A Wannier-Stark Optical Lattice Clock With Extended
Coherence TimesbyTobias Bothwell
B.S., University of Arkansas, 2014

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2022

Committee Members:
Jun Ye, Chair
Prof. James Thompson
Prof. Adam Kaufman
Prof. Ana Maria Rey
Prof. Penina Axelrad

ii
Bothwell, Tobias (Ph.D., Physics)
A Wannier-Stark Optical Lattice Clock With Extended Coherence Times
Thesis directed by Prof. Jun Ye

The pursuit of ever improving accuracy and precision in atomic clocks is inextricably linked
to discovery. With each new decade we gain deeper insight into nature, probing ever smaller energy
scales. In this thesis we report a body of research advancing our 1D strontium optical lattice clock(Sr1) to the frontiers of accuracy, precision, and atomic coherence. We demonstrate a new record
for strontium clock fractional frequency inaccuracy of 2.0 × 10−18 . We then leverage this in a seriesof comparisons, first comparing Sr1 with the Al+ and Yb clocks at NIST to 18 digits of accuracy.
Intra-lab comparisons with the 3D Sr lattice clock demonstrate record low instability between two
independent clocks (3.5 × 10−17 at 1 s). High uptime characterization and steering of Si3 by theSr1 system further demonstrates a proof of principle all-optical timescale system.
To move into the unknown, we introduce the newest version of Sr1. Utilizing a large waist, invacuum build up cavity we radically increase the homogeneity within the clock system. Operation
at shallow trap depths allows us to realize a Wannier-Stark optical lattice clock. By tuning thedelocalization of atomic wavefunctions we demonstrate the so called ‘magic depth’, where theclock frequency is free of atomic interaction induced frequency shifts regardless of atom number.
Combining these advances in precision we demonstrate a fractional frequency uncertainty of 4.4 ×
10−18 at 1 s of operation and 8 × 10−21 after 90 hours of operation, demonstrating nearly a factorof 100 lower uncertainty than the previous record. These advances allow us to rapidly evaluate
gradients across our millimeter length atomic sample, resolving the gravitational redshift within asingle clock.

Dedication

To my best friend Alex, without whom this would have been impossible.

iv

Acknowledgements

It is impossible to thank everyone who has helped me along this journey, let alone to fully
acknowledge everyone’s contributions. Nonetheless, I will try.
First and foremost - Jun. Thank you for bringing me into the Ye Lab family and supportingme throughout my graduate career. I have learned more than I knew possible in your group, witheach year serving to remind me how much more there is to explore. Leaving, I can’t help but be
excited for the next adventure!
My time on Sr1 has allowed me the privilege of working with fantastic colleagues. When I
first joined the group Xibo Zhang and Sarah Bromley welcomed me. Xibo is an incredibly curious
man. Sarah is a dedicated, thorough researcher who can always be relied on to ensure a project’s
success. The addition of Shimon Kolkowitz’s deep scientific thinking and knack for focused problem
solving further strengthened the team.
As the original crew left Sr1 I was joined by Dhruv Kedar and Colin Kennedy. Dhruv is a
careful, thoughtful scientist who will undoubtedly be a boon wherever he goes. Combined with
Colin’s go-getter attitude it was quite the energetic environment for physics research. For a period
we were joined by the Stable Lasers gurus - Eric Oelker and John Robinson. Any technology
sufficiently advanced is indistinguishable from magic, and Eric’s mastery of electronics and controls
is close. John is one of the most meticulous people I know, with an ability to focus on and solve
tasks that others would completely miss. Sr3 is sure to achieve success with him leading the charge.
Over the last year I have had the joy of training my replacement - ‘Alex Aeppli’. His ability andwillingness to take over the operation of the machine made the latter results in this thesis possible.

v
With the addition of Kyungtae Kim, Sr1 will surely continue to lead the way in the advancementof atomic clocks.
Sr2 has always been near in lab, providing valuable insight into science and life. Throughout
my graduate career I have overlapped with Ross Hutson, who seemingly always has a deep, nuanced
perspective to offer. I am still only beginning to fully appreciate the wisdom of Aki Goban’s advice.
Sara Campbell, Ed Marti, Travis Nicholson, Cristian Sanner, Lindsay Sonderhouse, and William
Milner all made lasting impressions on my development as a scientist. It will be exciting to watch
the results unfold in the coming years with Will taking over Sr2 and Yee Ming Tso and Maya
Miklos taking over Sr3.
The community is what makes JILA special. Collaboration through the years with the Reygroup has been vital to the research in this thesis. Ana Maria with Anjun Chu and Peiru He were
essential in the full realization of density-shift free Wannier-Stark optical lattice clock described inthis thesis. James Thompson has been a source of ideas and inspiration, always giving generouslyof his time. The machine shop, particularly James Uhrich and Hans Green, kept our results moving
foward. Jan Thomsen, Matt Norcia, Julia Cline, Tanya Roussy, and the whole KrB crew (you know
who you are) were all a joy to exchange ideas with. J.R., Corey Keasling, Brian Lynch, Dan Lewis,
and many others provided the support to make all of this possible.
Finally, family and friends kept this thesis going through the highs and lows. Jordan Stone,
Dave Reens, and Eddie Meyers all helped keep me sane. My parents (on both sides) were always afont of positivity. My brothers were my brothers. And most importantly - Alex was always there.
Also Heidi.

viContents

Chapter
1 An Introduction to the Wonderful World Of Clocks

1

1.1

Why Do We Want Better Clocks?

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2

1D 87 Sr Optical Lattice Clocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3

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

82 Record Accuracy For a Strontium Clock

11

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2

Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3

Systematic Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1

Blackbody Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.2

Density Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.3

Lattice AC Stark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.4

DC Stark Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.5

Second Order Zeeman Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.6

Background Gas Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.7

Doppler Shift and AOM Phase Chirp

2.3.8

Line Pulling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3.9

Servo Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

. . . . . . . . . . . . . . . . . . . . . . 33

2.3.10 Gravitational Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

vii
2.4

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5

Thermal Environment Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.6

2.5.1

Thermal Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5.2

Temperature Sensor Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5.3

BBR Dynamic Shift Correction . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Lattice Light Shift Evaluation - Model Uncertainty . . . . . . . . . . . . . . . . . . . 44

3 Comparisons and Demonstrations

48

3.1

Towards the Redefinition of the Second . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2

The Boulder Atomic Clock Optical Network . . . . . . . . . . . . . . . . . . . . . . . 50

3.3

Absolute Frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4

Intra-lab Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5

Towards Optical Timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.6

Searches For Ultralight Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 The New Sr1 Wannier-Stark Optical Lattice Clock
4.1

62

The New Sr1 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.1.1

The Build-Up Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.1.2

DC Stark Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.1.3

Temperature Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1.4

Expansion Port . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.2

Experimental Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3

Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4

Vacuum Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5 Wannier-Stark States, a Spin Model, and the ‘Magic Depth’

84

5.1

Bloch, Wannier, and Wannier-Stark Wavefunctions . . . . . . . . . . . . . . . . . . . 84

5.2

Spin-orbit Coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

viii
5.3

The Spin-Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.4

The ‘Magic Depth’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6 Resolving the gravitational redshift within a millimeter atomic sample
6.1

102

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.1.1

In-Vacuum Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.1.2

Atomic Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.1.3

Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.1.4

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.1.5

Atomic Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.1.6

Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7 Lattice Light Shifts

119

7.1

AC Stark Shift From Electric Dipole Interaction - π Transitions . . . . . . . . . . . . 120

7.2

Atom-Light Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.2.1

Matrix Elements of D̂E1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.2.2

Matrix Elements of D̂QM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.2.3

Quick Notes on Order of Magnitudes . . . . . . . . . . . . . . . . . . . . . . . 125

7.3

AC Stark Shift From E1, M1, and E2 . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.4

Harmonic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.5

BO+WKB Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7.6

Evaluating the Atomic Motional Distribution in the Harmonic Model . . . . . . . . . 136

7.7

Lattice Light Shift Evaluation in the Wannier-Stark Optical Lattice clock . . . . . . 138

8 Conclusions and Outlook
8.1

144

Towards Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

ix
Bibliography

147

Appendix
A Useful Derivations

158

A.1 Optical lattice Trap Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
A.2 Harmonic Oscillator Expectation Values . . . . . . . . . . . . . . . . . . . . . . . . . 159
A.3 Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

xTablesTable
2.1

Atomic Temperature Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2

Lattice light shift uncertainty contributions . . . . . . . . . . . . . . . . . . . . . . . 29

2.3

Sr1 systematic uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4

Sr1 uncertainty relative to ITRF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.5

Emissivities of vacuum chamber components

2.6

Specular steel reflectivities based on [11] . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.7

Vacuum chamber generalized exchange factors . . . . . . . . . . . . . . . . . . . . . . 40

6.1

Gradient Systematic Budget. Fractional frequency gradients and corresponding un-. . . . . . . . . . . . . . . . . . . . . . 39

certainties. Fractional frequencies denoted with ‘-‘ are corrected on a pixel-by-pixel
basis during initial data processing (Fig. 3a). The corrected gradient has known
systematics removed with uncertainty given by the quadrature sum of all correction
uncertainties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8.1

Sr1 systematic uncertainty from 2019 and new Sr1 projected uncertainty. ?s denote
optimal evaluations that are not guaranteed. . . . . . . . . . . . . . . . . . . . . . . 146

xiFiguresFigure
1.1

The SI Base units and defining constants. The second is defined by taking the fixed
frequency for the cesium hyperfine transition ∆ν, the meter by defining the speed oflight c, the kilogram by defining Planck’s constant h, and so forth. All other units
are derived from these seven base units. . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

2

Accuracy and precision. The top row shows marks on a target, the bottom row
frequency versus time of an oscillator that should be at frequency ν0 . Column a)shows high accuracy and high precision, b) high accuracy and low precision, c) low
accuracy and high precision, and d) low accuracy and low precision. . . . . . . . . .

1.3

4

Strontium energy levels. Even (odd) parity levels shown in black (red). The strong
dipole transition (461 nm) serves for initial cooling and trapping of hot strontium
atoms. A weak decay channel from 1 P1 to 1 D2 (purple line) leads to population in
3 P and 3 P (green lines). While 3 P decays to the ground state, 3 P is metastable12
1
2

and leads to significant loss. The addition of repumps at 679 and 707 nm ensures
cycling of metastable states back to the ground state, enhancing MOT operation andenabling excited state readout. Narrow-line MOT operation at 689 nm is cycling andenables efficient preparation of cold, dense samples of Sr.

. . . . . . . . . . . . . . .

7

xii
1.4

Clock spectroscopy of nuclear spin states [112]. Traditional stretched state spectroscopy is shown in blue, where the |g, mF = ±9/2i ↔ |e, mF = ±9/2i transition
is probed. Recent clock operation has become limited by magnetic field noise, necessitating operation with the least magnetically sensitive transition in strontium,|g, mF = ±5/2i ↔ |e, mF = ±3/2i. In each sequence (assuming a perfect atomic
reference) 1/2 of the sum of the nuclear spin state transition frequencies corresponds
to laser noise while 1/2 of the difference corresponds to magnetic field noise. During spectroscopy the frequency of each nuclear spin transition is found by probing
both the left and right of the lineshape, providing a frequency correction based on
excitation fraction differences and the corresponding lineshape. . . . . . . . . . . . .

9

xiii
2.1

Schematic view of the Sr1 clock. Ultrastable laser light is generated at 1542 nm
by referencing a diode laser to a crystalline silicon optical cavity operating at 124 K(red, dotted line). The stability of this laser is then transferred via an Er:fiber comb
to an external-cavity diode laser pre-stabilized by a 40 cm ULE cavity operating at
698 nm (blue, dotted line). An acousto-optic modulator (AOM 1) is then used tosteer the cavity light into resonance with the Sr clock transition. The excitation
fraction after probing the clock transition is detected by collecting fluorescence fromboth ground and excited state atoms. A frequency step applied to AOM 1 produces
an error signal for locking by alternately probing both sides of the | ± 9/2i stretchedstate transitions. Frequency corrections to the average of the | ± 9/2i frequencies areapplied to AOM 2 such that the cavity-stabilized light is steered onto the transition
frequency of the Sr atom. In addition, frequency corrections to the difference of the| ± 9/2i frequencies are applied to the AOM 1 frequency. An in-plane magnetic field,
B, providing a quantization axis for the atoms, is aligned to be collinear with boththe 1D optical lattice polarization, 813 , and the clock laser polarization, 698 . Outof-vacuum quadrant ring electrodes generate a DC electric field to cancel the ambient
field at the position of the atoms. Finally, a phase lock of the 813 nm trapping laser
to the Er:fiber comb stabilizes the frequency of the trapping light (green, dottedline). The trapping light is delivered to the atoms through a high power optical fiberand is intensity stabilized by actuating the RF power on AOM 3. . . . . . . . . . . . 14

xiv
2.2

Systematic shifts. (a) Plot of the time record of the systematic shifts. Changes
in atom number, ambient temperature, or magnetic field all result in corrections
to the clock frequency, and their total magnitude is shown over a six hour data
campaign. The clock achieves 98.9% uptime over the course of this single comparison
day and slight gaps in the data indicate brief periods where the laser is not locked
to the atoms. (b) The same data is plotted as a fractional instability normalized
to the Sr clock frequency. The individual contributions of density shift (blue), BBR(red) and second order Zeeman shift (yellow) are shown as the dashed curves. For
operation times up to 104 seconds, fluctuations in systematic offsets are bounded
below 4 × 10−19 . (c) Nonsynchronous comparison with the JILA 3D optical latticeclock demonstrates that the beat between the two clocks averages below the quoted
total systematic uncertainty. All error bars are derived from a white noise modeland the black line is a white noise τ −1/2 fit to the single clock instability. . . . . . . 16

xv
2.3

Active Temperature Control. (a) The optical table is divided into independently
controlled sections (grey and white shaded regions) and isolated from the room temperature by a laser curtain. Around the vacuum chamber is an additional black box.
Temperature sensors (TS) monitor each table half, allowing feedback (dotted lines)
for controlling the flow of cooling water through a water/air exchanger above eachtable half. Commercial HEPA filters pull room air into the box. (b) The Sr1 vacuum
chamber is contained in a black box to protect it from stray light and ensure thermal
homogeneity. Heat sources that are temperature controlled are shown in red: Zeeman window (ZW), Zeeman slower (ZS), and MOT coils (MC). Temperature controlof vacuum viewports is shown in blue: water temperature controlled copper tubes
around the top and bottom 6” CF flanges (CF) and thermoelectric cooler (TEC)
controlled 2.75” CF viewport faces (VF). The oven (not shown in the figure) is located past the Zeeman slower. The nozzle (6 mm in diameter and 575◦ C) is the only
heated oven component visible to the atoms. (c) The temperature at the locationof the atomic sample is directly measured using a thin-film platinum resistance thermometer (TFPRT) sensor and compared to a model derived from ray-tracing andtemperature sensors mounted on the chamber, verifying the stability afforded by our
extensive thermal control. The measurements shown are binned into 10 minute intervals. (d) At several hours averaging times the TFPRT shows sub-mK level stability.
The difference between the TFPRT and model shows similar temperature stability,
providing verification that long-term fluctuations in the temperature experienced by
the atoms are accurately captured by the ray-tracing model. . . . . . . . . . . . . . . 17

2.4

Density shift evaluations. Density shift measurements were performed over fourweeks at the same trap conditions. The weighted mean of all measurements is shownwith a dashed, black line, with a reduced chi-squared of 1.07. . . . . . . . . . . . . . 25

xvi
2.5

Scaling of the light shift with trap depth U (a) Density shift evaluated at
different trap depths, scaled to a shift with 1000 atoms. The dashed red line is a fit
to the data with the expected scaling of the shift as U 5/4 . (b) Lattice Stark shifts
measured relative to a trap depth of 45Er . Four different lattice frequencies areshown: νL = 368.55452610 (blue), 368.55447610 (red), 368.55446610 (green), and368.55442610 THz (gold). The three curves with dashed lines are independently fit
using the model in Eqn. 2.4, obtaining a weighted mean of their β ∗ ’s. Using this∗β ∗ , the red curve is then fit for αclock, fully characterizing our AC Stark shifts forclock operation at the red curve. Vertical error bars are obtained from fits to the
Allan deviation of each evaluation, extrapolated to the total measurement time, andscaled by the lever arm of the measurement. Horizontal error bars are uncertainties
on our determination of the trap depth, obtained from axial sideband scans. . . . . . 26
2.6

Evaluation of second order Zeeman coefficient. A history of evaluations of
the Sr second order Zeeman coefficient completed by the PTB [45], SYRTE [145],
and JILA [30, 17, 109] Sr OLCs. The dashed black line is a weighted mean of all six
measurements and the inset shows the three most recent evaluations. . . . . . . . . . 31

2.7

JILA fixed-point realizations. Data from the text(a) water ice melting pointand (b) gallium melting point realization. (c) Allan deviation of the gallium melt
curve (red) and the ice melt curve (blue), showing the capability of averaging down
to below 100 µK on each fixed-point in 104 seconds or less. . . . . . . . . . . . . . . 41

xvii
2.8

Calibration of the in-vacuum probe. (a) Immersion error data in the water
bath comparison. We measure the difference between the high vacuum resistanceRvac and the He back-filled case RHe as a function of axial gradient Tf lange − Tbath .
The data is fit to a linear function, and the fitted offset at zero axial gradient is (0.3
± 0.4) mΩ. This can be converted to temperature by using the sensitivity of the
TFPRT of 2.57 Ohms/Kelvin. (b) Breakdown of uncertainties stemming from the
NIST calibration. The green is the systematic error in the immersion error offset
from panel (a). The cyan color is the maximum temperature gradient observed in
the bath. The blue and orange curves are the fit interpolation errors from the slope
and offset respectively. The red line is the quadrature sum of several minor errors
relating to the calibration. The black dashed line is the quadrature sum of the errors. 43

2.9Model Error Colormap showing the standard deviation between simulated AC
Stark data (Eqn. 2.11) and a simple linear and quadratic fit to the data (Eqn. 2.4).
The average standard deviation is 3.3×10−19 which we take to be our model error. . 47

3.1

Overview of the Boulder Atomic Clock Optical Network (BACON). Network locations at NIST (JILA) are shown in the left (right) of the image. The two institutes
are linked both by a 3.6 km fiber link as well as free-space link. NIST is home to
the Al+ ion clock, Yb OLC, and OFCs used for measuring the 3 clock ratios. The
Sr clock is located at JILA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2

Time record of ratio measurements. Frequency ratios were measured from November
2017 to June 2018. Results are shown as fractional offsets from the reported values.
Left error bars represent pure statistical uncertainty, whereas error bars on the rightinclude systematic uncertainties added in quadrature to statistical uncertainties. . . 53

3.3

Example of daily ratio measurement. Uncertainties for each day are derived fromoverlapping Allan deviations of the corresponding beat. Instabilities for network
elements are shown, all well below the measured instabilities of the atomic ratios.

. 54

xviii3.4

Si3 stability. Long-term stability data of Si3 as measured by the Sr1 system showing
nearly thermal noise limited stability of just under 4 × 10−17 from 10 to 10,000
seconds.

3.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Sr1/Sr2 synchronous comparison. Each clock operated with 600 ms π-pulses and 570
ms deadtime, triggering the clock pulse at the same time. The redline corresponds√
to a fractional frequency instability of 3.5 × 10−17 / τ , consistent with the expected
QPN contribution of each clock.

3.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Si3 stability. Si3 was compared against Sr1 and NIST’s AT1 MASER array. For
timescale below 105 Sr resolves Si3 instability well below the current timescale.
Around 105 Si3 stability sufficiently degrades such that it becomes comparable toAT1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.7

Modeled fractional frequency stability of different potential timescales. Si3 steered
daily by Sr1, even for one hour, outperforms MASER based timescales. Both Si3models support fractional frequency uncertainties below 1 × 10−17 at less than one
year of averaging.

3.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Comparison of α sensitivity of the Sr clock transition, Si3 cavity frequency, and H
Maser frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.1

Post bake Sr1 system. The system is supported by black anodized aluminum supports. To the left is the AOSense atomic beam system (circled in red) for providing
a steady stream of cooled strontium into the MOT. Center is the Kimball Physics 6”
Spherical Octagon, with the 6” CF viewport facing the camera. All viewports were
mounted using silver coated threaded studs, washers, and nuts for use in mounting
important clock hardware. The large white G7 piece (blue arrow) mounted to theraw aluminum legs is the MOT coil that required construction and mounting before
baking. Optics on the table were used to check the cavity alignment post bake.

. . 64xix
4.2

Image of clock retro box without cover. The assembly is attached to the threadedrods of the bottom 2 3/4” viewport, ensuring a rigid connection between chamber(lattice retro) and box (clock retro). Lattice light (green arrows) is introduced to thecavity from above, with leakage below used for intensity stabilization. Clock light(red arrows) comes from the table and is partially reflected by a wedged mirror in theThor Labs mirror mount enabling fiber noise cancellation nearly to the viewport. A
slot enables various neutral density filters for clock power attenuation. A high quality
dichroic beamsplitter separates lattice light (813 nm) from clock light (698 nm).

4.3

. . 65

Installed mezzanines around the Sr1 chamber. Red arrows show installed bias coils.
The copper attachment within the front MOT coil assembly contains the Bx bias
coil, with water cooling lines for temperature control. The top mezzanine supports
optics for the 813 nm lattice system while cavity transmission optics, clock optics,
and MOT optics are on the table.

4.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Cavity mirror pictures. Top left: Top mirror assembly, with bottom stainless steel
plate for mounting to groove grabbers, mirror glued to grounded copper piece, andpiezo sandwiched between copper and steel. This ensures charges from applied voltages to the piezo do not charge the mirror. Top right: Groove grabber cavity
mirror support structure inside 2 3/4” CF flange. Bottom left: Installed top mirror assembly. Bottom right: Picture of Faraday shielding cage between atoms andmirror. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.5

Cavity ring-down measurement. Transmitted intensity through the cavity is measured on a PD. The light is extinguished by an acousto-optic modulator (AOM) andthe decaying exponential fit, finding a decay time constant of 183 ns. From thisdecay and measured cavity length a finesse of F = 1020(60) is found, with the large
uncertainty arising from the sensitivity of the fit to the number of data points cut
before fitting.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

xx
4.6

Comparison of hyperpolarizability and birefringence induced flock shifts as a function of trap depth as bounded by worst case ellipticity. Results are reported for
mF =±9/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.7

DC Stark plate construction. Top left: Example of a quadrant plate split in half,showing groove grabber assembly for both assembling all four quadrants together
and mounting to the chamber. Note the use of alumina to electrically isolate eachquadrant. Top right, bottom left: Picture of the plates installed into the Kimball
Physics chamber, showing wiring. Bottom right: Assembly for routing quadrant
plate wires around 2 3/4” CF port. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.8

DC Stark evaluation along X. On the left is the Allan deviation of the evaluatedbackground shift for voltages biases of ±80 V applied to each plate (160 V difference).
A background shift of 6(23) × 10−20 was found, consistent with zero. The right plotshows a histogram of background shifts. . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.9

Temperature control/optics mounting pieces for the 2 3/4” CF flanges (viewports).
Top left: The main control piece is a copper cylinder wrapped with hollow copper
tubing soldered in place. The tubing allows temperature controlled water to flow
around the piece, stabilizing the temperature. Holes for mounting to the chamber(bottom left), top piece installation (top right), and thermistor installation are
included. Bottom right:

An installed assembly including Thor Labs SM2 lens

tube for mounting of MOT optics is shown. Jack screws enable fine alignment of
the retros but are not needed thanks to the Kimball Physics chamber and carefulmachining of copper pieces. A thin layer of high thermal-conductivity paste is used
between the piece shown in the top left and the vacuum chamber, ensuring goodthermal conduct and compliance for any slight imperfections. . . . . . . . . . . . . . 77
4.10 Chamber temperatures from 8/14/21. Monitoring thermistors are mounted to thechamber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

xxi
4.11 Viewport temperature from 8/14/21. Each sensor is attached to a copper control
piece mounted to the respective viewport. . . . . . . . . . . . . . . . . . . . . . . . . 78
4.12 Doppler scans after initial loading and cooling at 300 Er (purple curve) and after
ramping to an operational trap depth (15 Er shown in green). . . . . . . . . . . . . . 80
4.13 Axial clock scans. a) Axial scans at 300 Er and after adiabatically ramping to 10
Er with atoms prepared in the excited clock state. The green line has been offset
by 0.5 from an excitation fraction of 1.0 for plotting. Note the near absence of
red sideband indicating nz ≈ 0. Additionally note the spiky structure within the
sideband structure - these correspond to spectroscopy of varying nz states in the
purple curve, now resolved thanks to the < 1 µK radial temperatures at 300 Er . b)
Enhanced view of the red sideband. c) Enhanced view of the blue sideband. . . . . . 81
4.14 Calibration of fluorescence probe scattering rate throughout the atomic sample. To
evaluate the scattering rate per pixel, a series of atom pictures is taking with varying
probe power, allowing calculation of the saturation parameter (ratio of intensity tosaturation intensity), and therefore maximum scattering rate.

. . . . . . . . . . . . 82

4.15 Vacuum lifetime versus oven temperature. The trend is well reproduced by a linear
fit of slope -0.167 s/◦ C and offset of 89.8 s at 0◦ C. . . . . . . . . . . . . . . . . . . . 83
5.1

Bloch band structures of optical lattices for varying trap depths. Each plot showsenergy versus quasimomentum for the first Brillouin zone. Purple, green, and yellow
correspond to the ground, first, and second bands respectively. Note the decreasing
bandwidth as trap depth increases, corresponding to decreased tunneling energy. . . 88

5.2

Wavefunctions (ψ) and probability densities (|ψ|2 ) for Bloch wavefunctions of trap
depths 8 and 20 Er . The real part of the wavefunction is plotted. . . . . . . . . . . . 89

5.3

Example Wannier state probability distributions as a function of lattice position ata trap depth of 3 Er . m gives the lattice index for each plotted Wannier function.
The optical lattice potential is plotted in black for reference. . . . . . . . . . . . . . . 91

xxii
5.4

Wannier-Stark state probability distributions for a variety of trap depths. All states
are centered at m = 0, making clear the increasing off-site contributions as trap
depth is decreased. This delocalization of the wavefunction provides the ability totune the overlap of wavefunctions between lattice sites, creating a tunable channel
for s-wave interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.5

Spin-orbit coupling in a horizontal 1D lattice. Left: Ground Bloch bands for 10 Er
for both ground (g) and excited (e) clock states with the energy of the clock photon
subtracted. The excited band has been shifted by the momentum of the clock photon.
Due to this, the energy difference between e and g depends on quasimomentum,
coupling spin to momentum (SOC). Right: A histogram of energy difference withevenly sampled quasimomentum. The turning points of the energy difference inthe left plot correspond to an increased number of quasimomentum points at large
detunings - the Van Hove Singularities.

5.6. . . . . . . . . . . . . . . . . . . . . . . . . 93

Fractional excitation versus detuning for varying depths in the horizontal lattice [76].
As the trap depth decreases, the carrier clearly broadens and shows the emergenceof Van Hove singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.7

A π-pulse on the Bloch sphere. A resonant monochromatic field (laser) applies afield along x̂ (black arrow). A spin initially at |gi (red arrow) then begins to precess
around the bloch sphere (blue dots), ending at |ei (blue arrow) for a π pulse. . . . . 95

5.8

π pulses with atomic interactions, with the spin starting in |gi (red arrow). Ωx = 1(black arrow), χ=0, and C=0.1. Left: π-pulse evolution for N = 10. The magenta
arrow shows Ω while the green dots and arrow show the evolution of the spin for aresonant π-pulse. Right: Same as left, but for N =0,10, and 20 corresponding toblue, green, and cyan arrows. For a system of traps with differing atom number eachsite’s collective spin will evolve differently.

. . . . . . . . . . . . . . . . . . . . . . . 98

xxiii
5.9

Density shift per atom per site in fractional frequency units versus trap depth. At
large depths both curves follow the U 5/4 scaling of Chapter 2, but deviate rapidly
at shallow depths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.10 ‘Magic depth’ operation at 12 Er . Over a 10 hour measurement the fractional density
shift was found to be 5.0 (2.8)×10−21 per atom per lattice site.
6.1. . . . . . . . . . . 101

Experimental system and quantum state control. a, A millimeter length sample
of 100,000 87 Sr atoms are trapped in a 1D optical lattice formed within an invacuum cavity. The longitudinal axis of the cavity, z, is oriented along gravity.
We probe atoms along the 1 S0

 P transition using a clock laser locked to an
30ultrastable crystalline silicon cavity [112, 97]. b, Rabi spectroscopy with a 3.1 spulse time. Open purple circles indicate data with a corresponding Rabi fit in green.
c, Neighboring lattice sites are detuned by gravity, creating a Wannier-Stark ladder.
Clock spectroscopy probes the overlap of Wannier-Stark states between lattice sitesthat are m sites away with Rabi frequency Ωm . d, Rabi spectroscopy probes WannierStark state transitions, revealing wavefunction delocalization of up to 5 lattice sites.
The number of lattice sites is indicated above each transition, with blue(red) denoting
Wannier-Stark transitions to higher(lower) lattice sites. . . . . . . . . . . . . . . . . 103

xxiv
6.2

Atomic coherence. We use Ramsey spectroscopy with a randomly sampled phase forthe second pulse to determine the coherence time of our system [93]. a, We measure
the excitation fraction across the cloud, shown in purple for a single measurement,
and calculate the average excitation fractions in regions p1 and p2 , separated by 2
pixels. b, Parametric plots of the excitation fraction of p1 versus p2 in purple for 6s, 30 s and 50 s dark time demonstrate a phase shift between the two regions andcontrast decay. Using a maximum likelihood estimator, we extract the phase andcontrast for each dark time with the fit, shown in green. c, Contrast decay as afunction of time in green is fit with an exponential decay in gold, giving an atomic
coherence decay time of 36.5(0.7) s and a corresponding quality factor of 3.6 × 1016 .
After accounting for the finite radiative decay contribution, we infer an additional
decoherence time constant of 52.8(1.5) s. . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.3

Evaluating frequency gradients. a, For each measurement we construct a microscopic
frequency map across the sample, with raw frequencies shown in green. The second
order Zeeman correction is shown as a dashed gold line. Processed frequencies shownin purple include both density shift corrections and second order Zeeman corrections,with uncertainties arising from the quadrature sum of statistical, density shift correction, and second order Zeeman correction uncertainties. To this we fit a linear
function, shown in black. b, Over the course of 10 days, we completed 14 measurements. For each measurement, we create a corrected frequency map and fit a linear
slope as in a. This slope is plotted for each measurement, as well as a weighted mean
(black) with associated statistical uncertainty (dashed black) and total uncertainty
as reported in this chapter’s table (dotted black). The expected gravitational gradient is shown in red. All data is taken with Rabi spectroscopy using a 3.1 s π-pulse
time except for 08/13 which used a 3.0 s pulse time. The reduced chi-square statistic
is 3.0, indicating a small underestimation of error variances entirely consistent withthe additional systematic uncertainties in Table 1. . . . . . . . . . . . . . . . . . . . 108

xxv
6.4

In-situ synchronous clock comparison. a, The cloud is separated as in Fig. 2a. The
gravitational redshift leads to the higher clock(blue) ticking faster than the lower
one(red). The length scale is in millimeters. b, Allan deviation of the frequency
difference between the two regions in a over 92 hours. Purple points show fractionalfrequency instability fit by the solid green line, with the quantum projection noise
limit indicated by the dashed black line. We attribute the excess instability of themeasurement relative to QPN to detection noise. The expected single atomic region
instability is shown in gold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.1

Wannnier-Stark ground band calculations. a) Expectation values calculated for numerically evaluated WS states in 1D. Notice the rapid increase in hsin2 (kz)i at decreasing trap depths - this is the origin of the nonlinear ∆αQM behavior. b Latticelight shift contribution from ∆αQM versus trap depth. Combined with a) we noticeboth the nonlinear behavior and small magnitude of the shift. For self-comparison
stabilities and systematic control resolution of the ∆αQM shifts presents a formidable
challenge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.2

Operational regimes at the magic depth (u = 12 Er ). a) For accuracy, operation at
the inflection point provides a robust lattice light shift against imperfect experimental
conditions. Varying temperatures and atomic wavefunctions modify effective trap
depths. Operating in this regime provides the prospect of lattice light shift stabilityof ≈ 1 × 10−19 over a wide range of trap depths. For this simulation the inflection
point was found for ∆αE1 ≈ −0.09∆αQM . b) For atomic coherence and quantum
information uses, operating with the minimal lattice light shift can be ideal. For∆αE1 ≈ −0.18∆αQM the lattice light shift goes to zero at the operational depth.
Notice the increased slope compared to a). Both regimes are impacted by a mistakein ∆αQM equally. Both simulations include hyperpolarizability contributions at thelevel of 1.5 × 10−19 .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

xxvi
7.3

∆αQM lattice light shift evaluation. To gain greater sensitivity to ∆αQM , varying
both high and low trap depths enables nearly constant ∆αE1 lattice light shift contributions with varying ∆αQM contributions. We explore this by simulating data for∆αE1 = −∆αQM . For a high depth of 40 Er we compare to a low depth of 4 Er fora ∆u = 36 Er . As the high depth is increased, the low depth is also increased nearly
equally (maintaining ∆u), allowing the ∆αE1 contribution to remain nearly constantin each differential evaluation while the ∆αQM contribution varies, providing a degree of organizational between the two. For the shown data, with uncertainties of
1 × 10−18 , a modest uncertainty of δ∆αQM = ∆αQM /2 is achieved.

