed from the outlet of the all-metal gate valve to our main chamber is calculated to be 0.01 L/s.
For vacuum pumping, we employ a 150 L/s Starcell ion pump from Varian, a titanium sublimation
pump (TSP), and a 200 L/s NEG. The TSP gives us the largest pumping speed, in excess of 1000
L/s for hydrogen in the chimney area of the chamber. We note that when one calculates capacity
for the TSP, one would expect the pump to saturate on the year timescale at 10−11 Torr level.
However, we have not noticed a rise in pressure from the ion gauge, and so we have not refired
the TSP. Interestingly, the story is the same with Sr2. Since the TSP does not provide efficientpumping of inert gases (argon, xenon, etc), we add the ion pump. This is especially important sinceanytime we replace the oven, one dominant gas will the be argon from the back-filling process.

5.1.2

Assembly and bake-out procedure

Given careful cleaning of the in-vacuum components, our chamber should mostly be limitedby the level of hydrogen outgassing of the chamber walls and the achieved pumping speed. Basedon an initial 150 C degree vacuum bakeout of the empty chamber (all steel blanks, no payload),
we estimated our initial steel outgassing rate to be 3 × 10−12 mbar L s−1 cm−2 . It has been shownthat vacuum baking steel vacuum chambers at a temperature greater than 400 C for a long periodof time can reduce the hydrogen outgassing rate by as much as a factor of ten [125]. We removed
all viewports and vacuum pumps except for the turbo-molecular pump (TMP), and proceeded tobake the chamber at 400-450 C for 25 days. At this elevated temperature, we observed the partialpressure of hydrogen to decrease from 1.5 × 10−5 Torr down to 3 × 10−6 Torr, indicating an overall
reduction of the hydrogen outgassing rate of nearly a factor of three.
With the steel in the main chamber thoroughly baked, next we needed to ensure that all

80
objects placed in vacuum were properly cleaned. The surface area of the MOT bucket is not
negligible, and all of the inner walls that are exposed to vacuum are steel. Since these parts had
been fabricated in the shop, we opted to sonicate the MOT bucket upside down in an Alconox
water bath for 30 minutes, followed by a solvent rinse. Since the MOT bucket has several invacuum welds, we combined a vacuum test with a pre bake of the MOT bucket. The MOT bucket
was attached to a brand new 10 inch diameter nipple, and was subsequently vacuum baked at 150C
for 7 days. The final pressure reading of this MOT bucket prebake reached the low 10−10 Torrlevel, limited by the RGA noise floor.
Our science cavity design employs viton for vibration decoupling. The LIGO collaboration
has found that viton can outgas a significant amount of hydrocarbons under vacuum, especiallyabove a temperature of 180◦ C. Since this viton is relatively close to our high finesse mirrors, wetook extra care in preparing these pieces for our system. For vacuum cleaning the viton, we followed
the LIGO cleaning procedures with some modifications :1. Ultrasonic bath in Alconox for 15 minutes
2. Ultrasonic bath in de-ionized water for 15 minutes
3. Vacuum bake at 180 C for 72 hours
The cavity pieces that are machined out of copper were cleaned ultrasonicly with Alconox,
and then in the “usual three solvents” of isoproponal, acetone and methanol. In order to ensurecleanliness and to increase the emissivity of the copper, we then air-baked at 150C for approximately
3 hours. This air-baking made the cavity spacer a reddish-brown color - indicating the formationof a Cuprous Oxide coating, which has a higher emissivity than polished copper. This will increase
the relative contribution of the copper to the black-body radiation environment of the atoms, thus
giving higher control and knowledge of the thermal environment. The final assembly procedure
was an involved effort of stringing various kapton wires through the cavity assembly. The electrical
feedthrough we used is part number : IFTRG327013, which uses a MIL-C-26482 style connector
with 32 total pins. Of these 32 pins, we use 4 for the two PZTs, and 10 total for the in-vacuum
thermal sensors.

81

(a)

(b)
MOT coils

Cooling fixtures

Figure 5.1: (a) Angle-view of the MOT bucket. The smaller MOT bucket pieces are welded
together on a 10 inch conflat flange that is bolted to the bottom of the chamber. (b) Side cut-viewof the MOT bucket. The MOT coils and the cooling fixtures inside the MOT bucket are labelled.
In total, there are 5 in-vacuum welds.

5.2

MOT bucket design
The MOT bucket was designed to provide a space inside the vacuum chamber to produce

the ultracold atomic sample for cavity QED. The MOT coils are placed 1.1 inches apart with an
overall diameter of 2.5 inches. We used solid-core, kapton coated 13 AWG wire from S&W wire
company, with a specified operating temperature up to 250 C. The MOT coils consist of 13 turnsof wire in two vertical layers, potted in a high-temperature non-ferromagnetic epoxy. This wireconfiguration was chosen to make the coils compact, while simultaneously maximizing the surface
area contact with the watercooling fixtures that are affixed to one side of each coil (see Figure5.1b). Four roughly calibrated sensors are installed inside the cooling fixtures, so in the future thethermal environment can be monitored for accuracy studies.
Installing magnetic field coils with so much surrounding metal, especially the copper coolingflanges can be worrisome in terms of generating significant Eddy currents. We measured the
responding magnetic fields inside the MOT bucket when we rapidly switch off the MOT coils, and82
confirmed that we could switch the magnetic fields on the ms level. The presence of the copper
cavity further complicates the issue of Eddy currents, which we discuss and present measurements
of in section 5.8.1.

5.3

Accuracy considerations

5.3.1

Faraday shielding - DC Stark

One of the potential drawbacks of implementing a cavity QED clock is the electric field
produced by the cavity PZTs. In order to shield the atoms from the PZT voltage, we make use of
the cavity itself as a Faraday shield. The cavity spacer is directly grounded to the vacuum chamber
through the thermal braid, ensuring a good ∆V = 0 ground plane relative to the chamber, ensuringthat the cavity should not build up any significant electrical charge. The aperture cut through thecavity for the cavity mode is the same diamter as the inner diameter of the PZT (6 mm). In
order to evaluate the efficacy of this design, we perform a COMSOL simulation of the electric field
distribution. The boundary conditions for the voltage are set by grounding the cavity spacer, andalso by an applied voltage to the PZT electrodes. The simulated electric field along the axis of
the cavity is shown in Figure 5.2. We consider the possibility of a surface charge on the surface ofthe mirror of 1 nC. This amount of charge is considered to be a worst case scenario, as this was
the magnitude of charge seen by SYRTE that gave a large 10−13 shift with essentially no Faradayshielding [126].
This same approach was taken in the recently built Sr1 system (described in [84]), where the
Faraday shield had the same inner diameter as the PZT. In Sr1, the DC Stark effect was measured
to be less than 10−18 , and could be easily compensated down to the 10−20 level by applying asmall field. While we expect success with the Faraday shielding in Sr3, experiments will have to be
performed to confirm this. By mounting electrodes outside of vacuum, one should be able to apply
a larger enough electric field to check the dc Stark shift in Sr3.

83

10−16

Fractional frequency shift

10

−17

200V, 1nC on mirror
20V, 1nC on mirror
0V, 1nC on mirror

10−18
10−19
10−20
10−21
10−22
10−23
10−24
−20

−15

−10

−5

0

5

10

15

20

Postition relative from atoms (mm)Figure 5.2: COMSOL simulations of the DC electric field at the location of the atoms. We simulate
the effect of having 1 nC of charge distributed uniformly on the surface of one of the mirrors. Also
shown is the effect of an applied voltage on that same PZT. Even at the highest operating voltage,
the overall shift should be on the order of 1 × 10−18 .

84
5.3.2

In-vacuum calibrated thermal sensors

Looking ahead, the hope is that Sr3 becomes a well-calibrated and accurate clock that canmake use of entanglement. In order to achieve the desired accuracy, we require good knowledge ofthe thermal environment. For this reason, we calibrated four PT100 thermal sensors and installed
them in the cavity space and cavity support mount. You can see the two sensors mounted in thecavity spacer in Figure 5.3a. The sensors are housed in a ceramic rod, where the wires are glued tothe ceramic rod, but the sensor itself is free floating to avoid strain related shifts of the resistancecalibration.
The sensors were first culled by performing repeated thermal cycles up to 150 C and backdown to room temperature. After each thermal cycle, the sensors were then checked by realizing
an ice melting point, by mixing doubled sitilled water with doubly distilled ice (frozen water). The
sensors that exhibited small jumps in the ice melting point value were subsequently calibrated. We
used a dry-block comparison method, where we couple the NIST calibrated sensors to a copper
block, and then couple our PT100 to that same copper block. We evaluated temperature gradients
across the block by comparing the temperature read by each reference sensors, and made sure thatthe gradient was below 1 mK. Example calibration data for Rgrey is shown in Fig. 5.3b, with thefit to the resistance data given by a quadratic function. The corresponding uncertainty resulting
from the fits, converted to temperature, is shown in Fig. 5.3c.

5.4

Laser systems and cooling

5.4.1

461 nm laser system and blue MOT

The blue laser system is a combination of extended cavity diode lasers (ECDLs) and injectionlocked lasers. Over time, the blue system in the Ye lab has become more centralized. The common
source is a blue laser locked to the atomic spectrometer. From there, a secondary Toptica 461
nm system serves as seed light for both Sr1 and Sr3. This secondary laser is offset locked to thespectrometer laser. We then seed injection locked lasers. In the past, our lab has routinely used

85

(b)

Resid.(mΩ)

RgreyResistance(Ω)

(a)

110
105
100

0.5
0.0
−0.5

0

15
Temperature (C)

30

Rblue

Uncertainty (mK)

(c)
10

Rblue
Rgrey

5

0
15.0

17.5

20.0
22.5
Temperature (C)25.0

Figure 5.3: (a) Top view of the optical cavity. Shown are the two thermal sensors that are installedin the cavity spacer. (b) Calibration of Rgrey in a dry-block comparison with three reference sensors.
(c) Calibration uncertainty for the two sensors as a function of operating temperature.

86
injection locked lasers from Nichia operating near 100 mW. For Sr3, we opted to try out a new
500 mW blue laser diode as the injection locked laser [127] . By doing so, we could simplify theblue laser system to only require two injection locked lasers to operate the experiment. The first
500 mW diode provides light for the Zeeman slowing. We find diminishing returns with increased
power in the Zeeman slowing beam and that 75 mW at the experiment is a healthy and reasonableamount. Thus, to avoid the damage of the optical fiber, we attenuate the laser before fiber coupling
to the experiment. The second diode is split into two overall paths, one serving the 2D MOT andthe other for the 3D MOT. On the experiment side, the 2D MOT has about 20 mW total. Each
3D MOT beam has between 5-10 mW of power.

5.4.2

Repump lasers

Initially we utilized two home-built ECDLs for the 679 nm and 707 nm transition. These lasers
utilized anti-reflection coated laser diodes, neither of which were close to the desired wavelength atroom temperature, thus necessitating heating and cooling near the maximum rated temperatures.
We have since replaced these lasers with commercial ECDLs from AOSense. In either case, thelasers are roughly set at the correct frequency using a wavemeter, and then frequency modulated byabout 1 GHz in order to address all of the hyperfine states. As a technical note - we generally findthat the blue MOT atom number is most critically dependent on the tuning of the repump lasers.
This indicates that a better scheme for the repump lasers would be to stabilize them either to an
optical cavity or an atomic reference (Sr hollow cathode lamp [128]). Then, one would directly
modulate the light with an EOM placing sidebands at precisely the correct frequencies, making
much more efficient use of the optical power.

5.4.3

689 nm lasers

To address the narrow, dipole forbidden transition of 1 S0 →3 P1 , we require reasonable quiet
lasers with minimal frequency drift. We use two 689nm lasers, both homebuilt interference filter
ECDLs, which were originally designed by Ben Johnson in Adam Kaufman’s group, though the87rough design for interference filter lasers have been around for quite some years. Josie Meyer helped
us out by building a few of these ECDLs for the 689 nm laser system. We have one laser each for
the F=9/2 and F=11/2 transitions of 1 S0 →3 P1 in 87 Sr. Each laser is phase-locked to the 689 nm
cavity system, which is an old “Ludlow” cavity that has an expected performance on the 1-10 Hz
level, well below any of our noise requirements for cQED experiments. To control the frequency ofeach 689nm laser, we put the reference synthesizer for the phase lock into frequency modulation
mode. Then, we can either drive the FM port with an arbitrary waveform generator (AWG), toachieve arbitrary modulation. For simplicity, we use an RF switch to change the modulation source
to a DAC channel, so we perform more simple linear ramps without having to reprogram the AWG.
Just before transferring the atoms from the blue MOT to red MOT, we ramp down the blue MOT
laser intensity to about 1/3 the initial value. This results in a colder blue MOT and optimizes thetransfer efficiency to the red MOT.
To capture atoms from the blue MOT to the red MOT, we need to broaden the laser linewidthof the red trapping lasers in order to match the Doppler broadening of the ≈ mK temperature blue
MOT atoms. For this initial “broadband red mot”, we frequency modulate both lasers with sinewave modulation, and the modulation depth is ramped linearly along with the intensity of the red
MOT beams in order to adiabatically transfer atoms from a broadband red MOT to the single
frequency red MOT. In order to reach the lowest possible red MOT temperature, the final intensity
setpoint is ≈ 0.1 × Isat . Our final red MOT temperature is approximately 1 µK, as measured bytime of flight. One special note is that our F = 11/2 laser also serves as our cavity probe laser.
The detailed optics layout for the implementation of that laser as the cavity probe is shown in Fig.
5.12.

5.4.4

Optical lattice and transport

We trap the atoms in a vertically-oriented optical lattice formed by the interference of two
counter-propagrating 813 nm beams. The lattice beams are split at the laser breadboard. From
there, each beam goes through their own AOM and then independently fiber-coupled to the exper-

88
iment. After the fiber outcoupler, the laser is picked off and intensity servoed. In order to stabilize
the relative phase of the two lattice beams, we form a Mach-Zender interferometer, using a dichroicon the top breadboard as one of the beamsplitters. When the lattice beams are well-aligned, we
observe an interference fringe, and can use this fringe for locking the relative phase of the twolattice beams. One issue is that the observed interference phase is not exactly the same phase thatthe atoms will experience in the vacuum chamber. Through geometric arguments, one can see that
there is a differential phase. Thus, despite stabilizing the interference fringe phase, we must be
careful to shield the open path length as best we can so that the atoms experience a stable lattice.
As a bare minimum, we evaluate the inloop error of the phase-stabilization, shown in Figure 5.4.
We compare the inloop error of the phase stabilization to the laser noise model of Si3 and of MJM.
For the synthetic laser that probes the cavity, we adopt the Si3 stability below ≈ 100 Hz, and follow
MJM above ≈ 100 Hz. We approach the MJM noise model around 1 kHz, and are otherwise safely
below both laser noise models [129, 17].
Further improvements to this stabilization scheme can be considered. One drawback ofdetecting the relative phase of the lattice at DC is that laser intensity noise is indistguishable
from phase noise. This means that we need to do a careful job at controlling the laser intensity(something we want to do anyway, for lattice heating considerations). While we don’t show thetraces here, the out-of-loop laser intensity dominates over the phase noise for Fourier frequencies
above 10 kHz. This is the primary reason for not pushing the bandwidth of the phase stabilization
with this setup, as we will actually write the laser intensity noise onto the phase of the opticallattice. Experimentally, we find that if we continue to increase the bandwidth, we start to sufferfrom atom loss. Using an EOM to sense the phase difference at an RF frequency in a similar
configuration as described in [130] might allow for tighter phase control. Another consideration isthat there can be out-of-loop phase fluctuations. For this, shortening the out-of-loop path (although
it is already as short as possible given the optics layout), and better shielding of the environment
from pressure and tempertuare fluctuations is always the way to go.
After the atoms are loaded into the lattice, we transport the atoms in this movable lattice

89

lattice phase noise freerun
lattice phase noise locked
MJM model
Si3 model
photodiode noise

Phase noise (rad2 /Hz)

10−2

10−4

10−6

10−8

10−10

10−12 0
10

101

102
103
Frequency (Hz)

104

105

Figure 5.4: Lattice phase noise measurements. The free running lattice phase noise is the red
trace, and the locked inloop error is the black trace. For comparison, the laser noise models for thetwo clock lasers MJM and Si3 are shown as the purple and cyan dashed curves. The photodiodenoise floor is the gray trace.

by ramping the frequency of the AOM in the bottom-up lattice beam. The motion of the lattice is
induced by changing the relative frequency of the two lattice beams and is given by∆~vlat =

λlat
∆ν,
2

(5.1)

where ∆~vlat is the velocity of the lattice and ∆ν is the relative frequency detuning of the lattice
beams. The transport waveform is shown in Figure 5.5. One important detail is that we mix threefrequency synthesizers together to form the AOM drive for the bottom-up lattice beam. In thisway, we can trigger three independent frequency ramp profiles corresponding to three transportdistances. This allows us to do smaller vertical manipulations of the cloud position relative tothe cavity, enabling the probing of multiple ensembles with the cavity. Future upgrades to the

1.2

0.5

1.0

0.4

0.8

0.3

0.6
0.2
0.4

Velocity (m/s)

Relative detuning (MHz)

90

0.1

0.2
0.0

0.0
0.00

0.02

0.04

0.06
0.08
Time (s)

0.10

0.12

0.14

Figure 5.5: The left axis indicates the relative frequency detuning of the optical lattice beams
for transport, and the right axis is the corresponding velocity of the lattice. The total transportdistance is approximately 42 mm.

RF synthesis will allow for more arbitrary configuration of ramp profiles, including more gentle
Gaussian ramps instead of the linear ramps we employ here. Another consideration for future
upgrades is the fact that during our transports, we allow the lattice phase to free run. One worries
about this, especially in the differential clock comparisons of Chapter 7. One can imagine that
small phase fluctuations lead to displacement noise of the atomic cloud, and if a measurement
is sensitive to this position (i.e. with respect to the clock laser phase), this can be a source of
unwanted noise. More sophisticated servo control can keep the lattice phase servoed during thetransport operations. We also wish to have a focus-tunable lens so that the tightest confinement
can be applied to the atoms in both the MOT bucket and the science cavity. This would allow forhigh radial trap frequencies, more easily driving the clock transition in the resolved radial sideband
regime Ω << ωR .

91

5.5

Clock laser
The clock laser is critical to the operation of the Sr3 experiment. We discussed in detail theSi4 and Si6 ultrastable laser systems in Chapter 2. Here we will discuss some of the critical aspectsof the clock laser as it applies to the Sr3 experiment.