. . . . . . . . . 142

Chapter 1

An Introduction to the Wonderful World Of Clocks“The recent discoveries due to the Kinetic theory of gases and to Spectrum analysis(especially when it is applied to the light of the heavenly bodies) indicate to us
natural standard pieces of matter such as atoms of hydrogen, or sodium, readymade in infinite numbers, all absolutely alike in every physical property. The timeof vibration of a sodium particle corresponding to any one of its modes of vibration,is known to be absolutely independent of its position in the universe, and it willprobably remain the same so long as the particle itself exists. The wavelength
for that particular ray, i.e. the space through which light is propagated in vacuo
during the time of one complete vibration of this period, gives a perfectly invariable
unit of length...”(Lord Kelvin [136])
Central to our modern world is an international definition of units - Le Systéme internationald’unités (SI). Consider your daily interactions with units that must be agreed upon for your life toproceed smoothly. You wake up to an alarm on your phone, based on an international timescale(the second). You buy food based on volume (the meter) or mass (the kilogram). Your electricity
bill is tied to amperes, the speed limit to meters per second, the brightness of your phone to lumens,
and your health to pascals. Like it or not, units define our lives!
The International Bureau of Weights and Measures (BIPM) is responsible for directly maintaining and distributing the SI units. The base units of the SI are shown in Figure 1.1 - all otherunits are derived from them. As of 2019, each base unit is now a defined quantity based on quantum
phenomena - our job as metrologists is to then realize these standards with the utmost precision.
Importantly, the more precisely we establish the base units the more precisely we realize the derived
units.
To illustrate this consider the SI unit of time, the second. It is defined by 9,192,631,770

2Figure 1.1: The SI Base units and defining constants. The second is defined by taking the fixed
frequency for the cesium hyperfine transition ∆ν, the meter by defining the speed of light c, thekilogram by defining Planck’s constant h, and so forth. All other units are derived from these sevenbase units.

oscillations of the ground-state hyperfine transition in cesium 133, an idea hearkening back to the
opening quote. The fractional uncertainty with which modern clocks define the second has reachedthe low 10−16 level, the most accurate realization of any unit in the SI system. The meter, defined as1
the length traveled by light in vacuum during 299792458
of a second, relies on our ability to measure

the second. Fortunately, the uncertainty in the second is several orders of magnitude smaller thanthe uncertainty of our ability to determine the refractive index of the medium in which the meter isrealized. This illuminates a key motivation for continually improving clocks - we want the accuracyof all units dependent on the second not to be limited by our best realizations. For now this remains
true, but improvement of the fractional frequency uncertainty of the world’s best cesium clocks has
stalled. This suggests we should reconsider the definition of the second, specifically the atomic
species and transitions.
So, what is an atomic clock in the simplest sense? Consider a standard clock. An oscillator(pendulum) ticks, the ticks are counted, and the number of ticks readout. An atomic clock canconceptually be broken down into similar pieces. Rather than a mechanical oscillator, our oscillator

3
is a laser, where the frequency of the electric field provides the ticks. Of course a laser, being a
man-made device, is prone to drift so we require a reference. By shining the laser on carefully
controlled atoms, we may steer the laser to constantly be at the correct frequency to cause theatoms to vibrate. Should those atoms be cesium (and the laser actually a microwave device) wewould be realizing the definition of the second! Finally, we need some way to take our corrected
laser and distribute its now highly accurate frequency across the electromagnetic spectrum, frommicrowave to optical. This is the role of the frequency comb.
Atomic clocks based on optical transitions are the natural candidates for the redefinition of
the second. A key figure of merit for a clock is accuracy (Figure 1.2). We parameterize this by aclock’s fractional frequency inaccuracy as given by ∆νν where ν is the clock’s frequency (numberof ticks per second) and ∆ν the absolute uncertainty of the clock’s frequency. ∆ν is usuallycomprised of two contributions: the statistical uncertainty of the clock (how well we know thefrequency after measuring for a period of time) and systematic uncertainty (our limited knowledgeof perturbations to the clock’s frequency). From the definition of fractional frequency inaccuracy
we may understand the advantage of optical clocks in comparison to traditional microwave clocks.
For the same absolute frequency inaccuracy, the fractional frequency inaccuracy is decreased bynearly 5 orders of magnitude by moving the clock frequency from the microwave to the opticaldomain!
Our discussion of ∆ν highlights the importance of the second key figure of merit for a clock- precision (Figure 1.2). In atomic clocks the fundamental limit of precision is given by the socalled quantum projection noise (QPN)[68] - the inherent quantum noise arising from projecting
atoms in a superposition of ground and excited clock states into either ground or excited state uponmeasurement. The fractional frequency instability of QPN scales as

S
σQP N ∝
ν0 Tc

sTc + Td,
N m(Tc + Td )(1.1)

where S is a prefactor dependent on contrast and spectroscopy type (Rabi or Ramsey), Tc the clock4

Frequency

a)

b)

0

0

c)

0

d)

0

Time

Figure 1.2: Accuracy and precision. The top row shows marks on a target, the bottom row frequency
versus time of an oscillator that should be at frequency ν0 . Column a) shows high accuracy andhigh precision, b) high accuracy and low precision, c) low accuracy and high precision, and d) low
accuracy and low precision.

interrogation time, Td the time spent preparing the atoms for interrogation (atom preparation),ν0 the clock frequency, N the number of atoms, and m the number of measurements of the clock
frequency. This emphasizes the importance of low instability for clocks - our knowlege of the clock√
frequency only improves as 1/ m. Consider the most accurate clock to date - the Al+ clock withfractional frequency inaccuracy of 9.4×10−19 [24]. With the Al+ instability slightly worse than1 × 10−15 it would take more than 106 seconds to reach their reported accuracy! Clearly, for therealistic ability to evaluate these state-of-the-art clocks precision is vital.
Optical lattice clocks (OLCs) are the only platform with sufficient precision to provide a
useful frequency reference at these levels of accuracy. Just like ion clocks, OLCs enjoy the benefitsof long lived clock states enabling extended interrogation times. In contrast with ion clocks, OLCs
can interrogate more than 10,000 atoms simultaneously, reducing the fundamental QPN limit of
the clocks by a factor of more than 100. The fundamental motivation of the work in this thesis is
the quest to push the 1D Sr OLC to uncertainties at the 10−19 level while simultaneously operating

5
in a regime of high precision. Equation 1.1 tells us that the tools we have at our disposal areextending laser interrogation times and increasing the number of atoms. In the chapters to comewe will discuss the research carried out during my graduate career that has culminated in a levelof precision capable of evaluating a clocks fractional frequency to 19 digits in 10 s. Critically,
this result was obtained in an operational regime fundamentally compatible with accuracy wellinto the nineteenth decade, successfully wedding next generation accuracy and precision into one
experimental platform.

1.1

Why Do We Want Better Clocks?
Let us briefly address the question of why we want better clocks. First and foremost, wewant to redefine the second to an optical transition. Doing so will enable orders of magnitudeimprovement in the accuracy of our realization of the second, ensuring that the second plays norole in the uncertainty of derived units. Additionally, optical atomic clocks have orders of magnitude
lower fractional frequency instability, meaning a given level of precision can be attained in ordersof magnitude less time! Comparison to an ideal frequency standard should be accurate and rapid,
with statistics not limited by the standard. OLCs excel in both metrics.
Improved clocks will allow us to probe the fundamental assumption of the SI units - theimmutable nature of fundamental constants. Many beyond Standard Model theories do not guarantee the stability of our nicely defined units. By making better clocks, we can compare them andtest these assumptions, ruling out new theories or perhaps some day discovering new physics [123].
Such research requires different atomic species, or at the least different clock transitions within
one atom. Comparing two clocks drifting in the same way would reveal nothing, but comparing
transitions with differential sensitivity to fundamental constants can open the door to testing avariety of new physics, such as changes in the fine structure constant.
An exciting, evolving frontier of clocks is space based applications. Centimeter scale accuracy
in an updated global navigation satellite system wouldn’t just improve navigation, but would allow
accurate mapping of the Earth’s geoid [99]. Deep space navigation is currently hampered by the6ability to send timing signals to distant space vehicles - integration of on-board, highly accurate
atomic clocks would remove this limitation. Improvement in clock precision may even some day
allow for space-based clocks to detect gravitational waves [77].
Beyond timescales and tests of fundamental constants, improved clocks will have valuableroles in terrestrial applications. Comparison of such clocks on Earth will provide a critical tool for
relativistic geodesy at the sub-centimeter level [47]. Advanced communications networks, entangled
networks of clocks [78], improved very-long-baseline interferometry - the list goes on!

1.2

1D 87 Sr Optical Lattice Clocks
This thesis discusses two generations of the Sr1 project at JILA, both using the 1D 87 Sr opticallattice clock (OLC) architecture. We will quickly review the operation of a 1D Sr OLC, referring
the extremely curious reader to detailed theses for further in depth discussion [21, 88, 146, 95, 25],
with experimental details of the new version discussed in Chapter 4.
Sr is in many ways the Goldilocks of the Alkaline-Earth elements. With atomic number 38, theoutermost valence electrons completely fill the 5s2 configuration, giving rise to the energy structure
shown in Figure 1.3. The two valence electrons may align their spins such that S=0 (singlet state
1 S ) or S=1 (triplet states 3 P ).
0
JThis has immediate relevance for creating a frequency standard

since dipole selection rules in LS coupling [39, 40, 63, 4] tell us that, for an E1 excitation, ∆S = 0.
This means in pure LS coupling transitions from 1 S0 ⇔3 PJ are forbidden.
Let’s turn our attention to the clock transition at 698 nm. This transition is also forbidden by the angular momentum triangle rule (J=06⇔J’=0), a significantly more rigorous rule than∆S = 0 (check the linewidth of the 3 P1 versus 3 P0 transitions in bosonic strontium). So, our clocktransition is doubly-forbidden! How do we drive the clock transition? LS-coupling is an approximation, neglecting spin-orbit coupling (SOC). SOC weakly perturbs the 3 P0 state, mixing in small
contributions of 1 P1 and 3 P1 [21]. Even with SOC mixing we can’t reliably probe the ultra-narrowbosonic clock linewidth, so we must either 1) apply a large magnetic field to further induce theclock transition or 2) work with the fermionic isotope. In fermionic strontium, 87 Sr (I=9/2), the7Figure 1.3: Strontium energy levels. Even (odd) parity levels shown in black (red). The strong
dipole transition (461 nm) serves for initial cooling and trapping of hot strontium atoms. A weak
decay channel from 1 P1 to 1 D2 (purple line) leads to population in 3 P1 and 3 P2 (green lines).
While 3 P1 decays to the ground state, 3 P2 is metastable and leads to significant loss. The additionof repumps at 679 and 707 nm ensures cycling of metastable states back to the ground state,enhancing MOT operation and enabling excited state readout. Narrow-line MOT operation at 689
nm is cycling and enables efficient preparation of cold, dense samples of Sr.

hyperfine interaction further perturbs LS coupling, creating an extremely narrow but usable clock
linewidth of 1.3 mHz which we use to form our frequency reference [21].
So, we have a doubly forbidden clock transition between two J=0 states, ensuring SU(N)symmetry [12, 25] and suppressing vector related clock shifts. How do we leverage our clocktransition as a frequency reference? First we must prepare our atomic sample [21, 88], all within
ultra-high vacuum. A small piece of strontium is heated to around 400◦ C, causing the sample toeffuse strontium atoms. The atomic flux is collimated, cooled, and loaded into a magneto-opticaltrap (MOT) first operating on the 461 nm transition and then a second MOT stage on the 689
nm transition (Figure 1.3). After the second MOT stage the ∼ µK atoms are loaded in an optical

8lattice where further cooling and nuclear spin-polarization may be performed. Atoms are tightly
confined to spatial dimensions much less than the wavelength of the lattice light, enabling Dopplerfree clock spectroscopy (the so called Lamb-Dicke regime [21, 88]). Critically, the lattice wavelengthis such that the ground and excited clock state polarizabilities are identical, preventing deleterious
frequency shifts to the clock transition.
Clock spectroscopy proceeds as in Figure 1.4 [88, 111, 112]. Every experiment corresponds
to loading atoms, preparing a single nuclear spin, probing the atoms with clock light, and finallyreading out the excitation fraction using the 461 nm transition. This is performed on the left andright side of the Rabi lineshape for each nuclear spin transition. Excitation fractions are converted
to frequency corrections, which provide feedback to the clock laser. The average frequency of the twonuclear spin transitions rejects vector shifts and is used for tracking the atomic frequency, while thedifferential frequency between opposite nuclear spin states tracks vector frequency perturbations.

1.3

Thesis Outline
This dissertation discusses experiments probing the frontiers of accuracy, precision, and ap-

plications of atomic clocks. All work was performed as part of the Sr1 group at JILA (the 1D
strontium optical lattice clock experiment) in close collaboration with the full Ye Lab’s Sr team,consisting of Sr2, Sr3, and Stable Lasers.
In Chapter 2 we present the current (2022) record accuracy evaluation for a strontium clock
and third most accurate clock demonstrated to date, documenting improvements from the previous
Sr1 fractional frequency inaccuracy of 5.3×10−17 [17] and Sr2 (1D) inaccuracy of 2.1 × 10−18 [109]to a new inaccuracy of 2.0 × 10−18 in 2019. We notably include an updated accounting of latticelight shifts to account for corrections beyond the dipole approximation.
Chapter 3 gives a brief overview of the studies enabled by this accuracy evaluation. Ouradvances in accuracy were motivated by our participation in an extensive clock comparison between
the Sr1 system and Al+ and Yb clocks at NIST. With Sr1’s improved accuracy and Sr2’s precisionexpertise we were able to demonstrate record low instability inter-clock comparisons between the

9
Laser

1/2

+

3

1

P0

-9/2

S 0 -9/2

-7/2

-7/2

1/2

-5/2

-5/2

-3/2

-3/2

Magnetic Field
-1/2

-1/2

Laser

+

1/2

1/2

3/2

3/2

5/2

5/2

7/2

7/2

9/2

9/2

Magnetic Field

Figure 1.4: Clock spectroscopy of nuclear spin states [112]. Traditional stretched state spectroscopy
is shown in blue, where the |g, mF = ±9/2i ↔ |e, mF = ±9/2i transition is probed. Recent clock
operation has become limited by magnetic field noise, necessitating operation with the least magnetically sensitive transition in strontium, |g, mF = ±5/2i ↔ |e, mF = ±3/2i. In each sequence(assuming a perfect atomic reference) 1/2 of the sum of the nuclear spin state transition frequencies corresponds to laser noise while 1/2 of the difference corresponds to magnetic field noise.
During spectroscopy the frequency of each nuclear spin transition is found by probing both the left
and right of the lineshape, providing a frequency correction based on excitation fraction differences
and the corresponding lineshape.

Sr1 and Sr2 clocks. Sr1 was further used to help characterize the labs newest clock laser, Si3,
enabling demonstrations of an all-optical timescale and searches for dark matter.
In Chapter 4 we switch gears, discussing the newest iteration of the Sr1 experiment. By

10utilizing a large waist in-vacuum build-up cavity for the optical lattice the atomic density is reducedby nearly an order of magnitude, supporting advances in precision owing to large atom-number
operation. The ability to initially load and cool the atomic sample in a deep trap enables a new
regime of temperatures for the 1D lattice clock architecture, building on the lessons learned fromthe 3D experiment [33].
Chapter 5 introduces the concept of the Wannier-Stark OLC [81]. Operation at depths near
10 lattice recoil energies results in the delocalization of atomic wavefunctions, enabling engineeringof atomic interactions. We demonstrate the ability to operate nearly free of atomic-interaction
induced frequency shifts by engineering on- and off-site atomic interactions to cancel, eliminating
the largest impediment to precision in the 1D clock platform.
Chapter 6 discusses advancements enabled by combining atomic coherence as demonstrated
in the 3D and tweezer platforms [33, 154] with the proven accuracy of the 1D platform [98, 19].
In-situ imaging of the atomic cloud [93] with differential systematic characterization allows us
to resolve the frequency gradient arising from the gravitational redshift across a millimeter scaleatomic sample. This observation is enabled by record precision, demonstrated by comparing two
regions of the cloud for nearly 90 hours allowing frequency resolution of 21 digits.
Finally, Chapter 7 revisits the fundamental limiting systematic of Chapter 2 - the lattice lightshift. We discuss the origin of higher order terms in the light shift and outline a possible plan of
attach for reducing their uncertainties.

Chapter 2

Record Accuracy For a Strontium Clock

The research in this chapter is reported in the publication: Bothwell et al., ‘JILA Sr1 opticallattice clock with uncertainty of 2.0×10−18 ,’ Metrologia (2019) [19].
Atomic clocks based on optical transitions have led to rapid advances in clock accuracy. In
2010 the NIST Al+ clock was the first system to reach a fractional frequency inaccuracy below
1 × 10−17 , evaluating the system to 8.6 × 10−18 [36]. The 1D version of Sr2 became the first OLC toreach this milestone in 2014, achieving an inaccuracy of 6.4 × 10−18 [17]. A year later, in 2015, theyfurther reduced their inaccuracy to 2.1 × 10−18 . Concurrently, Riken performed a comparison of
two cyrogenic 1D Sr OLCs with inaccuracies of 7.2 × 10−18 , solidifying Sr as the leading platform.
In 2018, the Yb OLC team at NIST similarly compared two clocks with inaccuracies of 1.4 × 10−18
and differential inaccuracy below 1 × 10−18 [98]. PTB’s Yb ion clocks have never been far behind,evaluating a clock inaccuracy of 3.0 × 10−18 in 2016 [66] and further improving to 2.7 × 10−19 in
2019. Finally, NIST’s Al+ system reported the current record inaccuracy of 9.4 × 10−19 in 2019[24].
While the leading clock platform for accuracy has been traded back and forth between ion
and OLC, OLCs are the superior platform for precision. The Yb OLC team at NIST demonstrated√
fractional frequency instability below 1 × 10−16 / τ (τ being averaging time in seconds) in 2016.
A year later, the 3D lattice version of Sr2 compared two regions of their atomic cloud with an√
instability of 3.1 × 10−17 / τ [33], a record held until the work presented in Chapter 6. A record√
low instability comparison between Sr1 and Sr2 of 3.5 × 10−17 / τ is reported in Chapter 3. These12advances in precision are critical for the ability to faithfully evaluate clock uncertainties of <1 × 10−18 .

2.1

Introduction
The unprecedented levels of stability and accuracy of modern clocks has enabled their use

in a variety of applications ranging from the proposed redefinition of the SI second [86, 45, 13],
to searches for variations of fundamental constants [43, 144, 122, 74], and increased capabilities
for positioning, navigation, and timing applications [44]. However, in order to fully realize thepotential for these applications, controlling the temporal drift of systematic offsets poses a potential
barrier to control of the system at and below the 10−18 level. While it is possible to evaluate thesystematic uncertainty of optical clocks at this level, active control of all systematics is now urgently
needed [53, 65] to realize a robust optical frequency reference that maintains this level of uncertainty
over long time periods. A systematic shift can be measured to an extremely high precision but if
it varies significantly over time the clock frequency will appear to drift unless real-time frequency
corrections are applied. An alternative approach is to instead control the atom environment to
the level where the temporal variations of systematic effects are well within the clock’s uncertainty
budget and frequency post-corrections need not be applied. This is the guiding principle taken
in upgrading the JILA Sr1 optical lattice clock. Here, we demonstrate clock performance wherefluctuations in systematic offsets are routinely bounded below 4×10−19 for times up to 104 seconds.
In implementing these upgrades the systematic uncertainty of the Sr1 clock, last evaluatedat 5.3×10−17 in 2013 [17], has been improved by more than a factor of 20 to 2.0×10−18 . Thissurpasses the uncertainty record for a strontium optical lattice clock and places this clock among
the most accurate clocks in the world [109, 98, 24]. The active control of clock operating conditions,
including the thermal environment, allows the clock to run in a robust manner without needingreal-time frequency corrections to average to the level of its accuracy. In this paper, we detail thecontrol of each major systematic effect in JILA Sr1.

13

2.2

Experimental Methods
The experimental sequence begins with a collimated atomic beam of Sr generated from aeffusive oven. Atoms of 87 Sr in the beam undergo 2D transverse cooling and Zeeman slowing using
the broadband 461 nm 1 S0 - 1 P1 transition (Γ = 2π × 32 MHz). The slowed atoms are then loaded
into a magneto-optical trap (MOT) operating on the same transition, and cooled to mK-level
temperatures. After 100 ms of loading, the atoms are transferred to a second narrow-linewidth(Γ = 2π × 7.5 kHz) MOT operating on the dipole-forbidden 1 S0 - 3 P1 line at 689 nm, cooling theatoms to a temperature of 3 µK. The sample is then transferred to a one-dimensional, 813 nm reddetuned optical lattice, which exhibits minimal differential polarizability for the two clock states,
and is optically pumped into one of the |F = 29 , mF = ± 92 i stretched states of the 1 S0 manifold.
The lattice is initially loaded at a depth of 180 Er to maximize the number of captured atoms and isthen adiabatically ramped down to a nominal operating depth of 45 Er (where Er is the recoil energy
associated with the 813 nm trapping light) to minimize the effects of systematic shifts associatedwith the trapping light.
With state preparation complete, clock spectroscopy is performed on the ∼ 1 mHz naturallinewidth, 698 nm 1 S0 - 3 P0 transition using a narrow-linewidth ultrastable laser. A bias magnetic
field of 57 µT (570 mG) splits the | 92 , ± 92 i states by 556 Hz, and the clock transition is interrogatedwith a 600 ms long π-pulse with polarization collinear with the quantization axis. After interrogation with the clock laser, the resulting excitation fraction is measured by first detecting the groundstate population via fluorescence on the 461 nm transition, and then detecting the excited state
population by repumping atoms back to the ground state and again collecting fluorescence on the
461 nm transition. This procedure is performed for detunings from resonance of ± Γ2 , one-half of
the Rabi linewidth, and the resulting difference in excitation fraction specifies the detuning of theclock laser from resonance. By alternately probing the two stretched states, we reject fluctuationsin the first-order Zeeman shift arising from magnetic field noise, further details of which are offeredin [112].

14

Figure 2.1: Schematic view of the Sr1 clock. Ultrastable laser light is generated at 1542 nm byreferencing a diode laser to a crystalline silicon optical cavity operating at 124 K (red, dotted line).
The stability of this laser is then transferred via an Er:fiber comb to an external-cavity diode laser
pre-stabilized by a 40 cm ULE cavity operating at 698 nm (blue, dotted line). An acousto-optic
modulator (AOM 1) is then used to steer the cavity light into resonance with the Sr clock transition.
The excitation fraction after probing the clock transition is detected by collecting fluorescence fromboth ground and excited state atoms. A frequency step applied to AOM 1 produces an error signal
for locking by alternately probing both sides of the | ± 9/2i stretched state transitions. Frequency
corrections to the average of the | ± 9/2i frequencies are applied to AOM 2 such that the cavitystabilized light is steered onto the transition frequency of the Sr atom. In addition, frequency
corrections to the difference of the | ± 9/2i frequencies are applied to the AOM 1 frequency. An inplane magnetic field, B, providing a quantization axis for the atoms, is aligned to be collinear withboth the 1D optical lattice polarization, 813 , and the clock laser polarization, 698 . Out-of-vacuum
quadrant ring electrodes generate a DC electric field to cancel the ambient field at the position of
the atoms. Finally, a phase lock of the 813 nm trapping laser to the Er:fiber comb stabilizes thefrequency of the trapping light (green, dotted line). The trapping light is delivered to the atoms
through a high power optical fiber and is intensity stabilized by actuating the RF power on AOM
3.

15
As depicted in Fig. 2.1, the ultrastable laser used to probe the clock transition consists ofa pre-stabilized 698 nm laser locked to a commercial Er:fiber comb which is phase-stabilized to amaster 124 K silicon cavity. A frequency step applied to an acousto-optic modulator (AOM 1 inFig. 2.1) alternately probes either side of the | 29 , ± 29 i transitions and generates an error signal. A
digital servo filter (PI2 D) is applied to this error signal to apply frequency feedback to AOM 2 inFig. 2.1. This loop configuration has the effect of stabilizing both the light after the AOM 2 andthe frequency of each comb tooth to the spectroscopic precision and accuracy of clock operation.
Under typical operating conditions (i.e. sample preparation time of 570 ms, interrogation time of√
600 ms, and an atom number of N = 1000) the clock achieves a stability of 4.8 × 10−17 / τ [112].

2.3

Systematic Evaluation
Accurate determination of the unperturbed 87 Sr clock transition frequency requires charac-terization of all systematic effects which produce energy shifts between the 1 S0 ground state andthe 3 P0 metastable excited state. These effects range from the interaction of a single atom withan external field to two-particle collisions and many-body effects. For each shift, the perturbing
effect is either directly measured - as in the case of the thermal electric field produced by room
temperature radiation - or the effect is inferred by modulation of the applied field and the corresponding measurement of a frequency shift - as in the case of the lattice light shift. Subsequently,
the systematic shifts relative to the unperturbed atomic transition frequency can be extrapolated
to daily operating conditions using well-characterized theoretical models of each effect.
For shifts evaluated using the lock-in technique, the record-low clock instability demonstrated
in Ref. [112] allows the rapid determination of these systematic offsets - without the need to apply
a large lever arm - at the 1 × 10−18 level in less than one hour. However, the capability to
rapidly evaluate shifts does not preclude the need for applying real-time frequency corrections tocompensate for non-stationary systematics. A systematic offset can have temporal variations if its
source (e.g. atom number) or calibration (e.g. fluorescence to atom number conversion) fluctuates
throughout the day. In the following subsections, we show how, by active stabilization of the thermal

16
environment and careful control of the operating parameters, drifting systematic offsets can remain
below 4 × 10−19 fractional instability over six hours of operation (Figs. 2.2a, b). Consequently, afrequency comparison of the Sr1 clock against a stable reference (the JILA 3D clock [33]), Fig. 2.2c,
can average well into the 10−19 decade without systematic effects impacting the clock stability.

b)
10 18

2.246
2.247

10 19

2.248
2.249 0c)
Quad Zeeman
Density
BBR
Total

SrI - SrII Fractional
Frequency Instability

2.245

Fractional Systematic
ShiftsNet Systematic
Shifts / Hz

a)

1

2

3

4

Time / Hr

5

6

10 20

101

102

Time / s

103

104

10 16
10 17
10 18
10 19 101

102

103

Time / s104

Figure 2.2: Systematic shifts. (a) Plot of the time record of the systematic shifts. Changes
in atom number, ambient temperature, or magnetic field all result in corrections to the clock
frequency, and their total magnitude is shown over a six hour data campaign. The clock achieves
98.9% uptime over the course of this single comparison day and slight gaps in the data indicate brief
periods where the laser is not locked to the atoms. (b) The same data is plotted as a fractional
instability normalized to the Sr clock frequency. The individual contributions of density shift(blue), BBR (red) and second order Zeeman shift (yellow) are shown as the dashed curves. For
operation times up to 104 seconds, fluctuations in systematic offsets are bounded below 4 × 10−19 .
(c) Nonsynchronous comparison with the JILA 3D optical lattice clock demonstrates that the beat
between the two clocks averages below the quoted total systematic uncertainty. All error bars arederived from a white noise model and the black line is a white noise τ −1/2 fit to the single clock
instability.

2.3.1

Blackbody Radiation

The frequency shift induced by blackbody radiation (BBR) is the largest systematic shift anda dominant source of uncertainty in state-of-the-art optical lattice clocks. Aside from cryogenic
systems [139], the BBR-induced clock shift for strontium is approximately 5 × 10−15 at room
temperature. The BBR shift of a thermal electric field distribution characterized by a temperature,
T , may be expressed as:

∆νBBR (T ) = νstat

T
T0

4


+ νdyn

T
T0

6



T
+O
T0

8 
(2.1)

17

Air
ZW
VF
MC

ZS

CF

Oven

Figure 2.3: Active Temperature Control. (a) The optical table is divided into independently
controlled sections (grey and white shaded regions) and isolated from the room temperature bya laser curtain. Around the vacuum chamber is an additional black box. Temperature sensors(TS) monitor each table half, allowing feedback (dotted lines) for controlling the flow of coolingwater through a water/air exchanger above each table half. Commercial HEPA filters pull room
air into the box. (b) The Sr1 vacuum chamber is contained in a black box to protect it from straylight and ensure thermal homogeneity. Heat sources that are temperature controlled are shown in
red: Zeeman window (ZW), Zeeman slower (ZS), and MOT coils (MC). Temperature control of
vacuum viewports is shown in blue: water temperature controlled copper tubes around the top andbottom 6” CF flanges (CF) and thermoelectric cooler (TEC) controlled 2.75” CF viewport faces(VF). The oven (not shown in the figure) is located past the Zeeman slower. The nozzle (6 mm indiameter and 575◦ C) is the only heated oven component visible to the atoms. (c) The temperature
at the location of the atomic sample is directly measured using a thin-film platinum resistancethermometer (TFPRT) sensor and compared to a model derived from ray-tracing and temperature
sensors mounted on the chamber, verifying the stability afforded by our extensive thermal control.
The measurements shown are binned into 10 minute intervals. (d) At several hours averaging times
the TFPRT shows sub-mK level stability. The difference between the TFPRT and model shows
similar temperature stability, providing verification that long-term fluctuations in the temperatureexperienced by the atoms are accurately captured by the ray-tracing model.

18
where νstat = −2.13023(6) Hz [100], νdyn = −148.7(7) mHz [109], and T0 = 300 K.
Characterizing the room temperature BBR shift at the low 10−18 level requires absoluteknowledge of the thermal environment of the atoms to within 50 mK. To date this challenging
technical requirement has been met using two approaches. In the JILA Sr2 clock, NIST-calibrated
in-vacuum sensors were used to directly measure the thermal environment of the atoms to an
uncertainty of 5 mK [109]. In the Yb OLC at NIST an in-vacuum radiation shield was characterized using precision thermometry and thermal modeling [6]. Here we take components of both
approaches by actively stabilizing and monitoring the thermal enclosure of the atoms while alsoutilizing in-vacuum thermometry.
Our primary objective is to create a frequency reference with low 10−18 level systematic
uncertainty that does not require point-by-point corrections to attain a similar instability. A
homogeneous thermal environment is essential for accurate and precise characterization of the surroundings with a finite array of temperature sensors and for avoiding complications which arise
when evaluating the dynamic BBR correction of a non-thermal spectrum of the electric field driven
by temperature gradients [109]. Furthermore, actively maintaining a constant operating temperature leads to a more stable and reproducible BBR shift that is crucial for post-correction-free
operation of the clock at the low 10−18 level.
For our evaluation of the BBR shift in Sr1, we utilized both a Monte-Carlo ray-tracing based
thermal model and an in-vacuum thermal probe to determine the temperature at the atoms in twocomplementary ways. One method involves actively stabilizing and carefully measuring the thermal
environment around the vacuum chamber using 16 servo loops and over 30 sensors. The temperature
at the atoms can then be computed from this array of temperature monitors using a detailed thermal
model based on ray tracing as described in Appendix 2.5.1. We then verify the first approach byinstalling a PT100 thermal sensor in-vacuum to directly measure the BBR environment. Excellent
agreement between the thermal model and the in-vacuum temperature measurement demonstrates
that our model accurately predicts changes in the temperature experienced by the atoms fromour external vacuum sensors readings at sub-mK levels. Our technical efforts focus on (1) active

19
temperature stabilization of our system, (2) creating an array of accurate temperature sensors tocharacterize the thermal boundary conditions, and (3) calibration of an in-vacuum temperatureprobe. To this end, we improve the overall thermal homogeneity of JILA Sr1 by more than a factorof 10, limiting thermal gradients across the chamber to less than 100 mK.
We construct a model of the temperature Ti at the location of the atoms, modeled by asmall spherical surface i, by mapping the measured temperatures on the surrounding surfaces ofthe vacuum chamber to Ti by:

Ti4 =

X

Fi→j Tj4 ,

(2.2)

j

where the index j enumerates the different surfaces of the vacuum chamber surrounding the atoms
and Fi→j is the exchange factor defined as the fraction of total energy emitted by surface i that isabsorbed by surface j directly or by reflection [103]. A description of our exchange factors is given
in Appendix 2.5.1. In the limit where the chamber temperature is nearly uniform, the uncertainty
in Ti can be expressed as

δTi ≈

X

Fi→j δTj .

(2.3)

j

This limit is important for understanding how to prioritize temperature control of the experimental
apparatus: surfaces with large exchange factors dominate the thermal environment of the atoms
and are subsequently the most important to control. A useful heuristic for identifying the largest
exchange factors, and therefore the most critical surfaces for thermal control, is to find the mosthighly emissive surfaces in the vacuum system subtending the largest solid angles at the locationof the atoms. Often, and in our case, these surfaces are large vacuum windows.
To achieve the required temperature uniformity and stability, we began by stabilizing the
air temperature around the experimental apparatus (Fig. 2.3a). The optical table is partitioned
to isolate the vacuum chamber from the auxiliary laser systems and the entire table is surroundedby curtains to isolate it from the room air temperature fluctuations. Around the vacuum chamber

20
a black box is installed to provide additional thermal homogeneity and to block stray laser light.
The box is made of readily available black hardboard, black acrylic, and aluminum framing. Bothsides of the table are actively temperature stabilized by controlling their inlet air temperature. Ourchamber has several heat sources including the MOT coils, Zeeman slower, heated Zeeman window,
and oven (Fig. 2.3b). Each heat source is accompanied with water cooling such that the relative
heating/cooling rate allows us to actively control the steady-state temperature of each.
The equilibrium temperature at the atoms is determined from Eqns. 2.2 and 2.3, which are
dominated by the contributions from the fused silica viewports due to their high emissivity and largesolid angles with respect to the atomic sample. Therefore, stabilization and accurate measurementof the viewport temperatures is of paramount importance. To stabilize their temperature, we fix
the thermal boundary conditions of all viewports (Fig. 2.3b). The stainless steel flanges of the6” viewports are wrapped in copper tubing carrying temperature controlled water to stabilize thetemperature of the steel around the glass. A copper ring with two embedded temperature sensors
and lined with thermally conductive silicon matting is pressed against the outer edge of the glass(leaving a small aperture in the middle for optical beams to pass through) to achieve thermal
homogeneity across the surface and allow direct monitoring of the glass temperature. All 2.75”
viewports are controlled using custom, TEC temperature controlled aluminum attachments backed
by water cooling.
This unique approach to active temperature control creates a stable thermal environment
where temperature-controlled viewports and flanges act as thermal reservoirs with boundary conditions set by the servo setpoints. All setpoints for temperature control are set to 22◦ C. To characterize the homogeneity of this thermal environment, 30 witness sensors with accuracy of 50 mK wereplaced around and on the chamber. Each viewport has an independent witness sensor, with thelarger 6” viewports each having two. With the data from this sensor array and a detailed 3D modelof the vacuum chamber as inputs to our thermal model, one can now compute the temperature
seen by the atoms using Eqn. 2.2.
Upon completion of clock operation, an in-vacuum thermal probe is inserted into the vacuum

21chamber in order to directly measure the temperature at the position of the atoms. Placementof the probe at the position of the atoms is ensured by aligning the sensor to overlap with boththe clock and the Zeeman slowing laser beams and the temperature is measured under identicalrun conditions to clock operation. This measurement allows for both the direct verification of thethermal model connecting temperatures measured on the vacuum chamber to the temperature seenby the atoms as well as the reduction of uncertainties associated with the emissivity of different
vacuum components. The in-vacuum thermal probe consists of a 60 cm evacuated glass tube witha calibrated (±1.4 mK) PT100 sensor epoxied at the end. The sensor design and calibration details
can be found in Appendix 2.5.2. After installing the in-vacuum sensor, a small static offset of
19.1 mK was discovered between the temperature measured by the zero power resistance of theprobe and the temperature derived from the thermal model using parameters given in literature,
shown in Fig. 2.3c. This offset is attributed to a limited knowledge of material emissivities aswell as calibration uncertainties of the thermistors (50 mK) of the majority of the sensor array.
The in-vacuum probe measurement allows for the characterization of any static offset between
the thermal model and the directly measured temperatures. Active stabilization of the thermal
environment allows this offset to remain stable over clock operation where the thermal model
successfully captures all fluctuations in the temperature to better than 1 mK over >10,000 s ofaveraging as indicated by the Allan deviation of the difference between the model and the measured
temperature shown in Fig. 2.3d. All parameters of the thermal model and a more detailed discussionof the construction of the model can be found in Appendix 2.5.1.
The total temperature uncertainty quoted in Table 2.1 has contributions from the calibration
uncertainty of the in-vacuum sensor, immersion error, self heating, modification of the thermal
environment by the sensor (insertion error), and statistical error of the agreement between theprobe and thermal model. Immersion error in the Sr1 chamber was measured by changing thebase flange temperature and monitoring the in-vacuum sensor, giving a slope of (0.65 ± 0.62)
mK/K. During clock operation, this flange is controlled to 0.2 K, and so we assign the total
immersion error uncertainty to be the quadrature sum of the overall offset and the coefficient22
uncertainty, arriving at a 1.8 mK uncertainty. Insertion error is bounded to 1.5 mK from previous
work comparing the temperature measured at one thermistor to the temperature measured atanother as the position of the second is translated away from the chamber center [109]. The total
uncertainty in the temperature of blackbody radiation seen by the atoms is then evaluated to beδT = 2.9 mK corresponding to an uncertainty of 2.0×10−19 . Uncertainty in the combined static anddynamic BBR coefficients accounts for the atomic response contribution to our BBR uncertainty,
at 1.49 × 10−18 .
Table 2.1: Atomic Temperature Uncertainty
Shift
Sensor calibration
Self heating
Immersion error
Sensor - model
Insertion error
Total

2.3.2

Correction (mK)
0
-1.4
0
20.5
0
19.1

Uncertainty (mK)
1.4
0.3
1.8
1.0
1.5
2.9

Density Shift

The high-degree of stability demonstrated by Sr1 and more generally by optical lattice clocks
is due to the ability to interrogate thousands of atoms simultaneously and read-out the measured
clock frequency with high signal-to-noise at the limit set by quantum projection noise (QPN) [112].
However, the presence of multiple spin-polarized atoms per lattice site introduces systematic frequency shifts due to p-wave interactions. The different triplet collision channels between groundand excited atoms have been shown to have different scattering lengths which subsequently produce
density-dependent differential clock shifts. These effects have been studied and characterized in
87 Sr [95, 155].

During clock operation, the gas is spin-polarized by optical pumping into either the | 29 , 29 i or
the | 92 , − 29 i ground hyperfine state before interrogation in order to suppress frequency shifts due
to s-wave collisions. However, s-wave interactions can still occur if impurity atoms remain present
after the optical pumping process. To mitigate this effect - and suppress temporal variation of this23
effect due to fluctuations in the efficiency of optical pumping - we perform clock spectroscopy on
impurity nuclear spin states and optimize the optical pumping process to reduce the population inother nuclear spin states below the detection threshold of our system. In this regime, the dominantdensity-dependent frequency shift is a collective p-wave interaction between identical fermions.
√
To reduce the QPN-limited clock stability to the low-10−17 / τ level, we initially prepare
more than 1000 atoms in the optical lattice. For these atom numbers in the deepest optical lattice
depths ∼180 Er , on-site densities are high enough that sub-percent-level changes in the trap depth
or atom distribution can produce variations of the shift at the 10−18 level. The effect of thesefluctuations is suppressed to the 10−19 level and below by lowering the lattice depth to 45 Er
thereby reducing the peak atomic density during clock interrogation. Vertical orientation of the
1D lattice ensures suppression of tunneling due to the difference in gravitational potential energy
between lattice sites.
In addition, fluctuations in the atomic density and the resulting density shift can be driven bychanges in the laser-cooling process which transfers the atoms from the narrow-line red MOT into
the lattice. This loading process is influenced by the relative frequency, intensity, polarization, andalignment stability of the red MOT lasers. To control these processes, the lasers are stabilized to an
ultra-stable cavity to a linewidth of 1 Hz. Temperature stabilization of the experimental apparatus- as discussed in the previous section - serves the dual purpose of stabilizing the alignment of
the red MOT and enables robust, alignment-free operation of the clock over many consecutive
months. After implementing these measures, we find both the sample temperature and the spatial
distribution of atoms in the lattice are insensitive to daily linewidth-level frequency drifts andalignment drifts. Linear scaling of the temperature with the loading lattice depth indicates that
final temperatures in the lattice are dominated by the large AC Stark shift of the 813 nm latticeon the 3 P1 cooling transition. Since the lattice depth is actively stabilized, the atomic distribution
within the lattice remains static over time leading to a density shift with stability at the 10−19level.
To measure and confirm this stability, the density-dependent shift is determined by modu-

24
lating the atom number by varying the loading time of the blue MOT which then varies the atomic
density in the lattice. Care is taken to ensure that, at the highest atom numbers, the spectroscopiclineshape remains Fourier-limited with high peak excitation fraction unblocked by a many-bodyexcitation gap [95]. A model of the density shift which is linearly dependent on atom number isapplied to extrapolate the measured shift to the atom number in the lattice on any particular experimental realization. To demonstrate the stability of the shift, repeated density shift evaluations
were performed at our nominal clock lattice depth of 45 Er over the course of several weeks, witha weighted standard error of 3.9 × 10−19 and χ2 = 1.07 for all these measurements (Fig. 2.4).
Finally, by verifying that the radial and axial trap frequencies in the lattice scale with trap
depth as expected for a thermal gas, we confirm the shift scaling of ∆νdensity ∝ U 5/4 as reported
in [132]. Fig. 2.5a shows the result of this evaluation where each point has error bars evaluated to
the low 10−18 level. The data fits well to a model with the functional form: a + bU 5/4 , where U is
the trap depth and a and b are fit parameters. Though we independently measure the density shift
for a particular lattice depth each day during clock operation, this scaling becomes highly usefulwhen evaluating shifts at several trap depths, as done during the AC Stark evaluation described in
the next section.

2.3.3

Lattice AC Stark

To eliminate Doppler shifts, the atoms are tightly confined in a one-dimensional optical lattice
to perform clock spectroscopy in the Lamb-Dicke limit. This confinement induces a differential AC
Stark shift between the ground and excited clock states. To minimize this effect, we operate ourlattice near the so-called magic frequency where the differential polarizability between the groundand excited states is nearly zero.
To constrain the lattice Stark shift at the 10−18 level, recent work has highlighted the importance of accounting for higher-order effects such as hyperpolarizability and the magnetic dipole andelectric quadrupole terms [140, 117]. These manifest in light shifts scaling with nonlinear powers
in trap depth. Recent work [27] has demonstrated that, given a thermal scaling of axial and radial25

Density Shift per 1000
atoms × 10 17

0.0
0.5
1.0
1.5
2.0

2.5
11 1-14 1-15 1-21 1-22 1-28 1-31 2-04 2-05 2-07 2-08
1
0 0 0 0 0 0 0 0 0 0 0
19- 19- 19- 19- 19- 19- 19- 19- 19- 19- 19Date
Figure 2.4: Density shift evaluations. Density shift measurements were performed over fourweeks at the same trap conditions. The weighted mean of all measurements is shown with a dashed,black line, with a reduced chi-squared of 1.07.

26

0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75

b)

Relative AC Stark Shift(× 10 16)

Density Shift at 1000
atoms (× 10 16)

a)

50

100

150

Trap Depth / Er

1.5
1.0
0.5
0.0
0.5
1.0
1.5

50

75

100

125

High Condition Trap Depth / Er

Figure 2.5: Scaling of the light shift with trap depth U (a) Density shift evaluated at different
trap depths, scaled to a shift with 1000 atoms. The dashed red line is a fit to the data with the
expected scaling of the shift as U 5/4 . (b) Lattice Stark shifts measured relative to a trap depthof 45Er . Four different lattice frequencies are shown: νL = 368.55452610 (blue), 368.55447610(red), 368.55446610 (green), and 368.55442610 THz (gold). The three curves with dashed lines are
independently fit using the model in Eqn. 2.4, obtaining a weighted mean of their β ∗ ’s. Using this∗, fully characterizing our AC Stark shifts for clock operation atβ ∗ , the red curve is then fit for αclock
the red curve. Vertical error bars are obtained from fits to the Allan deviation of each evaluation,
extrapolated to the total measurement time, and scaled by the lever arm of the measurement.
Horizontal error bars are uncertainties on our determination of the trap depth, obtained from axial
sideband scans.

27modes with trap depth U , these nonlinear shifts can be greatly simplified to a linear and quadratic
term, expressed as:νLS' α∗ U + β ∗ U 2 .
ν

(2.4)

Eqn. 2.4 shows that characterization of the lattice light shift requires the measurement ofexperiment-specific coefficients α∗ and β ∗ . A discussion of the uncertainty associated with the useof this model in the Sr1 system can be found in Appendix 2.6. Note that α∗ and β ∗ not only
depend on atomic coefficients, but also on the atomic distribution in both the radial and axial
directions. As such, care should be taken to evaluate these coefficients under reproducible sample
temperatures and lattice trapping frequencies.
To evaluate the lattice light shift a series of differential measurements of the light-induced
frequency shift between a series of lattice depths ranging from 34 Er to 141 Er and a reference
depth of 45 Er is performed. These measurements are repeated for four different lattice frequencies,νL , encompassing both positive and negative detunings from the magic frequency. This enables
accurate determination of νL dependent α∗ ’s and a νL independent β ∗ . The resulting data is shownin Fig. 2.5b. We perform this evaluation by relying on measurement precision as opposed to alarge measurement lever arm, enabling us to stay close to the clock operational trap depths andatomic distributions. To remove trap depth dependent density shifts from the lattice light shift
evaluations, we perform a series of density shift measurements over a range of lattice depths towhich we fit a U 5/4 model, shown in Fig. 2.5a. The residual uncertainty of the U 5/4 model fit to
the density shift data in Fig. 2.5a is then propagated to the lattice light shift measurements andadded in quadrature with the statistical uncertainty of the measurement.
Lattice lights shifts are evaluated at four different lattice frequencies, shown by the four
different curves in Fig. 2.5b. The determination of the lattice light shift at the operational latticefrequency proceeds in two steps. First, data from three different lattice frequencies (blue, green,and yellow points) is used to determine the wavelength-independent quantity β ∗ by least-squares

28
fitting to the model in Eqn. 2.4. Taking the weighted mean and weighted standard error of themean of the three β ∗ values from these fits results in β ∗ = 1.93(20) × 10−21 . Second, β ∗ is then∗used to fit αclock
to the data taken at the operational lattice frequency (red points), resulting in∗
αclock= −5.61(22) × 10−19 . The total AC Stark uncertainty is found in Tab. 2.2 and includes

uncertainties coming from α∗ and β ∗ as well as additional contributions from a 2% uncertainty intrap depth and a model uncertainty. In total, this procedure allows the evaluation of the latticelight shift with a total uncertainty of 1.2 × 10−18 .
To maintain this low uncertainty, we eliminate temporal variations of the lattice light shift.
To accomplish this, the trapping light is spectrally filtered and frequency stabilized. First, broadband amplified spontaneous emission (ASE) from the high power Ti:sapphire laser which generates
the trapping light is suppressed by reflecting the laser light off two volume Bragg gratings eachproviding in excess of 30 dB of suppression of ASE power outside the ∼10 GHz wide passband. The
Ti:sapphire laser is then phase-locked to an Er:fiber frequency comb which is stabilized to a cryogenic silicon reference cavity [96]. As a result, the drift of the lattice frequency is limited to the low
and well characterized drift rate of the cavity (-7.4 Hz/day at 813 nm), which produces negligible
drift of the lattice Stark shift. With this stabilization scheme, measurement of the cavity frequency
by routine clock operation allows determination of the absolute lattice frequency at the sub-Hz
level. In addition, alignment of the lattice polarization to the bias magnetic field and alternately
probing the stretched states suppresses the vector contribution and sensitivity to fluctuations in
the tensor term. Fluctuations in the relative direction of the magnetic field and the polarization
axis are additionally suppressed by the active temperature control of the experimental apparatus
with residual background field fluctuations observed at the 100 nT (1 mG) level, corresponding toa shift constrained below the 10−19 level. To suppress drifts in the lattice intensity, the laser poweris actively stabilized by monitoring the reflection of the incident light from the top surface of thevacuum window. To ensure consistent overlap of the in-going and retroreflected lattice beams, asmall amount of power from the retroreflection which is transmitted back through the optical fiberused for beam delivery is monitored to ensure stability of the lattice alignment. Finally, examining

29
Eqn. 2.4, we see that the light shift is sensitive to fluctuations in the atomic distribution. As
covered in the discussion of the density shift, several steps were taken to ensure robustness of bothtemperature and density and this is reflected in the high reproducibility of the density shift in
Fig. 2.4.
Table 2.2: Lattice light shift uncertainty contributions
Parameterα∗β∗
U (Er )
Model
Total

2.3.4

Value (Uncertainty)-5.61(22) ×10−19
1.93(20)×10−2145.0(9)Uncertainty (10−19 )
9.7
4.1
3.5
3.3
11.6

DC Stark Shift

Stray DC electric fields or patch charges on vacuum viewports can induce a frequency shift
due to the differential DC polarizability (α0 ) between the two clock states. To directly measure thiseffect with the atoms, a pair of ring electrodes are placed on the top and bottom viewports whichhave the closest proximity to the atoms. Each ring consists of four copper quadrants which caneach be independently biased. By applying a large lever arm of ±100V to the correct combinationof electrodes a shift can be measured along any of the three Cartesian axes.
Due to the quadratic dependence of the shift on the total electric field, the magnitude of the
background field along each direction can be determined by performing a two-point measurement.
We measure ∆ν+ , the frequency shift when a large field is applied along one direction and comparedto the reference case where both electrodes are grounded. The field direction is then reversed and thefrequency, ∆ν− , is recorded. The direction of the background field, along with its corresponding field
amplitude, can then be computed from these two measurements by noting that ∆ν± = − 21 α0 (Ea ±Er )2 for applied and residual fields Ea and Er . We observe a modest background field at the low
10−18 level in the vertical direction and no clearly resolvable field at the 10−20 level along thehorizontal axes. By applying (−4.2V, +4.2V) to the (top , bottom) electrodes, respectively, we30
cancel this background shift to an uncertainty of 2.5 × 10−19 .

2.3.5

Second Order Zeeman ShiftAs described in Section 2, the Zeeman shift is cancelled to first order by averaging frequency
measurements of the mF = ± 29 clock transitions. However, the Zeeman Hamiltonian contains aterm with a quadratic dependence on the magnetic field which is not suppressed by this technique.
The 57 µT (570 mG) bias field applied to resolve the hyperfine structure of the clock states induces
a second order Zeeman shift of ≈ 77 mHz. This shift is typically expressed as∆νB,2 = ξ (∆νB,1 )2(2.5)

where ξ is the quadratic Zeeman coefficient for mF = ± 92 and ∆νB,1 is the first-order Zeeman
splitting in Hz between the mF = ± 29 transitions due to the applied bias field. Previous uncertainty
in the value of ξ introduced a sizeable term in our error budget, motivating a more precise evaluationof the coefficient.
The determination of ξ is complicated by the fact that the observed splitting ∆ between the
mF = ± 92 transitions also contains a contribution from the vector AC Stark shift δν ≈ 0.4 Hz.
In analogy with ∆νB,1 , the vector AC Stark term is often thought of as arising from a synthetic
magnetic field. We evaluate ξ in the limit where the applied bias field is much larger than this
synthetic field so that changing the bias field magnitude does not significantly rotate the total
effective field vector. We validate that we are in the appropriate limit by measuring the lattice
vector shift directly in an AC Stark evaluation to determine the magnitude of the synthetic field. In
this limit, the second-order Zeeman shift can be expressed as ∆νB,2 = ξ(∆ − δν)2 . By performing
differential measurements of the clock transition frequency f0 at three different bias field values,
we may determine both ξ and δν from the following system of equations:

31

f02 − f01 = ξ(∆22 − ∆21 ) − 2ξδν(∆2 − ∆1 )(2.6)

Quadratic Zeeman
coefficient × 10 7 / Hz 1

f03 − f02 = ξ(∆23 − ∆22 ) − 2ξδν(∆3 − ∆2 )

2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0

2.45

PTB
JILA
SYRTE
2008

2010

2.50
2014

2012

Year

2014

2016

2016

2018

2018

Figure 2.6: Evaluation of second order Zeeman coefficient. A history of evaluations of the Sr
second order Zeeman coefficient completed by the PTB [45], SYRTE [145], and JILA [30, 17, 109]
Sr OLCs. The dashed black line is a weighted mean of all six measurements and the inset shows
the three most recent evaluations.
This measurement is repeated at three different sets of bias field values all yielding consistentmeasurements of ξ. We report the weighted mean of these three evaluations yielding a value ofξ = −2.456(3) × 10−7 Hz−1 . Fig. 2.6 shows this result along with historical evaluations showing
good agreement with previous measurements. Improved knowledge of this coefficient reduced theuncertainty associated with the second-order Zeeman shift by a factor of 5 to 2 × 10−19 .

2.3.6

Background Gas Collisions

Two-body collisions between a cold 87 Sr atom and a room temperature hydrogen molecule ora hot Sr atom emitted by the atomic beam source have the potential to cause a systematic shift. In
an uncorrelated thermal gas, a high-energy collision which removes an initially trapped atom from32
the sample does not produce a systematic shift. However, atom-light coherence can be affected bycollisions with low momentum transfer that produce phase-shifts while leaving the atom trapped.
In the process of a collision, the ground and excited states shift differentially as a function of the
distance between the Sr atom and its collision partner. When an atom is placed in a superpositionof ground and excited states during clock interrogation, the integrated energy difference over thetime of the collision produces an undesirable phase shift which appears as a systematic frequency
shift varying linearly as a function of the background gas density.
Under ultrahigh vacuum conditions, this frequency shift can be well-approximated by considering collisions between 87 Sr and hydrogen molecules [102]. The Sr1 vacuum chamber is stainlesssteel and contains no getter pump, so the dominant background gas contribution is primarily molecular hydrogen gas. The SYRTE collaboration recently measured a background gas collisional shiftof (−3.0 ± 0.3) × 10−17 /τ for a hydrogen-limited vacuum lifetime τ [2]. For the Sr1 clock, thelifetime of a dilute sample of 3 P2 atoms trapped in the quadrupole field of the MOT is measured
to be 8.1(2) s after correcting for BBR-induced decay [152]. The measured lifetime of ground state
atoms trapped in the optical lattice is the same as the lifetime in the quadrupole trap [95] andleads to a background gas collisional shift of (−3.7 ± 0.4) × 10−18 .
In addition, two-body collisions with hot Sr atoms emitted from the atomic beam source may
also cause a frequency shift. To evaluate this possible systematic shift, an atomic beam shutter isin place to block line of sight to the oven during clock interrogation. The oven flux is increased
above the normal operating condition by a factor of 21, and interleaved interrogation of the clock
frequency with and without the atomic beam shutter closed is performed. With this lever arm, no
significant shift is observed at a measurement precision of 1.0 × 10−18 , therefore bounding the shift
at normal operating conditions to below the 10−19 level. The shutter is not used for normal clock
operation due to intermittent failure of the shutter in the closed position. No difference in atom
number or lifetime in the trap is observed with the beam shutter closed versus open.

33
2.3.7

Doppler Shift and AOM Phase Chirp

For narrow line spectroscopy, Doppler shifts between the clock laser and atoms trapped in
the optical lattice are eliminated by active optical path length stabilization [91]. Ideally, the surface
which serves as a reference for the laser phase for clock interrogation is the same as the surface
which serves as the phase reference for the optical lattice. However, when the clock laser intensity
is suddenly changed from zero to a finite value to drive the clock transition, the response of theactive stabilization loop experiences transient phase shifts which may impart systematic frequency
shifts.
Simultaneous laser phase stabilization and atomic spectroscopy is accomplished with two
AOMs – one before an optical fiber that is the actuator for path length stabilization, and a second
after the fiber that is the actuator for steering the laser frequency to the atomic resonance. In the2013 systematic evaluation of the Sr1 clock the stabilization of the laser phase was accomplishedby retroreflecting the zeroth order of the second AOM from the mirror to which the optical lattice
is referenced while the first negative order was used for atomic spectroscopy. When the clock pulse
turns on, this stabilization method produces a differential path length change between the zerothand first-diffracted orders due to thermal changes in the AOM crystal. In addition, both differential
path length changes driven by air currents between the two diffracted orders and reflections fromthe tip of the optical fiber add phase noise above the level of the clock laser in the path lengthstabilization servo.
To circumvent the problems associated with thermal effects in the AOM crystal, fiber tip
reflections, and differential path length noise, the path length stabilization scheme is revised suchthat the first negative diffracted order from the AOM after the fiber is now used for both spectroscopy and path length stabilization – eliminating all differential optical path between the path
length stabilization light and the light used for atomic spectroscopy. Mounting a wedged beam
sampler to the back surface of the optical lattice retroreflector mirror allows 10% of the incident
clock light to be used for path length stabilization. Additionally, Rabi pulses are implemented by34
first turning on clock light at a large (1 MHz) detuning for 7 ms and then performing a 3.4 ms
ramp onto resonance. This method allows the clock intensity servo to settle before spectroscopy
and eliminates all thermally-induced differential path length transients in the AOM crystal. Furthermore, phase transients in the path length stabilization in-loop error signal that arise when
operating the servo in a pulsed fashion now occur when the clock light is detuned and the atomic
sensitivity function is zero. Eliminating sensitivity to these transients removes a potential source ofsystematic frequency offsets [46]. The fractional frequency shift due to the linear frequency ramp
to resonance is estimated with the Landau-Zener transition probability for the parameters above
and is calculated to be 2 × 10−22 for a 600 ms pi-pulse.

2.3.8

Line Pulling

In addition to collisional shifts, off-resonant excitation of an impurity spin population can
produce a systematic line pulling frequency shift. The impurity spin population is determined by
performing clock spectroscopy on impurity spin states and detecting the number of atoms promoted
to the excited state. The total impurity spin population is determined to be less than 0.5%, limitedby the detection threshold of the fluorescence measurement. The line pulling effect is subsequently
bounded by taking the maximum population in the ±|5/2i and ±|7/2i states after dark-state opticalpumping and calculating their maximum contributions to the excitation fraction observed on the±|9/2i transition. For a 600 ms pi-pulse and a 62 Hz splitting between the |9/2i and |7/2i clocktransitions, the maximum off-resonant excitation is given by the ratio of Clebsch-Gordon coefficients
0.49
squared and the off-resonant Rabi frequency: ( 0.82
) × (π/0.6)2 /((2π × 62)2 + (π/0.6)2 ) ≈ 1 × 10−4 .

Combining the upper bound of the atom number in impurity spin states (0.5%) with this excitation
fraction and converting to frequency units, an upper bound on the line pulling effect is set at the
6 × 10−22 level.

35
2.3.9

Servo Error

The servo error systematic arises from the linear frequency drift of the clock laser that serves
as a local oscillator, or from a drifting background magnetic field, producing a systematic bias ofthe excitation fraction error away from the desired lock point. To minimize this effect, a PI2 D
digital servo is used for locking the clock laser to the atomic transition. The servo loop is tuned by
optimizing the attack time of the lock error signal with respect to the steering control signal. As
a result, for a measurement time of 200000 seconds, a mean servo error of 4 × 10−20 is recorded,
which averages to an uncertainty of 2 × 10−19 .

2.3.10

Gravitational Redshift

The gravitational redshift of the clock transition does not effect the systematic uncertainty of
the local realization of the clock. However, for accurate comparison of a clock to any other frequency
reference the fact that each clock is generally realized at a different gravitational potential must be
taken into account as another source of systematic uncertainty. In the absence of another clock tocompare to in this work, the clock frequency can be reported relative to an agreed upon reference
frame, for example the International Terrestrial Reference Frame (ITRF). In Boulder, a precise
characterization of the geopotential of USGS marker Q407 found a clock shift of 179853(6)×10−16
relative to ITRF [115]. Local surveying in Boulder has evaluated the height of the JILA Sr1 in-lab
marker (S1B60V1) to be 9.7874(0.0025) m below Q407. Accounting for the atom’s height relative
to S1B60V1 and thus relative to Q407, we find a fractional shift of 178766.4(6.0)10−18 relative to
ITRF. Note that the uncertainty in the gravitational potential in Table 2.4 dominates over theclock systematic uncertainty, and suggests that terrestrial all-optical timescales of the future will
be limited by the current knowledge of Earth’s gravitational potential.