5.5.1

Fiber noise cancellation

In order to drive the clock transition in a low noise fashion, we need to transfer the laser lightfrom the distribution center to the experimental table without adding appreciable phase noise. The
distribution center houses the injection locked laser, which receives its seed from the fully stabilized
clock light. We employ fiber noise cancellation (FNC), where we have a reference plane (mirror)
at the distribution center, and another reference plane at the experiment where the light is retroreflected through the fiber. By interfering the light from these two reference planes, we directly
measure the relative phase noise introduced by the fiber. The 0th order of the second AOM at
the experimental table is retro-reflected as the reference light. We feedback to the AOM at the
distribution center, with a bandwidth of ≈ 100kHz.
Due to the fact that we stabilize the 0th order, and the 1st order goes to the atoms, any
differential noise between these two paths will be seen by the atoms. To evaluate the differential
noise, we temporarily insert a retro-reflector in the path of the 1st order beam, while stabilizing the0th order beam as usual. The 1st order beam will produce a beatnote at the photodetector that is
distinguished as a particular RF tone. If we move the retro-reflector and measure the phase noiseof this 1st order beat will tell us about the differential noise. The measured noise of this 1st order
beat is shown in Figure 5.6. When we place the 1st order retro close to the 0th order reference,
we observe the black trace, showing noise well below the laser noise model of Si3 (dashed line).
Moving the 1st order retro to just before the chamber, we see the noise is slightly larger, but stillat an acceptable level. We note that in order to achieve this noise performance, we cover the clock
optics with a crude enclosure of blackened aluminum foil, and close the enclosure around our entire

92
optics table.
Furthermore, any differential noise between the 1st order clock light and the optical lattice
will also appear as clock noise. We place the clock retro for fiber noise cancellation as close to theoptical lattice phase reference as possible. To minimize any potential relative path length noise, we
cover the optics in a secondary enclosure of blackened foil.

5.6

Atomic state preparation
Now it’s time to put all of the above-described laser systems into action. The overall ex-perimental sequence is displayed in Figure 5.7. The sequence starts with cooling and trapping
the atoms in a blue MOT. We send 13 A of current through our MOT coils, corresponding to ananti-Helmholtz magnetic field gradient of about 30 G/cm. A rough estimate of the blue MOT atom
number is 10 million. At the end of the blue MOT, we linearly ramp the intensity of the 3D MOT
beams down in an out-of-loop fashion, by about a factor of 10. This leads to increased transfer
efficiency from the blue MOT to the broadband red MOT, giving ≈ 2x more atoms.
The red MOT operates on the dipole-forbidden transition of 1 S0 →3 P1 [131]. We transfer
the atoms from the blue MOT to a broadband red MOT by simultaneously turning off the blue
MOT beams and turning on the red MOT lasers at full intensity (≈ 1000I/Isat each). During
the initial broadband red MOT, the 689 nm trapping lasers are frequency modulated to ≈ 1
MHz wide for the 11/2 laser and ≈ 500 kHz for the 9/2 laser, respectively. The broadband tosingle frequency transition is realized by linearly ramping the modulation depth of the frequency
modulation amplitude down to zero over 250 ms. Simultaneous with the frequency modulation
ramp, we linearly ramp the laser intensity down to less than Isat . Going to this low intensity
results in a red MOT temperature of about 1 µK, as measured by time-of-flight spectroscopy.
During nominal operation, the vertical optical lattice is kept on during the entire blue andred MOT sequence, meaning that we are continually loading the lattice. We note that because the813 nm trapping lattice causes an ac Stark shift of the 1 S0 →3 P1 transition, we optimize with thisinhomogenous shift always present. We suspect that we get away with this given the size of the93Phase noise (rad2 /Hz)

10next to 0th order retro
just before chamber
Si3 model

−3

10−5

10−7

10−9

10−11

10−1

100

101
102
frequency (Hz)

103

104

Figure 5.6: Differential fiber noise between the 0th order and 1st order beams. The black trace
indicates when we place the 1st order retroreflection near the 0th order retro, and red is placing theretro as close to the vacuum chamber as possible, just before the viewport. For reference, we plot
the Si3 noise model, showing reasonably adequate noise cancellation.

Y

Cleanup pulse

Lattice ramp

Transport

Optical pumping

Red MOTBlue MOT

94

Clock pulses and
QND measurements

X

Z

Anti-helmholtz
field 30 G/cm

Anti-helmholtz
field 1 G/cm

BY=2 G

Ramp toclock field : BX=2 G

Figure 5.7: The top panel shows the experimental sequence roughly laid out in time. Below
we graphically indicate the location of the MOTs, and the subsequent loading of the lattice andtransport into the cavity. We also note the magnitude and direction of the relavant magnetic field
for each step.

95
optical lattice is ≈ 150 µm diameter at the MOT location, keeping the inhomogenous ac Stark shift
to an acceptable level. However, it might be possible to improve the red MOT by instead ramping
the optical lattice up at the end of the MOT sequence. This can involve frequency ramping thered cooling lasers to compensate for the ac Stark shift, to optimize cooling of the atoms inside thelattice.
After extinguishing the red MOT light, we perform optical pumping into the ground-state
hyperfine level 1 S0 , mF = −9/2 . We apply a bias magnetic field of about 2 G along the Y direction, and shine on an optical pumping beam tuned near the 1 S0 →3 P , F = +9/2
1transition

with circularly polarized light, in order to pump the atoms into mF = −9/2. This optical pumping is optimized by adjusting the frequency, intensity, and duration of the pumping, to achieve
approximately 70% of the atoms in mF = −9/2. We then move the cloud up into the cavity by
detuning the bottom-up lattice beam, as described in Section 5.4.4. Once there, we begin clockstate manipulation and QND probing. Future improvements will implement optical pumping intandem with sideband/radial cooling.

5.7

Cavity-stabilization 813nm laser
In the very early stages of the Sr3 experiment, we stabilized the cQED cavity to our M

Squared Ti:S laser. Over time we realized that the performance of this Ti:S was much noisier thanwhat we required for the cQED cavity stabilization. We would often see large amounts of frequency
noise manifest itself as intensity noise in the transmission of the cavity, and this signal was actuallya good diagnostic for when the Ti:S was misbehaving. We knew we could build a lower noise laser
than the Ti:S, albeit at lower power, so we decided to build an independent ECDL for the cavity
stabilization. This ECDL is based on the same design as our 689 nm lasers described in section
5.4.3. The 813 nm laser uses an anti-reflection coated laser diode from Eagleyard (EYP-RWE0840-06010-1500-SOT02-0000), and the interference filter is a 3 nm FWHM bandpass filter from
Iridian (MI000002). The free-running Lorentzian linewidth of this laser is ≈ 20 kHz, which is abouta factor of 10 better than the M-Squared Ti:S. One unfortunate note on this laser - we have had

C
g0 /2π (kHz)L (cm)
finesse, Fκ (Hz)
mode waist (µm)
T1 (ppm)
T2 (ppm)
total loss L

Cavity parameters
689 nm0.3
8.6
6.9
16,000157,000 (measured)71
280 (specified)
20 (specified)
100 (estimated)

96
813 nm
NA
NA
6.9
8,000
330,00075
200
200
100 (estimated)

Table 5.1: Key parameters of optical cavity

the PZT fail on this laser twice during power outages. We therefore warn any graduate student
reading this thesis to not use the cylindrical PZTs from Thorlabs in any critical application.

5.8

cQED cavity

5.8.1

Cavity design and construction

Perhaps the most unique and critical aspect to the Sr3 experiment is the optical cavity. The
cavity parameters are shown in Table 5.1. We designed the cavity to operate on the λ = 689
nm 1 S0 →3 P1 transition. The cavity geometry and the atomic transition properties determine thepeak atom-cavity coupling, which we calculate to be g0 /2π = 8.6 kHz. Correspondingly, the peaksingle-atom cooperativity is given by C = 4g02 /(κΓ) = 0.25. This enables us to enter the collective
strong-coupling regime (N C  1) with a few thousands of atoms.

The cavity is designed as a single-ended cavity at 689 nm, with specified mirror transimission
values of T1 = 280 ppm and T2 = 20 ppm. This means that intra-cavity light will preferentiallyexit through the input mirror, making that the optimal measurement port for quantum efficiency.
Before baking the system, we measured a cavity finesse of F = 20, 000, which was consistent withthe specified transmission values of the mirrors and negligible optical loss. We knew the bake out

97Transmission intensity (V)

8
6
4
2
0
0

2

4
6
Time (µs)

810Figure 5.8: Measurement of the cavity finesse at 689 nm by monitoring the transmission after
abruptly shutting off the light. The black data and fit are before the bake out, giving F = 20, 000.
The red data and fit are after the chamber bakeout, giving F = 16, 000. As a check, we temporarily
move the photodetector to before the cavity (gray data), to confirm our light shut-off time issufficiently short.

98
process can be a risky time for such high reflectivity mirrors. To try to minimize the thermal
gradients in the chamber, we heated up the chamber over the course of ≈ 24 hours. Although
the bakeout was successful in reaching the 10−11 Torr level, the cavity finesse ended up degrading
during the bakeout. We calculate that the cavity obtained an additional loss of L = 100 ppm.
We suspect that this increase of loss could have come from a deposition of residual hydrocarbons
in the chamber. The primary consequence of the increase in loss is a decrease in the first mirror
transmission rate κ1 relative to the total cavity power decay rate κ. This is quantified and referred toas the quantum efficiency for the cavity Qcav = κ1 /κ, which went from Qcav = 0.85 to Qcav = 0.75.
This results in a change in the overall quantum efficiency of just 5%. Given the nonzero chance thatcleaning the mirrors could actually make things worse, we opted to move foward with the slightly
degraded mirrors.
For cavity stabilization, the cavity mirrors also have high reflectivity at 813nm, with thecavity parameters at this wavelength shown in the second column. The cavity linewidth at 813nm
is 330 kHz. This relatively narrow linewidth at 813 nm of κ813 = 300 kHz was chosen to makecavity stabilization less susceptible to technical effects like RAM. However, the narrower linewidth
comes with the tradeoff of needing to deal with the problem of FM to AM conversion.
A closeup photograph of one of the cavity mirror assemblies is shown in Figure 5.9, and afull view of the cavity is shown in Fig. 5.10. The cavity employs two PZTs supplied by Noliac, one
having 13.2 µm travel (NAC2123-H10-A01) and the other having 3.3 µm travel (NAC2123-A01).
The idea is that the longer travel PZT could allow us to attempt to make the 813nm mode closer
to the magic wavelength, but for the experiments in this thesis we have not used the longer travel
PZT. We note one consideration to PZT stacks is that the conversion from voltage to displacement
is multiplied by the number of PZTs glued together, in our case 4x. For the 100 Hz linewidthgoal of our stabilization, the high voltage PZT driver can be a significant source of noise. The
PZTs are glued to the copper plate using a very thin layer of EpoTek 353ND epoxy, and then acopper “mirror cup” is glued to the top of the PZT with the same epoxy. The mirror cup has three
extrusions to gently center the mirror on the cup. The mirror is glued in the midplane of the mirror

99

mirrorgrounded cup
PZT stackplatespacerFigure 5.9: Up close photo of the mirror assembly with the PZT stack. You can also see two of the
Ceramabond epoxy joints holding the mirror inside the mirror cup. The whole epoxied assemblyis bolted to the cavity spacer with four vented screws, of which two are visible here. The othermirror, is identical apart from the overall height of the PZT.

100
using a ceramic based adhesive Ceramabond 853. Prior to choosing these epoxies for use in our
vacuum chamber, both epoxies were vacuum tested using a residual gas analyzer, demonstrating
outgassing limited in the test chamber at to the 10−10 Torr level. We note that when gluing cavity
mirrors for optical cavities - it’s critical to choose an epoxy that does not induce a large amount ofstress on the optic. During the first gluing of the mirrors for the Sr1 buildup cavity, we found that
353ND epoxy could go as far as to shatter the fused silica optic, which drove us to the more gentle
ceramic-based adhesives.
The mechanical support for the cavity is a double pendulum isolation scheme. We suspenda heavy intermediate piece of copper from the vacuum chamber with three short rods of viton
serving as the lossy spring. The optical cavity is then suspended from this intermediate piecewith three more pieces of viton. The idea of this double pendulum is that it will greatly attenuate
vibrations coupled to the cavity above the resonant frequencies. By simulating the coupled system in
COMSOL, we predicted that the resonant frequencies of this mechanical system would be clustered
between 10 - 50 Hz. In hindsight, while these mechanical resonances do indeed improve the higher
frequency noise of the optical cavity, it poses significant challenges when our cavity frequency
measurement window lasts 40 ms. Future experiments that further reduce the atomic motion
might permit measurements with shorter duration, taking further advantage of this mechanicalisolation scheme.
While copper has nice properties for vacuum and BBR, there is the downside of being capableof supporting significant Eddy currents. The largest change in magnetic flux will occur when we
switch the MOT field and when we ramp on the clock bias field. These changes in B field can
and will induce currents in the copper spacer, leading to time-dependent forces on the cavity
Furthermore, these Eddy currents could take considerable time to damp out. To mitigate thiseffect, we used a wire-EDM to cut a very small slot in the optical cavity and the supporting copper
piece. We measured the effect of including such a slot using a test spacer. We suspended the cavity
test spacer from the upper copper piece, and mounted these pieces above the MOT bucket. A
3-axis magnetometer was placed inside the cavity spacer, and we rapidly switched off the current

101

Figure 5.10: Full view of the cavity, as it would be seen from below after installation. In clear
view are the two mirrors. Also visible is the slot we cut in the cavity, to reduce Eddy currents.
The cavity is mounted onto an aluminum jig that was used to secure the cavity during installationof the various components.

102
in the MOT coil and measured the time response of the magnetic field (Figure 5.11). We checked
the response with no cavity present (orange), with the cavity but without the slot cut (blue), andthe cavity in with the slot cut (green).

5.8.2

Cavity stabilization

In this section we will describe the cavity stabilization scheme that we used for most of ourearly experiments in Sr3. The requirements for the frequency noise performance depends on several
factors. The primary one is the magnitude of the cavity frequency fluctuations due to quantum
projection noise. When we are operating in the slightly detuned regime, the QPN fluctuations
are roughly 2 kHz for approximately 10,000 atoms. If we want the technical noise floor comingfrom relative laser - cavity frequency noise to be 20 dB below QPN, we need to reach a frequency
resolution of 200 Hz. As we will discuss, the timescale of the probing will change what Fourier
frequency we are sensitive to, and this can impact the strategy for stabilization. To put this incontext for our experiment, we probe the cavity with a measurement time of 40 ms, meaning that
we need to focus on frequency noise in the range of 1 - 100 Hz.
The simple idea is to make the cavity length as stable as possible for cQED probing of theatom-cavity system. To this end, we first stabilize the 813 nm ECDL laser described in section 5.7
to the optical frequency comb by feeding back to the current at high frequency, and to the PZT ofthat ECDL at lower frequencies. We achieve a locking bandwidth of ≈ 300 kHz. This allows us tostabilize this laser down to the noise floor given by Si3+comb in addition to the uncancelled fiber
noise floor between the ECDL and the comb. We then send this 813 nm light to the cQED cavity,and stabilize the cavity length to the ECDL via a PDH lock, feeding back to the cavity PZT. For
the experiments in this thesis, we keep the intra-cavity lattice at a low level so as not to significantly
trap the atoms along the cavity. To this end, we turn the incident power down to ≈ 10 µW, andactively stabilize the power in transmission of the cavity. After the bakeout the cavity has a finesseof ≈ 6300 at 813 nm, leading to a power buildup of ≈ 1200. We estimate that our operational
trap depth in the cavity is ≈ 0.1Er . The relative large finesse for 813nm allows us to more easily

103

Bz spacer in
Bz spacer out
Bz cut spacer incurrent

Magnetic field (G)

0.4
0.2
0.0
−0.2
0

20

40
60
Time (ms)80

100Figure 5.11: Response of the magnetic field when rapidly switching off the current through the
MOT coils. The magnetometer is placed inside a test cavity spacer having roughly the same
geometry as our actual spacer. We measure the response without the spacer (orange), with thespacer (blue) and with the spacer but with a slot cut into it (green). We show a dramatic effect ofincluding the slot in the spacer, showing a faster damping time constant of induced Eddy currents
in the spacer.

104

intensity control
feedback to AOM-3

Pre-stabilization
synthesizer

cavity PDH lock
comb
servo

813
ECDL

AOM-3

iso

Pre-stabilization
synthesizer

Cavity
servo

-

AOM-2
probe
LO

iso

AC

AOM-1

689
ECDL

EOM

DC
amp

servo

Picoscope

VCO
Carrier-LO phase stabilization

Figure 5.12: The optics layout and stabilization scheme of the cavity QED setup. The 813nm
laser is pre-stabilized, and the cavity is locked to that pre-stabilized laser via the cavity PDH lock.
The 689nm laser is pre-stabilized to the red cavity system. The 689 ECDL is split into the LO andprobe path. The leakage light from the cavity is detected via homodyne detection.

Laser frequencies

Cavity modes

105LO-75.6MHz-75.6MHz

AOM-2

AOM-1

EOM
Figure 5.13: Frequency diagram of the cavity modes and of the different laser light tones. The barecavity frequency ωc is detuned from the bare atomic resonance ωa by δc /2π = 1 MHz. The cavity
is shifted by the presence of atoms in the ground state |↓i. The relavant frequencies are shown.
The carrier is synthesized by frequency shift of AOM-2, and the sideband that probes the cavity
is generated by the EOM drive. The LO is correspondingly frequency shifted to be on resonancewith the sideband for homodyne detection.

stabilize the cavity length, since a narrow cavity resonance makes issues like RAM less worrisome.
The tradeoff is with FM → AM conversion, making the implemenation of an intra-cavity lattice
more challenging.
To measure the noise floor set by the relative fluctuations of the laser and the cavity, we probe
the cavity with a relatively large optical power (nW or so). We do this so that the photon shot noise
limit is negligible, and we can unveil the technical frequency noise floor that does not depend on
optical power. The relevant noise measurements are shown in Figure 5.14. For reference, the freerunning cavity noise is the purple trace, showing enormous noise at lower frequencies. The inloop
error of the 813nm lock to the comb is the blue trace, showing more than sufficient stabilization.
We note that this is an optimistic trace, as there is a 30 m optical fiber that is currently not fiber

106689 probing locked cavity
inloop of cavity lock to 813
inloop of 813 lock to comb
Freerunning cavity

Frequency noise (Hz2 /Hz)

108
106
104
102
100
10−2 010101102Frequency (Hz)

103

104

Figure 5.14: Noise measurements of cavity QED stabilization scheme. The blue line is the inloop
error of the 813 ECDL lock to the comb. The gray line is the inloop error of the PDH lock of thecavity to the 813 laser. The 689nm pre-stabilized laser then probes the cavity - serving as the outof loop noise measurement. We can see the frequency bands where we are still limited by the inloop
lock of the cavity.