36

Table 2.3: Sr1 systematic uncertainty
Systematic
BBR (environment)
BBR (atomic)
Density
Lattice AC Stark
DC Stark
Probe AC Stark
1st order Zeeman
2nd order Zeeman
2nd order Doppler
Servo error
Line pulling
Background gas
AOM phase chirp
Total

Shift (10−18 )-4974.1
0
-12.3
-21.3
0
0
0
-176.9000-3.7
0-5188.3

Uncertainty (10−18 )
0.2
1.5
0.4
1.2
0.3
<0.1
<0.1
0.2
<0.1
0.2
<0.1
0.4
<0.1
2.0

Table 2.4: Sr1 uncertainty relative to ITRF
Systematic
Local systematic uncertainty
Gravitational shift to ITRF surface
Total shift to ITRF surface

Shift (10−18 )-5188.3
178766.4
173578.1

Uncertainty (10−18 )
2.0
6.0
6.2

37

2.4

Summary
In summary, we demonstrate a significant advance in the characterization of strontium opticallattice clocks resulting in a low systematic uncertainty of 2.0 × 10−18 . In conjunction with its highuptimes as well as the highly predictable frequency evolution of the cryogenic silicon cavity, sucha clock will be a core component of an optical timescale. Indeed, the JILA Sr1 clock has alreadybeen used in an atom-cavity frequency intercomparison which highlights the long term stabilityof cryogenic crystalline cavities for timescales applications [101] and the search for time-variationof fundamental constants [123]. As illustrated in Fig. 2.2, fluctuations in systematic offsets arebounded below 4 × 10−19 over 104 seconds of operation. We additionally detail a powerful newtechnique to combat blackbody radiation shifts that provides a stable thermal environment in which
in-vacuum thermometry is only needed as calibration of sophisticated temperature modeling. As
clocks are pushed into the next decade of accuracy, the ability to remove the temporal variationof systematics will be a highly powerful tool. With the main limitations to our uncertainty being
insufficient knowledge of atomic coefficients, this system sets the path for developing strontium
optical lattice clocks at the 10−19 level.

2.5

Thermal Environment Evaluation

2.5.1

Thermal Modeling

Radiative heat transfer between specular greybody radiators is a well-studied problem in avariety of systems ranging from ovens to satellites [103]. To characterize the thermal radiation
experienced by the atoms we must understand the radiative contributions from each surface of thevacuum chamber. We start by modeling the atoms as a small spherical surface, hereafter referred
to as the probe, in thermal equilibrium with the vacuum chamber. The temperature Ti of this38
surface, or equivalently of the radiation bathing the atoms, is given by

i σTi4 Ai =X

j σAj Fj→i Tj4

(2.7)

j

where j enumerates the surrounding vacuum chamber surfaces, j is the surface’s emissivity, σ is
the Stefan-Boltzman constant, Aj the surface area, and Fj→i the exchange factor. The exchange
factor Fj→i is defined as the fraction of total energy emitted by surface j that is absorbed by surface
i, either directly or after reflection from any intermediate surfaces. In this language, Equation 2.7
is telling us that at thermal equilibrium, the total power radiated by our modeled atom surface
must be equal to the total incoming power from the surrounding vacuum chamber. Similarly, theexchange factors in Equation 2.7 tell us what portion of radiated power from each surrounding
surface j is incident on the probe surface.
Using our understanding of the system’s behavior at thermal equilibrium we can simplifyP
Eqn. 2.7 significantly. First, energy conservation requires that j Fi→j = 1, meaning all emitted
energy must be absorbed by the surrounding surfaces. Second, reciprocity demands that i Ai Fi→j =j Aj Fj→i [103]. This can be understood by considering all of the paths that emitted BBR (rays)
can propagate between surfaces i and j - whether the ray goes from i to j or j to i the paths
connecting the surfaces are the same. With these two conditions we can simplify Eqn. 2.7 as

Ti4 =

X

Fi→j Tj4 .

(2.8)

j

Note that the probe temperature no longer depends on the probe surface’s emissivity - consistent
with Kirchoff’s radiation law. Also note that the exchange factors now consider radiation propagating from the probe surface. Once we have the exchange factors we then have the connection
between the array of sensors on our vacuum chamber and the temperature experienced by theatoms.
Calculation of the exchange factors is performed using Monte Carlo ray tracing. This begins
with a 3D CAD replica of our vacuum system which is broken into several pieces to address the39different temperatures, emissivities, and reflective qualities (specular and diffusive) of each chambercomponent. From the location of the atoms we propagate rays with random orientations to evenly
sample the 4π steradians around the atoms. For each ray we find the intersection of that ray with thevacuum chamber boundary. Each surface has a probability to absorb (emissivity), diffusively reflect,or specularly reflect the incident ray. If a ray is absorbed, the location and surface is recorded.
For reflected rays, the new ray direction is found and the process repeated until every ray has
been absorbed by a boundary surface. We perform this process for 50 million initial rays to ensuresufficient ray intersections with all surfaces. Specifically, this ensures that the least interceptedsurface, the oven nozzle, absorbs more than 1000 rays for sufficient simulation convergence. From
this record we calculate what fraction of the total number of rays are absorbed by each surface.
This fraction yields the exchange factor for a given component of the vacuum chamber.
The emissivity and specular reflectivity values used in the ray tracing analysis are given
in Table 2.5 and Table 2.6 respectively. Uncertainties of these parameters may result in the small
static offset between the temperature determined by a ray-tracing model and the in-vacuum sensor;
however, using the calibrated sensor to account for this offset removes the need to propagate
uncertainties associated with both emissivities and reflectivities. Our calculated exchange factors
are given in Table 2.7. With these values we can evaluate the temperature at the atoms using
Eqn. 2.8. We find a stable offset between our model and the temperature measured with ourin-vacuum thermometer as illustrated in Fig. 2.3d.
Table 2.5: Emissivities of vacuum chamber components
SurfaceViewports (glass)
Vacuum chamber (stainless steel)
Sapphire
Oven nozzle

Emissivity
0.91 [135]0.08 [5, 148]
0.54 [150]0.82 [119]40
Table 2.6: Specular steel reflectivities based on [11]Component
Polished Chamber
Other Steel Surfaces

Specular Reflectivity
0.95
0.1

Table 2.7: Vacuum chamber generalized exchange factors
Component
Heated sapphire window
2.75” CF viewport glass - extended flanges
2.75” CF viewport glass - direct flanges
6” CF viewport glass
Glass cell
Metal chamber, tubing, and slower
Oven nozzle

2.5.2

Exchange Factor
8.81 × 10−4
1.47 × 10−2
1.18 × 10−1
5.39 × 10−1
1.08 × 10−3
3.26 × 10−1
2.15 × 10−5

Temperature Sensor Calibration

In order to calculate the temperature at the atoms using Eqn. 2.8, an array of sensors tomonitor the temperature of various points on the vacuum chamber is required. This array iscomposed of commercially available 50 kΩ negative temperature coefficient thermistors (US Sensor
PR503J2), specified by the manufacturer to be accurate to within 50 mK. Due to the relative
importance of the 6” CF viewports (exchange factor of 0.539) to the thermal model, the calibrationof the thermistors on these viewports is improved to a 13 mK uncertainty via comparison to athin-film platinum resistance thermometer (TFPRT) calibrated to fixed-point realizations of theice melting point and the gallium melting point.
Figs. 2.7a and 2.7b show typical melt curves and associated Allan deviations for both thegallium ice and water ice phase transitions as measured by a TFPRT. The ITS-90 temperature scale
defines these melting plateaus to be at temperatures of 29.7646◦ C (302.9146 K) and 0◦ C (273.15
K), respectively, allowing for the calibration of the TFPRT - up to systematic uncertainties - atthese points. The 2 mK offset from 0◦ C in our ice point cell is a consequence of performing ourcalibration measurements at an elevation of 1650 m above sea level [61]. Dissolved air in waterserves to suppress the ice point temperature at sea level, where the ice melting point corresponds

41
to 0◦ C by definition. Upon applying this known correction, the dominant uncertainty at 22◦ C isthen the unknown non-linearity of the TFPRT between the two fixed point temperatures. Aftercalibration of these crucial thermistors, this array of 50 kΩ thermistors is used as inputs to thethermal model. Using these inputs in conjunction with literature values for the different emissivitiesof the materials which compose the vacuum chamber, shown in Table 2.5, a real-time prediction
for the temperature seen by the atoms is generated.

Figure 2.7: JILA fixed-point realizations. Data from the text(a) water ice melting point and(b) gallium melting point realization. (c) Allan deviation of the gallium melt curve (red) and theice melt curve (blue), showing the capability of averaging down to below 100 µK on each fixed-pointin 104 seconds or less.

As discussed in the main text, in order to evaluate the accuracy of the prediction generated
by the ray-tracing-based thermal model, a TFPRT is installed at the position of the atoms in thevacuum chamber. This in-vacuum thermometer allows the characterization and removal of a static
offset between the thermal model and the measured temperature and also serves as verificationthat the thermal model successfully captures all fluctuations in the temperature below the level of
1 mK. In order to achieve an accurate measure of the in-vacuum temperature from the TFPRT,

42
the probe was hand-carried to NIST Gaithersburg for calibration in the Sensor Sciences Division
facilities. There, a water bath comparison calibration with Standard Platinum Resistance Thermometers (SPRT) traceable to the ITS-90 temperature scale enabled the in-vacuum thermometerto be calibrated to an uncertainty of 1.4 mK.
Fig. 2.8b shows the different contributions to the final sensor uncertainty. Immersion error isthe largest systematic effect, arising from heat flow between the room temperature environment andthe probe, producing a systematic temperature difference between the TFPRT and the bath. To
characterize this effect, a second TFPRT is mounted on the upper flange of the test chamber, whichlies just above the water line of the bath. To evaluate immersion errors in the water bath calibration,
two sets of measurements were undertaken, one under vacuum conditions (< 3 × 10−6 Torr), andanother under 30 Torr of back-filled helium. Figure 2.8a shows the measured difference in zero power
resistance for these two conditions as a function of the temperature gradient between the flange andthe bath, Tf lange − Tbath . The immersion error is well-described by a linear function of temperature,with a fitted Tf lange − Tbath = 0 systematic offset of (0.3 ± 0.4) mΩ, which can be converted to(0.69 ± 1.1) mK. Since the offset is consistent with zero, we simply take the standard uncertainty
of 1.1 mK as the immersion error contribution, plotted as the green line in Figure 2.8b. We also
include the bath homogeniety as the largest observed gradient as indicated by the two SPRT’s in
the bath of ± 0.5 mK, shown as the cyan line. The calibration coefficient uncertainties are shown as
the blue and orange curves. The red line shows the sum of all other minor errors, including SPRT
calibration, bridge nonlinearity, resistance standard and bath stability [134]. We also account forthe offset of 3.0 mK between ITS-90 and the definition of thermodynamic temperature, and it’scorresponding uncertainty of 0.3 mK is included in the minor errors [134].

2.5.3

BBR Dynamic Shift Correction

The dynamic BBR shift is described by∆νdynamic = νdyn

T
T0

6
.

(2.9)

43

Figure 2.8: Calibration of the in-vacuum probe. (a) Immersion error data in the water bathcomparison. We measure the difference between the high vacuum resistance Rvac and the He backfilled case RHe as a function of axial gradient Tf lange − Tbath . The data is fit to a linear function,and the fitted offset at zero axial gradient is (0.3 ± 0.4) mΩ. This can be converted to temperatureby using the sensitivity of the TFPRT of 2.57 Ohms/Kelvin. (b) Breakdown of uncertainties
stemming from the NIST calibration. The green is the systematic error in the immersion error
offset from panel (a). The cyan color is the maximum temperature gradient observed in the bath.
The blue and orange curves are the fit interpolation errors from the slope and offset respectively.
The red line is the quadrature sum of several minor errors relating to the calibration. The black
dashed line is the quadrature sum of the errors.

44
Care must be taken since the often reported value of νdyn contains higher order terms in T (T 8 andT 10 ) [124]. The reported coefficient νdyn therefore requires corrections for temperatures deviating
away from T0 = 300 K. For our temperature near 22◦ C, we find we must correct our dynamic BBR
shift by an amount of 1.48 × 10−18 .

2.6

Lattice Light Shift Evaluation - Model Uncertainty
Optical lattice clocks have reached an accuracy level where higher order polarizability terms

require careful consideration. Two separate approaches have dealt with this to high accuracy. At
NIST, the Yb clock group [27] demonstrated that in a system where the spatial modes have certain
scalings with trap depth the lattice light shifts can be well characterized by a so-called thermal
model, containing effective terms that are linear (α∗ ) and quadratic (β ∗ ) in U :∆νAC = α∗ U + β ∗ U 2 .

(2.10)The dipole polarizability αE1 and multipolarizability αQM are contained in α∗ while the hyperpolarizability β is contained in β ∗ . This approach is robust as it does not require explicit knowledgeof atomic coefficients or the axial and radial atomic mode numbers. A second approach is to workon a microscopic model, where for a given radial temperature and axial mode occupation a lightshift can be calculated [140]. While a microscopic model offers the prospect of accuracy below the10−18 level, it requires significantly more input information, a potential source of systematic errors
in reporting atomic coefficients. Recent work [107] has highlighted this importance by illustratingthat different methods of preparing the atomic sample can result in different measurement valuesof β in Yb.
For determining the operational light shift in the Sr1 clock the thermal model is employed.
This avoids reliance on atomic coefficients and characterizes an experiment-specific light shift model.
The characterization in the main text discusses the results of the fitting of the thermal model tothe data. The limitation to this analysis is that it requires addressing model error - deviations

45
from linear and quadratic behavior in light shifts that are described by the microscopic model.
To address the deviations from linear and quadratic scalings in the Sr1 light shifts we adopt theapproach in [27], writing the light shift as:∆νAC= −An αE1 U − Bn αQM U − Cn βU 2
νclock

(2.11)

where An , Bn , and Cn are spatial averages given by+*2(x2 + y 2 ) 2cos (kz) n
An = n exp −w02*+
2(x2 + y 2 )  2
Bn = n exp −sin (kz) nw02*+
4(x2 + y 2 ) 
4Cn = n exp −cos (kz) n .
w02
The e12 beam radius is given by w0 , the lattice wavenumber by k, and |ni = |nx , ny , nz i where ni
is spatial coordinate i’s mode number. To address gravitational sag resulting from the θ = 19◦
tilt with respect to vertical in the lattice, the component of gravity (g) along x̂ is included bysubstituting x → x − mg sin(θ)/ωr2 where ωr is the radial trapping frequency.
The gravitational tilt lifts the radial degeneracy so An , Bn , and Cn are broken into orthogonalbases such that An = Anx Any Anz , Bn = Bnx Bny Bnz , and Cn = Cnx Cny Cnz , expanding each to46
fourth order. The system is then described by following series of equations:γ=

g 2 sin2 (θ)
w02 ωr4√
2
3√ (nx + 1/2)(1 − 6γ) + 2 2 (n2x + nx + 1/2) − 2γ + 2γ 2Anx = 1 −2k
w0 U
w0 k UBnx = Anx√
32 2
√ (nx + 1/2)(1 − 6γ) + 2 2 (n2x + nx + 1/2) − 4γ + 4γ 2
Cnx = 1 −kw
w0 k U0U√
2
3√ (ny + 1/2) + 2 2 (n2y + ny + 1/2)
Any = 1 −2k
w0 U
w0 k U
Bny = Any√32 2
√ (ny + 1/2) + 2 2 (n2y + ny + 1/2)
Cny = 1 −kw
w0 k U0U(nz + 1/2) (n2z + nz + 1/2)√
Anz = 1 −+2UU2(nz + 1/2) (nz + nz + 1/2)
√
Bnz =
−
2U
U
2
2(nz + 1/2) 5(nz + nz + 1/2)√
Cnz = 1 −.
+2UU

(2.12a)(2.12b)(2.12c)(2.12d)(2.12e)(2.12f)(2.12g)(2.12h)(2.12i)(2.12j)In the above equations U is the trap depth in units of the photon recoil energy Er .
To quantify the model error, Eqn. 2.11 is utilized with input data from measured ni values
and αQM from [140]. The axial and radial mode occupation numbers are derived from carrier
sideband asymmetry measurements, radial trapping frequencies, and transverse clock spectroscopy[14]. The final two inputs for the model, αE1 and β, are evaluated at each point in a two dimensional
space of input parameters spanned by plausible values of αE1 and β. Limits on these parameters
are chosen to encompass the full range of αE1 values from the four curves in Fig. 2.5b. Similarly,
the range for β is chosen to encompass the weighted mean of values from published literature[109, 145, 140] to one which is consistent with what is observed in Fig. 2.5b. For each pair of
αE1 and β in this two dimensional space of values, Eqn. 2.4 is used to simulate lattice light shifts
over the range of trap depths used in the experimental data. The thermal model from Eqn. 2.4 isthen fit to this simulated data. For each set of simulated and fitted curves, the mean and standard

47
deviation of the difference between the two models is calculated. All mean differences are below
1 × 10−19 , ensuring that the standard deviation of the differences is not overlooking a constantoffset between curves. The resulting average of all standard deviation values shown in Fig. 2.9 is
3.3 × 10−19 . This number is thus interpreted as the model error. We note that use of the theory
value for αQM [117] provides a lower estimate of model uncertainty.
We additionally investigate the effect of a running wave in our system arising from reflectivity
on the viewports. We find that the only nonlinear and nonquadratic additional term provides anegligible 2 × 10−20 level effect.

Figure 2.9: Model Error Colormap showing the standard deviation between simulated AC Stark
data (Eqn. 2.11) and a simple linear and quadratic fit to the data (Eqn. 2.4). The average standard
deviation is 3.3×10−19 which we take to be our model error.

Chapter 3

Comparisons and Demonstrations

In 2017, we decided to perform clock comparisons between JILA and NIST. At the time,
the original JILA Sr2 system with uncertainty of 2.1 × 10−18 had been dismantled and replacedwith the 3D lattice clock. The Sr1 system, having been evaluated in 2014 to an uncertainty of
5.3 × 10−17 , was chosen to participate in the frequency comparisons. This motivated the upgrade
and systematic reevaluation of the original Sr1 clock as described in Chapter 2, achieving a record
low uncertainty of 2.0 × 10−18 for a strontium optical lattice clock and third lowest uncertainty forany atomic clock to date. This prepared the Sr1 system to perform clock comparisons with the twomost accurate clocks in the world - the Yb OLC and Al+ ion clock at NIST. An immediate benefit
of systematically characterizing and comparing a state-of-the-art clock is the ability to bootstrap
additional research. Sr1 served as a vetted reference for record precision inter-clock comparisons
between Sr1 and Sr2 as well as a high up-time reference for an all-optical timescale. This chapter
summarizes these results, motivating the idea that accuracy and precision go hand in hand. In thelanguage of quantum information, coherence follows careful quantum state control.

3.1

Towards the Redefinition of the Second
As metrologists a primary objective in the coming decade is to redefine the second to an

optical transition. The accuracy and precision of today’s best optical clocks are now orders ofmagnitude better than that of microwave clocks [98, 24, 19, 126, 33, 93, 154], encouraging discussion
about how such a redefinition may be done. In 2018 a CIPM working group established a set of

49
potential criteria for such a redefinition, summarized below [121].
(1) A minimum of three different optical clocks (different labs or species) have validated fractional frequency uncertainties two orders of magnitude lower than the best Cs clocks.
(2) A minimum of three independent measurements of at least one frequency standard from
(1) between different institutes at a level of ∆ν/ν < 5 × 10−18 .
(3) Three independent measurements of any frequency standard from (1) compared with three
independent Cs primary clocks, with uncertainty limited by the Cs clocks.
(4) Candidate frequency standards regularly contribute to international timescales.
These criteria are designed to ensure that a frequency standard is reproducible, consistent
with the current Cs definition, and capable of operating as a timescale. An additional criteria waslisted in [121]: (5) sufficient frequency ratio measurements between different frequency standards at
the level of ∆ν/ν < 5 × 10−18 . This criteria is essential to ensure closure relationships. A frequency
closure relationship between clocks a,b, and c is defined as


C=

νa
νb



νb
νc




νc
,
νa

(3.1)

where the level at which one measures 1 provides demonstration of the validity of the standards.
Due to the large number of optical secondary frequency standards being considered for a newprimary frequency standard, (5) is meant to ensure a smooth transition to a new definition that
does not void other standards or lead to inconsistent frequencies at the 5 × 10−18 level.
Criteria (5) is in some sense a political necessity. Today there are numerous optical standards
in development, with no standard so far ahead that a redefinition should clearly favor it. Whichfrequency standard should be chosen? Should a system based on ratios or multiple transitions
be used? These are the questions CIPM must debate in the coming decade. Before transportableclocks regularly begin comparing at the 10−18 level, such discussions are speculative. Let us instead
focus on fulfilling these criteria. In Chapter 2 we fulfilled (1), let’s discuss the progress towards (5).

50

3.2

The Boulder Atomic Clock Optical Network
The Boulder Atomic Clock Optical Network (BACON) is an extensive collaboration com-prised of members from both the National Institute of Standards (NIST) and JILA. NIST has
two major research clocks (Yb and Al+ ), state of the art optical frequency combs (OFCs), anda UTC traceable maser network (Figure 3.1). JILA contains Si3 (critical for connecting the fiberand free-space links), Sr1, Sr2, and the JILA frequency comb. This provides a unique opportunity
owing to three of these systems being the three most accurate clocks in the world [98, 24, 19], allwithin several kilometers. This establishes Boulder as a premier location for high accuracy clock
comparisons between three different next-generation frequency standards.
Clock comparisons compare clock frequencies through an optical comb, allowing frequency
ratios to be measured. This is critical since the absolute frequency of any of the three BACON clocks
would be limited by the realization of the Cs standard, but the ratios of the three clock frequencies
can be realized with precision below the accuracy limits of three clocks. To compare Sr to the NIST
based clocks, 1542 nm light (from the Si3 clock laser system) is transmitted through a 3.6 km fiber
link through the ‘Boulder Research and Administration Network’ (Figure 3.1). This is a shorter
version of similar fiber networks between other national labs, such as in Europe. NIST’s expertiseprovided an additional novel connection between clocks - a free-space link [18]. Using optical twoway time-frequency transfer (O-TWTFT) techniques, signals from optical frequency combs at two
locations (NIST and Duane tower at CU) are sent both directions in the link, allowing calibrationof common-mode noise from air turbulence. The NIST O-TWTFT OFC is locked to Yb, while the
JILA O-TWTFT OFC is locked to Sr, enabling the Yb/Sr ratio to be compared through free-spacein addition to fiber, a first for optical comparisons [38].
The systematic tables of the three clocks (and combs) are well described in [38], with the Srclock being slightly worse than reported in [19]. This is owing to incomplete upgrades/characterizations
performed for the accuracy paper, published in 2019, from the ratio data as taken in 2018 for Sr.
For the clock comparison, we briefly review the differences. The density shift uncertainty was larger

51

Figure 3.1: Overview of the Boulder Atomic Clock Optical Network (BACON). Network locations
at NIST (JILA) are shown in the left (right) of the image. The two institutes are linked both by
a 3.6 km fiber link as well as free-space link. NIST is home to the Al+ ion clock, Yb OLC, andOFCs used for measuring the 3 clock ratios. The Sr clock is located at JILA.

52
than in [19], owing to a less stable system as well as higher operational trap depth, increasing thedensity shift per atom. This higher operational trap depth (and uncertainty in daily depth) resulted
in a larger uncertainty in the lattice light shifts for Sr1. Thermal conditions were not as stable
during the clock comparison, necessitating a modest increase in BBR uncertainty dependent upon
the thermal stability during the course of each measurement. Additional temperature uncertainty
for aging thermistors (relative to the calibration performed in [19]) was included. Finally, the probe
chirp systematic was larger owing to traditional zeroth order retro-reflection, causing an abrupt
thermal shock upon Rabi excitation [109].
Over the course of Fall 2017 through Spring 2018 the three clocks were compared. Sr1 wasfound to have an uncontrolled density shift systematic that prevented inclusion of the Fall 2017
data. The results for each day’s clock comparison are shown in Figure 3.2. An example of the data
analysis for instability is shown in Figure 3.3.
Analysis of the data shown in Figure 3.2 presented a significant hurdle. Datasets were nonoverlapping owing to the downtime of the various clocks. Standard reduced chi-squared analysis
presented issues. The Yb/Sr ratio was over scattered, owing to the ability of the lattice clocks
to average below their combined uncertainties each day. Meanwhile the Al/Sr ratio was significantly under scattered, an abnormal situation in clock comparisons. To resolve these challenges, acomprehensive Bayesian framework was developed as detailed in [38]. The framework incorporates
a fluctuating dark uncertainty, accounting for under reported or unaccounted systematic errors
with the assumption of a normal distribution about zero. This allows a Bayesian estimate of thelikelihood of observing the measured ratios for a given statistical uncertainty. The results andcorresponding uncertainties are:

νAl+ /νY b = 2.162887127516663703(13),νAl+ /νSr = 2.611701431781463025(21),(3.2)

νY b /νSr = 1.2075070393433378482(82).
The result of the collaboration was the measurement of three ratios with fractional uncer-

53

Al+/Yb Ratio

v1
4
No
v1
5
No
v1
6
Fe
b1
6
Fe
b2
8
Ma
r1
Ma
r5
Ma
r6
Ma
r9
Ap
r3
Ma
y3
Ma
y9
Jun
14

Yb/Sr Ratio

b1

Fe 6
b2
8
Ma
r1
Ma
r5
Ma
r6
Ma
r9
Ma
r2
3
Ap
r3
Jun
14

Fe

No

v9

No

v8

No

No

C

b2
8
Ma
r1
Ma
r5
Ma
r6
Ma
r9
Ma
r2
8
Ap
r3
Jun
14

No

Al+/Sr Ratio

Fe

Fractional Offset from
Ratio (10 17)

5.0 B
2.5
0.0
2.5
5.0

v7

10.0
7.5
5.0
2.5
0.0
2.5
5.0

v6

Fractional Offset from
Ratio (10 17)

A

Figure 3.2: Time record of ratio measurements. Frequency ratios were measured from November 2017 to June 2018. Results are shown as fractional offsets from the reported values. Lefterror bars represent pure statistical uncertainty, whereas error bars on the right include systematic
uncertainties added in quadrature to statistical uncertainties.

Fractional Instability

54

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

10 16
Fre

10 17

e-S
p

ace

Fiber Li

Lin

k

nk

Net

wo
rk L

oop
ba

ck

Com

10 18

bs

101

102
103
Averaging Time (s)104

Figure 3.3: Example of daily ratio measurement. Uncertainties for each day are derived fromoverlapping Allan deviations of the corresponding beat. Instabilities for network elements areshown, all well below the measured instabilities of the atomic ratios.

55
tainties of 8 × 10−18 or lower. While this fell just fell of the goal of (5), it represented a massive
step forward from the previous best ratio measurement of 5 × 10−17 of Yb and Sr in 2015 [108].

3.3

Absolute Frequency
Clock comparison data campaigns provide a wealth of data that can be analyzed for purposes

beyond frequency ratios (see the dark matter bounds in [38]). Data taken during the comparison, as
measured by NIST’s OFCs, links the frequency of each clock to International Atomic Time (TAI),
providing a SI traceable absolute frequency measurement of each clock [83, 82]. This absolute
frequency calibration work linked the Sr1 clock to Cs through TAI, fulfilling criteria (3). The Sr
frequency was found to be [83]

νSr = 429, 228, 004, 229, 873.19(0.15) Hz.

3.4

(3.3)

Intra-lab Stability
Concurrent with the clock comparison was a major effort by the Stable Lasers team to bring

the next-generation cavity system online. In December of 2017, the 21 cm silicon cavity sytem(Si3) became the Ye Lab’s newest workhorse. Utilizing a custom commercial comb, the previous
laser system dubbed ‘MJM’ at 698 nm is steered at low frequencies to the Si3 system at 1542 nm[129]. Not only did this radically improve clock precision within the lab, but the 1542 nm lightfacilitated clock comparisons through the BRAN link owing to the ultra-low loss in the telecom
C-band. Throughout the clock comparison, Sr1 not only compared frequencies with NIST, butalso served to characterize the long-term stability of the new Si3 system, allowing upgrades suchas improved thermal shielding and intensity servos. An example of exemplary Si3 stability data isshown in Figure 3.4.
As the clock comparison finished during the summer of 2018, a major focus became a demonstration of the enhanced stability afforded by the Si3 system. An additional goal was to verify thefractional frequency stability measured via the Sr2 self-synchronous imaging technique (3 × 10−17

56

Modified Allan deviation

10 16

10 17 101

102

103

Averaging time (s)104

Figure 3.4: Si3 stability. Long-term stability data of Si3 as measured by the Sr1 system showing
nearly thermal noise limited stability of just under 4 × 10−17 from 10 to 10,000 seconds.

57
at 1 s [33, 93]). Critical to this work was combining the vetted accuracy and long term stability of

Total Allan deviation

the Sr1 system with the record precision expertise of the Sr2 team.

10 17

10 18
101

102

103

Averaging time (s)Figure 3.5: Sr1/Sr2 synchronous comparison. Each clock operated with 600 ms π-pulses and 570
ms deadtime, triggering the clock pulse at the same time. The redline corresponds to a fractional√frequency instability of 3.5 × 10−17 / τ , consistent with the expected QPN contribution of eachclock.

Initial comparisons between Sr1 and Sr2 uncovered several issues. On both systems, fiber
noise cancellation was not conducive to stability at < 10−16 at 1 s, owing to the differential air
path between retro-reflected zeroth order beams and the diffracted clock beam used for clockinterrogation. An updated technique, where the clock laser is turned on with a MHz detuning andthen swept to resonance, allowed direct retro-reflection of the clock beam used for interrogating theatoms, minimizing differential path noise between FNC and atoms. On Sr1, magnetic field noise
was found to limit clock stability, requiring a dedicated effort to find several noisy culprits, mostnotably a Thor-Labs shutter with moving magnets. Sr2 had similar magnetic field issues, but opted
to use the least magnetically sensitive clock transition (see Chapter 1). Finally, Sr2 upgraded their
thermal control to prevent flickering from an uncontrolled thermal environment.

58With these upgrades, Sr1 and Sr2 were able to compare both asynchronously and synchronously (Figure 3.5). The asynchronous comparison, sensitive to laser noise, was consistent
with the Si3 noise model as developed from 3-corner hat comparison with the MJM and Si4 laser√
systems [112]. The synchronous fractional frequency instability (3.5 × 10−17 / τ ) set a new record
for instability between independent clocks, capable of resolving a gravitational redshift of 1 cm in
15 minutes. This improved stability importantly allows fulfillment of the majority of the criteria
discussed for the reevaluation of the second in less than 100 s.