107
noise cancelled, and can easily be a contributor to noise at the 100 Hz2 /Hz level. The inloop errorof the cavity lock to the 813 nm laser is shown as the gray trace. When we then probe the cavity
with the pre-stabilized 689nm laser, we measure the frequency noise spectrum shown as the red
curve. We can see the servo bump at around 1 kHz, which is limited by the ringing up of mechanicalresonances starting at 1.2 kHz and 2.5 kHz.
To measure the cavity frequency, we employ homodyne detection of the probe laser. For thecavity QED experiments described later in this thesis, we will be doing repeated cavity frequency
shift measurements, where the cavity resonance has been shifted by the presence of atom in the
ground state. We will refer to the repeated probe of the cavity resonance as “two window probing”.
The transfer function for a two-window measurement is
T (f ) =4 sin2 (πf (Tm + τm )) sin2 (πf Tm ).
(πf Tm )2(5.2)

We can multiply the measured noise spectrum (red trace from Figure 5.14) with this transfer
function and predict the noise of the two window probing. When we perform this calculation,
we find that we can push the technical noise of two repeated cavity frequency measurements toapproximately 200 Hz rms for a measurement time of Tm = 40 ms.
Looking forward, one can imagine various ways to improve the cavity stabilization system.
The relative frequency noise between the 689 nm laser and the locked optical cavity is largely
limited by the inloop stabilization of the cavity in the band from 30 Hz to 1 kHz. Since there
are already strong correlations between the frequency noise at 689 nm and the inloop PDH error
signal at 813 nm from 20 Hz - 10 kHz, one could imagine implementing feedforward with no needfor any changes to the hardware. As an alternative to feedforward, one could directly improve
the relative noise by implementing a more sophisticated loop transfer function that compensates
for the mechanical resonances, allowing for a higher bandwidth. Either of these approaches would
improve the ultimate technical noise floor of the cavity frequency measurement system. Further
improvement to the low frequency noise below 20 Hz would also be beneficial. A primary suspect for
the excess low frequency noise is residual amplitude modulation in the 813 nm cavity stabilization

108
lock.

5.9

Homodyne interferometer
The homodyne interferometer is shown in the lower right corner of Figure 5.12. This iswhere we will measure the leakage light from the atom-cavity coupled system to perform QND
measurements. The primary goal is to achieve a photon shot noise limited homodyne detection of
the optical tone probing the atom-cavity system.
Performing homodyne detection combined with our somewhat long probe time of 40 ms means
we are increasingly sensitive to low frequency noise. We found that pointing instability of the LO
beam on its own (with the signal port blocked) could give rise to significant low frequency noisein the output of the homodyne photodetector. We confirmed the source of the low frequency noiseby picking off a small fraction of the beam near the homodyne PD and detect it with a quadrant
PD. There were significant correlations between the homodyne noise and the beam position as
measured by the quadrant PD. We then reconfigured the interferometer to keep the path as shortas was reasonably possible, and intentionally tilted optics to prevent optical etalons. Furthermore,
we enclosed the optics surrounding the cavity in a secondary box, although with an imperfect seal,
and found that this greatly improved our low frequency noise performance.
The homodyne fringe is detected using a home built balanced photodetector. The photodetector operates in current-subtracting mode, where the two photodiodes are wired back to backsuch that the sensed current is the difference between the two photocurrents. The technical darknoise of the photodetector is ≈ 31 dB below the LO photon shot noise for an LO power of 80µW. This photodetector design has advantages in simplification of the electronics, but means that
there is not a convenient way to electronically balance the two ports for optimal intensity noise
suppression. While one can fine tune the balance of the beamsplitter with polarization/alignment,
we add a safety cushion by adding active intensity control to both the LO and cavity probe beams,
The combination of active stabilization of the LO intensity and the common-mode rejection of the
LO intensity noise allows the homodyne detection to be nearly photon shot noise limited.

109
There are two primary sources of technical noise on top of the photon shot noise. We send the
LO beam through the homodyne interferometer, and measure the output noise (Figure ??). Notehere that we do not add any extra gain at the output of the photodetector, so the photocurrent
noise is amplified by the transimpedance gain of 106 Ω. In Figure ??, we can see the noise rises
with a 1/f scaling, and fit a noise model to the data shown as the orange line. The expected photon
shot noise is the red dashed line, and the dark noise is the solid blue line.
If one looks at the noise of the LO alone (signal port blocked) through the homodyne beamsplitter and detection, there are significant peaks in the noise spectrum (Figure 5.15). We placed an
accelerometer on the table near the homodyne interferometer, and discovered strong correlations
with the homodyne output voltage at certain frequencies. The distinct frequencies at ≈20 Hz and≈30 Hz come from air conditioning motor vibrations. There is another cluster of peaks in the300-400 Hz range that we identified to be the mechanical resonances of the optical breadboard thathouses all homodyne optics. Our understanding of the coupling between vibrations of the opticalbreadboard and the homodyne output voltage is that the vibrations drive differential pointing noiseof the LO beam through the interferometer. There can easily be differential etalons between thetwo paths after the beamsplitter to each photodiode. Such a differential etalon could give a varying
intensity due to mechanical fluctuations. Interestingly, we have found that the amplitude of thevibration induced peaks in the homodyne spectrum depends on the precise alignment of the LO
beam. This needs to be kept in mind when optimizing noise performance of the homodyne detection. Thinking toward the future, it will be advantageous to make the homodyne interferometereven more compact. An improved design would involve mounting all of the optics to a common
chunk of metal in a very compact fashion. The optics would be completely enclosed so as to prevent
air currents from giving time-varying etalon effects. The one downside is the compactness of the
optics can make it difficult to avoid spurious reflections, but should still be manageable.
When we are making measurements of the cavity frequency, we synchronously record thehomodyne signal and the accelerometer output on an oscilloscope (Picoscope), and perform subtraction of the vibration trace in order to reach the photon shot noise limit. The only free parameter110
in the subtraction is the arbitrary scale factor between the homodyne voltage and the accelerometer
voltage - there is no significant phase shift between the signals.

5.9.1

Path length stabilization

As seen in Figure 5.12, we split the 689 nm laser into the probing path and the LO path. Any
relative path length fluctuation between the two paths will lead to relative phase noise between
the probe light and the LO light. When probing with heterodyne, this phase noise can impact themeasurement, but slow overall relative phase drifts are less of a concern, since the heterodyne signal
spends equal time in the I and the Q quadrature. However, when probing the cavity on resonancewith homodyne detection, we need to actively control and set the relative phase. This is because
the most sensitive quadrature on resonance is Q, and any deviation will distort the homodyne
signal and reduce the signal to noise ratio. We thus implement a path length servo, shown in thebottom of Figure 5.12. We detect the beatnote between the carrier and the LO beam, which are
shown in frequency space in Figure 5.13. This beatnote is stabilized to a reference synthesizer by
feeding back to AOM-1 in the LO path. We note in passing that this path-length beatnote appears
at the same exact frequency as the EOM drive.
We mention one minor technical detail regarding the RF driving. While one could simply
split the RF drive for the EOM and use that synthesizer as the phase lock reference, we opt to use
two independent synthesizers that are phase locked to the same 10 MHz microwave reference. Thisway, one can change the relative phase by adjusting the synthesizer settings, instead of dealing
with delay lines, or other ways of setting the relative phase. The downside is that the relative
phase noise is not common-mode at higher frequency. Therefore we opt to use a low phase noise
synthesizer SRS SG382, such that the contributing phase noise of the synthesizer is well below thetechnical noise floor of the laser-cavity system. Another consideration for the path length servo
is the photon shot noise. The power in the carrier is approximately 150 nW. This relatively highcarrier power compared to the probe sideband means the photon shot noise limit will not degrade
the signal to noise of the detection of the sideband. One could imagine turning down the carrier

111

0.004

voltage (V)

0.002

0.000

−0.002−0.004

2.2

2.4

2.6
time (s)2.8

3.0

power spectral density (V 2 /Hz)

10−5
10−6

homodyne voltage
accelerometer
difference

10−7
10−8
10−9
10−10 010101102frequency (Hz)103

Figure 5.15: (a) Measured output of the homodyne voltage (blue) and the accelerometer (orange)
placed on the optical table near the homodyne interferometer. Both signals have been bandpass
filtered in software to make the correlations clearly visible. (b) The same signals in the frequency
domain. After we subtract the accelerations from the homodyne voltage, we can see strong cancellation of the noise peaks at 20 Hz and 30 Hz, without impacting the homodyne noise at otherfrequencies. We note here that the photodiode voltage is amplified by a voltage factor of 25, on
top of the 106 gain of the transimpedance amplifier.

112
power by a factor of 10 without facing serious consequences.

Chapter 6

Generating and measuring entanglement on the Sr clock transition

With this newly built Sr3 experiment, we can now ask: can we directly observe an opticalclock ticking with fluctuations below the fundamental limits set by the quantum projection noise?
Before even dreaming of obtaining that result, there were many aspects of the system to
implement and iron out. In this Chapter, we discuss performing state-of-the-art clock spectroscopy
in our system, and use it to understand the limitations imposed by atomic motion. In Section 6.2, weestablish coupling between the atoms and the cavity mode, entering the strong collective coupling
regime. In Section 6.3, we directly measure the QPN, allowing us to carefully calibrate the effective
atom cavity coupling gef f . We then demonstrate QND measurements with measured spin noise
reduction of -4.7 dB below QPN. Together with measurements of the Ramsey fringe contrast loss,
we measure a spin squeezed state that has phase resolution 1.7 dB below the Standard Quantum
Limit, verifying the presence of entanglement.

6.1

Clock spectroscopy
We first explore the temperature of our atomic sample by probing the atoms with clock

spectroscopy. As the simulations from Chapter 3 suggest, and by intuition, we know that we aregoing to be happier with a colder atomic sample. We take a brute force approach to prepare agas with temperatures less than 1 µK. After the initial state preparation described in Section 5.6,
we remove the higher temperature atoms by applying a “lattice ramp”, which involves ramping
the optical lattice to ≈ 7Er over 10 ms, hold there for 20 ms, and then back up to 20Er in 10

114

0.5

Excitation fraction

0.4

lattice ramp
sideband coolingno cooling/ramp

0.3

0.2

0.1

0.0
−40

−30

−20

−10
0
10
Clock detuning (kHz)

20

30

40

Figure 6.1: Sideband scans of the clock transition. Without any ramping of the lattice or additionalcooling (green), with sideband cooling (red), and with the lattice ramp along (black). The latticeramp gives a sample prepared primarily in the ground motional band hnz i = 0, and a narrower
blue sideband (radial temperature). The carrier signal is heavily saturated in this clock scan.

ms. The ramp is adiabatic with respect to the fast motion along the tightly confined direction,and quasi-adiabatic with respect to the radial motion. There is no phase-space compression in thislattice ramp, it instead lets the hotter atoms to simply leave the trap. It is a bit of a shortcut
compared to implementing efficient radial and sideband cooling. At the expense of throwing away
about half of our atoms, it easily leads to a cold sample of 20,000 atoms.
We perform sideband clock spectroscopy in order to look at the population of higher motional
bands (Figure 6.1). The sideband spectrum just after transport and optical pumping but without
further cooling strategies is the green trace, giving a temperature of 1.2 µK. We compare justsideband cooling vs just the lattice ramp alone in the red and black traces, respectively. Both leadto hnz i → 0, but the lattice ramp also leads to a colder radial sample as seen by the width of theblue sideband.

115
Extracting the radial temperature from the blue sideband is not a reliable method. To
carefully measure the radial atomic temperature, we perform radial scans of the clock transition.
For this, we send the clock laser along the radial direction of the trapped pancakes in the 1D lattice.
In this scenario the clock laser probes the atoms well outside the Lamb-Dicke regime, and will besensitive to the Doppler shifts of the radially moving atoms. By scanning the frequency of theclock laser, we directly access the Doppler limited linewidth. This measurement is shown in Figure
6.2, where the red data points is without the lattice ramp, and the black data points are with thelattice ramp. The extracted radial temperature is 447(20) nK without lattice ramp, and 291(22)nK with the lattice ramp.
With the atomic temperature nice and cold, we proceed by zooming in on the optical transition. The background magnetic field is nulled to zero at the location of the cavity by performing
clock spectroscopy with all 10 ground-state spin states present. When the applied bias field exactly
cancels the background magnetic field, all 10 π-transitions will be degenerate. By adjusting thefields in the X, Y, and Z direction and iteratively zooming in on the carrier by probing with longer
π-pulses, we achieve a Fourier limited linewidth of 5 Hz. We do not push this further as we do notplan a long interrogation time at the first stage of spin squeezing experiments.
To remove the atoms in hyperfine spin states other than mF = −9/2, we apply a “cleanup
pulse”. This involves driving a clock pulse on the magnetic sub-level transition (mF = −9/2 →mF = −9/2), and then blowing away all of the remaining ground-state atoms that were either notpromoted to the excited state using a pulse of 461 nm light. With the pure initial excited state,
we are finally ready for spectroscopy. In the same bias magnetic field, we can drive the atoms andobserve the Rabi flopping between the ground and excited clock states.
For example, we probe the atoms via Rabi flopping with a Rabi frequency of 12 Hz in Figure
6.3a and 111 Hz in Figure 6.3b. We observe dephasing of the Rabi flopping. This is particularly
relavant for squeezing experiments where we need to apply several pulses to manipulate the atomic
state. To attempt to understand our loss of contrast, we explore the model presented in [132]. In
that work, a finite misalignment angle between the clock laser and the optical lattice is considered.

116

Excitation fraction

0.20

0.15

0.10

0.05
without lattice ramp
with lattice ramp

0:00
20000

10000
0
10000
Clock laser frequency (Hz)20000

Figure 6.2: Radial Doppler scans of the clock transition. Red data points are without the latticeramp, black points are with the lattice ramp. The radial temperature extracted from the Gaussianfits (corresponding dashed lines) are T=290(10) nK for with lattice ramp, and T = 447(20) nK
without the lattice ramp.

117ΩR = 111Hz

Excitation fraction

1.0
0.8
0.6
0.4
0.2
0.0
0.000

0.005

0.010
0.015
Pulse time (s)

0.020

0.025

ΩR = 12Hz

Excitation fraction

1.0
0.8
0.6
0.4
0.2
0.0
0.00

0.05

0.10

0.15 0.20 0.25
Pulse time (s)

0.30

0.35

0.40

Figure 6.3: Rabi flops (a) Measured excitation versus the Rabi pulse time in seconds, with adrive Rabi frequency of ΩR /2π = 111 Hz. (b) Same as (a), with the drive Rabi frequency set toΩR /2π = 12 Hz.

118

1.0

Ramsey contrast

0.8

0.6

0.4

0.2

π/20 − T − π/2φ

π/20 − π/290 − T − π/2φ
0.0
10−3

π/20 − π0 − π/290 − T − π/2φ
10−2

10−1
Dark time (s)100

Figure 6.4: Ramsey contrast for increasingly complex pulse sequences. The pulses are labelled witha sub-script, indicating the relative rotation angle. π/290 refers to an “align” pi/2 pulse, required
for squeezing. The introduction of the π0 is the π pulse we need to perform the pre-measurementin our squeezing experiments.

Due to this finite angle, there is a projection of the radial motion along the axial clock, modifying
and reducing the Lamb-Dicke parameter. We fit our measured data to a model from [132], plugging
in our measured radial temperature, and those fits are shown as the solid red lines in Figure 6.3.
These fits suggest an overall angle between the clock and lattice of 2.2 mrad.
We also perform Ramsey spectroscopy experiments, and explore the Ramsey contrast versus
the dark time T for different pulse sequences. For the simplest Ramsey type experiments, wemeasure Ramsey contrast of 90 to 95%, as seen by the black points in Figure 6.4. However, for
our spin squeezing experiments, we will be required to apply several clock pulses to manipulate
the atomic state in the clock manifold. These clock pulses will be separated in time according to
the time duration of the population measurements. One can imagine that radial motion will again
play an important role in these pulse sequences. As shown in Figure 6.4, we can see by adding

119
the necessary clock pulses reduce our contrast to the 70% level. This is further reduced by adding
dwell time between the π/20 and the π0 , which is necessary for the QND measurements.
One other crucial takeaway for Ramsey contrast is in relation to our previous discussionsof atomic temperature. We have found that the Ramsey contrast can be negatively impactedwith hotter atoms. This again suggests that atomic motion is playing an important role in theatomic coherence. The clear way forward is to freeze out the motion by implementing more atomic
confinement with a 2D or 3D optical lattice.

6.2

Entering the collective strong-coupling regime
To realize the coupled atom-cavity system, we first need to align the lattice beam to thecavity mode. Most cavity QED experiments trap the atoms in the cavity mode, but we have a moreunconstrained geometry without strong trapping in the cavity. Rough alignment is obtained byimaging the atoms through the second mirror (T≈0.5 at 461 nm) onto a CCD camera. By observing
the transmitted cavity mode on the same CCD, rough alignment can be obtained. However, if thecavity mode is not well centered on the imaging lens, there can be small errors in this technique,and thus we look directly for the vacuum Rabi splitting as a measure of successful alignment. To
more easily obtain the first signal, we operated with all 10 spin states in a nominally zero magnetic
field, so all of the different spin states added together in the vacuum Rabi splitting, and we cantake advantage of the higher atom number.
Since we wish to avoid Raman spin flips, we choose to operate on the nominally cyclingtransition |1 S0 , F = 9/2, mF = −9/2i → |3 P1 , F = 11/2, mF = −11/2i. To couple to thistransition, we apply a 2 G bias magnetic field along the cavity optical axis (X direction), andprobe the system with σ− polarization. With the bare cavity tuned on resonance, we measure atypical vacuum Rabi splitting of 750 kHz, as shown in Figure 6.5. Measuring the ground state atompnumber amounts to measuring the vacuum Rabi splitting Ω↓ = 2g N↓ . Operating on resonanceoffers several technical advantages. By simultaneously probing the two vacuum Rabi modes, andtaking the difference, one becomes insensitive to relative fluctuations between the laser and the

0.02
0.00

atom
Ω=2gN1/2

0.02
0.04
500 0
500
ωprobe/2π (kHz)-10

Homodyne signal (V)

ty
vi
ca

0.04

EnergyHomodyne signal (V)120

1Atom-cavity detuning, δc/2π (MHz)

0.10
0.05
ω↓

0.00
ω↓

0.05
0

250

500

ωprobe/2π (kHz)

Figure 6.5: Center : we show the eigenvalue spectrum of the atom-cavity coupled system versus theatom-cavity detuning δc /2π. Left : When we tune the cavity onto the bare atomic resonance, we
observe the vacuum Rabi splitting, shown in the left plot. This demonstrates entering the collective
strong-coupling regime. Right : For the QND measurements, we operate in the slightly detuned
regime.

121
cavity. This is because laser-cavity fluctuations will lead to an overall translation of the eigenvalue
spectrum in Figure 6.5a. One technical tradeoff for operating in this regime is the signal-to-noise
versus the loss of atomic coherence from the probing. This is because atom number fluctuations
cause the coupled modes to vary more than κ/2, which forced us to perform frequency sweeps
instead of fixed-tone probing. This could in principle be avoided by implementing an active servoof the probe tones onto the coupled modes, but due to complexity we opted to not pursue that asa first experiment.
Instead, we opt to operate in the slightly detuned regime. There are multiple trade-offs when
deciding on an operating detuning. As the operating atom number increases, one can afford to
detune the cavity further from the bare atomic resonance, if you want to fix the technical noise
floor relative to the QPN. For our experiment we set the atom - cavity detuning to δc /2π = 1 MHz.
We then observe the dispersive shift of the cavity mode due to atoms in the ground-state. At thisdetuning, the operational cavity frequency shift is anywhere between 100 - 400 kHz depending on
the exact atom number. A fixed-frequency laser is then parked on resonance of the cavity-like mode
to measure the dispersive shift ω↓ . For our highest atom numbers, the far-detuned approximation
for the frequency shift begins to break down, and we need to include relatively significant higher
order corrections to the usual g 2 /δc scaling. We can use the exact expression of the eigenvalues toobtain an expression for the number of atoms in |↓i in terms of ω↓ asδcN↓ = ω↓ 2
g



ω↓
1+
.
δc

(6.1)

An optical π-pulse is then applied to swap the population between |↑i and |↓i. The frequency shift
is measured again to determine the excited state population N↑ . To zoom ahead, it is by measuring
these frequency shifts in a non-destructive way that we will generate entanglement via conditional
spin squeezing. After an initial π/2, we can make two repeated measurements of Jz , and since thetwo measurements will contain highly correlated QPN, taking their difference allows us to subtract
the common quantum noise [111, 31]. By using that differential information, we will project theatoms into a spin-squeezed-state (SSS).