3.5

Towards Optical Timescales
Dissemination of current timescales is based on well developed microwave technology linked

to Cs standards. While impressive, the accuracy and precision are far off the levels promised by
optical clocks. Just as the best clocks now operate in the optical domain, future timescales will also
require all-optical operation to avoid degradation of the accuracy and precision of future primary
optical frequency standards. Such advances will also benefit microwave stability thanks to thetechnology of the frequency comb [106].
The development and maturation of cryogenic, crystalline silicon cavity based lasers overthe last 10 years [75, 97] has opened the possibility for proof-of-principle demonstrations of alloptical timekeeping. With the consistent, high-uptime operation of Sr1 a long-term record of Si3’s
frequency and drift rate began to develop. Spurred on by datasets like that shown in Figure 3.4,
we realized that Si3 was essentially an ‘optical’ MASER, with sufficiently well behaved frequency
drifts that it could serve as the frequency flywheel for periods where an optical clock isn’t running.
To demonstrate the use of silicon cavities as tools for all-optical timescales, the Sr1 clock was
operated nearly daily over a 34 day interval, with 25% uptime. By tracking the Si3 frequency bothwith Sr1 as well as a TAI referenced maser array at NIST (AT1), we were able to compare Si3’s
performance against the current US civilian timescale. As shown in Figure 3.6, Si3 outperforms
AT1 at timescales under 105 s, suggesting daily corrections to Si3 using Sr1 should outperform Cs
steered MASER arrays.

Overlapping Allan Deviation

59

10

Si-AT1
Si-87 Sr
Si Model

−15

10−16

10−17

101

102

103
104
Averaging time (s)

105

106

Figure 3.6: Si3 stability. Si3 was compared against Sr1 and NIST’s AT1 MASER array. For
timescale below 105 Sr resolves Si3 instability well below the current timescale. Around 105 Si3stability sufficiently degrades such that it becomes comparable to AT1.

60
From Sr1 and AT1, the performance of a Si3 based timescale can be modeled and compared
to MASER based timescales (see [101] for details). As shown in Figure 3.7, Si3 steered by even 1
hour of Sr1 per day outperforms all MASER based timescales, even with Sr1 steering the MASERs.
Importantly, fractional frequency uncertainties below 1 × 10−17 become realizable within 1 year of
averaging with an optical timescale. A few additional observations from Figure 3.7 are apparent.
Daily interrogation results in a significant bump in the Allan deviation at the day timescale. Considering the nature of the Sr1 machine at the time of this data, obvious gains could come fromincreased uptime. A dedicated effort to automate relocking of lasers and improve several experimental weaknesses could extend Sr uptime significantly and would be well within the realm ofa national lab’s expertise. An advantage of Si based timescales would be the ability to instantly
improve timescale performance with continued advances in optical clock precision, accuracy, andModified Allan Deviation

reliability.

10−15

10−16
Maser →133 Cs 24 Hours/Day
Maser →87 Sr 1 Hour/Day
Maser →87 Sr 12 Hours/Day
Si →87 Sr 1 Hour/Day
Si →87 Sr 12 Hours/Day

10−17

10−18

103

104

105
106
Averaging time (s)107

Figure 3.7: Modeled fractional frequency stability of different potential timescales. Si3 steered
daily by Sr1, even for one hour, outperforms MASER based timescales. Both Si3 models support
fractional frequency uncertainties below 1 × 10−17 at less than one year of averaging.

61

3.6

Searches For Ultralight Dark Matter

Figure 3.8: Comparison of α sensitivity of the Sr clock transition, Si3 cavity frequency, and H
Maser frequency.

The integration of Sr1 into the BACON network provided a unique opportunity to placebounds on the existence of ultralight dark matter owing to varying α sensitivities between Sr1,
Si3, and the hydrogen masers at NIST (Figure 3.8). Hypothetically, ultralight dark matter may
couple to the fine structure constant α, causing variation in α as a function of time. By performing
frequency comparisons of references with energies scaling to different powers of α, oscillations in
the frequency record at Compton frequency corresponding to the dark matter candidate mass canbe searched for. While our search yielded a result consistent with no detection, it established afurther impact of continued development and operation of clock networks.

Chapter 4

The New Sr1 Wannier-Stark Optical Lattice Clock

Let’s briefly set the stage for the remainder of this thesis. The last two chapters discussed
the original workhorse of Jun’s Sr group - the old Sr1 machine. The machine had been used forearly MOT studies [151, 87], some of the original absolute frequency measurements of the 87 Srclock transition [31] , demonstrations of SU(N) symmetry [155], and increasingly impressive clock
comparisons [90, 16, 112, 38]. Unfortunately the machine was showing its age, from poorly AR
coated viewports to challenging magnetic field stabilization issues. Meanwhile clock precision wasadvancing in the 3D lattice machine (Sr2) and the lab’s dream of spin squeezing was growing more
vivid (Sr3).
In the spring of 2019 a successful pitch was made by T. Bothwell to separate what was
dubbed ‘SrE’ (spin squeezing, accuracy, near zero deadtime) into a rebuilt Sr1 and new Sr3 project(a dedicated spin-squeezing machine). By rebuilding Sr1 separately, the lab would have a highuptime, accuracy focused machine. Chapter 3 made clear how vital such a machine is for a large
clock group. Let’s outline some of the key design goals of a newly rebuilt Sr1.
(1) High uptime. A clock that runs robustly and ideally continuously provides a constantreference, be it for a timescale or one of the other 5 Sr projects in JILA. It also accelerates
the data taking rate, making both adviser and students happy.
(2) Improved vacuum lifetime. The old Sr1 machine had a vacuum lifetime of 8 seconds,
limiting interrogation times and increasing the background gas collision systematic.

63(3) Vertical lattice. Sr1 had run for years with a lattice tilted with respect to gravity at
18◦ , leading to an inability to operate at < 25 Erec before atoms were lost. This createsproblems for Raman scattering and lattice light shifts. Exploration of shallow trap depths
was a longstanding goal.
(4) 813 nm build-up cavity inside vacuum. The retro-reflected lattice setup in Sr1 was amajor hindrance to clock operation. Alignment drifts required regular correction, limiting
uptime and operational stability. To reduce atomic interactions with a reasonable trap
waist, a high power Ti:sapphire laser system was required which regularly broke leading
to weeks of unpredictable downtime every year. A build-up cavity enables a solid state
diode based lattice system, provides geometrical stability to the system, and supports asignificantly larger trap waist for reduction of atomic interactions. To enable reasonablefinesse for power enhancement and ensure a pristine wavefront the cavity must be invacuum.
(5) Improved thermal stability. The original Sr1 had numerous heat sources, from poorly
cooled MOT coils to the Zeeman slower. These presented a challenge both in terms of characterizing the BBR environment of the atoms as well as placing demands on environmental
control in order to robustly operate the machine. Incorporation of temperature monitoringand stability during construction would be beneficial.
(6) Increased atom-light interaction times. There was no physical reason that suggested
the 1D architecture should be limited to the 600 ms pulses used for the Sr1/Sr2 comparisons.
We had tried longer pulses, but Rabi spectroscopy at the second timescale was unreliable in
the Sr1 machine. An important clue however, was the increase in contrast and interrogation
times when we switched to larger trap waists for Chapter 2, enabling reduced atomic
densities.
To accomplish these goals Sr1 was rebuilt. Let’s introduce the new system.

64

4.1

The New Sr1 System

Figure 4.1: Post bake Sr1 system. The system is supported by black anodized aluminum supports.
To the left is the AOSense atomic beam system (circled in red) for providing a steady stream ofcooled strontium into the MOT. Center is the Kimball Physics 6” Spherical Octagon, with the 6”
CF viewport facing the camera. All viewports were mounted using silver coated threaded studs,
washers, and nuts for use in mounting important clock hardware. The large white G7 piece (bluearrow) mounted to the raw aluminum legs is the MOT coil that required construction and mounting
before baking. Optics on the table were used to check the cavity alignment post bake.

At the heart of the system is the original Sr1 Kimball Physics 6” spherical octagon (Figure4.1). The chamber is oriented with a pair of 2.75” CF flanges coaxial with gravity and the 6” CFflanges perpendicular. This allows incorporation of the in-vacuum build-up cavity into the groove
grabbers on the 2.75” axis and improved imaging and conductance through the 6” CF flanges.
Vacuum quality is significantly improved compared to the previous Sr1 system [95]. Vacuum
conductance through a pipe scales as (diameter)3 , giving a factor of 10 improvement in the new
design by pumping through the 6” versus 2.75” CF flange. The ion pump is upgraded from a 40
L/s to 150 L/s Starcell design for enhanced pumping of noble gases. The addition of two 400 L/s

65
non-evaporable getters provides pumping for getterable gases (particularly H).
The chamber is supported by milled aluminum pieces (Figure 4.1). The ion pump is supported
by a solid block of aluminum. All pieces are coated black to prevent light scattering. Compliance
between supports and chamber is ensured through the use of Viton, providing additional dampening.

Figure 4.2: Image of clock retro box without cover. The assembly is attached to the threaded rodsof the bottom 2 3/4” viewport, ensuring a rigid connection between chamber (lattice retro) andbox (clock retro). Lattice light (green arrows) is introduced to the cavity from above, with leakage
below used for intensity stabilization. Clock light (red arrows) comes from the table and is partiallyreflected by a wedged mirror in the Thor Labs mirror mount enabling fiber noise cancellation nearly
to the viewport. A slot enables various neutral density filters for clock power attenuation. A highquality dichroic beamsplitter separates lattice light (813 nm) from clock light (698 nm).

Magnetic fields are controlled by the Anti-Helmholtz (AH or MOT) coils and 3 pairs ofHelmholtz (bias) coils. A particular nuance in this design is the necessity to build and install the
AH coil in the middle of the system before before baking the chamber. The AH coils are built using
G7, a commercial composite built of glass fabric and temperature resistant resin, allowing baking

66

Figure 4.3: Installed mezzanines around the Sr1 chamber. Red arrows show installed bias coils.
The copper attachment within the front MOT coil assembly contains the Bx bias coil, with watercooling lines for temperature control. The top mezzanine supports optics for the 813 nm latticesystem while cavity transmission optics, clock optics, and MOT optics are on the table.

67
up to 200◦ C. The MOT coils are constructed out of hollow core copper tubing with temperature
resistant insulation potted in a temperature resistant epoxy. The coils are supported by hollow,
square aluminum pieces filled with lead shot. All components of the MOT coil assembly can be
baked up to 200◦ C. After baking, the second MOT coil was installed. The bias coils are mounted
to the MOT coil assembly as shown in Figure 4.3. The bias coils were wound in place (using equal
lengths of wire to ensure equal numbers of turns) and then potted to provide structural robustness.
The Bx bias (axis through the 6” CF flanges of the main chamber) is built into 6” copper pieces
used for temperature control of the chamber. More details of temperature control will be specified
in the following sections.
A major improvement in the rebuild of Sr1 is the removal of degrees of freedom. Threadedrods were used for the mounting of all viewports, enabling attachment of system pieces directly thechamber (Figures 4.2 and 4.9). This was a valuable lesson learned during the work to modernize theSr1 system for the clock comparison (Chapter 3) - if it can be adjusted it can drift. Each MOT axis
incorporates a temperature control apparatus (itself attached to the threaded rods) threaded using
standard Thor Labs SM threads. This enables MOT waveplates, retros, and irises for alignment
to be attached to the chamber with excellent precision. Since initial construction and optimization(1.5 years ago), none of these have been touched.
One final improvement in the new system is the use of mezzanines around the chamber for
optics (Figure 4.3), eliminating the prevalence of periscopes prone to drift in the previous machine.
Each mezzanine is a standard 24”×48” honeycomb breadboard. Legs are black coated aluminum
tubing filled with lead shot for vibration dampening [32]. Note - it is important to use non-magnetic
steel or aluminum. We found during the studies of Chaper 6 that we had magnetically charged theoriginal Thor Labs breadboard next to the 6” imaging window. Exchange of this breadboard to analuminum version reduced the background magnetic field gradient across the atomic sample nearly
an order of magnitude.

68

Figure 4.4: Cavity mirror pictures. Top left: Top mirror assembly, with bottom stainless steel
plate for mounting to groove grabbers, mirror glued to grounded copper piece, and piezo sandwiched
between copper and steel. This ensures charges from applied voltages to the piezo do not charge
the mirror. Top right: Groove grabber cavity mirror support structure inside 2 3/4” CF flange.
Bottom left: Installed top mirror assembly. Bottom right: Picture of Faraday shielding cage
between atoms and mirror.

69
4.1.1

The Build-Up Cavity

At the heart of the new Sr1 system is the in-vacuum build-up cavity for the lattice laser 1 .
The basic design criteria are:(1) Vertical mounting. We want the lattice to be parallel to gravity, enabling the lowest
operational trap depth possible without atom loss. This is critical for reduction of both
Raman scattering and lattice light shifts.
(2) High quality 698 nm AR coating. We need quality clock spectroscopy, meaning no
significant standing wave at 698 nm.
(3) Large build-up factor. For lattice light shift evaluations trap depth operation over 500
Er is desirable. Commercial diodes offer around 500 mW of power at 813 nm and would
be a far more robust solution for the optical lattice than a commercial Ti:sapphire system(and much cheaper!). Finally a large build-up factor would enable a larger trapping waist.
(4) A large waist. The previous sytem used a 80 micron waist. Increasing this by a factorof three would increase trapping volume by nearly a factor of 10, reducing the density
shift systematic significantly. Furthermore, a large waist corresponds to a large Rayleighrange, which when combined with a modest finesse (near 1000) ensures homogeneous lattice
trapping conditions throughout the atomic sample.
(5) Faraday shielding. Patch charges may build up on the cavity mirrors, and the piezo forcavity stabilization may provide additional fields [85]. The atoms require shielding fromthese electric field sources to avoid deleterious DC Stark shifts.
To address (1) the cavity mirrors are mounted using Kimball Physics groove grabbers in the
2 3/4” CF flanges oriented along gravity (Figure 4.4). By using the precisely machined, one-piece
1
Special shout-out to John Robinson and Eric Oelker. The robust, well designed in-vacuum build-up cavity is
the heart of the new Sr1 system and would not be the same without their cavity expertise. John led the design,
in-vacuum construction, and initial testing. Eric put together the locking infrastructure and careful mode-matchingsolution. John’s original notes are the starting motivation for this section.

70
vacuum chamber as the cavity spacer we gain a robust, stable lattice that forms the physical basis
for our experiments.
Mounting the cavity within the groove grabbers set the spacing between mirrors to ∼ 15 cm.
The waist for a symmetric resonator in vacuum is given by [105]

λ
ω02 =
π

r

dR d2
−
2
2

(4.1)

where λ is the cavity wavelength, d the cavity length, and R the radius of curvature of the mirrors.
For standard mirrors and our dimensions we settled on R = 1 m, giving ω0 ≈ 260 micron, a 3x
increase compared to the lattice from Chapter 2. This corresponds to a Rayleigh range of zr =26 cm. Measurements of the free spectral range of the cavity (νF SR = c/2d) performed
after completion gave a cavity length of d = 16.9 cm, corresponding to a waist of 262µm.
To ensure sufficient power enhancement of the cavity for deep trap depths while using a diodelaser system, an over-coupled design was chosen. The power enhancement factor is given by [128]

Icirc
4δ1
≈
Iinc
(δ1 + δ2 )2(4.2)

where δi = ln(1/Ri ) with Ri being the reflectivity of mirror i. Equation 4.2 assumes ideal mirrors√
with no loss besides transmission. The corresponding finesse for the coefficient of reflection ri = Ri
is given by [105]√π r1 r2F =.
1 − r1 r2

(4.3)

np.
To optimize power buildup while ensuring sufficient transmission through the low loss mirror
for intensity control mirrors were chosen to have a transmission of T1 = 1 − R1 = 6000 ppm andT2 = 1 − R2 = 100 ppm, corresponding to B = 643 and F = 1027.

71
4.1.1.1

Finesse Measurement

To measure the finesse of the cavity recall that the average time light remains inside thecavity is given by t0 = 2πc/dF where c is the speed of light, d the standing wave resonator length,and F the finesse. For reasonably high finesse, t0 is found by quickly turning off the light incident
on the cavity and measuring the exponential decay of transmitted power P as a function of time t,
as given by [105]P (t) = P0 e−t/t0 = P0 e−(2πc/dF )t .

(4.4)

The results of a post-bakeout finesse measurement are shown in Figure 4.5. The finesse
was found to be F = 1020(60), with the uncertainty due to the pulling of the fit based on initial
cuts of data points. The associated decay time constant (Figure 4.5) is quite fast for an AOM,
requiring exclusion of datapoints at the onset of extinction of the ingoing cavity light. The results
are consistent with the designed mirror specifications previously listed.

4.1.1.2

Birefringence Measurement

For rejection and/or control of differential AC Stark shifts to the ground and excited clockstate (lattice light shifts) it is usually ideal to operate with linear polarization for the lattice light.
Circular polarization introduces several complications, most notably by altering the tensor shift ofthe lattice. A small mixture of non-circular polarization acts a fictitious magnetic field, changing
the alignment of lattice polarization relative to the quantization axis set by a bias field. This
perturbation is given by [127]∆νvt = −γvt |~ · ~B |2 ξ cos θU02 .

(4.5)

Here γvt = 11 µHz/E2r for a 1 G field. ~ describes the lattice polarization and ~B the unit vector of
the bias field. θ describes the angle between the lattice light wavevector and bias field and U0 the
trap depth. ξ is the ellipticity of the lattice light.

Amplitude (arb)

0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0.05

72

Ring down measurement

6.0

6.5

7.0

7.5

8.0

Time ( s)

8.5

9.0

9.5 10.0

Figure 4.5: Cavity ring-down measurement. Transmitted intensity through the cavity is measured
on a PD. The light is extinguished by an acousto-optic modulator (AOM) and the decaying exponential fit, finding a decay time constant of 183 ns. From this decay and measured cavity length afinesse of F = 1020(60) is found, with the large uncertainty arising from the sensitivity of the fit
to the number of data points cut before fitting.

Equation 4.5 tells us that we have a scaling with U02 , the same as the hyperpolarizability
coefficient. We would therefore like to make sure that our ellipticity 1) is such that evaluations of
the hyperpolarizability coefficient will not be polluted and 2) provides no meaningful contribution
on our path towards accuracy at the 19th digit.
The ellipticity parameter is given by

r
ξ=

Pmin
Pmax

(4.6)

where Pmin (Pmax ) is the minimum (maximum) power transmitted through a rotated polarizer after
the optic in question. To ensure pure polarization before entering the cavity (including viewports),a polarizer was placed before the chamber. A polarizer placed on the opposite side was rotated 360◦
to probe the transmitted light, noting the minimum and maximum power through the polarizer. themeasurement was repeated a second time after rotating the initial clean-up polarizer 45◦ to bound73
a worst-case ellipticity should the cavity be probed off the eigen-axis of birefringence. We bounded
the extinction ratio to 3.2 × 10−5 (ξ = 5.7 × 10−3 ), likely limited by the achievable extinction ratiosof the given polarizers.
For our system, even with applied fields as small as 200 mG, we anticipate ∆νvt contributing
at < 1/100 of the level of hyperpolarizability.

Hyperpolarizability
200 Hz Splitting
600 Hz Splitting
1 kHz Splitting

10 15
10 16

Fractional Shift

10 17
10 18
10 19
10 20
10 21

0

200

400

600

Trap Depths (E_rec)

800

1000

Figure 4.6: Comparison of hyperpolarizability and birefringence induced flock shifts as a functionof trap depth as bounded by worst case ellipticity. Results are reported for mF =±9/2.

4.1.2

DC Stark Plates

In the early days of Sr OLC research, SYRTE was a major proponent of in-vacuum build-up
cavities [29]. Unfortunately they found large DC Stark induced systematic shifts (10−13 level), hindering accuracy and requiring extensive work to cancel [85]. Care must be taken when attempting
to cancel a DC electric field across an atomic sample as the DC Stark shift is quadratic (Chapter 2),

74
meaning a global measurement of the DC Stark shift (and elimination) may still be contaminatedby a gradient, resulting in a residual shift.

Figure 4.7: DC Stark plate construction. Top left: Example of a quadrant plate split in half,showing groove grabber assembly for both assembling all four quadrants together and mounting to
the chamber. Note the use of alumina to electrically isolate each quadrant. Top right, bottom
left: Picture of the plates installed into the Kimball Physics chamber, showing wiring. Bottom
right: Assembly for routing quadrant plate wires around 2 3/4” CF port.

To avoid the above issues, it is always best to begin with as close to zero DC Stark shift
as possible. From the previous Sr1 machine we know that viewports don’t spontaneously develop
patch charges - one must polarize them. To avoid patch charges from the cavity and residual fields
from the cavity piezo, the atoms are shielded from each mirror assembly by a grounded copper
Faraday shield. The shield of copper is 5 mm thick with an ID of 6 mm, matching the piezo ID.
COMSOL simulations showed that the design would provide sufficient attenuation of any reasonable75
patch charges and fields from the piezo.
In the previous machine, quadrant plates for applying an electric field across the atoms
enabled probing and elimination of DC Stark shifts in all three spatial dimensions. A constantworry was the the issue of developing charges around the chamber from application of large electricfields near dielectrics (we did temporarily create a background field but it dissipated quickly [25]).
To avoid this issue in the previous machine the applied fields were regularly switched during thesequence (atom preparation period), enabling a nearly zero time average for the applied field. In
the new machine we incorporated bias plates inside the vacuum chamber. By reducing the distance
between the plates we readily increase the strength of applied field. By building the plates invacuum, worries about charging dielectrics is removed.
The DC Stark bias plates in the new system are shown in Figure 4.7. They are built into the6” CF flange groove grabbers of the Kimball Physics chamber, with each side having an assemblyof 4 copper plates. They are electrically isolated from each other by the use of alumina spacers.
Each plate has a separate electrical connection out of the chamber, allowing each quadrant to becontrolled separately. The wires pass though a grounded braid to eliminate bias fields from thewires themselves.
For initial DC Stark evaluations we applied voltages of ±80 V to the X, Y, and Z directions.
Recall X is the axis of the 6” CF flanges and Z the axis of the cavity. Figure 4.8 shows the resultsfor X, with no statistically meaningful field found at the 2 × 10−19 level. Y and Z additionally
showed no statistically meaningful field found at the 2 × 10−19 level. After installation of imaging,
DC Stark gradients were also not found at any meaningful level for accuracy purposes (Chapter 6).

4.1.3

Temperature Control

Extensive temperature control of the new system was installed during construction, eliminating the need for painful installation of a thermistor array after the full rebuild. Baseline thermal
control is ensured by utilizing temperature controlled air pulled through a HEPA filter unit onto
both the chamber and laser side of the table (portioned for light control and independent temper-

76

Figure 4.8: DC Stark evaluation along X. On the left is the Allan deviation of the evaluatedbackground shift for voltages biases of ±80 V applied to each plate (160 V difference). A background shift of 6(23) × 10−20 was found, consistent with zero. The right plot shows a histogram ofbackground shifts.

ature control), as in Chapter 2. MOT coil temperatures are controlled by a powerful recirculating
chiller, itself water chilled ensuring removal of heat from the lab. Each MOT coil, while electricallyconnected, is split into two in regards to water connections allowing significantly improved coolingcompared to the previous Sr1 design. The blue MOT is additionally operated with <1/2 of thegradient of the previous machine, further reducing the heatload.
50 mK interchangeable thermistors (PR503J2) for monitoring the chamber and air temperatures were mounted after baking using thermally conductive epoxy. Building on the lesson ofChapter 2, each viewport has a dedicated temperature control apparatus installed, providing bothtemperature control and an insulating air pocket over the airport for improved homogeneity (Figure4.9). The pieces are constructed of copper to provide maximum thermal homogeneity to minimize
any temperature gradients.
A major improvement to the new machine is the elimination of heatloads present in theprevious design. The AOSense atomic beam system uses permanent magnets, eliminating current
controlled bias coils - a large negative of the previous machines Zeeman slower design. The AOSense
system furthermore steers the atomic beam around a corner, eliminating line of sight between heated
Zeeman window and atoms. The camera used for imaging is water cooled. The summation of these
improvements and controls is a stable, robust system. Alignment is robust over the timescale of

77

Figure 4.9: Temperature control/optics mounting pieces for the 2 3/4” CF flanges (viewports). Top
left: The main control piece is a copper cylinder wrapped with hollow copper tubing solderedin place. The tubing allows temperature controlled water to flow around the piece, stabilizing
the temperature. Holes for mounting to the chamber (bottom left), top piece installation (topright), and thermistor installation are included. Bottom right: An installed assembly includingThor Labs SM2 lens tube for mounting of MOT optics is shown. Jack screws enable fine alignmentof the retros but are not needed thanks to the Kimball Physics chamber and careful machining ofcopper pieces. A thin layer of high thermal-conductivity paste is used between the piece shownin the top left and the vacuum chamber, ensuring good thermal conduct and compliance for any
slight imperfections.

many months, radically reducing the maintenance required in the previous version. Examples of
the temperature control stability on the chamber (viewports) are shown in Figure 4.10 (4.11).

78
Sr1 Table Chamber Side
22.6

6 Way Cross E
6 Way Cross W
Oven Valve
Chamber HEPA - Servo
Lattice Fiber Top
PMT Andor
X MOT Retro Table
X MOT Input Window
X Input Table
TOP PDH PD Table
Absorption Fiber Middle TableNo Probe XArm Input Table
Ion Pump
Ion Pump Table
813 Transmission PD

Temperature (C)

22.4
22.2
22.0
21.8
21.6
21.4

0

5

10

Time (Hours)

15

20

25

Figure 4.10: Chamber temperatures from 8/14/21. Monitoring thermistors are mounted to thechamber.

Sr1 Chamber Viewports
22.10

Temperature (C)22.08
XMOT Probe Retro Window
XMOT Retro Window
X MOT Probe Input Window
X MOT Input Window
Cavity Top Window
6 CF Window Front Left
6 CF Window Back Left
Chamber Yoke

22.06
22.04
22.02
22.00
21.98

0

5

10

Time (Hours)

15

20

25

Figure 4.11: Viewport temperature from 8/14/21. Each sensor is attached to a copper control piece
mounted to the respective viewport.

79
4.1.4

Expansion Port

Figure 4.1 shows the expansion port present on the Sr1 system. An MDC gate valve wasinstalled for use in the next accuracy evaluation (10−19 level). The plan is to add a custom chamber
for installation of an in-situ thermometer for probing the temperature (BBR) at the atoms. A 30
cm linear push-pull device will be used to insert/retract the probe. The expansion chamber will
include additional pumping (25 L/s ion pump and 50 L/s NEG) as well as a residual gas analyzer
to document the composition of the vacuum chamber gas load for the background gas systematic.

4.2

Experimental Operation
The Sr1 system largely follows the operation described in [21, 88, 95, 25].
Standard blue MOT operation proceeds for ∼ 100 ms, though at a lower gradient (∼ 20G/cm) compared to the old machine (50 G/cm). At the end of the blue MOT sequence the MOT
gradient is ramped to ∼ 6 G/cm over 10 ms and the blue intensity is reduced, leading to a colder,less compressed blue MOT. The magnetic gradient is then set to ∼ 3 G/cm for the broadband(BB) and single frequency (SF) red MOT operation. With atoms < 10 µK loaded into the 300
Er optical lattice an additional cooling stage is performed. Atoms are simultaneously nuclear spinpolarized, doppler cooled (< 1µK, Figure 4.12), and axially cooled (nz ≈ 0, Figure 4.13). This isenabled by reducing the intensity of the cooling light provided by the 9/2 to 11/2 cooling laser by> 10, 000.
After cooling, the lattice is adiabatically ramped to the operational trap depth. Dependingon the experiment, a series of rapid clock laser pulses is used to change the nuclear spin state of thepolarized sample. Standard clock spectroscopy [21, 88] proceeds, after which the lattice is ramped
back to 300 Er and the excitation fraction is readout by fluorescence spectroscopy. Imaging photons
are collected on both a standard PMT for center of mass clock locking as well as an Andor iXoncamera for in-situ imaging [93].

80

0.175

790.0 nK

Excitation Fraction

0.150
0.125

166.0 nK

0.100
0.075
0.050
0.025
0.000
−50

−25

0

25

50

Detuning (kHz)

Figure 4.12: Doppler scans after initial loading and cooling at 300 Er (purple curve) and after
ramping to an operational trap depth (15 Er shown in green).

4.3

ImagingStandard fluorescence imaging is used in the system. The calibrated camera pixel size (in-cluding imaging system) is evaluated to be 6 micron based on free-fall time of flight measurements[25].
Important for in-situ imaging is the consistent scattering rate of imaging photons across theatomic sample. In the Sr1 system, the density shift systematic leads to inconsistent scattering rates
creating erroneous frequency shift corrections. To avoid this, we shape the probe beam to have an
aspect ratio of nearly 3 to 1, with the long axis matched to the millimeter atomic sample (Chapter
6). The scattering rate throughout the atomic sample is consistent at the 1% level (Figure 4.14).

81

Excitation Fractiona) 0.510 Erec

0.4
No Axial Cooling
Axial Cooling
Adiabatic Ramp

0.3
0.2
0.1

300 Erec

0.0
−100

−50

0

50

100

Detuning (kHz)b)

Red Sideband @ 300 Er c)

0.125

Excitation Fraction

Excitation Fraction

0.150

0.100
0.075
0.050
0.025
0.000

0.150

Blue Sideband @ 300 Er

0.125
0.100
0.075
0.050
0.025
0.000

−120

−100

−80

Detuning (kHz)

80

100

120

Detuning (kHz)

Figure 4.13: Axial clock scans. a) Axial scans at 300 Er and after adiabatically ramping to
10 Er with atoms prepared in the excited clock state. The green line has been offset by 0.5
from an excitation fraction of 1.0 for plotting. Note the near absence of red sideband indicatingnz ≈ 0. Additionally note the spiky structure within the sideband structure - these correspond tospectroscopy of varying nz states in the purple curve, now resolved thanks to the < 1 µK radial
temperatures at 300 Er . b) Enhanced view of the red sideband. c) Enhanced view of the blue
sideband.

82

Figure 4.14: Calibration of fluorescence probe scattering rate throughout the atomic sample. To
evaluate the scattering rate per pixel, a series of atom pictures is taking with varying probe power,
allowing calculation of the saturation parameter (ratio of intensity to saturation intensity), andtherefore maximum scattering rate.

4.4

Vacuum Lifetime
The vacuum lifetime leads to an atom loss mechanism, potentially increasing QPN at longinterrogation times through reduction of maximum atom number. Background gas collisions also
lead to a systematic shift of the clock frequency.
Evaluation of the vacuum lifetime is readily performed by lifetime measurements of 3 P2 atoms
trapped in the AH magnetic field. Loading a blue MOT without repumps over the course of several
seconds loads a significant population into magnetically trapped 3 P2 states. With a magneticallytrapped population, the lasers may be extinguished for a varied period of time. After holding, theatoms are repumped and the population is measured via fluorescence. The atomic population (N )as a function of hold time is then fit to the following equation

N (t) = N0 exp(−t × (1/τ + 1/τBBR )) + C,

(4.7)

where N0 is the initial population, t is the varied hold time, τ the vacuum lifetime, and C an

83
offset. τBBR = 104 s is the lifetime of the 3 P2 state at room temperature [152]. Figure 4.15 shows
the vacuum lifetime as evaluated at a variety of oven temperatures of the AOSense oven. It was
found that as the oven pressure was increased the vacuum pressured degraded. We find for an oventemperature of 375◦ τ = 27 s, just shy of a factor of four improvement compared to the previous

Vacuum Lifetime (s)system despite the addition of significant in-vacuum hardware.

30
25
20
15
10
400

420 440 460
Oven Temp. C

480

Figure 4.15: Vacuum lifetime versus oven temperature. The trend is well reproduced by a linear
fit of slope -0.167 s/◦ C and offset of 89.8 s at 0◦ C.

Chapter 5

Wannier-Stark States, a Spin Model, and the ‘Magic Depth’The study of atomic interactions within optical lattice clocks has a rich history, with dedicatedexperimental and theory collaboration critical for developing the spin model characterizing theinteractions [94, 95, 118]. The ability to understand the atomic interactions in our system using
the spin model has not only benefited clock evaluations, but enabled studies of SU(N) symmetry[155] and synthetic gauge fields [76, 26].
In this chapter we explore atomic interactions in a Wannier-Stark (WS) ladder [143]. We
begin by reviewing the calculation of the WS wavefunctions, illuminating the origin of a new s-wave
interaction in the clock. We briefly connect this work with our earlier spin-orbit coupling (SOC)
studies, emphasizing the power of analyzing our system utilizing the tools of solid state physics[81, 67]. With then turn our attention to the spin model, qualitatively discussing the effect of
atomic interactions on Rabi spectroscopy. With this understanding we finally discuss the ‘magic
depth’ operation in the Sr1 system [1], demonstrating an operational regime where precision andaccuracy are combined as demonstrated by Chapter 6.

5.1

Bloch, Wannier, and Wannier-Stark Wavefunctions
The physics of independent electrons moving in a lattice is well known [3, 49], providing an

excellent basis for the study of atoms in an optical lattice. We begin with the Hamiltonian for a

85single particle in a periodic potential, given by

H=

p~ 2
+ V (~r)2m(5.1)

where p~ is the momentum operator, m the particle’s mass, V the potential, and ~r the position
operator. Consider the case of a constant potential. In this situation H is invariant under any
translation. As p~ is the generator of translation, this means[H, p~ ] = 0,

(5.2)

and thus eigenstates of p~ (plane waves) are eigenstates of H. Now instead consider a periodic
potential

V (~r + ~a ) = V (~r),(5.3)

where ~a corresponds to a primitive lattice vector (in the 1D Sr clock, λ/2). The Hamiltonian is
now only invariant under translation by a primitive lattice vector (T~aj ). Bloch’s theorem [49] tells
us that the general solution to Equation 5.1 with a potential of the form of Equation 5.3 is givenbyψq~ (~r) = ei~q·~r/~ uq~(~r)(5.4)

where ~q is crystal (lattice) quasimomentum, being a good quantum number for the Hamiltonian
under study. Note the solid state literature typically uses crystal momentum ~k = ~q/~, but here weuse ~q to both be consistent with the ultracold community and avoid additional use of the letter ‘k’.
The translational symmetry of the potential is carried by uq~(~r):

uq~(~r + ~aj ) = uq~(~r).

(5.5)

86
To proceed, consider the result of the momentum operator applied to Equation 5.4:

p~ψq~(~r) = ei~q·~r/~ (~q + p~)uq~(~r).

(5.6)

We may then find the modified Schrodinger equation to be




1
2
Hq~un~q =(~p + ~q) + V (~r) un~q
2m
(5.7)

= En~q un~q

where En~q is the eigen-energy and n is the so called ‘band index’ arising from restricting oursolutions to the first Brillouin zone [49].
Let us now consider the 1D problem, denoting 1D position z, anticipating connection to our
1D optical lattice. Thanks to the Bloch theorem both our optical potential and functions un~q havethe same periodicity, allowing us to write both as a discrete Fourier sum [54]:V (z) =

X

Vb ei2bkz

and un,~q =bX (n,q)cl ei2lkz .

(5.8)

l

It follows that

V (z)un~q =XXbVb ei2(b+l)kz cn,qllX n,q i2(l+1)kz
i2(l−1)kz
= −V0
cl
e
+e

(5.9)

l

where we exploit the expansion of the potential,

V (z) = −V0 cos(kz)2


1
2ikz
−2ikz
= − V0 e+e+2 .
4Similarly, we expand the kinetic energy term,(5.10)

87

1
1 X
(n,q)
(p + q)2 un~q =(p + q)2 e2ilkz cl
2m2ml(5.11)

1 X(n,q)=(2~kl + q)2 e2ilkz cl .
2mlWith these in hand, we may finally rewrite Equation 5.7 asX  (~k)2 
q 2 (n,q) V0 (n,q)(n,q)(n+q) 2iklz
2l +cl− [cl−1 + cl+1 ] − En cl
e
= 0.
2m
~k
4

(5.12)lEquation 5.12 may be written as an infinite square matrix of dimension l × l, though forcalculations l is typically truncated to around 20 lattice sites. The truncated, tridiagonal matrix isof the form [54]:

X

(n,q)

Hl,l0 · cl
(2l + q/~k)2 Erec





and Hl,l0 = −V0 /4







0

(n,q)

= Eqn cll

if l = l0
if |l − l0 | = 1(5.13)else.

For 5 lattice sites the Hamiltonian matrix takes the form

















−V0
4

0

−V0
4

(−2 + q/~k)2

−V0
4

0

0

0

−V0
4

(q/~k)2

−V0
4

0

0

0

−V0
4

(2 + q/~k)2

−V0
4

0

0

0

−V0
4

(4 + q/~k)2(−4 + q/~k)2

0

0

(n,q)

which when diagonalized gives the eigenergies and Fourier coefficients cl







.







. The corresponding

Bloch states can then be readily found using the evaluated coefficients and Equation 5.8. Solving
for the eigenergies of different bands allows calculation of the tunneling rate (J) as the width ofeach band corresponds to 4J. Energy solutions and Bloch wavefunctions are shown in Figures 5.1
and 5.2, respectively.

88

4 Er

0 Er

10

12 Er

24 Er

Energy (Er)

8
6
4
2
0
1

0

q ( k)

1

1

0

1

1

0

1

1

0

1

Figure 5.1: Bloch band structures of optical lattices for varying trap depths. Each plot showsenergy versus quasimomentum for the first Brillouin zone. Purple, green, and yellow correspondto the ground, first, and second bands respectively. Note the decreasing bandwidth as trap depth
increases, corresponding to decreased tunneling energy.

89

8 Er

8 Er

1.5

2.5

1.0

2.0

0.0

| |2

Re( )

0.5

1.0

0.5
1.0

q=0
q= k/2
q= k

1.5

0.5
0.0

20 Er

2.0

20 Er
3.5

1.5

3.0

1.0

2.5

0.5
0.0

| |2

Re( )

1.5

0.5

2.0
1.5
1.0

1.0
1.5

0.5

2.0

0.0
1.0

0.5

0.0

z ( lat)

0.5

1.0

1.0

0.5

0.0

z ( lat)0.5

1.0

Figure 5.2: Wavefunctions (ψ) and probability densities (|ψ|2 ) for Bloch wavefunctions of trap
depths 8 and 20 Er . The real part of the wavefunction is plotted.

90
As shown in Figure 5.1, the bandwidth decreases with increasing trap depth, analogous to
the Lamb-Dicke regime in the spectroscopy community [80, 89]. As trap depth increases, eventually
J << Ω where Ω is the Rabi frequency. In the language of atoms trapped in an optical lattice, thiscorresponds to the atomic wavefunctions becoming increasingly localized. Wavefunctions in thisregime may be treated at harmonic oscillators, or building on the Bloch states, Wannier states.
The Wannier function for band n on site i in the 1D lattice is given by [49, 54]w(z − zi ) = N −1/2Xe−iqzi /~ ψqn (z).

(5.14)

q

See Figure 5.3 for examples.
In clocks, tunneling broadens lineshapes and leads to detrimental systematics. This may be
avoided by operation at deep trap depths, or better yet, by providing an energetic offset between
lattice sites to suppress tunneling. This is a well known problem in solid state physics with thefollowing Hamiltonian:

H=

p~ 2
+ V (~r) + F ~r,2m(5.15)

the solutions of which are the well known Wannier-Stark (WS) states [51]. The Hamiltonian isreadily realized in a lattice clock by orienting the lattice along gravity [81]. In the Tight-Binding
approximation, the WS state for lattice site l may be found to be [50]|Ψn,l i =

X


Jm−l

m

=

X
m


Jm−l

∆n
2dF


|mi ,

4J0
M gλlat(5.16)

|mi .

Note the second line is valid for the ground band of the lattice. Here the WS state of lattice sitem is given by |mi, Ji is the Bessel function of the first kind for order i, ∆n is the bandwidth of theband n, d the distance of the lattice spacing, and F the applied force as given in Equation 5.15.
This simplifies to the second line for the Sr1 system where M is the mass of the Sr atom, g gravity,

91
h p
i
√
1/4 3/4
λlat = 813 nm, and J0 ≈ (4/ π)Er V0 exp −2 V0 /Er ) (valid for the ground band)[34, 1].
Numerical results are plotted in Figure 5.4.

| |2

m=0
m=-1
m=1
Vlat

2.0

1.5

1.0

0.5

0.0

z ( lat)

0.5

1.0

1.5

2.0

Figure 5.3: Example Wannier state probability distributions as a function of lattice position at a
trap depth of 3 Er . m gives the lattice index for each plotted Wannier function. The optical latticepotential is plotted in black for reference.

5.2

Spin-orbit Coupling
The solid state tools used in the previous section for calculating wavefunctions provide a

powerful analytical tool for the study of atomic clocks. Historically, the mature field of ion clocks
enabled rapid progress in optical lattice clocks using the framework of ions in harmonic traps

92

101

5 Er
10 Er
20 Er

100

| |2

10 1
10 2
10 3
10 4

1.5

1.0

0.5

0.0

z ( lat)

0.5

1.0

1.5

Figure 5.4: Wannier-Stark state probability distributions for a variety of trap depths. All states
are centered at m = 0, making clear the increasing off-site contributions as trap depth is decreased.
This delocalization of the wavefunction provides the ability to tune the overlap of wavefunctions
between lattice sites, creating a tunable channel for s-wave interactions.

93[80, 89]. In contrast, studies of many-body physics in the 1D lattice clock [95, 155, 25] and thedevelopment and study of the 3D lattice clock [33, 52] exemplify the importance of the solid state
approach. A great example of this is the Sr2 proposal paper discussing increased lattice constantsfor operation of shallow trap depths without motional dephasing [67].

Energy vs Quasimomentum

0.08

Histogram
g
e
e-g

0.06

60
50

0.04

Counts = DOS

Energy

0.02
0.00
0.02
0.04

30
20
10

0.06
0.08

40

1.0

0.5

0.0

q ( k)

0.5

1.0

0

Energy (Arb)

Figure 5.5: Spin-orbit coupling in a horizontal 1D lattice. Left: Ground Bloch bands for 10 Er
for both ground (g) and excited (e) clock states with the energy of the clock photon subtracted.
The excited band has been shifted by the momentum of the clock photon. Due to this, the energy
difference between e and g depends on quasimomentum, coupling spin to momentum (SOC). Right:
A histogram of energy difference with evenly sampled quasimomentum. The turning points of theenergy difference in the left plot correspond to an increased number of quasimomentum points atlarge detunings - the Van Hove Singularities.

To exemplify this point further we briefly review the origins of spin-orbit coupling (SOC) in
the Sr1 system with a horizontal (perpendicular to gravity) lattice [76, 25]. We return to the Blochbands of Figure 5.1. Immediately we note that the energy associated with a particle is dependent
on its quasimomentum. In a magic trap (the same potential for ground and excited clock state) the
bands are the same, but we must account for the change in quasimomentum from absorption of aclock photon (Figure 5.5). We find that this intricately links the energy for the |gi to |ei transition
to the particles quasimomentum - giving rise to SOC in a neutral atom system (and connecting to

94
the field of synthetic gauge fields [84]).
Clock spectroscopy of such a system serves as a probe of the density of states (DOS). This isreadily understood by connecting clock detuning to quasimomentum and excitation fraction to the
DOS at that quasimomentum. Clock spectroscopy of the SOC Sr1 system revealed sharp featuresin the lineshape (Figures 5.5 and 5.6), an observation of Van Hove singularities [142, 49].

Figure 5.6: Fractional excitation versus detuning for varying depths in the horizontal lattice [76].
As the trap depth decreases, the carrier clearly broadens and shows the emergence of Van Hove
singularities.

5.3

The Spin-Model
We now turn our attention for the remainder of the chapter to a qualitative discussion of

the spin model using the Bloch sphere. The reader is referred to the extensive literature for a full
treatment [94, 95, 118]. We begin by reviewing the effect of monochromatic radiation on a two-level

95

|e

y

x
|g

Figure 5.7: A π-pulse on the Bloch sphere. A resonant monochromatic field (laser) applies a fieldalong x̂ (black arrow). A spin initially at |gi (red arrow) then begins to precess around the bloch
sphere (blue dots), ending at |ei (blue arrow) for a π pulse.

96
atom using the Bloch sphere picture. The Bloch vector Û on the Bloch sphere evolves as

~
dU
~ ×U
~
=Ω
dt

(5.17)~ = (Ω1 , 0, 0) for monochromatic radiation (our laser) with no detuning. Recall that this iswhere Ω
analagous to a spin in a magnetic field, which we will connect to the spin model. An example ofa π pulse is given in Figure 5.7, illustrating the single particle physics of resonantly addressing aground state atom with a π pulse.
We now want to understand how this picture changes with atomic interactions. We begin bytaking advantage of the fact that the Hamiltonian describing the spin model (atomic interactions)in our system has been mapped to the XXZ Heisenberg spin model (we will call this the XXZ model
to avoid confusion) given by [48]:

N
X

N
X
z z
x x
y y
Snz ,
HXXZ = −J[Sn Sn+1 + Sn Sn+1 + ∆Sn Sn+1 ] − 2h
n=1
n=1
N
N
N
X
X
X
z
= −J
Sn · Sn+1 − J(∆ − 1)
Snz Sn+1
− 2h
Snz .
n=1
n=1
n=1

(5.18)

Here n denotes the index of a spin, h is a magnetic field, J the strength of spin-spin interactions,and Snα = 12 σnα with σ α being the Pauli matrices. In the Sr system this will be a large, collective
spin with S = NL /2 (NL being the number of atoms on site L). ∆ sets the strength of theanisotropy of the spin-spin interaction. This is a 1D spin system with a longitudinal field that can
exhibit ferromagnetic, paramagnetic, and anti-ferromagnetic order depending on ∆ and h [48]. The
magnitude of ∆ relative to J sets whether the spin-spin interactions are dominated by radial or
longitudinal interactions (x̂ and ŷ versus ẑ on the Boch sphere).
Notice that in Equation 5.18 the second line is written such that the first summation corresponds to an isotropic spin-spin term. This is a critical for our understanding of atomic interactions
with clock spectroscopy. We wish to understand the effect of Equation 5.18 on the evolution of acollection of atoms exposed to a π pulse (Figure 5.7). Assuming conservation of total spin S and97
a mean-field picture, the isotropic spin-spin interaction does not affect the evolution of the systemmeaning we are concerned only with the second and third summations in Equation 5.18 with contributions along ẑ. In our Bloch sphere picture this may be understood as a fictitious longitudinalmagnetic field. In the spin model for the clock, it has been shown that in the mean field treatmentof our system that we have a fictitious magnetic field [95, 94, 34, 1]

Bz (N ) = N C + 2χhSz i
where N is the number of atoms on a lattice site and hSz i the expectation value of Sz =(5.19)
jj=1 Sz .

PN

For a pulse area of θ, hSz i = −N/2 cos(θ) [95]. C and χ are non-trivial interaction parameters. In
the presence of spin-orbit coupling χ is defined by a mixture of s- and p-wave two-body interactions(only p-wave in the absence of SOC) while C is dominated by p-wave interactions. The additionof this effective magnetic field serves to perturb the axis the spin of our clock precesses about.
Also note the dependence of Equation 5.19 on N - each lattice site with a different atom number
will experience a different Bz and therefore different detuning, a lineshape broadening mechanism
limiting coherence times.
~ from Equation 5.17 to be Ω~ ∝ (Ω1 , 0, Bz (N )).
To qualitatively understand this we can modify Ω
We may then numerically model a π pulse as before, but with the addition of the effective magnetic
field arising from interactions. For a resonant pulse, the addition of atomic interactions serves to
tip the axis preccessed about, leading to reduced rotations in ẑ, as shown in Figure 5.8.
Figure 5.8 motivates the traditional approach to solving the density shift - make Bz smaller!
This can be done by 1) operating with less atoms and/or 2) reducing density through larger trapping
volumes or lower trap depths, reducing χ and C.

5.4

The ‘Magic Depth’
The rebuilt Sr1 system immediately allowed shallow trap depth operation below 10 Er . As webegan to explore frequency gradients throughout our system using imaging techniques we also began

98

|e

|e

y
x

y
x

|g

|g

Figure 5.8: π pulses with atomic interactions, with the spin starting in |gi (red arrow). Ωx = 1(black arrow), χ=0, and C=0.1. Left: π-pulse evolution for N = 10. The magenta arrow shows Ωwhile the green dots and arrow show the evolution of the spin for a resonant π-pulse. Right: Sameas left, but for N =0,10, and 20 corresponding to blue, green, and cyan arrows. For a system oftraps with differing atom number each site’s collective spin will evolve differently.

99
to track the density shift per atom per lattice site ∆νatom . Originally operating from |ei → |gi wecontinued to explore shallower depths, maintaining high pulse fidelity even for multi-second pulses.
Naturally we continued to go shallower, reducing Raman scattering, lattice light shifts, and thedensity shift.
Measurements of ∆νatom versus trap depth are shown in Figure 5.9. To our disappointment(both for |ei → |gi and |gi → |ei) ∆νatom diverged at shallow depths. Initially we took just the|ei → |gi curve, choosing to operate at the minima of |∆νatom |.

Density Shift Per Atom Per Site (frac.)

1e 18
|g |e
|e |g

4
2
0
2
4
0

20

40
60
80
Lattice Depth(Erecoil)100

Figure 5.9: Density shift per atom per site in fractional frequency units versus trap depth. At large
depths both curves follow the U 5/4 scaling of Chapter 2, but deviate rapidly at shallow depths.

Fortunately we have the information we need in this chapter to understand Figure 5.9. As wehave established lowering trap depth is a good way to reduce p-wave interactions between atoms(and residual s-wave interactions from nuclear spin impurities). However, as we continue decreasing
the trap depth the atomic wavefunctions delocalize (Figure 5.4), enabling WS states centered on
different lattice sites to appreciably overlap. This introduces a s-wave component that actually

100
increases relative to the p-wave contribution as trap depth continues to decrease.
Why do the curves in Figure 5.9 diverge different ways? The qualitative argument comesfrom Equation 5.19. The expectation value hSz i goes from -1 to 1, so starting in |gi versus |ei andperforming a π pulse changes the sign. Thus when χ >> C the density shift diverges with opposite
signs.
Most importantly, Figure 5.9 informs us of a different approach to reducing Equation 5.19.
Tuning the trap depth to where ∆νatom = 0 for the |gi → |ei curve of Figure 5.9 corresponds tozero fictitious magnetic field

0 = N C + 2χhSz i,(5.20)= N [C − χ cos(θ)].
We see that we can cancel the interaction parameters by tuning a combination of pulse depth toadjust the cosine contribution or trap depth to adjust C and χ. In our case, rather than tune thepulse area we adjust the trap depth to find this zero crossing. We call this the ‘magic depth’ as
the density shift vanishes regardless of atom number - it is an experiment specific condition. An
example evaluation is shown in Figure 5.10. With the ‘magic depth’ we now have the ability to
combine the accuracy of the 1D system with previously unheard of precision.

101

Figure 5.10: ‘Magic depth’ operation at 12 Er . Over a 10 hour measurement the fractional density
shift was found to be 5.0 (2.8)×10−21 per atom per lattice site.

Chapter 6

Resolving the gravitational redshift within a millimeter atomic sample

The research in this chapter is reported in the publication: Bothwell et al., ‘Resolving thegravitational redshift within a millimeter atomic sample,’ arXiv (2021) [20].
Modern atomic clocks embody Arthur Schawlow’s motto to ‘never measure anything but
frequency.’ This deceptively simple principle, fueled by the innovative development of laser scienceand quantum technologies based on ultracold matter, has led to dramatic progress in clock performance. Recently, clock measurement precision reached the mid-19th digit in one hour [33, 112], andthree atomic species achieved systematic uncertainties corresponding to an error equivalent to lessthan 1 s over the lifetime of the Universe [109, 98, 24, 19]. Central to this success in neutral atom
clocks is the ability to maintain extended quantum coherence times while using large ensembles ofatoms [33, 112, 93]. The pace of progress has yet to slow. Continued improvement in measurement
precision and accuracy arising from the confluence of metrology and quantum information science[116, 73, 78, 154] promises discoveries in fundamental physics [123, 74, 38, 77].
Clocks fundamentally connect space and time, providing exquisite tests of the theory of general relativity. Hafele and Keating took cesium-beam atomic clocks aboard commercial airliners
in 1971, observing differences between flight-based and ground-based clocks consistent with special
and general relativity [58]. More recently, RIKEN researchers compared two strontium optical lattice clocks (OLCs) separated by 450 m in the Tokyo Skytree, resulting in the most precise terrestrial
redshift measurement to date [133]. Proposed satellite-based measurements [79, 137] will provide
orders of magnitude improvement to current bounds on gravitational redshifts [?]. Concurrently,

103

a

b 1.0

E la ttic e

CountsSi cavity

Excitation Fraction

0.8

B
g

0.6

0.4

0.2

z

0.0
−2.0 −1.5 −1.0 −0.5

E c loc k

d 1.0
…

…

|e,+m

|e,-m

Ω-m

Ω

…

Ω+m

…
|g,+m

1.0

1.5

2.0

-2

+2

0.6

0.4

0.2

-3

|g,0
|g,-m

0.5

+1

-1

0.8

|e,0
Excitation Fractionc0.0Frequency (Hz)

0.0

-5
−4

+3
+4 +5

-4
−2

0

2

Lorem ipsum
Frequency (kHz)4

Figure 6.1: Experimental system and quantum state control. a, A millimeter length sample of
100,000 87 Sr atoms are trapped in a 1D optical lattice formed within an in-vacuum cavity. The
3P
longitudinal axis of the cavity, z, is oriented along gravity. We probe atoms along the 1 S00transition using a clock laser locked to an ultrastable crystalline silicon cavity [112, 97]. b, Rabi
spectroscopy with a 3.1 s pulse time. Open purple circles indicate data with a corresponding Rabi
fit in green. c, Neighboring lattice sites are detuned by gravity, creating a Wannier-Stark ladder.
Clock spectroscopy probes the overlap of Wannier-Stark states between lattice sites that are msites away with Rabi frequency Ωm . d, Rabi spectroscopy probes Wannier-Stark state transitions,revealing wavefunction delocalization of up to 5 lattice sites. The number of lattice sites is indicatedabove each transition, with blue(red) denoting Wannier-Stark transitions to higher(lower) latticesites.



104
clocks are anticipated to begin playing important roles for relativistic geodesy [56]. In 2010 Chouet al. [35] demonstrated the precision of their Al+ clocks by measuring the gravitational redshift
resulting from lifting one clock vertically by 30 cm in 40 hours. With a decade of advancements,
today’s leading clocks are poised to enable geodetic surveys of elevation at the sub-centimeter level
on Earth, a result unobtainable with other current techniques [138].
Atomic clocks strive to simultaneously optimize measurement precision and systematic uncertainty. For traditional OLCs operated with one-dimensional (1D) optical lattices, achieving low
instability has involved the use of high atom numbers at trap depths sufficiently large to suppress¬

tunneling between neighboring lattice sites. While impressive performance has been achieved,

effects arising from atomic interactions and AC Stark shifts associated with the trapping lightchallenge advancements in OLCs. Here we report a new operational regime for 1D OLCs, bothresolving the gravitational redshift across our atomic sample and synchronously measuring a fractional frequency uncertainty of 7.6 × 10−21 between two uncorrelated regions. Our system employs
100,000 87 Sr atoms at 100 nK loaded into a shallow, large waist optical lattice, reducing both AC
Stark and density shifts. Motivated by our earlier work on spin-orbit coupled lattice clocks [76, 26],
we engineer atomic interactions by operating at a ‘magic’ trap depth, effectively removing collisional frequency shifts. These advances enable record optical atomic coherence (37 s) and expected
single clock stability (3.1 × 10−18 at 1 s) using macroscopic samples, paving the way toward lifetime
limited OLC operation.
Central to our experiment is an in-vacuum optical cavity (Fig. 1a and Methods) for power
enhancement of the optical lattice. The cavity (finesse 1100) ensures wavefront homogeneity of our
1D lattice while the large beam waist (260 µm) reduces the atomic density by an order of magnitude
compared to our previous system [19]. We begin each experiment by trapping fermionic 87 Sr atoms
into the 1D lattice at a trap depth of 300 lattice photon recoil energies (Er ), loading a millimeterscale atomic sample (Fig. 1a). Atoms are simultaneously cooled and polarized into a single nuclear
spin before the lattice is adiabatically ramped to an operational depth operational depth of 12 Er
and prepare atoms into a single nuclear spin. Clock interrogation proceeds by probing the ultra-

105
narrow 1 S0 (g) P (e) transition with the resulting excitation fraction measured by fluorescence
30

spectroscopy. Scattered photons are collected on a camera, enabling in-situ measurement with 6µm resolution, corresponding to ∼15 lattice sites (Fig. 1a).
Quantum state control has been vital to recent advances in atom-atom and atom-light coherence times in 3D OLCs and tweezer clocks [33, 93, 154]. Improved quantum state control isdemonstrated through precision spectroscopy of the Wannier-Stark states of the OLC [149, 81].
The 1D lattice oriented along gravity has the degeneracy of neighboring lattice sites lifted by thegravitational potential energy. In the limit of shallow lattice depths, this creates a set of delocalizedstates. By ramping the lattice depth to 6 Er , much lower than in traditional 1D lattice operations[109, 98, 19], clock spectroscopy probes this delocalization (Fig. 1d). The ability to engineer theextent of atomic wavefunctions through the adjustment of trap depth creates an opportunity tocontrol the balance of on-site p-wave versus neighboring-site s-wave atomic interactions. We utilize
this tunability by operating at a ‘magic’ trap depth [1], where the frequency shifts arising from onsite and off-site atomic interactions cancel, enabling a reduction of the collisional frequency shifts
by more than three orders of magnitude compared our previous work [19].
Extended atomic coherence times are critical for both accuracy and precision. An aspirational
milestone for clock measurement precision is the ability to coherently interrogate atomic samples
up to the excited state’s natural lifetime. To evaluate the limits of our clock’s atomic coherence,
we perform Ramsey spectroscopy to measure the decay of fringe contrast as a function of the freeevolution time. By comparing two uncorrelated regions within our atomic sample, we determine
the contrast and relative phase difference between the two sub-ensembles (Fig. 2). The contrast
decays exponentially with a time constant of 37 s (quality factor of 3.6 × 1016 ), corresponding to an
additional decoherence time of 53 s relative to the 3 P0 natural lifetime (118 s [104]). This represents
the longest optical atomic coherence time measured in any spectroscopy system to date.
We utilize Rabi spectroscopy in conjunction with in-situ imaging to microscopically probe
clock transition frequencies along the entire vertically oriented atomic ensemble. With a standard
interleaved probing sequence using the |g, mF = ±5/2i to |e, mF = ±3/2i transitions for minimal

106

a 1.0
p2

30 s

50 s

p2

p1

b 1 6s0.8

c 1.0 0

0.6

p1

1

0.8
0.4

Contrast

Excitation Fraction

0

0.2

0.6
0.4

τ=36.5(0.7) s

0.2
0.0

−0.5

0.0

z (mm)

0.5

0.0

0

20

40

60

Dark Time (s)Figure 6.2: Atomic coherence. We use Ramsey spectroscopy with a randomly sampled phasefor the second pulse to determine the coherence time of our system [93]. a, We measure theexcitation fraction across the cloud, shown in purple for a single measurement, and calculate theaverage excitation fractions in regions p1 and p2 , separated by 2 pixels. b, Parametric plots ofthe excitation fraction of p1 versus p2 in purple for 6 s, 30 s and 50 s dark time demonstrate aphase shift between the two regions and contrast decay. Using a maximum likelihood estimator, we
extract the phase and contrast for each dark time with the fit, shown in green. c, Contrast decay
as a function of time in green is fit with an exponential decay in gold, giving an atomic coherence
decay time of 36.5(0.7) s and a corresponding quality factor of 3.6 × 1016 . After accounting for thefinite radiative decay contribution, we infer an additional decoherence time constant of 52.8(1.5) s.

107
magnetic sensitivity, we reject the first order Zeeman shifts and vector AC Stark shifts. The in-situ
imaging of atoms in the lattice allows measurement of unprocessed frequencies across the entire
atomic sample (Fig. 1a and Methods). The dominant differential perturbations arise from atomatom interactions (residual density shift contributions after we operate at the ‘magic’ trap depth)
and magnetic field gradients giving rise to pixel-specific second order Zeeman shifts. Using thetotal camera counts and mF -dependent frequency splitting, we correct the density and second order
Zeeman shift at each pixel. These corrections result in the processed frequencies per pixel shown in
Fig. 3a, with error bars representing the quadrature sum of statistical uncertainties from the center
frequency, the density shift correction, and the second order Zeeman shift correction. Additional
systematics are described in the Methods. This approach demonstrates an efficient method for
rapid and accurate evaluation of various systematic effects throughout a single atomic ensemble.
Unlike traditional 1D OLCs where systematic uncertainties are quoted as global parameters, we
now microscopically characterize these effects.
This new microscopic in-situ imaging allows determination of the gravitational redshift within
a single atomic sample, probing an uncharacterized fundamental clock systematic. Two identicalclocks on the surface of a planet separated by a vertical distance h will differ in frequency (δf ) as
given by

δf
ah
= 2,
f
c

(6.1)

with f the clock frequency, c the speed of light, and a the gravitational acceleration. The gravitational redshift at Earth’s surface corresponds to a fractional frequency gradient of −1.09×10−19 /mm
in the coordinate system of Fig. 1a. Measurement of a vertical gradient across the atomic sample
consistent with the gravitational redshift provides an exquisite verification of an individual atomic
clock’s frequency control.
Our intra-cloud frequency map (Fig. 3a) allows us to evaluate gradients across the atomic
sample. Over 10 days we performed 14 measurements (ranging in duration from 1-17 hours) to108

Fractional Frequency (10-18)

0.75
0.50
0.25
0.00
−0.25
−0.50
−0.75
−1.00

−0.4

−0.2

0.0

z (mm)

0.2

0.4

3
20

2
1

10

0

0

−1

−10

−2

−20

−3

Gravitational Acceleration (m/s2)

Fractional Frequency Gradient (10-19/mm)

b

−30

08
/
08 13
08 /14/08 15 A/1
08 5 B
/
08 16
/1
08 7
/
08 18
/1
08 9
08 /20/08 21 A/2
08 1
/ B
08 21 C/
08 22 A/2
2
B

a 1.00Measurement Date

Figure 6.3: Evaluating frequency gradients. a, For each measurement we construct a microscopic
frequency map across the sample, with raw frequencies shown in green. The second order Zeeman
correction is shown as a dashed gold line. Processed frequencies shown in purple include bothdensity shift corrections and second order Zeeman corrections, with uncertainties arising fromthe quadrature sum of statistical, density shift correction, and second order Zeeman correction
uncertainties. To this we fit a linear function, shown in black. b, Over the course of 10 days, we
completed 14 measurements. For each measurement, we create a corrected frequency map and fit
a linear slope as in a. This slope is plotted for each measurement, as well as a weighted mean
(black) with associated statistical uncertainty (dashed black) and total uncertainty as reported inthis chapter’s table (dotted black). The expected gravitational gradient is shown in red. All data istaken with Rabi spectroscopy using a 3.1 s π-pulse time except for 08/13 which used a 3.0 s pulse
time. The reduced chi-square statistic is 3.0, indicating a small underestimation of error variances
entirely consistent with the additional systematic uncertainties in Table 1.