122

6.3

QPN and the effective atom-cavity couplingWith the strong-coupling regime in our wheelhouse, we now seek to perform conditional spinsqueezing. For that, we will make repeated measurements of the Bloch vector projection along z,
and take direct advantage of the correlated QPN in those two measurements. Here, we will makefull use of the low-noise cavity probing system we described in Chapter 5.8.2. It is therefore criticalto know the level of QPN so we can properly benchmark our QND measurements. It turns out,
and we will show derivations below, that the atom-cavity coupling constant g serves as the bridge
between QPN in terms of population fluctuations and cavity frequency shift fluctuations. We cansee this with a few lines of math and considerations. In the far-detuned limit, the shift of thecavity-like mode isω↓ =The noise from projection noise is ∆N↓ =

√

g2
N↓
δc

(6.2)

N /2. Since we infer the atom number from the shift,

one can show that the noise of the frequency shift isr∆ω↓ = g

ω↓
.
2δc

(6.3)

We can then see in a straightforward way that measuring the QPN using our cavity QED system
is directly connected to the atom-cavity coupling g. A stronger atom-cavity coupling means that
QPN converts to larger frequency fluctuations, which makes good physical sense. We note that wewrite g, but we really mean the effective quantity.

6.3.1

QPN in slightly detuned regime

We operate in the slightly detuned regime, where we have set a detuning of δc /2π = 1 MHz.
Even at our lower atom numbers, the shift is 100 kHz or greater. To fit g to our measured noise of
the differential cavity frequency shift ∆(ω↓ − ω↑ ), we require an expression for this noise in termsof the sum shift ωsum = ω↓ + ω↑ . The eigenvalues for the |↓i and |↑i states areqδc + δc2 + Ω2↓,↑,ω↓,↑ =
2

(6.4)

123
where the vacuum Rabi splitting for each spin state are Ω↓ = 2gppN↓ and Ω↑ = 2g N↑ . The

QPN fluctuation of the frequency shift is obtained by propagating the QPN noise ∆N↓ throughthe expression
dω↓
∆N↓ .
dN↓

∆ω↓ =

(6.5)

For a CSS of N total atoms prepared by a π/2 pulse on the equator of the Bloch sphere with state
population N↓ = N↑ = N/2, one can express N in terms of the sum of measured frequency shifts
ωsum = ω↓ + ω↑
1 ωsum
1+
.
(6.6)
2 δc
√
QPN for the two spin states is ∆N↓ = ∆N↑ = N /2. We can calculate the projection noiseδc
N = ωsum 2gfluctuations of the frequency shift using the derivative of the eigenvalue expression. Projection
noise fluctuations of N↓ and N↑ are perfectly anti-correlated, and therefore ∆(ω↑ − ω↓ ) is twice ofthe fluctuations of ω↓ ,√g2 N∆(ω↑ − ω↓ ) = 2∆ω↓ = q.
δc2 + Ω2↓

(6.7)

Using Eq. 6.6 and Eq. 6.7, the expression for characterizing g based on the measurement on QPN
fluctuations of the initial CSS iss∆(ω↑ − ω↓ ) = g

6.3.2

2
ωsum
/2 + δc ωsum
(ωsum + δc )2(6.8)

Measuring QPN

We will now measure the QPN of our ensemble by looking at the fluctuations of (ω↓ − ω↑ )versus the total shift ωsum = ω↓ + ω↑ . The pulse sequence for measuring QPN is relatively simple
as shown in Figure 6.6a. We perform an initial π/2 pulse to prepare a CSS on the equator of the
Bloch sphere. We measure the shift of the cavity-like mode for the ground state atoms, indictaedby ω↓ , perform a π pulse to swap populations, and measure ω↑ . As we will now go into detail we need to be careful about the influence of technical noise in the clock rotations or the cavity

124

(a)

π
2

ω↑

π

4

∆(ω↓ − ω↑ )/2π (rms kHz)

(b)

ω↓

3

2

1

0

0

50

100

150
200
250
(ω↓ + ω↑ )/2π (kHz)

300

350

400

Figure 6.6: (a) Pulse sequence for measuring QPN. Black open pulses correspond to clock rotations,and the red pulses are cavity frequency shift measurements. (b) QPN measurement. The red
datapoints have no rotation noise subtracted, and the model including a linear term is fit (redline) giving g/2π = 5230(213) Hz. The open blue circles have an estimated clock rotation noise
contribution subtracted, and is fit without a linear term giving g/2π = 5127(192) Hz.

125
frequency shift measurements, as these can both mimic QPN, and give incorrect normalizations forour conditional spin squeezing.

6.3.3

Noise in QPN measurement

When we perform a measurement of QPN, we have to be aware of systematic or technicaleffects that can influence our determination of g. There is noise from both rotation noise andthe cavity shift measurements. At low atom number where the influence of QPN is small due√to the N scaling, we run into our technical noise floor of the homodyne measurement. Whenmeasuring QPN, we can feel free to turn up the optical power so that we can minimize the impactof photon shot noise. Nonetheless, the combination of photon shot noise, backaction on the atoms,
and technical laser - cavity noise will all contribute, as seen by the finite offset at ω↓ + ω↑ = 0.
At higher atom number, as the length of the Bloch vector gets longer, rotation noise can start tobecome noticeable in the measurements. In principle, the noise can come from amplitude or phasenoise of the clock local oscillator.
The pulse sequence for measuring QPN is relatively simple, as shown in Figure 6.6a. Let’s
first consider amplitude noise during the clock pulses in the QPN measurement sequence. We needto consider the noise introduced in the π/2 pulse and the π pulse, since this added noise would add
in quadrature with the technical measurement noise and the true QPN. Let’s consider shot-to-shot
fluctuations  in the clock drive Rabi frequency, and assume it is correlated between the π/2 andπ pulse for each shot of the experiment. If we tune the Rabi frequency to Ω0 to nominally give a
π/2 pulse with a pulse time of tπ/2 , we can write the modified Rabi frequency asΩ1 = Ω0 (1 + )

(6.9)

A Bloch vector undergoing evolution with a Rabi frequency Ω1 gives a Bloch vector angle ofθ1 = π/2 + π/2. Since we are treating the  error as correlated, the π-pulse will result in a further
angular deviation of π, and so the two pulses together give the angular error of this pulse sequence

126
is
∆θ =

3π
2

(6.10)

This can be understood intuitively since we are treating the error as correlated, so the total pulse
area of the combined π/2 + π pulse is 3π/2.
Now we consider the pulse sequence we utilize to benchmark rotation noise, which is π/2 ω↓ - 2π - ω↓ . Since we always take the difference between the two ω↓ measurements, and the initial
π/2 pulse is common to both measurements, this pulse sequence is sensitive only to the noise of
the 2 π pulse. Again, given the shot-to-shot error in the Rabi frequency of , the angular error ofthis 2π pulse is∆ω = 2π(6.11)

Thus, to estimate the amount of rotation noise present in the QPN pulse sequence, we multiply the
benchmarked noise by the ratio (1.5/2) and subtract that in quadrature (taking the linear atom
number dependence into account).
We also take another approach where we allow for a linear term in our non-linear fit, andsee how large that term is, and how it impacts the extraction of gef f . This is potentially an
appealing approach, since the rotation noise subtraction can be sensitive to your assumed modelof the rotation noise (amplitude versus phase, correlated versus uncorrelated noise). By allowing
for a linear term in the model, we allow for a generic classical rotation noise source. Fitting for g
this way gives g/2π = 5230(213) Hz, which is in good agreement with the method of subtracting
the measured rotation noise. We treat this as a consistency check, since the rotation noise term is
not perfectly linear in the shift. Our final result is taken as the value obtained by the subtraction
method, giving g = 2π × 5.1(2) kHz as shown in Figure 6.6b, and the bare cavity noise offset of
2π × 0.76(5) kHz.

127

6.4

Conditional spin squeezing

6.4.1

Spectroscopic enhancement

The first method of generating useful spin-squeezing we pursued is conditional spin squeezing.
This involves making two repeated measurements of Jz = (N↓ − N↑ )/2. In principle, we can makethis differential measurement with precision well below the level of QPN [34], and how far below
QPN is referred to as spin noise reduction R,
R=∆(Jz,f − βJz,p )∆Jz,QP N

2,(6.12)

where the numerator is the difference between the pre and final measurement, with β acting as
a Bayesian optimal estimator. The β deweights the pre-measurement, and will typically take ona value from 0 - 1, depending on the strength of the pre-measurement. An important concept toremember is that we have prior measurement about the state we prepare (Jz ≈0), and in principle
the noise of that prior distribution is set by the QPN. If the noise of the pre-measurement was
higher than the QPN level, and you imagined getting a result for Jz that was far from the expectedvalue near the equator of Jz = 0, you would clearly want to deweight that measurement, since you
know it is exceedingly unlikely that the actual state is far from Jz = 0 (ignoring real outliers due toissues on the experiment). A further reason for a non-unity β is the fact that the pre-measurement
is taken at a lower probe power than the final measurement due to the backaction. One therefore
should deweight the pre-measurement accordingly.
To certify that the measured spin noise reduction arises from entanglement, one must weigh
R against the loss of coherence induced by the probe as measured by the Ramsey fringe contrast. In
other words - if the information leaks out to the universe via free space scattering into 4π sterradian,then the system loses it’s entanglement.
The Wineland parameter, ξ, serves as a criterion for the generation of entanglement, and was
originally expressed as [108]
ξ=R

1
,
Cf2

(6.13)

128
where we weigh the spin noise reduction below QPN against the loss of signal induced by thesqueezing operation. This formulation of the Wineland parameter assumes an initial contrast of
Ci = 1. We further note that we have defined the QPN assuming unity contrast (all of the atoms√
contribute to the cavity signal and corresponding noise), ∆Jz,QP N = N /2. For our experiment,
we want to express the Wineland parameter taking into account an initial contrast Ci < 1. When
we have Ci < 1, the initial length of the Bloch vector is shorter, making the ∆θSQL worse by√a factor of Ci . The Wineland parameter may be rewritten with the initial contrast taken into
account asξ==R∆(Jz,f − βJz,p )∆Jz,QP N

2

Ci
,
Cf2

Ci
Cf2

(6.14)(6.15)

where Ci and Cf are the initial and final Ramsey fringe contrast, respectively. Another equivalent
way to think about ξ is by looking at the measured phase resolution relative to the SQL limit ofphase resolution,

ξ=

∆θm
∆θSQL

2
(6.16)√
where ∆θm is the measured phase resolution and ∆θSQL = 1/ Ci N with Ci being the initial
Ramsey fringe contrast. This formulation has the advantage that it shows us exactly what we needto measure in our experiment, which is the phase resolution of our state. The phase resolution
can then be easily mapped to Ramsey spectroscopy and how one gains enhancement of the clock
performance. Further, it is this relative phase resolution that intuitively shows why ξ is a witness
for entanglement.

6.4.2

Calibrating incident photon number

Here we include a short aside on how we calibrate and estimate the number of incident
photons on the cavity for a given QND measurement. While the practical method of benchmarking
the QND strength is by the loss of Ramsey fringe contrast, determining the incident photon number
will allow us to compare our measured spin noise reduction and contrast loss against theoretical

129
predictions. To get our initial bearings, we first want to independently determine the optical powerin the sideband. For this we turn up the EOM drive such that we put at least 10% of the opticalpower into the first order sideband. We can then measure the total optical power in the beamincident on the cavity (in front of the viewport just before the cavity, loss in the viewport is on the% level) with a power meter. By observing the relative size of the reflection dip on the cavity mode
for the carrier and the sideband, we can reliably determine the total amount of sideband opticalpower incident on the cavity in units of Watts, labelled as Pinc,tot,cav . We label with “tot” because
this includes the part of the beam that is not actually modematched to the cavity. So we know
how much optical power is in that sideband, and we can track the optical loss from the incident
viewport on the chamber to the homodyne photodiode, and the fraction of power that makes itto the photodetector is the path efficiency. We then know how much sideband optical power isincident on the photodiode, and we label this as Pinc,P D . The power in the LO incident on the PD
is labelled as PLO . The expected peak-to-peak voltage assuming unity mode matching between thereflected beam off of the cavity mirror and the LO is
Vp2p,ideal = 4SP D Gp
PLO Pinc,tot,P D(6.17)

where SP D is the sensitivity of the photodiode (SP D = 0.48 A/W), G is the transimpedance gain of
the photodetector (G = 1MOhm). However, the finite mode matching between the reflected beamand the LO leads to a smaller observed peak-to-peak Vp2p,obs . The ratio F = Vp2p,obs /Vp2p,ideal
depends on the quality of various mode-matching and alignment, and can typically be 0.6. To be
explicit, we rewrite the signal as
Vp2p,obs = F Vp2p,ideal = 4F SP D Gp
PLO Pinc,tot,P D(6.18)

and use this expression to infer Pinc,tot,P D . Using the path efficiency we can determine the incident
power on the cavity, but only a fraction of that incident beam is actually mode matched to the
TEM00 mode. We measure this mode matching by back-propagating a beam through the cavity,and observe the back-coupling efficiency through the launch fiber. In the experiments shown in130this thesis, the mode matching of the incident beam to the TEM00 mode was 0.75. With thosevalues, we can deduce the amount of optical power incident on the cavity in the sideband that is
actually mode matched to the cavity mode. This allows us to map measured homodyne signal sizes
to incident photon number, and use them to compare against predictions of photon scattering into
free space.

6.4.3

Spin noise reduction

We now demonstrate repeated measurements of Jz with differential resolution well below
QPN. After an initial π/2, we make a pre-measurement denoted by Jz,p , wait a dwell time of 20
ms, and perform the final measurement Jz,f (Figure 6.7a). While the pre-measurement fluctuates√with standard deviation ∆Jz,QP N = N /2, the final measurement contains highly correlated QPN,
as seen by the strong correlations between the two measurements (Figure 6.7). We turn up thephoton number for the measurement of Jz,f to optimize the signal to noise of that final readout
measurement. The effect of β is to de-weight the pre-measurement so that we optimize the resulting
standard deviation of the difference. We directly observe spin-noise reduction R = −4.6(6) dB
relative to QPN at the optimal squeezing photon number of 2.3 × 104 per measurement window.
We can ask the question - after taking into account the technical noise of the final-measurement,
what is the full spin noise reduction R that we can infer? To answer this, we assume that the readout
noise in the final measurement is uncorrelated with the pre-measurement. Due to our measurement noise roughly matching our expected photon shot noise (see Figure 6.11), this is a somewhat
reasonable assumption to make. We then calculate the photon shot noise present in the final two
measurements, and subtract this off from our measured spin noise reduction. For example, for the
spin noise reduction shown in Figure 6.8, both measurements are taken at the highest probe powerof 4.2 × 104 photons per measurement. In this case the final measurement would contribute thesame amount of noise as the pre-measurement, and the inferred spin noise reduction is R = −8.5
dB. From a practical perspective, this inferred R is not what we could make use of in a real metrological measurement. However, it tells us how much we could improve our conditional squeezing if

131

100

0

(a)
−100

π
2

π
N↓,p

N↑,p

−100
0
100
Pre-measurement, Jz,p

200

τ

π
N↑,f

N↓,f

−200
−200

(b)

Final measurement, Jz,f
200

100

0

−100

−100

−200

0

−100
0
100
Pre-measurement, Jz,p

100

−200

200
Final measurement, Jz,fFinal measurement, Jz,f

200

200

−200
−200

−100
0
100
Pre-measurement, Jz,p200

Figure 6.7: (a) Pulse sequence used for measuring spin noise reduction. After an initial π/2, we
measure Jz,p , wait a dwell time τ , and then measure Jz,f . (b) The high-degree of correlations areshown between Jz,p and Jz,f .

132

200

Jz,f − βJz,p

Jz

100
0
−100
−200

0

25

50

75

100
0
25
Experimental shot

50

75

100

Figure 6.8: The left panel shows the pre-measurement (red points) and the final-measurement(open cyan points). The incident photon number per measurement is 4.2 × 104 . The optimal
subtraction Jz,f − βJz,p is shown in the right panel. The dashed lines indicate the calculated QPN,
and the solid lines indicate the calculated standard deviation of the measured data. The spin noise
reduction here is R = −6.3(4) dB.

we could turn up our probe power arbitrarily high to make the final measurement noise completely
negligible. At the moment, we observe an increase of the noise at even higher power - indicating
extra noise from the light action on the atoms.

6.4.4

Ramsey contrast

For our conditional spin squeezing we measure the initial Ramsey contrast Ci without QND
probing using the pulse sequence shown in Figure 6.9a. We then add QND probing and measure
the final contrast Cf using the sequence in Figure 6.9b. To map out the fringe, we vary the phaseof the final π/2 pulse and record the excitation fraction which is defined as
fe =

N↑
.
N↑ + N↓(6.19)

Examples for the measured Ramsey fringe is shown in Figure 6.9c. The initial contrast with no
QND probing is the open green squares. We note that our initial contrast is degraded primarily by
the optical manipulations of the π pulse and the π/2 align pulse. When we add the QND probe,133
we see the contrast come down accordingly (black and red points). We note that we do not see anappreciable phase shift of the Ramsey fringe. This leads one to believe that we have sufficiently
echoed away the ac Stark shift induced by the QND probe. The maximum phase shift that we
observe is 50 mrad.
We plot the measured fractional loss of contrast as Cf /Ci versus the incident photon number
per population measurement as the black points in Figure 6.10. The expected loss of coherence
from free-space scattering is
Cf = Ci e−Ms /N(6.20)

where Ms is the number of scattered photons. We estimate the expected number of scatteredphotons by solving for the steady-state atomic excitation |hbi|2 . The rate of scattered photons isthen Ṁs = Γ|hbi|2 (see Appendix C for more details). We clearly observe more contrast loss thanour free space scattering estimate. In principle this could be explained by a miscalibration of thephoton number, but the discrepenacy seems too large to be explained by this. It is also possiblethat there is a second-order in Ms effect playing a role to the contrast loss. We need to further
explore our loss of contrast - and whether it depends on any other parameters in the system, like theduration of the measurement window, π-pulse time for the clock rotations, etc. Another possible
excess loss of contrast could come from additional inhomogneous effects induced on the atoms dueto the QND probe. Better atomic confinement could lead to improved Ramsey contrast in boththe initial state and the final probed atoms.