109
search for the gravitational redshift across our sample. For each dataset we fit a linear slope andoffset, reporting the slope in Fig. 3b. From this measurement campaign we find the weighted mean
(standard error of the weighted mean) of the frequency gradient in our system to be −1.00(12) ×
10−19 /mm. We evaluate additional differential systematics (see Methods) and find a final frequency
gradient of −9.8(2.3) × 10−20 /mm, consistent with the predicted redshift.

z (mm)

-0.5

0.0

0.5

Fractional Frequency Uncertaintyba4 . 4 × 1 0 − 1 8 /√ τ

10-18

3 . 1 × 1 0 − 1 8 /√ τ
2 . 6 × 1 0 − 1 8 /√ τ

10-19

10-20

102

103

Time (s)104

105

Figure 6.4: In-situ synchronous clock comparison. a, The cloud is separated as in Fig. 2a. The
gravitational redshift leads to the higher clock(blue) ticking faster than the lower one(red). The
length scale is in millimeters. b, Allan deviation of the frequency difference between the two regions
in a over 92 hours. Purple points show fractional frequency instability fit by the solid green line,with the quantum projection noise limit indicated by the dashed black line. We attribute theexcess instability of the measurement relative to QPN to detection noise. The expected single
atomic region instability is shown in gold.

The ability to resolve the gravitational redshift within our system suggests a level of frequency control beyond previous clock demonstrations, vital for the continued advancement of clock110
accuracy and precision. Previous fractional frequency comparisons [154] have reached uncertainties
as low as 4.2 × 10−19 . Similarly, we perform a synchronous comparison between two uncorrelated
regions of our atomic cloud (Fig. 4a). By binning ∼100 pixels per region, we substantially reduce
instability caused by quantum projection noise [89]. Analyzing the frequency difference between√
regions from 92 hours of data, we find a fractional frequency instability of 4.4 × 10−18 / τ (τ is theaveraging time in seconds), resulting in a fractional frequency uncertainty of 7.6 × 10−21 for fullmeasurement time, nearly two orders of magnitude lower than the previous record. From this mea√
surement we infer a single region instability of 3.1 × 10−18 / τ . Dividing the fractional frequency
difference by the spatial separation between each region’s center of mass gives a frequency gradientof −1.30(18) × 10−19 /mm. Correcting for additional systematics as before results in a gradient of−1.28(27) × 10−19 /mm, again fully consistent with the predicted redshift.
In conclusion, we have established a new paradigm for atomic clocks. The vastly improved
atomic coherence and frequency homogeneity throughout our sample allow us to resolve the gravitational redshift at the submillimeter scale, observing for the first time the frequency gradientfrom gravity within a single sample. We demonstrate a synchronous clock comparison between two
uncorrelated regions with a fractional frequency uncertainty of 7.6 × 10−21 , advancing precisionby nearly two orders of magnitude. These results suggest that there are no fundamental limitations to inter-clock comparisons reaching frequency uncertainties at the 10−21 level, offering newopportunities for tests of fundamental physics.

6.1

Methods

6.1.1

In-Vacuum Cavity

Central to our system is an in-vacuum lattice buildup cavity oriented along gravity (Fig.
1a). Two mirrors with radius of curvature of 1 m are separated by ∼17 cm, achieving a mode
waist of 260 µm. Our over-coupled cavity has a finesse at the lattice wavelength (813 nm) of 1100
and a power buildup factor of 700 (ratio of circulating to input intensity). This enables lattice

111
depths in excess of 500 Er (lattice photon recoil energy) using a diode-based laser system. The
dimensional stability of the cavity combined with the simplified diode laser system enables robust
operation compared with our previous Ti:Sapphire retro-reflected design [19]. The cavity mirrors
are anti-reflection coated at the clock wavelength of 698 nm.
One cavity mirror is mounted to a piezo for length stabilization while the other mirror isrigidly mounted for phase reference for the clock laser. Grounded copper shields between atoms
and mirrors prevent DC Stark induced shifts due to charge buildup on the mirrors and piezo [29, 85].
Each shield (5 mm thick) has a centered hole of 6 mm diameter to accommodate the optical latticebeam, with shielding performance verified through evaluation of the DC Stark shift systematic.

6.1.2

Atomic Sample Preparation

87 Sr atoms are cooled and loaded into a 300 Eroptical lattice using standard two stage

magneto-optical trapping techniques [19]. Once trapped, atoms are simultaneously nuclear spinpolarized, axially sideband cooled, and radially doppler cooled into a single nuclear spin state at
temperatures of 800 nK. The lattice is then adiabatically ramped to the operational trap depth of
12 Er , where a series of pulses addressing the clock transition prepares atoms into |g, mF = ±5/2i.
Clock spectroscopy is performed by interrogating the |g, mF = ±5/2i to |e, mF = ±3/2i transition,
the most magnetically insensitive 87 Sr clock transition [112].

6.1.3

Imaging

The clock excitation fraction is read out using standard fluorescence spectroscopy techniques[33, 93, 80]. Photons are collected on both a photo-multiplier tube for global readout and electron
multiplying charge coupled device camera for an in-situ readout of clock frequency. Camera readout
is performed in full vertical binning mode, averaging the radial dimension of the lattice. This
provides 1D in-situ imaging for all synchronous evaluations.
We use a 25 µs fluorescence probe with an intensity of I/Isat ∼ 20 (Isat being the saturation
intensity), ensuring uniform scattering across the atomic sample. Before imaging, the optical

112
lattice is ramped back to 300 Er to decrease imaging aberration resulting from the extended radial
dimension at 12 Er .

6.1.4

AnalysisStandard clock lock techniques and analysis are used [112, 93, 89], with differences in excitation fraction converted to frequency differences using Rabi lineshapes. Each dataset is composed ofa series of clock locks, tracking the center of mass frequency of the atomic sample. A clock lock isfour measurements probing alternating sides of the Rabi lineshape for opposite nuclear spin transitions. Frequency corrections based on excitation fraction become ambiguous when the excitation
fraction measured is consistent with the Rabi lineshape at multiple detunings. To avoid erroneous frequency corrections, we remove clock locks with excitation fractions above (below) .903×C(.116×C), where C is the Rabi contrast. From each clock lock, a pixel specific center frequency fi
and frequency splitting between opposite mF states ∆i are calculated, creating an in-situ frequency
map of the 1D atomic sample. This allows rejection of vector shifts on a pixel-by-pixel basis andprobes the magnetic field induced splitting of mF transitions. The atom weighted mean frequency
is subtracted from every lock cycle to reject common mode laser noise.
For each dataset we approximate the atomic profile with a Gaussian fit, identifying a center
pixel and associated Gaussian width (σ). All analysis is performed within the central region of±1.5σ which demonstrates the lowest frequency instability. Identifying a center pixel for dataprocessing ensures rejection of any day-to-day drift in the position of the cloud due to varying
magnetic fields modifying MOT operation on the narrow line transition. The density shift coefficient(see Density Shift section) is derived from the average center frequency per pixel. Using thiscoefficient, we correct fi and ∆i for the density shift. second order Zeeman corrections using these
updated frequencies are then applied.
Gradient analysis is based on the processed center frequencies per pixel. A linear fit to
the frequencies as a function of pixel is performed using least squares, with uncertainty per pixel
arising from the quadrature sum of statistical frequency uncertainty, statistical second order Zeeman

113
uncertainty, and density shift correction uncertainty.
For the two-clock comparison (Fig. 4b), all data from 8/14-8/22 was taken with the same
duty cycle and π-pulse time (3.1 s). Data was processed relative to a fit center pixel as discussed
and concatenated. Two equal regions extend from the center of the sample to a width of ±1.5σ ,with two empty pixels between regions to ensure uncorrelated samples. Each region is processed
for the atom weighted mean frequency, enabling a synchronous frequency comparison between two
independent clocks.

6.1.5

Atomic Coherence

We use a Ramsey sequence to measure the atomic coherence. We prepare a sample in the|g, mF = +5/2i state and apply a π/2 pulse along the |g, mF = +5/2i to |e, mF = +3/2i transition.
After waiting for a variable dark time, we apply a second π/2 pulse with a random phase relative
to the first. We then measure the excitation fraction.
Two regions, p1 and p2 , are identified using the same technique as in the synchronous instability measurement. For each experimental sequence, we find the average excitation fraction in
p1 and p2 . A mean frequency shift across the sample primarily due to a magnetic field gradientcreates a differential phase as a function of time between p1 and p2 . We create a parametric plotof the average excitation in p1 and p2 for each dark time and use a maximum likelihood estimator
to fit an ellipse to each dataset, calculating phase and contrast [154, 93]. To estimate uncertainty
in the contrast for each dark time a bootstrapping technique is used [93]. Fitting the contrast as afunction of dark time with a single exponential returns an effective atomic coherence time.

6.1.6

Systematics

6.1.6.1

Imaging

We calibrate our pixel size using standard time of flight methods: we observe an atomic
sample in freefall for varying times to determine an effective pixel size along the direction of gravity.
Immediately after our 10-day data campaign we measured our effective pixel size to be 6.04 µm.

114
Due to thermal drift of our system, the pixel size can vary by up to 0.5 µm/pixel over months
which we take as the calibration uncertainty.
Spatial correlations may limit imaging resolution. We measure these correlations by placing
atoms into a superposition of clock states. Any measured spatial correlation is due to the imaging
procedure. In our system we find no correlations between neighboring pixels [93]. The opticalresolution of our imaging lens is specified at 2 µm.
Lattice tilt from gravity will modify the measured gradient. We find the lattice tilt in theimaging plane to be 0.11(0.06) degrees, providing an uncertainty orders of magnitude smaller thanthe pixel size uncertainty. We are insensitive to lattice tilt out of the imaging plane.

6.1.6.2

Zeeman Shifts

First order Zeeman shifts are rejected by probing opposite nuclear spin states [112]. The
second order Zeeman shift is given by∆νB,22 = ξ(∆νB,1 )2 ,(6.2)

where ∆νB,1 is the splitting between opposite spin states and ξ the corresponding second order
Zeeman shift coefficient. For stretched spin state operation (mF = ±9/2), ξ = −2.456(3) × 10−7
Hz−1 . Using known atomic coefficients [22] we find the second order Zeeman coefficient for the|g, mF = +5/2i to |e, mF = +3/2i transition to be ξop = −1.23(8)×10−4 Hz−1 , with the uncertainty
arising from limited knowledge of atomic coefficients.
The second Order Zeeman corrections are made for every clock lock (analogous to the insitu density shift corrections). For a typical day (8/13) the average second order Zeeman gradientis −7.0 × 10−20 /mm, corresponding to a splitting between opposite nuclear spin states of 12.7
mHz/mm (0.291 mG/mm). We include an error of 4 × 10−21 /mm in the below table to account for
the atomic uncertainty in the shift coefficient.

115
6.1.6.3

DC Stark

Electric fields perturb the clock frequency via the DC Stark effect. We evaluate gradients
arising from this shift by using in-vacuum quadrant electrodes to apply bias electric fields in all
three dimensions. We find a DC Stark gradient of 3(2) × 10−21 /mm.

6.1.6.4

Black Body Radiation Shift

The dominant frequency perturbation to room temperature neutral atom clocks is black bodyradiation (BBR). Similar to our previous work [19], we homogenize this shift by carefully controlling
the thermal surroundings of our vacuum chamber. Attached to the vacuum chamber are additionaltemperature control loops, with each vacuum viewport having a dedicated temperature control
system. This ensures our dominant BBR contribution – high emissivity glass viewports – are all
the same temperature to within 100 mK.
To bound possible BBR gradients, we introduce a 1 K gradient between the top and bottomof the chamber along the cavity axis by raising either the top or bottom viewports by 1 K. We
compare these two cases and find no statistically significant changes in the frequency gradient acrossthe entire sample. Accounting for uncertainty in linear frequency fits for each case, we estimate an
uncertainty of 3 × 10−21 /mm. This finding is supported with a basic thermal model of the vacuum
chamber.

6.1.6.5

Density Shift

Atomic interactions during Rabi spectroscopy lead to clock frequency shifts as a functionof atomic density [95]. For each gradient measurement, we evaluate the density shift coefficientχdens by fitting the average frequency f per pixel versus average camera counts per pixel N to an
equation of the formf (N ) = χdens N + B.

(6.3)

116
Here B is an arbitrary offset. Once χdens is known, we remove the density shift at each pixel.
Residual density shift corrections may lead to error in our linear gradient. To bound thiseffect, we compare the density shift coefficient and gradient from our data run with a separate
dataset at 8 Er . With the trap depth at 8 Er we found a linear gradient of s= −1.08 × 10−18 /mm
and a density shift coefficient of χ8 = −1.39 × 10−6 Hz/count. During our data run we had an
average density shift coefficient of χop = −2.43 × 10−8 Hz/count. We bound the uncertainty in our
gradient from density shift as σden,unc = |s × χop /χ8 | = 1.7 × 10−20 /mm.

6.1.6.6

Lattice Light Shifts

Lattice light shifts arise from differential AC Stark shifts between the ground and excitedclock states. An approximate microscopic model of the lattice light shift (νLS ) in our system isgiven by [140]hνLS (u, δL ) ≈ 1/2  E1 
u
δα
∆αE1
− ∆αQM
−
δL u,
δν
2
δν

(6.4)

where u is the trap depth in units of Erec, ∆αE1 the differential electric dipole polarizability, ∆αQM
the differential multi-polarizability, and δL = (νL − ν E1 ) the detuning between lattice frequency νLand effective magic frequency ν E1 . Our model has no dependence on the longitudinal vibrational
quanta since we are in the ground vibrational band. We neglect higher order corrections fromhyperpolarizability due to our operation at depths <60 Er . At our temperatures thermal averagingof the trap depth is a higher order correction (< 5%) that is also neglected.
We model the linear differential lattice light shift across the atomic cloud as δαE1( δν ) δαE1
δhνLS (u, δL )
δu
≈
−
δL
1/2
δz
δν
δz
4u

(6.5)

where z is the coordinate corresponding to the axis of the cavity along gravity. To evaluate ourdifferential lattice light shift at our operational depth we need δL and δuδz . We modulate our lattice
between two trap depths (u1 = 14 Er , u2 = 56 Er ) and find our detuning from scalar magic

117
frequency to be δL =7.4(0.6) MHz. To evaluate δuδz at our operational depth (uop ) we measure thelinear gradient across the atomic cloud at δL + 250 MHz and δL − 250 MHz, the difference given by E1
δhνLS (uop , δL+250M Hz ) δhνLS (uop , δL−250M Hz )
δuop
1
δα
−
≈
−1
δ500
,
1/2
δz
δz
δν
δz
4uop

(6.6)

where δ500 = 500 MHz. We find (δuop /δz)=0.0383/mm, which when combined with δL = 7.4 MHz,
gives us a fractional frequency gradient of −5 × 10−21 /mm. Accounting for error in our lattice
detuning and linear gradient gives us an uncertainty of 1 × 10−21 /mm.

6.1.6.7

Other Systematics

For a 3.1 s π-pulse the probe AC Stark shift7 is -3(2)×10−21 . A frequency scan of the|g, mF = −5/2i to |e, mF = −3/2i transition limits the variation of excitation fraction across theatomic sample to 1% or below, bounding any possible probe AC Stark gradient across the sample
to < 1 × 10−22 .

6.1.6.8

Known Redshift

The gravitational acceleration (rounding to 4 digits) within our lab was evaluated by a USGS
survey to be a=-9.796 m/s2 .

118

Systematic
Gradient
BBR
Density
Lattice light shift
DC Stark
Pixel Calibration
Second Order Zeeman
Other
Corrected Gradient
Known Redshift

Slope (10−20 /mm)
-10.0
0
-0.5
0.3
0
0
-9.8
-10.9

Uncertainty (10−20 /mm)
1.2
0.3
1.7
0.1
0.2
0.8
0.4
< 0.1
2.3
<0.1

Table 6.1: Gradient Systematic Budget. Fractional frequency gradients and corresponding uncertainties. Fractional frequencies denoted with ‘-‘ are corrected on a pixel-by-pixel basis during initial
data processing (Fig. 3a). The corrected gradient has known systematics removed with uncertainty
given by the quadrature sum of all correction uncertainties.

Chapter 7

Lattice Light Shifts

While the proposal of trapped ion clocks came as early as 1982 [41], the development of theneutral atom frequency standards came significantly later. In the late 1990’s, the quantum optics
community was beginning to wrestle with the issue of far off resonant traps (FORTs) providing
deleterious AC Stark shifts for different electronic states [153]. In tandem, interest was growing in
the use of forbidden transitions (specifically 1 S0 →3 P1 transitions) for frequency references. Out ofthese ideas came the proposal to trap strontium atoms in a 1D lattice at the ‘magic wavelength’,
where the perturbation to ground and excited clock states (now 1 S0 →3 P0 ) were equal. This
allowed Doppler-free confinement of neutral atoms without differential lattice induced AC Stark
shifts (lattice light shifts). The idea would be expanded to other Alkaline-Earth and Alkaline-Earthlike atoms, with an initial flurry and race amongst national labs to establish new atoms.
Ultimately the idea of a ‘magic wavelength’, as with all physics, is an imperfect approximation
to the world around us. While the magic wavelength condition may be satisfied in the dipole
approximation, higher order perturbations lead to an over-confined problem - for a thermal sample,
insufficient degrees of freedom exist to ensure each atom experiences no differential AC Stark
shift from the lattice. Magnetic dipole and electric quadrapole terms, as well as higher orderintensity terms (hyperpolarizability) lead to challenging trapping requirements in order to achieve
inaccuracy at the 10−18 level. In the 1D clock, evaluation of atomic wavefunctions is necessary toproperly account for the average potential experienced by the atoms and corresponding light shifts.
Unfortunately, this is far more complicated than previous linear light shift models [109].

120
The newest version of Sr1 has been built with the goal of studying lattice light shifts. Fromthe beginning, the Sr1 system was designed to utilize shallow trap depths (∼ 10 Er ), effectively
eliminating hyperpolarizability contributions. This places Sr1 in the position to carefully study√
lattice light shifts scaling as u and u (u being trap depth in units of Er ). Towards this we will
first review the origins of the lattice light shift, referencing the work of [28, 107]. We will thendiscuss possible approaches towards lattice light shift evaluations in the Sr1 system.

7.1

AC Stark Shift From Electric Dipole Interaction - π Transitions
We begin our discussion of the lattice light shift by first focusing on the AC Stark effect arisingfrom the differential electric dipole (E1) polarizability between the ground and excited state. The
corresponding clock shift in frequency for ∆mF = 0 transitions is given by [147]∆ν E1 = (∆κs + ∆κv mF ξ e~k · e~B + ∆κt β)U0 .

(7.1)

Here we have used irreducible tensors to explicitly give the scalar, vector, and tensor shift contributions as ∆κs , ∆κv , and ∆κt respectively. e~k and e~B are the unit vectors along the latticepropagation wavevector and quantization axis (as defined by an external bias field) respectively. ξis the degree of circularity and is given by ξ e~k = i~ × ~∗ where ~ is the complex polarization unit
vector. Finally β = (3|~ · e~B |2 − 1)[3m2F − F (F + 1)].
From Equation 7.1 we connect with the traditional approach of operating the 1D optical lattice clock. Consider the vector shift contribution. We ensure excellent rejection of this perturbation
three ways. First, during clock operation we probe opposite mF states, where the average frequencyof the two rejects the vector shift. Second, we use linearly polarized light (ξ ≈ 0). Third, we operatewith the bias field as close to perpendicular to the lattice wavevector as possible (e~k · e~B ≈ 0).
Note that Equation 7.1 must be modified for σ transitions owing to different mF values
between ground and excited clock states [127].

121

7.2

Atom-Light Hamiltonian
A goal of this chapter is to elucidate the origins of the various lattice light shift terms. We

first consider a single electron in an atom, with the atom’s center of mass at the origin. We begin
with the standard Hamiltonian for an electron interacting with radiation [64, 37, 23, 141],

Ĥ =

2
1
p̂ + eA + V.
2me

(7.2)

Here me is the reduced electron mass, p̂ = −i~∇ is the momentum operator, A the vector potential,
and V the Coloumb potential experienced by the electron due to the the atomic nucleus and otherelectrons. Expanding the square we can rewrite this asĤ = Ĥ0 + Ŵ1 + Ŵ2

(7.3)

p̂2
+ V,
2me
e
Ŵ1 = A · p̂
m
q2 2
A .
Ŵ2 =
2m

(7.4)

where

Ĥ0 =

Ĥ0 is the field-free portion of the Hamiltonian (and basis used for calculating the perturbationsof the Ŵ terms). Ŵ1 is the perturbation that corresponds to first order in A. The A2 term isincredibly weak (and a two photon term) - we will discuss this later and instead focus our attention
on Ŵ1 . Using the Coloumb gauge (∇ · A = 0), we then have
∇ · AΨ = A · ∇Ψ + ∇ · A Ψ

(7.5)


= A · ∇Ψ .

(7.6)

To proceed we will consider a plane wave interacting with the electron, with vector potential,
electric field, and magnetic field given by

122



A0
i(k·r−ωt)∗ −i(k·r−ωt)A(r, t) =
e
+ e
2


iE0
i(k·r−ωt)∗ −i(k·r−ωt)E(r, t) =
e
− e
2


iB0 k
i(k·r−ωt)∗ −i(k·r−ωt)B(r, t) =
× e
− e
2 k

(7.7)

∂
where we recall E(r, t) = − ∂tA(r, t) and B(r, t) = ∇ × A(r, t).  specifies the polarization of theplane wave. Following the notation of [64] we can write Ŵ1 in terms of the transition operator D̂.



eE0
−iωt
∗ +iωtD̂e− D̂ e
Ŵ1 =
2

(7.8)

where

eik·r
p̂ · 
ωme
1
1 + ik · r + ... p̂ · 
=
ωme


p̂ · 
i=+k · r p̂ · ωme ωme

D̂ =

(7.9)

= D̂E1 + D̂QM .
The electron orbit is much smaller than the wavelength of the optical light we are considering,
meaning k · r << 1. To this end we break D̂ into D̂E1 and D̂QM which correspond to dipole (E1)and electric quadrupole (E2)/magnetic dipole (M1) transitions respectively.
Our goal is to now explore the matrix elements corresponding to transitions between states.
These will in turn be used to give us our oscillator strength and polarizability for use in AC Stark
calculations.

7.2.1

Matrix Elements of D̂E1

We define the dipole transition matrix element between two states |ii and |ji as DE1,ij .

123

DE1,ij = hj|p̂ · 
|ii
ωij meωij me

me 

= hj|
r, Ĥ0 |ii ·i~
ωij me= hj| p̂ |ii ·(7.10)= −i hj| r |ii · 
We have used the Heisenberg equations of motion to handle p̂. Also note since we are calculating
the dipole transition matrix elements we are interested in the case of resonance radiation currently,
setting ω = ωij . This is the usual dipole transition, with a dipole moment coupling to the electricfield.

7.2.2

Matrix Elements of D̂QM

Just like for E1, we define the multipolar (QM) transition matrix element between two states|ii and |ji as DQM,ij .

k · r p̂ · 
DQM,ij = i hj|
|ii
mωij

(7.11)

Note this is not a perfect notation since as we will see the QM term is composed of two different
transition types, each with their own selection rules: the M1 transition and E2 transition.
A scalar quadruple product has the following relation in 3D Euclidean space:(a × b) · (c × d) = (a · c)(b · d) − (a · d)(b · c).

(7.12)

allowing us to rearrange (using the commutativity of p̂ and ).

(k · r)(p̂ · ) = (k × ) · (r × p̂) + (k · p̂)( · r).
We now focus on each part separately.

(7.13)

124
7.2.2.1

M1 Transition Elements

We define the transition matrix element for M1 transitions as

DM 1,ij = i hj|(k × ) · (r × p̂)
|ii .
mωij

(7.14)

We immediately note that L̂ = (r× p̂). Since we have neglected spin to this point in the calculation,0
we substitute L̂ → L̂ + 2Ŝ [23]. Additionally, for a plane wave note that (k × ) = k B
B0 (B0 being

the magnetic field of the plane wave). We can then rewrite Equation 7.14 as

DM 1,ij =


ik
B0
hj|
· L̂ + 2Ŝ |ii .
me ωijB0(7.15)

This makes it clear that the M1 transition matrix elements couple to the magnetic field.

7.2.2.2

E2 Transition Elements

We define the transition matrix element for E2 transitions as

DE2,ij = i hj|(k · p̂)( · r)
|ii
mωij

(7.16)

Accounting for all possible options in cartesian coordinates this can be rewritten as

DE2,ij =i X
km n hj| p̂m rn |ii .
mωij mn(7.17)

We stop here and note that for a situation like our lattice with a plane wave propagating along ẑ
and polarization parallel to x̂, we get

DE2,ij =ik hj| p̂z x |ii .
mωij(7.18)For our 1D lattice setup, this then corresponds to taking the gradient of the electric dipole operator.
We then see that while the E2 transition couples to the electric field, it does so to the gradient.

125
For completion, we can rearrange the Equation 7.17 to bring it in line with the form often
quoted in the literature. Recalling operator algebra, we note that p̂m rn ψ = (pn rn )ψ + rn (pm ψ).
Using this to rewrite Equation 7.17 and recalling im~ e [H0 , r] = p, we findDE2,ij =X

km n hj| rm rn |ii .

(7.19)

mn

7.2.3

Quick Notes on Order of Magnitudes

7.2.3.1

Order of Magnitude: E1 vs QM

In our derivation we found the QM transitions by including the next order expansion termk · r. We can estimate the ratio RQM/E1 of the QM to E1 transition matrix elements as

RQM/E1 = kr
≈

2π
a0
λ

(7.20)≈ 10−3where λ = 813 nm. Since the AC Stark shift corresponds to these transition matrix elements
squared, that means the QM terms are nearly order ≈ 10−6 relative to the E1 terms. This is apedagogical argument that is consistent with the 10−6 value found in the literature [7].
Importantly, this tells us we are safe to ignore the next order expansion term corresponding
to (k · r)2 , since for depths of ∼ 10 Er the lattice light shift from the QM term is around 1 × 10−18
meaning the next term is a shift at the 1 × 10−24 level or below.

7.2.3.2

Dropping the A2 Term

It is standard to drop the A2 term in the atom-light Hamiltonian. To see why we can ignore
A2 , we can estimate the ratio RA2 of eA to p̂.

126

eA
p
eE0 /ω≈~~Ω≈~ω

R A2 =(7.21)≈ 10−8
We find this estimate assuming a Rabi frequency of 10 MHz and optical frequency of 1 × 1015 . Even
compared to the M1/E2 term coming from k · r, RA2 is negligible and safe to ignore.
Finally, note that A2 has no atomic part and thus cannot lead to transitions [141].

7.3

AC Stark Shift From E1, M1, and E2
In the previous section we explored the origin of the E1, M1, and E2 transitions matrix

elements, essential for calculating oscillator strengths between atomic transitions. Now we areinterested in the question of the perturbation on the atomic energy levels arising from off-resonantradiation - the AC Stark effect. Recall that this is a second order perturbation in field, corresponding
to intensity.
The AC Stark shift is given by [64]α(ω) 2E04α(ω) I=−2 0 copt
e2 I X fji=−
2 − ω220 cmeωji∆EAC = −(7.22)

j6=i

opt
where I in the intensity of the radiation, given by E02 = 2I. α is the polarizability. fji
is given by
0c

[64]

opt
fji
=

2Wji Dji 2,
Eh a0(7.23)

127
where Eh is the Hartree energy and Wji = Ej − Ei . Finally combining all the relevant Dji
contributions we arrive at the AC Stark shift using the matrix elements we have calculated.

X Wji
Dji 2
e2 I
Eh 0 cmeω 2 − ω 2 a0
j6=i ji
X Wji  DE1,ji 2
DM 1,ji 2
DE2,ji 2e2 I=++
2 − ω2Eh 0 cme
a0
a0
a0
ωji

∆EAC =(7.24)

j6=i

Typically for the dipole approximation one only considers DE1,ji , but it is clear that we have M1and E2 contributions as well.
Recall the electric field for a standing wave along z (we’ll discuss the transverse confinementin the following sections):

E(z, t) = 2E0  cos(kz) cos(ωt).

(7.25)

We now need to account for the spatial position of our atoms along z, carefully considering how eachpolarizablity term couples. Considering the strength of E1 relative to the other terms, the atoms
are trapped at the potential maxima to excellent approximation. Thus, our spatial distribution isdefined by the E1 polarizability which couples to the potential corresponding to electric field (∝cos2 (kz)). The M1 polarizability, dependent on the magnetic field, and E2 polarizability, dependent
on the gradient of the electric field, are both ∝ sin2 (kz). Since the M1 and E2 have the same spatial
dependence in our lattice, we combine them into one QM polarizability term. Finally, we consider
the next order perturbation corresponding to 4 photon transitions. Since this is dominated by theE1 terms (and fourth order in field), the hyperpolarizability is then ∝ cos4 kz. Combining all of
this we arrive at the functional form for the lattice light shift in our 1D system for either the ground|gi or excited |ei clock state.

qm
AC
E1
Ug(e)
(I) = −αg(e)I cos2 kz − αg(e)I sin2 kz − βg(e) I 2 cos4 kz(7.26)

E1 is the dipole polarizability, αQM is the multipolarizability corresponding to M 1 andwhere αg(e)g(e)

E2, and βg(e) is the hyperpolarizability. Note that additional corrections for radial dimensions are

128
discussed next. In the 1D clock system, this corresponds to modifying the intensity terms by a a
Gaussian.
With Equation 7.26 we now have the general form of the lattice light induced AC Stark shiftessential for deriving lattice light shift models [107, 7].

7.4

Harmonic Model
It is important to have a model for our lattice light shift study that is readily understood byphysicists using standard analysis techniques. As such we will first approximate our optical lattice
as a harmonic trap. Such an approach is not new and has undergone theoretical development in[114, 71, 28]. For the reader who wishes to jump ahead, we will be deriving Equation 2 in [28], which
is what we will call the harmonic model. This section will detail exactly how to arrive at thisequation used in the aforementioned references, ensuring we can connect upcoming experimental
analysis with the results of Chapter 2 as well as the work of the NIST and Riken clock groups.
We first write the 1D optical potential Ulat (ρ, z) as

2 2Ulat (ρ, z) ≈ −U0 e−κ ρ cos2 (kz),(7.27)√

where κ = ω02 . We use cylindrical coordinates, with z the axial direction and ρ describing the radial
direction. We assume radial symmetry and a lattice waist ω0 . We rewrite Equation 7.26 using the
spatial form of the potential given in Equation 7.27. Recall that we are using g(e) to denote theground(excited) clock state. Also recall that we are still only describing the potential - we haven’t
included atomic wavefunctions or spatial expectation values.

qm
AC
E1
Ug(e)
(I) = −αg(e)I(ρ, z) cos2 kz − αg(e)I(ρ, z) sin2 kz − βg(e) I(ρ, z)2 cos4 kz(7.28)

2 2
2 2
2 2
qm
E1
= −αg(e)I0 e−κ ρ cos2 kz − αg(e)I0 e−κ ρ sin2 kz − βg(e) I02 e−2κ ρ cos4 kz

Here I0 is the peak intensity. We then expand in powers of z and ρ (see Appendix A.2), collecting
terms of equal powers.