6.4.5

Measured Wineland parameterWe systematically vary the number of photons incident on the cavity per measurement window, and measure the spin noise reduction R and the contrast loss Ci /Cf2 (Figure 6.11). For
example, turning on a QND probe with 2.3 × 104 photons per population measurement reduces thecontrast to Cf = 0.60(1). At this optimal probe power, the measured Wineland parameter reaches-1.7(7) dB (Figure 6.11), a direct verification of spin entanglement in our system. If we want toinfer the amount of entanglement in the system before the final measurement, we need to calculate

134

2 |y

π
2 |x

π|y

π
2 |φ

T

(b)
π
2 |y

π
2 |x

π|y
N↓

N↑

π
2 |φ

(c)
1.0
Excitation fraction

(a) π

0.8
0.6
0.4
0.2

T
0.0
360

180
0
180
Relative phase (deg)360

Figure 6.9: (a) Pulse sequence used for measuring the initial contrast with no QND probing
applied. (b) Pulse sequence for measuring final contrast Cf with the QND probing. (c) Open
green squares is the Ramsey fringe with no probing, black points is with 2.3×104 photons per
measurement, and red triangles is with 4.2 × 104 photons per measurement. We do not observe
any significant phase shift due to the probing.

135

1.0

Cf /Ci

0.9

0.8

0.7

0.6

0

10000
20000
30000
40000
Incident photon # per measurement

50000

Figure 6.10: Measured fractional contrast Cf /Ci versus the incident photon number per measurement. The solid line indicates the calculated expected loss of contrast from free-space scattering,
described in Appendix C.

136
the noise of the final measurement to subtract in quadrature (assuming uncorrelated detection
noise). Using the model given by the cyan line in Figure 6.11, we subtract the final measurement
noise, and infer a Wineland parameter of -3.7(7) dB. This inferred value is not the useful amountof entanglement, as real clock operation relies on the final measurement. It is insightful however,since we can infer how much more squeezing we could expect if we could improve the cavity shiftnoise at higher photon number.

137

10 log10 ( R, Ci /Cf2 , ξ)

5.0
2.5
0.0
−2.5
−5.0
−7.5

−10.0

104
Incident photon number per measurement

105

Figure 6.11: Relative spin noise reduction R (cyan circles), fractional contrast loss Ci /Cf2 (redcircles), and the corresponding Wineland parameter ξ (black circles). At the optimal photon
number, we directly measure a Wineland parameter of -1.7(7) dB. Expected R given our estimatedquantum efficiency of Q = 0.28 (cyan line), expected contrast loss via free-space scattering (redline), and the corresponding expected Wineland parameter (black line, see Methods).

Chapter 7

Squeezed clock comparisons

To quickly recap, we have built a new apparatus that is capable of state-of-the-art clock
operation with cQED. We then demonstrated that we can generate entanglement on the Sr clocktransition. The next question is - can we use this spin squeezing for a real clock measurement, anddirectly observe an enhancement beyond the QPN limit?
In this Chapter we describe our squeezed clock comparisons that we perform in order todemonstrate the direct advantage of implementing spin squeezing. We perform a differential clock
measurement, where we compare two sub-ensembles within the same atomic cloud. Original demonstrations of this technique by the Sr2 team made use of a high resolution imaging system in order
to divide their Fermi-degenerate gas into two sub-ensembles [28]. Instead of using imaging, we
use the selective nature of the cavity probing to sub-divide our ensemble. By using the movable
optical lattice, we can shuffle different parts of the cloud in and out of the cavity mode on demand.
In Section 7.1, we first define the QPN and SQL limit to a clock comparison with initial contrast
Ci < 1. Section 7.2 will provide a description of the differential clock measurements, and present
our results for both a classical clock and squeezed clock comparison. In the later sections, wehighlight various details that provide further experimental measurements that are relavant to thespin squeezing observation.

139

7.1

QPN and SQL
Neutral atom optical clocks probe many atoms to obtain the best signal-to-noise ratio. The

fundamental noise source in atomic clocks is the quantum projection noise that stems from theinherent population fluctuations associated with the projective measurement of N uncorrelated
atoms [20]. Operating at higher atom number is clearly advantageous from a signal-to-noise pointof view. However, technical noise from imperfect state readout, intrinsic atom-atom interactions,or aliased frequency noise of the interrogating clock laser pose challenges for observing clock performance at the QPN limit [133]. With precise engineering of quantum states and control of
atomic interactions [134], and using laser-noise mitigation techniques such as synchronous comparisons [28, 17], state-of-the-art optical clocks are currently approaching QPN-limited stability withup to 105 atoms [19]. It is an exciting prospect to imagine pushing these state-of-the-art clocks
past the QPN-limit, making use of quantum correlations to improve precision measurements.
We want to establish a clear definition for the standard quantum limit for a clock with a finite
initial contrast Ci < 1. The primary loss of atomic coherence in our squeezed clock experiments
comes from the multiple optical rotations needed in our sequence. Conceptually, we treat thesystem as having two components - a pure state with |Jz | = Ci N/2, and treating other (1 − Ci )atoms as no longer partipating in the same CSS. We thus need to calculate the standard quantum
limit (SQL) for the Ci atoms that participate in the CSS. In this case, we have∆θSQL =√Ci N /2
1=√
Ci N/2
Ci N(7.1)

We find it useful to define another limit, which we call the “QPN” limit. In our situation, thesequence of optical rotations leads to dephasing, but the atoms are not lost from the trap. The
dephased atoms still couple to the cavity mode, and will still contribute quantum projection noise.
Thus, the QPN limit is the ratio of the projection noise for all of the atoms divided by the shortened
Bloch vector,∆θQPN =

√

N /2
1
= √
Ci N/2
Ci N

(7.2)

140
We note here that operation of a clock with phase resolution below ∆θQP N is a real enhancement,in that no classically operating clock could go below this level. However, going below ∆θSQL goes
further as a metric for entanglement in the clock.

7.2

Self-synchronous clock comparison with cQED
The timing sequence for the differential clock comparison includes clock rotations, transports,

and QND measurements (Figure 7.1a). Clock rotations are shown as black pulses, with each pulse
labelled by the pulse area and axis of rotation. The atomic cloud is moved vertically by detuning
one of the lattice beams, and each transport operation is indicated by either a green or purple
pulse. The waveforms for the optical lattice detuning are linear ramps of the frequency over a totalof 5 ms, as shown below the pulse sequence. All clock pulses are applied with the lattice at thesame vertical location, so we do not have to take into account the varying clock laser phase. Afterthe pre-measurements, the π2 |x clock pulse rotates the spin-squeezed axis to the phase-sensitiveaxis. Our Ramsey dark evolution time is T = 14 ms, and the final π/2 pulse converts an acculated
phase to a tipping of the Bloch vector in the z direction. To minimize the impact of photon shotnoise, the final measurements use a higher probe photon number of ≈ 4.5 × 104 per measurement
window. This is safe since the final measurement is outside of the Ramsey interferometer, and ashortening of the Bloch vector at this point is harmless. The CSS - CSS comparison uses the exact
same pulse sequence, with the only exception that we do not apply any probe light during the
pre-measurement windows. Running the two comparisons with the same pulse sequence allows us
to see and understand the role of spin-squeezing.
The relative separation between the two ensembles is set such that they give the same shiftof (ω↓ + ω↑ ) ≈ 2π × 215 kHz, corresponding to ≈ 8500 atoms per independent ensemble. We
first measure the Ramsey fringe contrast for each ensemble, shown in Figure 7.1b. We plot Jz =(N↓ − N↑ )/2 versus the phase φ of the final π/2 pulse. Red is ensemble A and blue is ensemble B.
The open circles correspond to running the sequence with no probe light applied during the premeasurement windows. The solid circles are the Ramsey fringes with probing. The loss of contrast

141

B
Final Jz,f

500
0
−500
−500 0500
A
Final Jz,f(a)B
Final Jz,f

B
N↑,p

π
|
2 φ

z

z

z

νlatticeνlattice

Time

y

(c)
5000

y
x

B
N↓,f

A
N↓,f

y
x

(d)

1000
B
A

Final Jz,f

0
−2500

0

−1000

0
−500

−500

−250
0
250
Relative phase (deg)500
B
Final Jz,f

500

2500
Jz

B
N↑,f

Time

(b)

−5000

y
x

π

A
N↑,f

z

5 ms

x

500

A
N↑,p

0

−500

B
N↓,p

T

π
|
2 x

π
A
N↓,p

−500 0500
A
Final Jz,f

π
2

−500
0
Pre Jz,p−500 0500
A
Final Jz,fFigure 7.1: (a) Clock pulses are the black pulses, measurements of ensemble A are the red pulses,
and the transports are shown as the green and purple pulses. The Bloch spheres depict the spinstate distribution at various points during the sequence. (b) Ramsey fringe measured by varying
the phase of the final π/2 pulse. (c) Pre and final measurements of ensemble A and B. (d) The
final measurements of ensemble A and B show strong correlations, allowing for the subtraction of
the common-mode laser phase noise.

142
at our operating probe photon number is ≈ 5%. An ideal Ramsey fringe with unity contrast is
shown as the dotted curve, reaching the maximal values of Jz = ±N/2. The initial Ci and final
contrast Cf are given by the ratio of the measured fringe amplitude and the ideal Ramsey fringe
amplitude.
For each cycle of the experiment we simply observe the final tipping of each ensemble’s Blochvector. We do not apply feedback to the laser to steer onto the resonance. The phase of the
final π/2 pulse is set such that the final Bloch vector for each ensemble is near the equator of the
Bloch sphere, corresponding to Jz ≈ 0. This puts the Bloch vectors at the point of maximal phasesensitivity for Ramsey spectroscopy. The raw pre and final measurements are plotted for eachensemble in Figure 7.1c (red is ensemble A and blue is ensemble B). One can see the correlations
by eye, and the correlation coefficient between the pre and final measurement is ≈ 0.3. However, it
is clear that the laser noise is playing a significant role, as the correlation between the pre and final
measurement is far worse than what we see for the spin noise reduction measurements (Figure 6.7).
This is due to the addition of the Ramsey pulse sequence. We can look at the final measurements
for A and B (Figure 7.1d), which clearly shows rejection of the common-mode laser frequency noise.
A and J B for both classical and spin squeezed comparisons
The correlation coefficient between Jz,fz,fis ≈ 0.9. In Section 7.5 we measure the degree of common mode rejection, and estimate the residual
laser noise to contribute a noise floor 30 dB below QPN.
For each experimental shot, we need to combine the measurements to give the best estimateof the differential phase between the two ensembles. To be explicit, we will use the measured
amplitude of the Ramsey fringe α (Figure 7.1b) to convert Jz measurements to phase. Assuming
the small angle limit, this conversion is expressed asθ A − θB =A − β J A ) − β (J B − β J B )
(Jz,f
A z,p
D z,f
B z,p

α

(7.3)

Note that we use θ, because the final Ramsey π/2 pulse converts phase evolution φ to tipping in
the polar angle θ. The phase difference can then be converted to frequency units using the simple
relationship between phase and frequency ∆θ = ∆ωT = 2π∆νT .

143clock laser(a)clock laser

(a)

Jz,B

Jz,A
10−15

Total Allan deviation

(b)

10−16

10−17

101

102
Averaging time (s)103

Figure 7.2: (a) Schematic showing the transported atomic cloud for squeezing/readout of each ensemble. The clock laser propagates vertically, addressing both ensembles in a synchronous fashion.
(b) Fractional frequency stability given by the total Allan deviation for the CSS - CSS comparison(green open squares) and the SSS - SSS comparison (black circles). The green dashed line is theestimated QPN limit, and the dotted blue line√is the SQL. We note that we plot the observed
comparison stability, so we have not divided by 2 to estimate single clock stability.

144
We first compare classical clocks (CSS - CSS), where we run the pulse sequence without
QND probing. Making use of the eigenvalue expression for the atom-cavity system, we convert
these frequency differences to atom number, and then combine the measurements to attain thecorresponding Jz . To convert measured Jz for each ensemble directly to phase, we use the amplitudeof the Ramsey fringes (Figure 7.1b). The SSS - SSS comparison uses the same pulse sequence, with
the QND probe pulses applied.
We determine the stability for each comparison by fitting the first 5 points of the Allan
deviations plotted in Figure 7.2. The observed stability is 1.58(3) × 10−15 τ −1/2 and 1.25(2) ×
10−15 τ −1/2 for the CSS - CSS and SSS - SSS comparison, respectively. We directly observe an
enhancement of stability by 2.0(2) dB in the SSS - SSS comparison over that of CSS - CSS. We√
observe a deviation from 1/ τ averaging at longer averaging times. This arises due to a drift in theatom number which changes the conversion from measured homodyne voltage to cavity frequency
shift. In future experiments we will be able to eliminate this source of flicker noise by tracking thechange of the homodyne fringe slope. For this reason, we fit the Allan deviation for the first four
data points. The deviation is likely due to a drift in the slope of the homodyne signal, as the atom
number slowly drifts.
To put our results in the proper context, we seek to benchmark the observed SSS-SSS stability to both the practically achievable limit set by QPN and fundamental limit set by SQL. The
measurement of (Jz,A − Jz,B ) is limited by the quadrature sum of QPN arising from each subensemble. We first define the fundamental limits in the absence of any scale factor between theensembles (βD = 1). The QPN-limited phase estimation between the ensembles is (assuming thesame initial contrast Ci for each ensemble)∆θQPN =√

1

Ci NA + NB

(7.4)

using Ci as the slope of the Ramsey fringe to convert QPN into fractional frequency noise. However,the ultimate bound on the performance of an unentangled ensemble is the SQL. This strictest boundtreats the (1 − Ci ) fraction of atoms as no longer participating in the pure CSS, thus reducing the

145
magnitude of the projection noise, resulting in the SQL-limited phase sensitivity of1∆θSQL = p
Ci (NA + NB )(7.5)

The above limits must be adjusted for the optimal estimator βD . For example, the SQL for theclock comparison isr∆ (φA − φB )SQL =

1
1
2
+ βD
.
Ci NA
Ci NB

(7.6)

The observed stability of the CSS - CSS comparison is consistent with the QPN-limited
stability, and 2.6(3) dB above SQL. Implementing QND-based squeezing operation and accounting
for the final contrast of Cf = 0.50(1), the SSS - SSS comparison shows a 2.0(3) dB gain overthe QPN-limited stability, demonstrating practical metrological enhancement from the squeezing
operation. This result is above SQL by 0.6(3) dB, indicating a Wineland parameter near unity. We
therefore can not
This is the first direct observation of an operating optical clock below the QPN limit. The
direct observation of the clock comparison below the QPN limit with measurement precision averaging down to the 10−17 level is a crucial step towards improving the performance of the best
optical lattice clocks via entanglement. Such improvement in differential clock stability translates
directly into increased sensitivity for many applications of interest, including the measurement of
the gravitational redshift at ever-shorter length scale and the future development of clock networks
for fundamental physics [135, 113, 136, 44]. Further improvement will come from enhanced control of atomic motion, which will yield improved spin rotation fidelity and increased coupling to
the cavity mode. Larger atom number, improved single-atom cooperativity, and better quantum
efficiency will all lead to stronger spin squeezing.

7.3

Optimal estimators
In this Section we will discuss the optimal estimators used in the differential clock comparison

in more detail. To attain the ideal clock comparison given the measured quantities, we havethree optimal estimators in the estimated phase difference φA − φB in Equation 7.3. The optimal

146
estimators φA and φB represent the weighting of the pre-measurement. Due to the finite signalto-noise ratio, the pre-measurement contains imperfect information about the collective spin state.
This means that the optimal value of βA and βB are less than unity. A more subtle optimal estimatorfor our comparison is βD . During a particular measurement, we can look at the individual noiseof each ensemble on their own ∆(Jz,A,f ) and ∆(Jz,B,f ) For example, in the data presented in thischapter, ensemble B was noisier. We found that the relative noise between the two ensembles was
not exactly unity. This led us to include an additional optimal estimator βD to account for thedifferential technical noise.
The simplest analysis would be to simply simultaneously optimize the three estimators over
the whole dataset. We then asked several questions regarding these β values. What is the uncertainty of these optimal estimators; and how does it propagate to the final squeezing result?
Furthermore, how does the length of the dataset influence these parameters. To answer these questions, we temporarily break the data up into a “training set” and a “clock run”. We increase thelength of the training set to see how the optimal estimators evolve and eventually converge in Figure
7.3. Taking the second half of the data, we calculate the mean values (dashed line) and plot thestandard deviation (shaded region). The values are βA = 0.49(1), βB = 0.48(1) and βD = 0.907(5).
We take these values to be the optimal estimators for analyzing the whole clock run. To propagate
the uncertainty of the optimal estimators, we use a simple numerical method. We pull from normal
distributions for βA , βB , βD and gef f using the 1σ standard deviation for each quantity. For each
iteration, we calculate the phase resolution compared to the QPN and the SQL limits, and plotthese results as histograms.

7.4

Vertical extent of cloudIn the differential clock measurements, we squeeze and compare two ensembles within thesame atomic cloud. The final Z position of the atomic cloud can be precisely set by adjusting thetransport waveform. We vary the final position of the cloud and measure the mean shift of thecavity-like mode, and plot this in Figure 7.4. Using our knowledge of the cavity mode distribution,

147

0.50

βA

0.45
0.40
0.35
0.55

βB

0.50
0.45
0.40
0.35

βD

1.2
1.1
1.0
0.9
0

50

100
150
200
Experimental shot

250

300

Figure 7.3:
The optimal estimators calculated for the spin-squeezed clock comparison data.
(a) Optimal estimator βA for the pre-measurement of A. (b) Optimal estimator βB for the premeasurement of B. (c) Optimal estimator βD , which is the differential scale factor between thetwo ensembles. The shaded region indicates the ± 1 standard deviation for the datapoints beyond
experimental shot 150, and the dashed line is the corresponding mean value.

148

Mean cavity shift (Hz)100000
80000
60000
40000
20000
0
−400

−300

−200 −1000100200Vertical displacement of cloud (µm)

300

400

Figure 7.4: Measured shift of the cavity-like mode versus vertical displacement of the atomic
cloud. The green dashed line is the Gaussian fit of the form of Eq. 7.8, giving a fitted σz = 130µm.

149
we calculate the frequency shift using Eq. B.1 and Eq. B.2
ω↓ =

2 N
gef
f ef fδc=

Ntot hgi2 i
δc

(7.7)

Plugging in the appropriate distributions in hgi2 i, and considering a varying offset of the cloud inthe vertical direction Z0 , we find that the shift has the form,−2Z 20ω↓ (Z0 )2
2= e 4σZ +w0 ,ω↓ (0)(7.8)

where σZ is the rms extent of the cloud along the vertical direction. We then fit our measured
cavity frequency shift to Eq. 7.8. After plugging in our cavity mode waist of w0 = 71 µm, we
extract a vertical cloud size of σZ = 130 µm. This cloud size is in good agreement with images of
the cloud using fluoresence imaging.
One can imagine various different routes for improvement. If we could get an atomic cloudwith both a larger atom number and larger extend σz , we could easily probe more than 2 ensembles
within the same cloud. Another approach would be to create spatially seperated ensembles [87],each having σz < w0 . These more compressed clouds could give both an improved effective values
for atom number N and the atom-cavity coupling g.

7.5

Common-mode suppression of laser noise
Probing the two sub-ensembles with the same clock laser has the great advantage of allowing

the laser phase noise to be common between the evolution of the two Bloch vectors. In orderto check our level of common-mode suppression, we operate a two-ensemble Ramsey experimentand add in artificial laser noise. We raise the artificial noise level by adding a white frequency
noise term into the fiber noise cancellation reference synthesizer. This is achieved using an SRS
SG382 synthesizer, and using the internal noise modulation feature. The measured cavity shiftnoise versus added white noise is shown in Figure 7.5. It is clear the the added noise shows up ineach ensemble (red and blue circles), but when we take the difference (open green squares), we see agood common-mode rejection of noise. Looking at the largest noise, we can see a direct suppression

150

20000
Ensemble A
Ensemble B
Difference A-B∆(ω↓ − ω↑ )/2π (rms Hz)

17500
15000
12500
10000
7500
5000
2500
0

0.0

0.2

0.4
0.6
0.8
2
Added clock laser noise (Hz /Hz)

1.0

Figure 7.5: Measured cavity shift noise with artificial white frequency noise imposed on the clock
laser. Each ensemble (red and blue circles) becomes noisier as the magnitude of the added noise
is increased, but taking the difference (open green squares) gives a 10 dB common mode noise
reduction factor.

151
Table 7.1: Quantum efficiency contributions
Parameter
QE of cavity (κ1 /κ)Photodiode QE
Homodyne-cavity mode overlap
Path efficiency
Photodiode technical
Total

Value
0.68
0.88
0.75
0.62
0.99
0.28

of noise corresponding to 10 dB. The Si3 clock laser has a frequency noise power spectral densityof about 10−2 Hz2 /Hz. Our estimated contribution in terms of cavity frequency shift noise wouldbe ≈ 50 rms Hz, or 30 dB below QPN. We comment here that only 10 dB of this cushion comesfrom the common-mode suppression. The full 30 dB is only possible because of our advanced clock
laser technology.

7.6

Quantum efficiency
The overall quantum efficiency of the measurement plays a key role in QND-based spinsqueezing. In thinking about probing the atom-cavity coupled system, we define the overall quantum efficiency as the product of all of the independent sources of loss of photons, or decrease of
signal to noise ratio for the homodyne detection. The amount of attainable spin-squeezing is linearly proportional to the quantum efficiency [32]. Furthermore, Q < 1 leads to excess noise in
the anti-squeezed quadrature. We first estimate the overall quantum efficiency by considering thedifferent contributions, including the cavity quantum efficiency κ1 /κ = 0.68, mode overlap of thecavity leakage light and the homodyne LO beam of 0.75, quantum efficiency of the photodiode of
0.88, and finite path efficiency of 0.62, and other negligible sources. Multiplying these together
gives a quantum efficiency of Q = 0.28. We use this value when estimating the expected R (cyanline in Figure 6.11), showing reasonable agreement.

152

7.7

Spin state tomography
We perform state tomography to evaluate the amount of anti-squeezing introduced by the

QND measurement. The pulse sequence used for measuring the anti-squeezed quadrature is shownin Figure 7.6a. We make a pre-measurement of Jz , jump the clock laser phase by 90◦ to rotate
the state by an Ψ, and then perform a final measurement. For the rotation angles larger than 5
degrees, we use a swept readout for the final measurement, since the measurement noise is large
enough to depart from the linear regime of the Q response of the homodyne signal. To our surprise,
the observed anti-squeezing is well above the expected level given our estimated quantum efficiency
of Q = 0.28. This indicates additional technical noise in the anti-squeezing quadrature, or in themeasurement itself. The measured noise level at the anti-squeezed quadrature is 9 dB above whatis expected from the estimated quantum efficiency (Figure 7.6b). This indicates an additional noise
source contributing to the anti-squeezing, but it does not preclude observing the benefit of spinsqueezing. This is clearly an avenue for further investigation.

7.8

Independence of the two ensembles
There is an inherent limitation in the total atom number per sub-ensemble for a given total

atom number and vertical extent of the cloud. For the differential clock comparisons performedin this thesis, we tune the transport distances such that the cavity frequency shift is equal for thetwo ensembles (to within 10 - 20%). When tuning up the experiment and the relative distances,
one wants to confirm the independence of the sub-ensembles and determine the magnitude of thecorrection factor due to finite overlap between the ensembles and the cavity mode. To explore
this, we vary the vertical separation between the ensembles and evaluate the Pearson correlation
coefficient between the measured Jz for each ensemble. For no separation, we probe the sameensemble and expect high correlations between the measured QPN to within the measurement noise.
Then as we increase the separation distance, the correlation coefficient will decrease accordingly toa noise floor set by correlated noise (classical rotation noise). We first measure Jz,A for ensemble A,

153

(a)

π
2

(b)

π
N↓,p

Ψ
N↑,p

π
N↑,f

N↓,f

30

Noise relative to QPN (dB)

25
20
15
10
5
0
−5
−10

0

45
Ψ (degrees)

90

Figure 7.6: (a) Pulse sequence used for performing spin state tomography. (b) Relative spin noise
after a rotation angle of Ψ. We infer the spin noise just after the pre-measurement by assuming
the noise in the pre and final measurement is uncorrelated. The black dashed line is a unitaryspin squeezed state, and the dashed-dot line indicates our expected anti-squeezing for a quantum
efficiency of 0.28.

154
apply a vertical displacement of the cloud via the movable optical lattice to put ensemble B in thecavity, and subsequently measure Jz,B . The measured correlation coefficient between Jz,A and Jz,B
is shown versus the separation distance between the ensembles in units of the waist of the cavity
mode w0 (Figure 7.7a). The blue line is a numerical Monte Carlo simulation of the correlation
coefficient for two ensembles with varying mean separation. Our operational ensemble separation
Z0 is denoted as the dashed grey line, at Z0 = 150 µm.
While the correlation coefficient is a relatively easy quantity to calculate from experimental
data, it is not the quantity we really would like to know. Based on the finite overlap of the ensemble,
we would like to calculate the change of the combined QPN versus the separation distance. For
this we use both analytical and numerical methods, with both shown in Figure 7.7b. In the limitof a flat density distribution, the analytical correction to the QPN is straightforwardly calculatedas22d(∆θQPN ) = 1 − e−Z0 /w0(7.9)

We can calculate the full analytical expression taking into account the finite σZ for the ensemble.
From this, we perform a seriest expansion so we can see the second order term,−Z02 /w02d(∆θQPN ) = 1 − e


+

Z
√0
2 2σZ

2

2

2

e−Z0 /w0 .

(7.10)

At our vertical cloud size of σz = 130 µm, and a separation distance of Z0 = 150 µm, this higher
order term contributesa 0.2% correction. For the differential clock comparisons of Figure 7.2, aspatial separation of 150 µm (dashed grey line) is chosen to guarantee sufficient independence ofthe ensembles. At this separation, the adjustment to the QPN is 0.045 dB.

155

Correlation coefficient

0.8

numerical

0.6

0.4

0.2

0.0

numerical
analytical0dB change of QPN

2
4
6
8
10
12
14
0.0

0.5
1.0
1.5
2.0
2.5
3.0
3.5
Ensemble separation in units of mode waist w0

Figure 7.7: Independence of atomic ensembles (a) Measured correlation coefficient between Jz,Aand Jz,B versus the separation between the ensembles. (black circles). The blue line is a Monte
Carlo simulation. (b) Corresponding change of the QPN due to the finite overlap of the ensembles,with numerical Monte carlo simulation (blue) and analytical calculation (orange). At our operating
ensemble separation, the change to QPN is 0.05 dB.

Chapter 8

Conclusion and outlook

This Thesis has covered the improvement of optical clock technology on multiple fronts. We
developed some of the best stable lasers using cryogenic silicon cavities. This involved building
high finesse cavities that reduced the Brownian thermal noise from the optical coating, which isthe primary source of the noise floor of these optical interferometers. We then built a new opticalclock apparatus that implements spin-squeezing approaching the state-of-the-art level. For the firsttime, we were able to directly observe an optical clock comparison operating below the QPN limit.
One obvious goal is to continue to push the performance below the SQL. There are clear
improvements and aspects of the system to further understand. The aim is to directly leverageand highlight entanglement for enhanced clock performance. The initial contrast of the atomic
ensembles will be further improved by optimizing aspects of the optical lattice and the associatedatomic motion. Adding more confinement either along the cavity axis or out of the plane of thecavity could improve the atomic coherence, allowing for immediate improvements in the Winelandparameter. Another approach would be to generate entanglement in the ground-state manifold of
87 Sr similar to [35], which could avoid some of the detrimental clock pulses.

Another priority is to further understand the magnitude of our measured anti-squeezed
quadrature. Excess anti-squeezing can make us more sensitive to amplitude errors in the clock
rotations. Further, as we push the Ramsey dark time to longer intervals, the leakage of the antisqueezed quadrature will start to play a role [137].
Improving the amount of squeezing involves improving the readout noise of our QND mea-

157
surement. Since the measurement noise increases with high photon number, we could win 2-3 dB ofspin squeezing by understanding and mitigating this effect. We could also other ways of generating
entanglement, for example using one-axis twisting [109, 111]. Depending on the exact experimental
parameters, one might find OAT or QND the favorable approach.
In the longer term, the system demonstrated in this thesis will serve as a blue print forbuilding more advanced versions of entangled clocks, with prospects of two-mode squeezing for clock
comparisons and integrating squeezing into the best performing precision to enhance measurement
bandwidth.

Bibliography[1] H Gegier and Ernest Marsden. On a diffuse reflection of the α-particles. Proceedings of
the Royal Society of London. Series A, Containing Papers of a Mathematical and PhysicalCharacter, 82(557):495–500, 1909.
[2] Ernest Rutherford. The scattering of α and β particles by matter and the structure of theatom. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science,
21(125):669–688, 1911.
[3] Niels Bohr. I. on the constitution of atoms and molecules. The London, Edinburgh, andDublin Philosophical Magazine and Journal of Science, 26(151):1–25, 1913.
[4] I. I. Rabi, J. R. Zacharias, S. Millman, and P. Kusch. A new method of measuring nuclear
magnetic moment. Phys. Rev., 53:318–318, Feb 1938.
[5] Willis E. Lamb and Robert C. Retherford. Fine structure of the hydrogen atom by a microwave method. Phys. Rev., 72:241–243, Aug 1947.
[6] Norman F. Ramsey. A molecular beam resonance method with separated oscillating fields.
Phys. Rev., 78:695–699, Jun 1950.
[7] Norman F. Ramsey. Obiturary : I.I. Rabi, 1988.
[8] Harold Lyons. The atomic clock: A universal standard of frequency and time. The American
Scholar, 19(2):159–168, 1950.
[9] Norman F. Ramsey. Experiments with separated oscillatory fields and hydrogen masers. Rev.
Mod. Phys., 62:541–552, Jul 1990.
[10] L. Essen and J. V. L. Parry. An atomic standard of frequency and time interval: A cæsium
resonator. Nature, 176(4476):280–282, 1955.
[11] John L. Hall. Why it took so long for the laser and the optical comb to be invented: theunmarked trail from concept to experimental reality. J. Opt. Soc. Am. B, 34(2):338–346, Feb
2017.
[12] Martin M. Boyd, Tanya Zelevinsky, Andrew D. Ludlow, Seth M. Foreman, Sebastian Blatt,Tetsuya Ido, and Jun Ye. Optical atomic coherence at the 1-second time scale. Science,
314(5804):1430–1433, 2006.

159[13] A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda,
M. J. Martin, J. W. Thomsen, S. M. Foreman, J. Ye, T. M. Fortier, J. E. Stalnaker, S. A.
Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates. Srlattice clock at 1×10−16 fractional uncertainty by remote optical evaluation with a Ca clock.
Science, 319(5871):1805–1808, 2008.
[14] Tobias Bothwell, Dhruv Kedar, Eric Oelker, John M Robinson, Sarah L Bromley, Weston L
Tew, Jun Ye, and Colin J Kennedy. Jila Sr1 optical lattice clock with uncertainty of 2.0 ×
10−18 . Metrologia, 56(6):065004, 2019.
[15] W. F. McGrew, X. Zhang, R. J. Fasano, S. A. Schäffer, K. Beloy, D. Nicolodi, R. C. Brown,
N. Hinkley, G. Milani, M. Schioppo, T. H. Yoon, and A. D. Ludlow. Atomic clock performanceenabling geodesy below the centimetre level. Nature, 564(7734):87–90, 2018.
[16] S. M. Brewer, J.-S. Chen, A. M. Hankin, E. R. Clements, C. W. Chou, D. J. Wineland, D. B.
Hume, and D. R. Leibrandt. 27 Al+ quantum-logic clock with a systematic uncertainty below
10−18 . Phys. Rev. Lett., 123:033201, Jul 2019.
[17] E. Oelker, R. B. Hutson, C. J. Kennedy, L. Sonderhouse, T. , A. Goban, D. Kedar, C. Sanner,
J. M. Robinson, G. E. Marti, D. G. Matei, T. Legero, M. Giunta, R. Holzwarth, F. Riehle,
U. Sterr, and J. Ye. Demonstration of 4.8 × 10−17 stability at 1 s for two independent opticalclocks. Nature Photonics, 13(10):714–719, 2019.
[18] William Bowden, Alvise Vianello, Ian R. Hill, Marco Schioppo, and Richard Hobson. Improving the q factor of an optical atomic clock using quantum nondemolition measurement.
Phys. Rev. X, 10:041052, Dec 2020.
[19] Tobias Bothwell, Colin J. Kennedy, Alexander Aeppli, Dhruv Kedar, John M. Robinson,
Eric Oelker, Alexander Staron, and Jun Ye. Resolving the gravitational redshift across amillimetre-scale atomic sample. Nature, 602(7897):420–424, 2022.
[20] W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen, F. L. Moore,
M. G. Raizen, and D. J. Wineland. Quantum projection noise: Population fluctuations in
two-level systems. Phys. Rev. A, 47:3554–3570, May 1993.
[21] Giorgio Santarelli, Ph Laurent, Pierre Lemonde, André Clairon, Anthony G Mann, S Chang,
Andre N Luiten, and Christophe Salomon. Quantum projection noise in an atomic fountain:
A high stability cesium frequency standard. Physical Review Letters, 82(23):4619, 1999.
[22] G John Dick. Local oscillator induced instabilities in trapped ion frequency standards. In
Proceedings of the 19th Annual Precise Time and Time Interval Systems and Applications
Meeting, pages 133–147, 1989.
[23] G. Santarelli, C. Audoin, A. Makdissi, P. Laurent, G.J. Dick, and A. Clairon. Frequency
stability degradation of an oscillator slaved to a periodically interrogated atomic resonator.
IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 45(4):887–894,
1998.
[24] M. Martin. Quantum Metrology and Many-Body Physics: Pushing the Frontier of the Optical
Lattice Clock. PhD thesis, Boulder, CO 80309-0440, 2013-04 2013.

160[25] D. G. Matei, T. Legero, S. Häfner, C. Grebing, R. Weyrich, W. Zhang, L. Sonderhouse, J. M.
Robinson, J. Ye, F. Riehle, and U. Sterr. 1.5 µm lasers with sub-10 mHz linewidth. Phys.
Rev. Lett., 118:263202, Jun 2017.
[26] W. Zhang, J. M. Robinson, L. Sonderhouse, E. Oelker, C. Benko, J. L. Hall, T. Legero, D. G.
Matei, F. Riehle, U. Sterr, and J. Ye. Ultrastable silicon cavity in a continuously operatingclosed-cycle cryostat at 4 K. Phys. Rev. Lett., 119:243601, Dec 2017.
[27] John M. Robinson, Eric Oelker, William R. Milner, Wei Zhang, Thomas Legero, Dan G.
Matei, Fritz Riehle, Uwe Sterr, and Jun Ye. Crystalline optical cavity at 4 K with thermalnoise-limited instability and ultralow drift. Optica, 6(2):240–243, Feb 2019.
[28] G. Edward Marti, Ross B. Hutson, Akihisa Goban, Sara L. Campbell, Nicola Poli, and Jun
Ye. Imaging optical frequencies with 100 µHz precision and 1.1 µm resolution. Phys. Rev.
Lett., 120:103201, Mar 2018.
[29] Luca Pezzè, Augusto Smerzi, Markus K. Oberthaler, Roman Schmied, and Philipp Treutlein. Quantum metrology with nonclassical states of atomic ensembles. Rev. Mod. Phys.,
90:035005, Sep 2018.
[30] Monika H. Schleier-Smith, Ian D. Leroux, and Vladan Vuletić. States of an ensemble oftwo-level atoms with reduced quantum uncertainty. Phys. Rev. Lett., 104:073604, Feb 2010.
[31] Zilong Chen, Justin G. Bohnet, Shannon R. Sankar, Jiayan Dai, and James K. Thompson.
Conditional spin squeezing of a large ensemble via the vacuum Rabi splitting. Phys. Rev.
Lett., 106:133601, Mar 2011.
[32] Zilong Chen, Justin G. Bohnet, Joshua M. Weiner, Kevin C. Cox, and James K. Thompson.
Cavity-aided nondemolition measurements for atom counting and spin squeezing. Phys. Rev.
A, 89:043837, Apr 2014.
[33] Onur Hosten, Nils J. Engelsen, Rajiv Krishnakumar, and Mark A. Kasevich. Measurement
noise 100 times lower than the quantum-projection limit using entangled atoms. Nature,529(7587):505–508, 2016.
[34] Kevin C. Cox, Graham P. Greve, Joshua M. Weiner, and James K. Thompson. Deterministicsqueezed states with collective measurements and feedback. Phys. Rev. Lett., 116:093602,
Mar 2016.
[35] Edwin Pedrozo-Penafiel, Simone Colombo, Chi Shu, Albert F. Adiyatullin, Zeyang Li, Enrique Mendez, Boris Braverman, Akio Kawasaki, Daisuke Akamatsu, Yanhong Xiao, andVladan Vuletic. Entanglement on an optical atomic-clock transition. Nature, 588(7838):414–
418, 2020.
[36] G Vallet, E Bookjans, U Eismann, S Bilicki, R Le Targat, and J Lodewyck. A noise-immune
cavity-assisted non-destructive detection for an optical lattice clock in the quantum regime.
New Journal of Physics, 19(8):083002, aug 2017.
[37] Andrew D. Ludlow, Martin M. Boyd, Jun Ye, E. Peik, and P. O. Schmidt. Optical atomic
clocks. Rev. Mod. Phys., 87:637–701, Jun 2015.

161[38] Dieter Hils and J. L. Hall. Improved Kennedy-Thorndike experiment to test special relativity.
Phys. Rev. Lett., 64:1697–1700, Apr 1990.
[39] E. Wiens, A. Yu. Nevsky, and S. Schiller. Resonator with ultrahigh length stability as a probe
for equivalence-principle-violating physics. Phys. Rev. Lett., 117:271102, Dec 2016.
[40] The LIGO Scientific Collaboration. LIGO: the laser interferometer gravitational-wave observatory. Reports on Progress in Physics, 72(7):076901, 2009.
[41] T. L. Nicholson, M. J. Martin, J. R. Williams, B. J. Bloom, M. Bishof, M. D. Swallows,
S. L. Campbell, and J. Ye. Comparison of two independent Sr optical clocks with 1×10−17
stability at 103 s. Phys. Rev. Lett., 109:230801, Dec 2012.
[42] Y. Y. Jiang, A. D. Ludlow, N. D. Lemke, R. W. Fox, J. A. Sherman, L. S. Ma, and C. W.
Oates. Making optical atomic clocks more stable with 10−16 level laser stabilization. Nature
Photonics, 5:158, 01 2011.
[43] Y. V. Stadnik and V. V. Flambaum. Enhanced effects of variation of the fundamental
constants in laser interferometers and application to dark-matter detection. Phys. Rev. A,
93:063630, Jun 2016.
[44] S. Kolkowitz, I. Pikovski, N. Langellier, M. D. Lukin, R. L. Walsworth, and J. Ye. Gravitational wave detection with optical lattice atomic clocks. Phys. Rev. D, 94:124043, Dec
2016.
[45] M. J. Martin, M. Bishof, M. D. Swallows, X. Zhang, C. Benko, J. von Stecher, A. V. Gorshkov,
A. M. Rey, and Jun Ye. A quantum many-body spin system in an optical lattice clock. Science,
341(6146):632–636, 2013.
[46] Fritz Riehle. Towards a redefinition of the second based on optical atomic clocks. Comptes
Rendus Physique, 16(5):506 – 515, 2015.
[47] Hidekazu Hachisu, Fumimaru Nakagawa, Yuko Hanado, and Tetsuya Ido. Months-long realtime generation of a time scale based on an optical clock. Scientific Reports, 8(1):4243,
2018.
[48] Fritz Riehle, Patrick Gill, Felicitas Arias, and Lennart Robertsson. The CIPM list of recommended frequency standard values: guidelines and procedures. Metrologia, 55(2):188, 2018.
[49] Colin J. Kennedy, Eric Oelker, John M. Robinson, Tobias Bothwell, Dhruv Kedar, William R.
Milner, G. Edward Marti, Andrei Derevianko, and Jun Ye. Precision metrology meets cosmology: Improved constraints on ultralight dark matter from atom-cavity frequency comparisons.
Phys. Rev. Lett., 125:201302, Nov 2020.
[50] William R. Milner, John M. Robinson, Colin J. Kennedy, Tobias Bothwell, Dhruv Kedar,
Dan G. Matei, Thomas Legero, Uwe Sterr, Fritz Riehle, Holly Leopardi, Tara M. Fortier,Jeffrey A. Sherman, Judah Levine, Jian Yao, Jun Ye, and Eric Oelker. Demonstration of atimescale based on a stable optical carrier. Phys. Rev. Lett., 123:173201, Oct 2019.
[51] Matthew A. Norcia, Julia R. K. Cline, Juan A. Muniz, John M. Robinson, Ross B. Hutson,
Akihisa Goban, G. Edward Marti, Jun Ye, and James K. Thompson. Frequency measurementsof superradiance from the strontium clock transition. Phys. Rev. X, 8:021036, May 2018.

162[52] Judith Olson, Richard W. Fox, Tara M. Fortier, Todd F. Sheerin, Roger C. Brown, Holly
Leopardi, Richard E. Stoner, Chris W. Oates, and Andrew D. Ludlow. Ramsey-bordé matterwave interferometry for laser frequency stabilization at 10−16 frequency instability and below.
Phys. Rev. Lett., 123:073202, Aug 2019.
[53] R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, andH. Ward. Laser phase and frequency stabilization using an optical resonator. Applied Physics
B, 31(2):97–105, 1983.
[54] John M. Robinson, Eric Oelker, William R. Milner, Dhruv Kedar, Wei Zhang, Thomas Legero, Dan G. Matei, Sebastian Häfner, Fritz Riehle, Uwe Sterr, and Jun Ye. Thermal noiseand mechanical loss of SiO2 /Ta2 O5 optical coatings at cryogenic temperatures. Opt. Lett.,
46(3):592–595, Feb 2021.
[55] Kenji Numata, Amy Kemery, and Jordan Camp. Thermal-noise limit in the frequency stabilization of lasers with rigid cavities. Phys. Rev. Lett., 93:250602, Dec 2004.
[56] M J Martin and J Ye. High-precision laser stabilization via optical cavities. In Optical
Coatings and Thermal Noise in Precision Measurement. Cambridge University Press, 2012.
[57] Peter R. Saulson. Thermal noise in mechanical experiments. Phys. Rev. D, 42:2437–2445,
Oct 1990.
[58] Yu. Levin. Internal thermal noise in the LIGO test masses: A direct approach. Phys. Rev.
D, 57:659–663, Jan 1998.
[59] W. E. Gifford. The Gifford-Mcmahon cycle. In K. D. Timmerhaus, editor, Advances in
Cryogenic Engineering, pages 152–159, Boston, MA, 1966. Springer US.
[60] http://montanainstruments.com/. Commercial products are identified for technical clarity.
Such identification does not imply endorsement by authors.
[61] DG Matei, T Legero, Ch Grebing, S Häfner, Ch Lisdat, R Weyrich, W Zhang, L Sonderhouse, JM Robinson, F Riehle, et al. A second generation of low thermal noise cryogenic
silicon resonators. In Journal of Physics: Conference Series, volume 723, page 012031. IOP
Publishing, 2016.
[62] Hui Shen, Liufeng Li, Jin Bi, Jia Wang, and Lisheng Chen. Systematic and quantitative
analysis of residual amplitude modulation in Pound-Drever-Hall frequency stabilization. Phys.
Rev. A, 92:063809, Dec 2015.
[63] W. Zhang, M. J. Martin, C. Benko, J. L. Hall, J. Ye, C. Hagemann, T. Legero, U. Sterr,
F. Riehle, G. D. Cole, and M. Aspelmeyer. Reduction of residual amplitude modulation to1 × 10−6 for frequency modulation and laser stabilization. Opt. Lett., 39(7):1980–1983, Apr
2014.
[64] Alessandro Farsi, Mario Siciliani de Cumis, Francesco Marino, and Francesco Marin. Photothermal and thermo-refractive effects in high reflectivity mirrors at room and cryogenic
temperature. Journal of Applied Physics, 111(4):043101, 2012.

163[65] Kazuhiro Yamamoto, Shinji Miyoki, Takashi Uchiyama, Hideki Ishitsuka, Masatake Ohashi,
Kazuaki Kuroda, Takayuki Tomaru, Nobuaki Sato, Toshikazu Suzuki, Tomiyoshi Haruyama,
Akira Yamamoto, Takakazu Shintomi, Kenji Numata, Koichi Waseda, Kazuhiko Ito, and Koji
Watanabe. Measurement of the mechanical loss of a cooled reflective coating for gravitational
wave detection. Phys. Rev. D, 74:022002, Jul 2006.
[66] Eiichi Hirose, Kieran Craig, Hideki Ishitsuka, Iain W. Martin, Norikatsu Mio, Shigenori
Moriwaki, Peter G. Murray, Masatake Ohashi, Sheila Rowan, Yusuke Sakakibara, Toshikazu
Suzuki, Kouichi Waseda, Kyohei Watanabe, and Kazuhiro Yamamoto. Mechanical loss of amultilayer tantala/silica coating on a sapphire disk at cryogenic temperatures: Toward the
kagra gravitational wave detector. Phys. Rev. D, 90:102004, Nov 2014.
[67] Christian Hagemann, Christian Grebing, Christian Lisdat, Stephan Falke, Thomas Legero,Uwe Sterr, Fritz Riehle, Michael J. Martin, and Jun Ye. Ultrastable laser with averagefractional frequency drift rate below 5 × 10−19 /s. Opt. Lett., 39(17):5102–5105, Sep 2014.
[68] Dhruv Kedar, Jialiang Yu, Eric Oelker, Alexander Staron, William R. Milner, John M. Robinson, Thomas Legero, Fritz Riehle, Uwe Sterr, and Jun Ye. Frequency stability of cryogenic
silicon cavities with semiconductor crystalline coatings, 2022.
[69] Garrett D. Cole, Wei Zhang, Bryce J. Bjork, David Follman, Paula Heu, Christoph Deutsch,
Lindsay Sonderhouse, John Robinson, Chris Franz, Alexei Alexandrovski, Mark Notcutt,
Oliver H. Heckl, Jun Ye, and Markus Aspelmeyer. High-performance near- and mid-infrared
crystalline coatings. Optica, 3(6):647–656, Jun 2016.
[70] Garrett D Cole, Wei Zhang, Michael J Martin, Jun Ye, and Markus Aspelmeyer. Tenfold
reduction of brownian noise in high-reflectivity optical coatings. Nature Photonics, 7(8):644–
650, 2013.
[71] Ting Hong, Huan Yang, Eric K. Gustafson, Rana X. Adhikari, and Yanbei Chen. Brownian
thermal noise in multilayer coated mirrors. Phys. Rev. D, 87:082001, Apr 2013.
[72] Gregory M Harry, Andri M Gretarsson, Peter R Saulson, Scott E Kittelberger, Steven D Penn,
William J Startin, Sheila Rowan, Martin M Fejer, D R M Crooks, Gianpietro Cagnoli, Jim
Hough, and Norio Nakagawa. Thermal noise in interferometric gravitational wave detectors
due to dielectric optical coatings. Classical and Quantum Gravity, 19(5):897–917, feb 2002.
[73] Alex Amato, Gianpietro Cagnoli, Maurizio Canepa, Elodie Coillet, Jerome Degallaix, Vincent
Dolique, Daniele Forest, Massimo Granata, Valérie Martinez, Christophe Michel, Laurent
Pinard, Benoit Sassolas, and Julien Teillon. High-reflection coatings for gravitational-wave
detectors:state of the art and future developments. Journal of Physics: Conference Series,
957:012006, feb 2018.
[74] S. Gras and M. Evans. Direct measurement of coating thermal noise in optical resonators.
Phys. Rev. D, 98:122001, Dec 2018.
[75] Kieran Craig, Jessica Steinlechner, Peter G. Murray, Angus S. Bell, Ross Birney, Karen
Haughian, Jim Hough, Ian MacLaren, Steve Penn, Stuart Reid, Raymond Robie, Sheila
Rowan, and Iain W. Martin. Mirror coating solution for the cryogenic Einstein telescope.
Phys. Rev. Lett., 122:231102, Jun 2019.

164[76] Kazuhiro Yamamoto, Shinji Miyoki, Takashi Uchiyama, Hideki Ishitsuka, Masatake Ohashi,
Kazuaki Kuroda, Takayuki Tomaru, Nobuaki Sato, Toshikazu Suzuki, Tomiyoshi Haruyama,
Akira Yamamoto, Takakazu Shintomi, Kenji Numata, Koichi Waseda, Kazuhiko Ito, and Koji
Watanabe. Measurement of the mechanical loss of a cooled reflective coating for gravitational
wave detection. Phys. Rev. D, 74:022002, Jul 2006.
[77] Raymond Robie. Characterisation of the mechanical properties of thin-film mirror coating
materials for use in future interferometric gravitational wave detectors. PhD thesis, Universityof Glasgow, 2018.
[78] Tara Chalermsongsak, Frank Seifert, Evan D Hall, Koji Arai, Eric K Gustafson, and Rana X
Adhikari. Broadband measurement of coating thermal noise in rigid Fabry–Pérot cavities.
Metrologia, 52(1):17–30, dec 2014.
[79] Sebastian Häfner, Sofia Herbers, Stefan Vogt, Christian Lisdat, and Uwe Sterr. Transportableinterrogation laser system with an instability of mod σy = 3 × 10−16 . Optics Express,
28(11):16407–16416, 2020.
[80] Thomas Kessler, Christian Hagemann, C Grebing, T Legero, Uwe Sterr, Fritz Riehle, MJ Martin, L Chen, and J Ye. A sub-40-mhz-linewidth laser based on a silicon single-crystal opticalcavity. Nature Photonics, 6(10):687, 2012.
[81] C. Hagemann. Ultra-stable laser based on a cryogenic single-crystal silicon cavity. PhD thesis,
2013-04 2013.
[82] B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang,
S. L. Bromley, and J. Ye. An optical lattice clock with accuracy and stability at the 10−18
level. Nature, 506(7486):71–75, 2014.
[83] M. Schioppo, R. C. Brown, W. F. McGrew, N. Hinkley, R. J. Fasano, K. Beloy, T. H.
Yoon, G. Milani, D. Nicolodi, J. A. Sherman, N. B. Phillips, C. W. Oates, and A. D. Ludlow.
Ultrastable optical clock with two cold-atom ensembles. Nature Photonics, 11(1):48–52, 2017.
[84] Tobias Bothwell. A Wannier-Stark Optical Lattice Clock With Extended Coherence Times.
PhD thesis, Boulder, 2022-04 2022.
[85] S. L. Campbell, R. B. Hutson, G. E. Marti, A. Goban, N. Darkwah Oppong, R. L. McNally,
L. Sonderhouse, J. M. Robinson, W. Zhang, B. J. Bloom, and J. Ye. A Fermi-degenerate
three-dimensional optical lattice clock. Science, 358(6359):90–94, 2017.
[86] Martin M. Boyd, Tanya Zelevinsky, Andrew D. Ludlow, Sebastian Blatt, Thomas ZanonWillette, Seth M. Foreman, and Jun Ye. Nuclear spin effects in optical lattice clocks. Phys.
Rev. A, 76:022510, Aug 2007.
[87] Xin Zheng, Jonathan Dolde, Varun Lochab, Brett N. Merriman, Haoran Li, and Shimon
Kolkowitz. Differential clock comparisons with a multiplexed optical lattice clock. Nature,602(7897):425–430, 2022.
[88] David N. Schramm and Michael S. Turner. Big-bang nucleosynthesis enters the precision era.
Rev. Mod. Phys., 70:303–318, Jan 1998.

165[89] Gianfranco Bertone and Dan Hooper. History of dark matter. Rev. Mod. Phys., 90:045002,
Oct 2018.
[90] D. Antypas, A. Banerjee, C. Bartram, M. Baryakhtar, J. Betz, J. J. Bollinger, C. Boutan,
D. Bowring, D. Budker, D. Carney, G. Carosi, S. Chaudhuri, S. Cheong, A. Chou, M. D.
Chowdhury, R. T. Co, J. R. Crespo Lpez-Urrutia, M. Demarteau, N. DePorzio, A. V. Derbin,
T. Deshpande, M. D. Chowdhury, L. Di Luzio, A. Diaz-Morcillo, J. M. Doyle, A. DrlicaWagner, A. Droster, N. Du, B. Dbrich, J. Eby, R. Essig, G. S. Farren, N. L. Figueroa, J. T.
Fry, S. Gardner, A. A. Geraci, A. Ghalsasi, S. Ghosh, M. Giannotti, B. Gimeno, S. M. Griffin,
D. Grin, D. Grin, H. Grote, J. H. Gundlach, M. Guzzetti, D. Hanneke, R. Harnik, R. Henning,
V. Irsic, H. Jackson, D. F. Jackson Kimball, J. Jaeckel, M. Kagan, D. Kedar, R. Khatiwada,
S. Knirck, S. Kolkowitz, T. Kovachy, S. E. Kuenstner, Z. Lasner, A. F. Leder, R. Lehnert,
D. R. Leibrandt, E. Lentz, S. M. Lewis, Z. Liu, J. Manley, R. H. Maruyama, A. J. Millar, V. N.
Muratova, N. Musoke, S. Nagaitsev, O. Noroozian, C. A. J. O’Hare, J. L. Ouellet, K. M. W.
Pappas, E. Peik, G. Perez, A. Phipps, N. M. Rapidis, J. M. Robinson, V. H. Robles, K. K.
Rogers, J. Rudolph, G. Rybka, M. Safdari, M. Safdari, M. S. Safronova, C. P. Salemi, P. O.
Schmidt, T. Schumm, A. Schwartzman, J. Shu, M. Simanovskaia, J. Singh, S. Singh, M. S.
Smith, W. M. Snow, Y. V. Stadnik, C. Sun, A. O. Sushkov, T. M. P. Tait, V. Takhistov,D. B. Tanner, D. J. Temples, P. G. Thirolf, J. H. Thomas, M. E. Tobar, O. Tretiak, Y. D.
Tsai, J. A. Tyson, M. Vandegar, S. Vermeulen, L. Visinelli, E. Vitagliano, Z. Wang, D. J.
Wilson, L. Winslow, S. Withington, M. Wooten, J. Yang, J. Ye, B. A. Young, F. Yu, M. H.
Zaheer, T. Zelevinsky, Y. Zhao, and K. Zhou. New horizons: Scalar and vector ultralightdark matter, 2022.
[91] Thibault Damour and John F. Donoghue. Equivalence principle violations and couplings ofa light dilaton. Phys. Rev. D, 82:084033, Oct 2010.
[92] Asimina Arvanitaki, Junwu Huang, and Ken Van Tilburg. Searching for dilaton dark matterwith atomic clocks. Phys. Rev. D, 91:015015, Jan 2015.
[93] Aurélien Hees, Olivier Minazzoli, Etienne Savalle, Yevgeny V. Stadnik, and Peter Wolf. Violation of the equivalence principle from light scalar dark matter. Phys. Rev. D, 98:064051,
Sep 2018.
[94] Gretchen K Campbell, Andrew D Ludlow, Sebastian Blatt, Jan W Thomsen, Michael J Martin, Marcio H G de Miranda, Tanya Zelevinsky, Martin M Boyd, Jun Ye, Scott A Diddams,
Thomas P Heavner, Thomas E Parker, and Steven R Jefferts. The absolute frequency of the
87 Sr optical clock transition. Metrologia, 45(5):539–548, sep 2008.
[95] P Wcislo, P Ablewski, K Beloy, S Bilicki, M Bober, R Brown, R Fasano, R Ciurylo, H Hachisu,
T Ido, et al. New bounds on dark matter coupling from a global network of optical atomic
clocks. Science Advances, 4(12):eaau4869, 2018.
[96] Ken Van Tilburg, Nathan Leefer, Lykourgos Bougas, and Dmitry Budker. Search for ultralight
scalar dark matter with atomic spectroscopy. Phys. Rev. Lett., 115(1):011802, 2015.
[97] A Hees, J Guéna, M Abgrall, S Bize, and P Wolf. Searching for an oscillating massive scalar
field as a dark matter candidate using atomic hyperfine frequency comparisons. Phys. Rev.
Lett., 117(6):061301, 2016.

166[98] Joel Bergé, Philippe Brax, Gilles Métris, Martin Pernot-Borràs, Pierre Touboul, and JeanPhilippe Uzan. MICROSCOPE mission: First constraints on the violation of the weak equivalence principle by a light scalar dilaton. Phys. Rev. Lett., 120(14):141101, 2018.
[99] Stephan Schlamminger, K-Y Choi, Todd A Wagner, Jens H Gundlach, and Eric G Adelberger. Test of the equivalence principle using a rotating torsion balance. Phys. Rev. Lett.,
100(4):041101, 2008.
[100] N. Leefer, A. Gerhardus, D. Budker, V. V. Flambaum, and Y. V. Stadnik. Search for theeffect of massive bodies on atomic spectra and constraints on yukawa-type interactions ofscalar particles. Phys. Rev. Lett., 117:271601, Dec 2016.
[101] Pierre Touboul, Gilles Métris, Manuel Rodrigues, Yves André, Quentin Baghi, Joël Bergé,Damien Boulanger, Stefanie Bremer, Patrice Carle, Ratana Chhun, Bruno Christophe, Valerio
Cipolla, Thibault Damour, Pascale Danto, Hansjoerg Dittus, Pierre Fayet, Bernard Foulon,
Claude Gageant, Pierre-Yves Guidotti, Daniel Hagedorn, Emilie Hardy, Phuong-Anh Huynh,
Henri Inchauspe, Patrick Kayser, Stéphanie Lala, Claus Lämmerzahl, Vincent Lebat, Pierre
Leseur, Fran çoise Liorzou, Meike List, Frank Löffler, Isabelle Panet, Benjamin Pouilloux,
Pascal Prieur, Alexandre Rebray, Serge Reynaud, Benny Rievers, Alain Robert, Hanns Selig,
Laura Serron, Timothy Sumner, Nicolas Tanguy, and Pieter Visser. MICROSCOPE mission:
First results of a space test of the equivalence principle. Phys. Rev. Lett., 119:231101, Dec
2017.
[102] T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay,
R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury,
W. M. Itano, D. J. Wineland, and J. C. Bergquist. Frequency ratio of Al+ and Hg+ single-ion
optical clocks; metrology at the 17th decimal place. Science, 319(5871):1808–1812, 2008.
[103] Steven T Cundiff and Jun Ye. Colloquium: Femtosecond optical frequency combs. Reviewsof Modern Physics, 75(1):325, 2003.
[104] Holly Leopardi, Josue Davila-Rodriguez, Franklyn Quinlan, Judith Olson, Jeff A. Sherman,
Scott A. Diddams, and Tara M. Fortier. Single-branch Er:fiber frequency comb for precision
optical metrology with 10−18 fractional instability. Optica, 4(8):879–885, Aug 2017.
[105] N. R. Newbury, P. A. Williams, and W. C. Swann. Coherent transfer of an optical carrier
over 251 km. Opt. Lett., 32(21):3056–3058, Nov 2007.
[106] Holly Leopardi, Kyle Beloy, Tobias Bothwell, Samuel M Brewer, Sarah L Bromley, Jwo-Sy
Chen, Scott A Diddams, Robert J Fasano, Youssef S Hassan, David B Hume, Dhruv Kedar,
Colin J Kennedy, David R Leibrandt, Andrew D Ludlow, William F McGrew, William R Milner, Daniele Nicolodi, Eric Oelker, Thomas E Parker, John M Robinson, Stefania Romisch,
Jeff A Sherman, Lindsay Sonderhouse, Jian Yao, Jun Ye, Xiaogang Zhang, and Tara M
Fortier. Measurement of the 27Al+ and 87Sr absolute optical frequencies. Metrologia,
58(1):015017, jan 2021.
[107] Z. Chen. Breaking Quantum Limits with Collective Cavity-QED: Generation of Spin Squeezed
States via Quantum Non-Demolition Measurements. PhD thesis, Boulder, CO 80309-0440,
2013.

167
[108] D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and D. J. Heinzen. Spin squeezing
and reduced quantum noise in spectroscopy. Phys. Rev. A, 46:R6797–R6800, Dec 1992.
[109] Masahiro Kitagawa and Masahito Ueda. Squeezed spin states. Phys. Rev. A, 47:5138–5143,
Jun 1993.
[110] Jürgen Appel, Patrick Joachim Windpassinger, Daniel Oblak, U Busk Hoff, Niels Kjærgaard, and Eugene Simon Polzik. Mesoscopic atomic entanglement for precision measurements beyond the standard quantum limit. Proceedings of the National Academy of Sciences,106(27):10960–10965, 2009.
[111] Ian D. Leroux, Monika H. Schleier-Smith, and Vladan Vuletić. Implementation of cavity
squeezing of a collective atomic spin. Phys. Rev. Lett., 104:073602, Feb 2010.
[112] Ian D. Leroux, Monika H. Schleier-Smith, and Vladan Vuletić. Orientation-dependent entanglement lifetime in a squeezed atomic clock. Phys. Rev. Lett., 104:250801, Jun 2010.
[113] B. C. Nichol, R. Srinivas, D. P. Nadlinger, P. Drmota, D. Main, G. Araneda, C. J. Ballance,
and D. M. Lucas. An elementary quantum network of entangled optical atomic clocks. Nature,609(7928):689–694, 2022.
[114] Christian D. Marciniak, Thomas Feldker, Ivan Pogorelov, Raphael Kaubruegger, Denis V.
Vasilyev, Rick van Bijnen, Philipp Schindler, Peter Zoller, Rainer Blatt, and Thomas Monz.
Optimal metrology with programmable quantum sensors. Nature, 603(7902):604–609, 2022.
[115] W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik. Quantum noise limited and entanglement-assisted magnetometry. Phys. Rev. Lett., 104:133601,
Mar 2010.
[116] Graham P. Greve, Chengyi Luo, Baochen Wu, and James K. Thompson. Entanglementenhanced matter-wave interferometry in a high-finesse cavity. Nature, 610(7932):472–477,
2022.
[117] Serge Haroche. Nobel lecture: Controlling photons in a box and exploring the quantum toclassical boundary. Rev. Mod. Phys., 85:1083–1102, Jul 2013.
[118] Gerhard Rempe, Herbert Walther, and Norbert Klein. Observation of quantum collapse andrevival in a one-atom maser. Phys. Rev. Lett., 58:353–356, Jan 1987.
[119] C. J. Hood, M. S. Chapman, T. W. Lynn, and H. J. Kimble. Real-time cavity QED withsingle atoms. Phys. Rev. Lett., 80:4157–4160, May 1998.
[120] S Weyers, V Gerginov, M Kazda, J Rahm, B Lipphardt, G Dobrev, and K Gibble. Advancesin the accuracy, stability, and reliability of the PTB primary fountain clocks. Metrologia,
55(6):789–805, oct 2018.
[121] H J Kimble. Strong interactions of single atoms and photons in cavity QED. Physica Scripta,
1998(T76):127, jan 1998.
[122] T. Holstein and H. Primakoff. Field dependence of the intrinsic domain magnetization of aferromagnet. Phys. Rev., 58:1098–1113, Dec 1940.

168[123] Dorian Gangloff, Molu Shi, Tailin Wu, Alexei Bylinskii, Boris Braverman, Michael Gutierrez,
Rosanna Nichols, Junru Li, Kai Aichholz, Marko Cetina, Leon Karpa, Branislav Jelenković,Isaac Chuang, and Vladan Vuletić. Preventing and reversing vacuum-induced optical losses
in high-finesse tantalum (v) oxide mirror coatings. Opt. Express, 23(14):18014–18028, Jul
2015.
[124] Sara Campbell. A Fermi-degenerate three-dimensional optical lattice clock. PhD thesis,
Boulder, 2017-05 2017.
[125] Makfir Sefa, James A. Fedchak, and Julia Scherschligt. Investigations of medium-temperature
heat treatments to achieve low outgassing rates in stainless steel ultrahigh vacuum chambers.
Journal of Vacuum Science & Technology A, 35(4):041601, 2017.
[126] Jerome Lodewyck, Michal Zawada, Luca Lorini, Mikhail Gurov, and Pierre Lemonde. Observation and cancellation of a perturbing dc stark shift in strontium optical lattice clocks. IEEE
Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 59(3):411–415, 2012.
[127] Vladimir Schkolnik, Jason R. Williams, and Nan Yu. Generating 500 mW for laser coolingof strontium atoms by injection locking a high power laser diode, 2020.
[128] Matthew A. Norcia and James K. Thompson. Simple laser stabilization to the strontium 88sr
transition at 707 nm. Review of Scientific Instruments, 87(2):023110, 2016.
[129] M. Bishof, X. Zhang, M. J. Martin, and Jun Ye. Optical spectrum analyzer with quantumlimited noise floor. Phys. Rev. Lett., 111:093604, Aug 2013.
[130] Ichiro Ushijima, Masao Takamoto, Manoj Das, Takuya Ohkubo, and Hidetoshi Katori. Cryogenic optical lattice clocks. Nature Photonics, 9(3):185–189, 2015.
[131] Thomas H. Loftus, Tetsuya Ido, Martin M. Boyd, Andrew D. Ludlow, and Jun Ye. Narrow
line cooling and momentum-space crystals. Phys. Rev. A, 70:063413, Dec 2004.
[132] S. Blatt, J. W. Thomsen, G. K. Campbell, A. D. Ludlow, M. D. Swallows, M. J. Martin,
M. M. Boyd, and J. Ye. Rabi spectroscopy and excitation inhomogeneity in a one-dimensional
optical lattice clock. Phys. Rev. A, 80:052703, Nov 2009.
[133] Ali Al-Masoudi, Sören Dörscher, Sebastian Häfner, Uwe Sterr, and Christian Lisdat. Noiseand instability of an optical lattice clock. Phys. Rev. A, 92:063814, Dec 2015.
[134] Alexander Aeppli, Anjun Chu, Tobias Bothwell, Colin J. Kennedy, Dhruv Kedar, Peiru He,
Ana Maria Rey, and Jun Ye. Hamiltonian engineering of spin-orbit-coupled fermions in a
Wannier-Stark optical lattice clock. Science Advances, 8(41):eadc9242, 2022.
[135] P. Kómár, E. M. Kessler, M. Bishof, L. Jiang, A. S. Sørensen, J. Ye, and M. D. Lukin. A
quantum network of clocks. Nature Physics, 10(8):582–587, 2014.
[136] B M Roberts, P Delva, A Al-Masoudi, A Amy-Klein, C Bærentsen, C F A Baynham, E Benkler, S Bilicki, S Bize, W Bowden, J Calvert, V Cambier, E Cantin, E A Curtis, S Drscher,
M Favier, F Frank, P Gill, R M Godun, G Grosche, C Guo, A Hees, I R Hill, R Hobson,
N Huntemann, J Kronjger, S Koke, A Kuhl, R Lange, T Legero, B Lipphardt, C Lisdat,
J Lodewyck, O Lopez, H S Margolis, H Álvarez-Martı́nez, F Meynadier, F Ozimek, E Peik,169
P-E Pottie, N Quintin, C Sanner, L De Sarlo, M Schioppo, R Schwarz, A Silva, U Sterr,
Chr Tamm, R Le Targat, P Tuckey, G Vallet, T Waterholter, D Xu, and P Wolf. Search fortransient variations of the fine structure constant and dark matter using fiber-linked opticalatomic clocks. New Journal of Physics, 22(9):093010, sep 2020.
[137] Boris Braverman, Akio Kawasaki, and Vladan Vuleti. Impact of non-unitary spin squeezingon atomic clock performance. New Journal of Physics, 20(10):103019, oct 2018.
[138] Jiazhong Hu, Wenlan Chen, Zachary Vendeiro, Hao Zhang, and Vladan Vuletić. Entangled
collective-spin states of atomic ensembles under nonuniform atom-light interaction. Phys.
Rev. A, 92:063816, Dec 2015.
[139] C. W. Gardiner and M. J. Collett. Input and output in damped quantum systems: Quantum
stochastic differential equations and the master equation. Phys. Rev. A, 31:3761–3774, Jun
1985.

Appendix A

Vacuum noise limits of cavity frequency determination

In this thesis, we have estimated the frequency of our probe laser with respect to the cQED
cavity in various ways. One method involves sweeping the probe tone over the cavity and measuring
the full IQ response using heterodyne detection. When possible, we have also parked on resonanceand measured the Q response using heterodyne. In the latest work, we switched to homodyne
detection of the probe laser and performed both sweeps and fixed-tone probing, where the homodyne
detection phase has been chosen to detect the Q quadrature. In the following, we examine the fullI − Q response of the cavity leakage field, and use it to calculate the vacuum noise limit on
determining the frequency of the cavity (or cavity-like) mode.
The field that leaks from the cavity has the form


p
βr


I0 = Rinc 1 −
2 

1 + κpδ/2

 
δ
β
p
 r κp /2 
Q0 = Rinc 2 δ
1 + κp /2

(A.1)



(A.2)

Here Rinc is the rate of incident photons, κp is the probed FWHM of the cavity mode, and δ is thedetuning of the laser with respect to the cavity mode. It is convenient to shift the I quadraturesuch that the phasor is centered on the origin. We define our shifted I quadrature aspβrIp = I0 − Rinc 1 −
2

(A.3)

171
Note that the length of the phasor from the center of the centered circle to the edge is β2r√

Rinc T .

Also note that βr is the diameter of the circle, but for estimating the signal to noise, we require
the radius. Estimating the detuning of the cavity probe with respect to the cavity mode amounts
to estimating the angle of the phasorΨ = arctan

A.0.1


Q0
.
Ip

(A.4)

Heterodyne detection

We first consider performing heterodyne detection, and perform a sweep over the cavity
resonance. We consider the local sensitivity to how Ψ changes with a small change in the detuningδ by calculating the derivative,
4δκp
dΨ
=− 2
.
dδ
4δ − κ2p(A.5)

If the total number of photons incident is Minc = Rinc dt, then the radius of the circle√
is β2r Rinc dt. Quantum optics teaches us that the vacuum noise creates fluctuations that will
degrade our ability to estimate Ψ, having magnitude 1/2. So the uncertainty in Ψ is given by theratio of the noise to the overall length of the phasor,
1/2∆Ψ = βr √.
Rinc dt
2(A.6)

We can then calculate the uncertainty of the detuning as∆δ = −

4δ 2 + κ2p
∆Ψ
√
=
.
dΨ/dδ
4βr κp Rinc dt(A.7)

We will think about sweeps of the probe tone across the cavity resonance as taking weighted
averages. If we have several measurements, each with uncertainties σ1 , σ2 , ..., σn , then the weighted
averages is given by the inverse-squared sum σ21 = σ12 + σ12 + ... + σ12 . The swept measurement is
tot

1

2

n

thought of as spending a dwell time dt and making a frequency estimate, then we step the frequency
and repeat. The final estimate is taken as the weighted sum of all of the measurements. We thenexpress the final estimate as an integral, where we take the sweep rate as δ = αt,!−2 −1/2Z ∞r224(αt) + κp
κ2p
α


√
dt
=
.
∆δsweep =2κ1 πRinc κp
4βr κp Rinc
−∞

(A.8)

172
We can use this expression to estimate the frequency noise corresponding to a single sweep overthe cavity resonance. It requires careful calibration of the rate of incident photons Rinc . If we then√want to take into account a finite quantum efficiency, one simply includes a 1/ Q term, meaningthat a poor quantum efficiency will lead to a higher ∆δsweep .

A.0.2

Homodyne detection

Now let’s consider a homodyne measurement of the cavity frequency. We will get the mostinformation if we park directly on the resonance and measure the Q-response of the cavity leakage
field. Taking the local derivative dQdδ to lowest order in δ gives
κ2p
dQ
√
=
,
dδ
4κ1 Minc(A.9)

where Minc is the number of photons in a measurement window. The corresponding frequency
uncertainty is given by∆δresonant =

1/2
κ2
= √
.
dQ/dδ
4 Minc κ1(A.10)

Appendix B

Calculation of effective atom-cavity coupling

The effective atom-cavity coupling g is independently estimated as a consistency check on
our experimentally determined value from Figure 6.6a. We follow the convention from [138], where
the effective g is
g2 =hgi4 i,hgi2 i(B.1)

where gi is the atom-cavity coupling for the ith atom. The effective atom number is
N = Ntot

hgi2 i2
hgi4 i(B.2)

where Ntot is the total atom number. In this work, when we refer to g and N , we refer to theeffective quantities. Our coordinate system is defined such that X is the direction along the cavity
axis, Y is the other horizontal axis orthogonal to X, and Z is the vertical direction along gravity.
2

2

N
The atomic density distribution in Z is modeled as a Gaussian, ρZ (Z) = √2πσe−Z /(2σZ ) , with
Z

2

2

1
standard deviation σZ . The probability distribution along Y is PY (Y) = √2πσ
e−Y /(2σY ) is aY

Gaussian with a standard deviation σŷ set by the thermal cloud radius. We calculate σY fromthe radial temperature of Tr = 290(10) nK from a radial Doppler scan of the clock transition andthe radial trap frequency of 34(3) Hz, giving a σY ≈ 26µm. If we allow the atoms to sufficiently
time-average the standing wave along x, we havegi2 (Y, Z) =with peak coupling g0 = d0qωp20 ~Vg02 −2(Y2 +Z2 )/w02
e
2

(B.3)

= 2π × 8.6 kHz. Here, d0 is the dipole matrix element and ωp

is the angular frequency of the |↓i →|ei transition, and V = 14 πω0 L is the effective cavity mode

174
volume. The ensemble averages are evaluated as
hgi2 i =1
NZ

hgi4 i =1NZ

gi2 (Y, Z)ρZ (Z)PY (Y)dYdZ(B.4)

gi4 (Y, Z)ρZ (Z)PY (Y)dYdZ,

(B.5)

and

and when combined using Eq. B.1 gives the estimated value of g = 2π × 4.8(2) kHz.

Appendix C

Calculation of free space scattering

This section describes the calculation of free space scattering in the atom-cavity coupledsystem.
To describe the atom-cavity coupled system, we use an input-out formalism [139]. The
Heisenberg-Lengevin equations of motion are [32]
Ω↓
√
dhai
κ
= −(iδc + )hâi − i hĉi + κ1 ain
dt
2
2

(C.1)

√
dhbi
Γ
= − − i N ghb̂i
dt
2

(C.2)

where hai is the cavity photon operator and hbi is the atom operator. ain is the drive amplitude,pwith units of photons/sec. We note that the cavity operator has time-dependence as hai = |a|eiδt ,and similar for the atomic operator hbi = |b|eiδt . Here, δ is the detuning of the probe light withrespect to the bare atomic transition frequency. We wish to solve the equations of motion for thesteady-state values, and so take the derivative of each operator and obtainihaiδ = −(iδc +Ω↓√κ)hâi − i hĉi + κ1 ain
2
2

ihbiδ = −√Γ
− i N ghb̂i
2

(C.3)

(C.4)

We solve these coupled equations to determine the steady-state excited state population a. It thenfollows that the rate of scattered photons will beṀs = Γ|hai|2

(C.5)

176
The solution has the form|hai|2 =4c2i κ1 Ω↓(Γ2 + 4δ 2 )(4(δc + δ)2 + κ2 ) + 2Ω2↓ (Γκ − 4δ(δc + δ)) + Ω4↓(C.6)

We can then plug in the approriate vacuum Rabi splitting Ω↓ , κ1 , δc . One then plugs in the correct
detuning δ to the eigenvalue, assuming we are probing on resonance. We require a good estimate of
the incident photon number ci , which we estimate by detuning the laser from the cavity resonance,and measuring the resulting homodyne fringe amplitude. We can calibrate the sideband power,and determine the mode overlap between the reflected mode and the LO mode.