129

E1
AC
I0 − βg(e) I02(I) = −αg(e)Ug(e)+ k 2 z 2 (αg(e) I0 + 2βg(e) I02 )−k4 z 4(αg(e) I0 + 5βg(e) I02 )3(7.29)

E1
I0 + 2βg(e) I02 )+ κ ρ (αg(e)2 2−

κ4 ρ4 E1(αg(e) I0 + 4βg(e) I02 )2− κ2 k 2 ρ2 z 2 (αg(e) I0 + 4βg(e) I02 )E1 − αQM . For a discussion of the coupling of radial/axial terms (z and ρ) see
Note that αg(e) = αg(e)g(e)[14, 15, 7].
From Equation 7.29 we first perform standard normal mode analysis to find the radial andaxial trapping frequencies. We first find the axial trapping frequency ωz .

mωz2 =∂ 2 Ulat (ρ = 0, z)
∂z 2
z=0

(7.30)

2= 2k (αI0 + 2βI02 )Similarly, we find the radial trapping frequency ωρ .

mωρ2 =∂ 2 Ulat (ρ, z = 0)
∂ρ2
ρ=0
2

= 2κ (αE1(7.31)

I0 + 2βI02 )Notice that the two are quite similar but with the switch of κ for k and αE1 for α. The normal
mode analysis for the radial direction is performed at z = 0, the node of the αqm term. Hence forpurely radial terms αqm contributions are not present.
Now we wish to perform first order perturbation theory and find the motional energies associated with this potential. To make this clear, we write the Hamiltonian for this system.

130

0
harm
+ Ĥg(e)Ĥg(e) = Ĥg(e)


p̂2
1
harm
2
2
2
2
Ĥg(e) =+ m ωz,g(e) z + ωr,g(e) ρ
2m 2
E1
0
I0 − βg(e) I02= −αg(e)Ĥg(e)

2k 4 z 4 mωz,g(e)2−+ 3βg(e) I0
3
2k 2


2
κ4 ρ4 mωρ,g(e)2+ 2βg(e) I0
−
2
2κ2

2
mωz,g(e)2 2 2 22
−κ k ρ z (+ 2βg(e) I0
2k 2(7.32)For the harmonic oscillator we know the energies. For Ĥ 0 we use first order perturbation
theory operating on the harmonic basis.

harm0Eg(e) = Eg(e)+ Eg(e)1Eg(e) = ~ωz,g(e) (nz + ) + ~ωρ,g(e) (nρ + 1)
2
0
E1
Eg(e)
= −αg(e)I0 − βg(e) I02 mω 2
3~2 k 41z,g(e)
2
2−
nz + nz ++ 3βg(e) I02
2
2k 2
2m2 ωρ,g(e) mω 2


4 2 8
~2 κ4
ρ,g(e)
2− 2 2
n + nρ + 2+ 2βg(e) I02κ2m ωρ,g(e) 3 ρ 3




2
~2 κ2 k 2
1 mωz,g(e)2− 2nρ + 1 nz +(+ 2βg(e) I0m ωρ,g(e) ωz,g(e)2
2k 2(7.33)In the interest of being pedagogical, we ever so slightly rewrite the perturbed energy to make clear
the clock state dependent terms.

131

E1
0
I0 − βg(e) I02= −αg(e)
Eg(e)



βg(e) 2k 2 Erec 2
1m−I
nz + nz +
+3 2
m
2
2k 2
ωρ,g(e) 0βg(e) 2
~2 κ4 4 2 8
m
−
+2 2
I
n + nρ + 2
2m2 3 ρ 32κ2
ωρ,g(e) 0mωz,g(e)βg(e)
12κ2 Erecnρ + 1 nz +
( 2
+2
I02
−
m
2 2k ωρ,g(e)ωρ,g(e) ωz,g(e)(7.34)We have now written the energies associated with motional quanta in our trap, using firstorder perturbation theory, for the ground(excited) clock state. We will assume in our analysis thatthe harmonic quantum numbers (nz and nρ ) do not change. This is true assuming single particle
physics and no radial/axial coupling. We now wish to find the lattice light shift in our systemcoming from the difference in trapping conditions between the excited and ground clock state.
We will use ∆αE1 = αeE1 − αgE1 , ∆β = βe − βg , ∆αQM = αeQM − αgQM , ∆ωz = ωz,e − ωz,g , and∆ωρ = ωρ,e − ωρ,g .

∆E = Ee − Eg
1
∆E = ~∆ωz (nz + ) + ~∆ωρ (nρ + 1)
2
− ∆αE1 I0 − ∆βI02



βg
3I02 k 2 Erec 2
1βe−nz + nz +
− 2
2
m
2
ωz,e
ωz,g



2
4
βg
2I0 κ Erec 4 2 8βe−n + nρ + 2
− 2
2
mk 2
3 ρ 3
ωρ,e
ωρ,g




2
ωz,e ωz,gκ Erec1−nρ + 1 nz +−k22ωρ,e ωρ,g




βg
4I02 κ2 Erec1βe−nρ + 1 nz +−m2
ωρ,e ωz,e ωρ,g ωz,g(7.35)

According to [71], near the magic wavelength the dipole polarizablity αE1 is about 106 times
larger than the higher order polarizabilities as well as ∆αE1 . Additionally, αeE1 = ∆αE1 + αgE1 .
We thus expand and keep terms to first order in differential polarizablity parameters. From hereon out I will use αE1 ≈ αgE1 . We first find ∆ωz and ∆ωρ .

132

r



Erec I01
nz +(∆αE1 − ∆αQM + 2∆βI0 )
αE1
2
r


κ Erec I0~∆ωρ (nρ + 1) =nρ + 1 (∆αE1 + 2∆βI0 )k
αE11~∆ωz (nz + ) =2(7.36)

We similarly solve for the two lines containing n2z and n2ρ .



βg
1
βe
3
1
I0 Erec
3I02 k 2 Erec 2
nz + nz += (n2z + nz + )∆β E1
− 2
2
m
2
ωz,e ωz,g
2
2
α



βg
3
2I02 κ4 Erec 4 2 88βe
I0 Erec= 2 2 (n2ρ + 2nρ + )∆β E1
n + nρ + 2
− 2
2
mk 2
3 ρ 3
ωρ,e
ωρ,g
2
α
3k ω0

(7.37)

We finish by finding the mixed nz and nρ terms.




ωz,e ωz,g1
κ2 Erec
Erec ∆αqm
√
n
+
1
n
+
−
=
ρ
z
k2
2
ωρ,e ωρ,g
αE1 2kω0




βg
1
βe
4I02 κ2 Erec
∆β I0 Erecnρ + 1 nz +
−
= 4√
m
2
ωρ,e ωz,e ωρ,g ωz,g2kω0 αE1(7.38)

Finally, we redefine our polarizabilities to simplify our formula [140].

α̃E1 = ∆αE1 Erec /αE1
α̃qm = ∆αqm Erec /αE1
(7.39)β̃ = ∆β(Erec /α

E1 2

)

u = I0 αE1 /Erec
Combining all of the above, we arrive at the harmonic model [28], with each line expressing a
different power dependency on u, the trap depth in units of Erec .

∆E = n5 α̃qm u0+ [(n1 + n2 )α̃E1 − n1 α̃QM ]u1/2− [α̃E1 + (n3 + n4 + 4n5 )β̃]u+ [2β̃(n1 + n2 )]u3/2− β̃u2(7.40)

133√
As in [28] n1 = (nz + 1/2), n2 = [ 2/(kω0 )](nρ + 1), n3 = 3/2(n2z +nz + 1/2), n4 = [8/(3k 2 ω02 )](n2ρ +√2nρ + 3/2), and n5 = 1/( 2kω0 )(nz + 1/2)(nρ + 1).

7.4.0.1

Radial Treatment of the Riken Model

If a Taylor expansion for ρ in Equation 7.28 is not performed one arrives at the Riken
model [140, 107]. To account for the radial distribution in the optical trap Ushijima et al. utilize
an effective trap depth. For a harmonic trap the radial spatial distribution can be described bya temperature dependent Gaussian. They calculate the average trap depth experienced by thethermal distribution and use this in place of an expansion in ρ. Unfortunately, this means theytreat the radial dimension as a purely harmonic trap, excluding a quartic correction. As kz and κρare a similar magnitude this may be a poor approximation.
In Nemitz et al. [107] they recognize the shortcoming of this approach for samples that aren’t
sufficiently cold. They instead note that the axial trapping frequency is a probe of the trap depth
experienced by an atom at a particular radial position. They perform axial sideband spectroscopy
and numerically fit the sidebands for a range of axial trapping frequencies, giving them a numericalestimate of trap depths. This is a clever idea, but worryingly they only query the sidebands forms timescales to avoid fitting issues that they associate with dephasing effects. Their fitting of theaxial sidebands relies on the work done by Blatt [14] which we know does not fully capture thebroadening of the sidebands. It is unclear at what level the approach employed by Nemitz et al. is
accurate.
Recent theory work has sought to evaluate the error associated with these techniques [7]. It
is important to emphasize as well that lattice light shift evaluations are still routinely performed
at significantly deeper depths than actual clock operation. While useful for evaluation of atomic
coefficients, it is worrying that no depths near clock operation are used, raising question aboutmodel extrapolation to operational depth. This is a driving motivation behind the previous andfuture Sr1 evaluations where we evaluate over a range of trap depths including the operational trap
depth. An additional benefit of the new Sr1 system is significantly colder radial temperatures in134comparison with the original Sr1 system as well as other 1D OLCs.

7.5

BO+WKB Model
A challenge in the analysis of the atomic spatial wavefunctions of the 1D OLC system isthe treatment of radial and axial modes. Beyond the harmonic approximation the longitudinal andradial degrees of freedom become coupled. Here we give a brief overview of the BO+WKB model,
described extensively in [7], expanding as necessary. The first premise of the BO+WKB model isto invoke a Born-Oppenheimer approximation in our system. Consider our new Sr1 system, witha waist of ω0 ≈ 260 microns. In this system, the ratio of the radial trapping frequency νr to axial
1
trapping frequency νz is ννzr < 1000
. We thus first model the atom to be at a fixed radial position ρand solve the corresponding eigenvalue equation,~2 ∂ 2+ Ulat (ρ, z) Znz (ρ, z) = Unz (ρ)Znz (ρ, z),−
2m ∂z 2

(7.41)

where Unz reprsents the nz eigenvalue and Znz the nz eigenfunction. As mentioned, we wish tosolve for Znz as a function of ρ owing to the vastly different timescales of axial versus radial motion.
To proceed we rewrite Equation 7.27.

2 2Ulat (r, z) ≈ −U0 e−κ ρ cos2 (kz)
1 + cos(2kz)
2 2≈ −U0 e−κ ρ2(7.42)The second line of Equation 7.42, upon inserting into Equation 7.41, can be rearranged to match
the form of Mathieu’s differential equation,d2 y+ (a − 2q cos 2x)y = 0,dx2

(7.43)

with known solutions. From Mathieu’s functions, we can impose the additional physical constraintthat tunneling is forbidden on the timescales considered, and are then motivated to normalizeπ
the Mathieu functions to a single lattice site ( |z| ≤ 2k
where k is the lattice wavenumber).

135
Renormalizing the Mathieu functions and utilizing a few of their properties, we arrive at the
eigenfunctions and eigenvaluesrπ D(ρ)
2k,senz +1 kz + ,
π
2
4




D(ρ)
D(ρ)
Unz (ρ) = ER bnz +1−,
4
2

Znz (ρ, z) =(7.44)

where senz +1 and bnz +1 are the even Mathieu functions and their characteristic values. D(ρ) =2 2(U0 /ER )e−κ ρ where U0 is the trap depth and ER is the recoil energy. The eigenvalues Unz (ρ) arethen inserted into the radial eigenvalue equation,




~2 d2
~2 l2 − 1/4−+ Unz (ρ) +Rnρ lnz (ρ) = Enρ lnz Rnρ lnz (ρ),
2m dρ22m ρ2(7.45)

where l specifies the z component of angular momentum (units of ~). The eigenvalues are given byEnρ lnz and the eigenfunctions by Rnρ lnz (ρ). nρ is the radial quantum number.
As Equation 7.45 has no known general solution, Beloy et al. utilize a WKB approximation[7]. They introduce the WKB phase for energy E < 0 (corresponding to trapped states) asr
φlnz (E) =

Z s
~2 l2
2m
E
−
U
(ρ)
−
dρ
n
z
~2 R
2m ρ2

(7.46)

where R denotes taking the integral over real values. Associated with the WKB phase is thecondition
1φlnz (Enρ lnz ) = π nρ +.
2

(7.47)

Our goal is not to numerically solve for the eigenergies, but rather the density of states (DOS)associated with the radial modes. We identify the density of states of nρ for a given nz and l as
equal toφ0lnz (E), with the prime denoting the derivative.
πIntegrating over all l we arrive at the

136
density of states for a single value of nz :1
Gnz (E) =
2

r

2m
~2

Z Z

1
q
dρdl.
~2 l2
R
E − Unz (ρ) − 2m
2ρ

(7.48)

Following the integration procedures outlined in [7] we find thatGnz (E) =2
1 2m 
(E)
.
R
n
z
4 ~2

(7.49)

Rnz (E) is the inverse function of Unz (ρ). Hence Rnz (Unz (ρ)) = ρ. This relates the DOS to the
known eigenvalues Unz (ρ) that we found earlier, leading a numerically tractable method for finding
the DOS.

7.6

Evaluating the Atomic Motional Distribution in the Harmonic Model
Earlier we derived the lattice light shift harmonic model. We now outline a method forevaluating the three input parameters u, nz , and nρ in the harmonic model but now accounting forthe coupling of ρ and z. This amounts to refining the work done in [14].
To extract information about our trap we will focus on sideband spectroscopy in our opticallattice. Recalling that to excellent approximation our trapping conditions are given by the dipole
polarizablity αE1 , we rewrite Equation 7.29.

U AC (I) ≈ −U0 + k 2 z 2 U0 −

κ4 ρ4
k4 z 4
U0 + κ2 ρ2 U0 −U0 − κ2 k 2 ρ2 z 2 U0
3
2

(7.50)

Here U0 ≈ αE1 I0 . Following our approach in deriving the harmonic model and using AppendicesA.1 and A.2, we find the energies E E1 for our optical trap,

1
Erec 2
1
E E1 (U0 , nz , nρ ) ≈ −U0 + ~ωz (nz + ) + ~ωρ (nρ + 1) −(nz + nz + )222ωρErec ωρ2 4 2 8
1−( n + nρ + 2) − Erec (nρ + 1)(nz + ).
2 ωz2 3 ρ 3ωz2(7.51)

137
Seeking to be consistent with Equation G1 of [7], we note that only accounting for the dipole√√
polarizability (Appendix A.1) ωz = 2/~ U0 Erec , ωρ = 2κ/~k U0 Erec , and ωz /ωρ = κ/k. We findpp1
Erec 2
1
U0 Erec (nz + ) + 2 U0 Erec (κ/k)(nρ + 1) −(nz + nz + )
2
2
2
2κ2 2
κ
1
− Erec 2 (nρ + 2nρ + 3/2) − Erec (nρ + 1)(nz + )3kk2

E E1 (U0 , nz , nρ ) ≈ −U0 + 2(7.52)
This is consistent with Equation G1 of [7], except they exclude the radial/axial coupling term. Thisis done to maintain our usual notion of axial/radial temperature in the calculation of expectation
values.
We identify the radial and axial energies which we will need for calculating thermal expectations.

1
Erec 2
1
Ez (U0 , nz ) ≈ ~ωz (nz + ) −(nz + nz + )22
22
Erec ωρ 4 2 8( n + nρ + 2)
Eρ (U0 , nρ ) = ~ωρ (nρ + 1) −2 ωz2 3 ρ 3(7.53)

Full modeling of sideband spectra in our system remains an open question, with this section serving to highlight one approach. The probing Rabi frequency, probe time, variation in
Rabi frequencies owing to varying Lamb-Dicke parameters with varying trap depth, and more allserve to complicate the single particle physics we are discussing. Early attempts at modeling thissystem were met with modest at best modeling/experiment agreement in the new system. The
low temperatures that enabled resolution of individual nz states precludes the more readily modeled blurred sidebands of [14]. With the improvement of axial cooling demonstrated in the new
system (hnz = 0i), a degree of freedom is removed from modeling which should allow better modeling/experiment agreement. The realization of 100 nK radial temperatures at ∼ 10 Er trap depths(as directly measured via transverse Doppler spectroscopy with the clock laser) will further enhancemodel agreement. Such work will be critical for the upcoming lattice light shift evaluation.

138

7.7

Lattice Light Shift Evaluation in the Wannier-Stark Optical Lattice
clock

a)

b)

Expectation Values Vs Depths

0.9

1e 18

QM Shift Vs Depths

6

0.8
5

0.6

Fractional Shift

Expectation Value

0.7
cos2(kz)sin2(kz)cos4(kz)

0.5
0.4

4

3

0.3
0.2

2

0.1
5

10

15

20

Depth (Er)

25

30

5

10

15

20

Depth (Er)25

30

Figure 7.1: Wannnier-Stark ground band calculations. a) Expectation values calculated for numerically evaluated WS states in 1D. Notice the rapid increase in hsin2 (kz)i at decreasing trap depths- this is the origin of the nonlinear ∆αQM behavior. b Lattice light shift contribution from ∆αQM
versus trap depth. Combined with a) we notice both the nonlinear behavior and small magnitudeof the shift. For self-comparison stabilities and systematic control resolution of the ∆αQM shifts
presents a formidable challenge.

Shallow trap depth provides a range of benefits for clock operation, reducing lattice lightshifts, Raman scattering, density shift systematics, heating, and more. Naturally, a consequence ofshallow trap operation is less tightly confined particles in the axial direction, leading to particles
deviating further and further from the anti-node of the lattice light. This results in significant
non-linear behavior in the M1/E2 perturbations at shallow depths, readily describable using theWannier-Stark basis discussed in Chapter 5.
In Chapter 5 we derived the Wannier-Stark (WS) states for the 1D optical lattice. Let us

139
now use those to discuss lattice light shift evaluations in the WS OLC. For our discussion we will
explicitly focus on an operational trap depth of uop = 12 Er , corresponding to the magic depth
introduced in Chapter 5 where the density shift is ≈ 0.
Ignoring radial degrees of freedom we may write the lattice light shift term in the 1D latticeas∆hνLS = −∆αE1 cos2 (kz)u − ∆αQM sin2 (kz)u − cos4 (kz)∆βu2 .

(7.54)

Let’s consider the Wannnier-Stark states in the position basis z corresponding to the lattice position:|ψW S (u)i. Following Chapter 5, we calculate the wavefunctions at various depths for the groundband. With the wavefunctions in hand we readily calculate the expectation values hcos2 (kz)i,
hsin2 (kz)i, and hcos4 (kz)i with the results shown in Figure 7.1. We see the increasing expectation
value of hsin2 (kz)i at shallower depths. Figure 7.1 additionally shows the ∆αQM fractional shift√
versus trap depth, clearly showing the u scaling discussed in the harmonic model.
When OLCs operated with fractional inaccuracies just below 10−17 , linear models were used
for lattice light evaluations. Operating between two trap depths, ∆αE1 was tuned (via the latticefrequency) to eliminate any differential shift - the so called ‘magic wavelength’ regime. However,
the higher order light shift terms discussed in this chapter have forced experimentalists to movebeyond the linear model for accuracy at the 10−18 level.
Considering ∆αQM , there are two operational regimes worth discussing at the magic depth.
For accuracy, it is preferable to operate at an inflection point of the light shift (δνLS /δu ≈ 0).
Making the lattice light shift first order insensitive to changes in trap depth ensures robustness
against fluctuations in experimental conditions. Alternatively, the experimentalist may wish for themost homogeneous trapping conditions possible (for instance for quantum information purposes).
In this case one would tune ∆αE1 such that νLS ≈ 0. Both situations may be modeled as discussed
earlier, numerically finding the correct ∆αE1 for each. The results are shown in Figure 7.2. Noteeven in the second case, a 10% variation in trap depth still manifests itself at the 10−19 level.

140

a)

Lattice Light Shift Inflection Point

2.050 1e 18
2.025

Fractional Shift

2.000
1.975
1.950
1.925
1.900
1.875
1.850

b)

Lattice Light Shift Zero Crossing

1.0 1e 18

Fractional Shift

0.5
0.0
0.5
1.0
1.5

6

8

10

12

Trap Depth (Er)

14

16

18

Figure 7.2: Operational regimes at the magic depth (u = 12 Er ). a) For accuracy, operation at
the inflection point provides a robust lattice light shift against imperfect experimental conditions.
Varying temperatures and atomic wavefunctions modify effective trap depths. Operating in thisregime provides the prospect of lattice light shift stability of ≈ 1 × 10−19 over a wide range oftrap depths. For this simulation the inflection point was found for ∆αE1 ≈ −0.09∆αQM . b) For
atomic coherence and quantum information uses, operating with the minimal lattice light shift canbe ideal. For ∆αE1 ≈ −0.18∆αQM the lattice light shift goes to zero at the operational depth.
Notice the increased slope compared to a). Both regimes are impacted by a mistake in ∆αQM
equally. Both simulations include hyperpolarizability contributions at the level of 1.5 × 10−19 .

141Figure 7.2 demonstrates the power of the inflection point. At trap depths of u = 12 ± 6 Er
the lattice light shift varies at the 1 × 10−19 level, suggesting stability of the lattice light shift at
the operational depth of Sr1 supports accuracy evaluations well into the 19th decade. While theinflection point location is robust to small mistakes in ∆αQM , the value of the lightshift at the trap depth is not and needs to be evaluated.
Evaluation of the atomic parameter ∆αQM is challenging. Consider a traditional lattice shift
evaluation as done in Chapter 2, where the trap depth is modulated. Such an evaluation performed
at even shallow depths where the radio of M1/E2 to E1 shifts is best is incredibly challenging
owning to the minuscule magnitude of light shifts (Figure 7.1). Evaluating to uncertainties below
1 × 10−18 per comparison lacks the sensitivity for resolving ∆αQM for trap depths between 6 and
30 Er .
One way to gain sensitivity to ∆αQM is to operate in a regime where comparison between
two different depths results in different contributions from ∆αQM while the ∆αE1 contribution
remain the same - in effect orthogonalizing the two. Considering the ∆αE1 contributions are nearlylinear in trap depth, this evaluation method corresponds to changing both high (uH ) and low trap
depth (uL ) but maintaining nearly the same difference in trap depth (∆u = uH − uL ). The resultsof such an evaluation are shown in Figure 7.3.

7.7.0.1

Transparency Beam

Even with a careful attempt to decouple ∆αQM and ∆αE1 , the evaluation technique of Figure
7.3 resulted in a poor evaluation of ∆αQM . Considering the complications of the density shift atthese depths, this raises serious concerns for the viability of such a study. Nonetheless, the inflection
point of Figure 7.2 shows us that given knowledge of ∆αQM we are well on the way to the next
accuracy goal of 5 × 10−19 . In Chapter 6 we showed a level of precision and intra-cloud frequency
control sufficient to resolve the gravitational redshift across the atomic sample. How can we use
this to aid our evaluation?
One technique would be to use a transparency beam [131]. Application of a laser at 688 nm

142

1e 17
6.75
6.80

(fractional)

6.85
6.90
6.95
7.00
7.05
7.10
40

50

60

70

High Depth (Er)8090Figure 7.3: ∆αQM lattice light shift evaluation. To gain greater sensitivity to ∆αQM , varying both
high and low trap depths enables nearly constant ∆αE1 lattice light shift contributions with varying∆αQM contributions. We explore this by simulating data for ∆αE1 = −∆αQM . For a high depth of
40 Er we compare to a low depth of 4 Er for a ∆u = 36 Er . As the high depth is increased, the low
depth is also increased nearly equally (maintaining ∆u), allowing the ∆αE1 contribution to remain
nearly constant in each differential evaluation while the ∆αQM contribution varies, providing a
degree of organizational between the two. For the shown data, with uncertainties of 1 × 10−18 , amodest uncertainty of δ∆αQM = ∆αQM /2 is achieved.

slightly detuned from the 3 P1 to 3 S1 transition shifts the 3 P1 state sufficiently far that laser coolinglight at 689 nm is no longer resonant. Applying such a beam to a region of the 1D lattice could
be used to to allow one region of the trap to remain at the single frequency MOT temperature
while a second region can be cooled much further as discussed in Chapter 4. Such a technique143
allows 1) modulation of temperature in a thermal manner and 2) the ability to compare the two√√
synchronously. This would move the self comparison stability from 1×10−16 / τ to < 1×10−17 / τ ,allowing high precision evaluation of shifts at shallow depths where density shifts are tamable.

Chapter 8

Conclusions and Outlook

The work presented in this thesis details an extended exploration of the frontiers of atomic
clocks. To summarize, we discussed:(1) Record low fractional frequency inaccuracy of 2.0 × 10−18 for a Sr OLC and third lowest
inaccuracy to date.
(2) First clock ratios between different species with 18 digits of accuracy.
√(3) Record low instability comparisons between two independent clocks (3.5 × 10−17 / τ ).
√(4) Record low intra-clock instability (4.4 × 10−18 / τ ).
(5) Evaluation of a differential frequency uncertainty to 21 digits (previous record was 19
digits).
(6) First ever observation of the gravitational redshift within a single atomic clock.
(7) Demonstration of 1D OLC operation free of atomic-interaction induced frequency shifts.
Paving the way for these results was work performed by numerous colleagues, from clock
evaluations [109, 98] to breakthroughs in atomic coherence times [33, 93] to collaboration withtheorists [76, 26, 1, 34]. Perhaps most vital is the work towards making ultrastable lasers ever morestable [112].

145
The breakthroughs in precision described in Chapter 6 are only the beginning. Camera noise,
not QPN, dominates the readout noise at the current number of atoms (100,000). Improved laser
coherence times will immediately enable longer clock interrogation times, further decreasing QPN.
Finally, improvements to lattice lifetime will ensure high atom number operation as pulse times
move to 10 s and beyond.
Atomic coherence times as reported in Chapter 6 are likely under reported, warranting careful
study. Reducing or eliminating the atomic interaction induced dephasing present in the current
Ramsey coherence measurements may allow for a deeper understanding of the limits of atomic
coherence in the new Sr1 system. A steered Si3 combined with Sr1 may already be an exciting
platform for pushing the limits of light-matter interactions.

8.1

Towards Accuracy
A major goal of this thesis thesis is advancing the Sr1 system towards the 19th decade of

accuracy. Let’s discuss ideal projected systematics (Table 8.1) compared to Chapter 2.
The environmental BBR uncertainty was limited by thermometer calibration and will not bea source of improvement in the reevaluation. As discussed in Chapter 4, an expansion port is readyfor installation of a calibrated in-situ temperature sensor, enabling the opportunity reevaluate thecalibration over time. The atomic BBR uncertainty has in principle a factor of ∼ 5 improvement
available by further evaluation of the 3 D1 lifetime [111]. The use of the 6” viewport will readily aid
in photon collection, greatly increasing the signal to noise ratio of the measurement as comparedto the previous Sr2 evaluations [109, 111].
Many systematics are already sufficient for the next evaluation. As discussed in Chapters 5
and 6 the density shift systematic is removed at the level of Table 8.1. DC Stark is statisticallyconsistent with zero and electrodes enable removal of any residual field. Operation of the systemwith multi-second pulses further reduces the probe AC Stark effect as well as enables operation
with smaller bias fields while still reducing line pulling. This serves to further reduce the 2ndorder Zeeman coefficient. Servo error can in principle be reduced but for now we will assume

146
Table 8.1: Sr1 systematic uncertainty from 2019 and new Sr1 projected uncertainty. ?s denote
optimal evaluations that are not guaranteed.
Systematic
BBR (environment)
BBR (atomic)
Density
Lattice AC Stark
DC Stark
Probe AC Stark
1st order Zeeman
2nd order Zeeman
2nd order Doppler
Servo error
Line pulling
Background gas
AOM phase chirp
Total

Shift (10−18 )-4974.1
0
-12.3
-21.3
0
0
0
-176.9000-3.7
0-5188.3

Uncertainty (10−18 )
0.2
1.5
0.4
1.2
0.3
<0.1
<0.1
0.2
<0.1
0.2
<0.1
0.4
<0.1
2.0

Projected Uncertainty (10−18 )
0.2
0.3?
<0.1
0.2?
<0.1
<0.1
<0.1
<0.1
<0.1
0.2
<0.1
0.1
<0.1
0.5?

no improvement. The upgraded vacuum enables 27 second lifetimes for 375◦ C oven temperature
operation, reducing the background gas systematic.
Finally, the lattice AC Stark shift (lattice light shift) will be the other major limiting factor.
Uncertainty will be dominated by the ∆αQM contribution. Assuming we match the Riken measurement uncertainty of 5% (currently in contention with theory), this will lead to an uncertaintyof ∼ 2 × 10−19 at the magic depth. As a result, the new Sr1 system should fundamentally be able
to approach a fractional frequency inaccuracy of 5 × 10−19 (Table 8.1). Cheers!

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Appendix A

Useful Derivations

A.1

Optical lattice Trap Frequencies
We write down the trapping potential in our optical lattice as

2

2

Ulat (r, z) ≈ −U0 e−2r /ω0 cos2 (kz)

(A.1)

where U0 is the peak trap depth, r is the radial position (assuming radial symmetry), ω0 is thetrap waist, k is the lattice wavenumber, and z the axial position along the lattice.
Using normal mode analysis we can then find the axial trapping frequency ωz asmωz2 =∂ 2 Ulat (r = 0, z)
∂z 2
z=0

(A.2)

= 2k 2 U0
and radial trapping frequency ωρ asmωρ2 =∂ 2 Ulat (r, z = 0)
∂r2
r=0

(A.3)

4U0
= 2 .
ω0
2 2

k
Recall that we will usually write our trap depth as u in units recoil energy Er = ~2m. We

then find the form of ωz that we will use in our model.
√~k 2 u
ωz =
m

(A.4)

159

A.2

Harmonic Oscillator Expectation Values
We begin by identifying the position operator x̂ in the harmonic basis [55]:

r
x̂ =



~
†
a +a
2mωz

(A.5)

where a(a† ) are the standard lowering (raising) operator, m is the mass of the oscillator, and ωz is
the oscillation frequency. In the harmonic basis we then find the following relations.

~
1
nx +
mωx
2
2
1
3~
n2x + nx +hn| x̂4 |ni =
2
2
2m ωx
2
3

5~
3
2
hn| x̂6 |ni =
4n
+
6n
+
8n
+
3
x
x
x
8m3 ωx3
35~4
hn| x̂8 |ni =2n4x + 4n3x + 10n2x + 8nx + 3
4
416m ωxhn| x̂2 |ni =(A.6)

Application of these relations to the axial dimension z is straightforward. Now consider theradial dimension ρ, identifying the radial expectation number nρ = nx + ny . We start with the ρ2
term. We will assume a 2D isotropic harmonic oscillator, with radial frequency ωρ .

hn| ρ̂2 |ni = hn| x̂2 + ŷ 2 |ni~=nρ + 1)mωρ

(A.7)

The expectation for ρ4 is more tricky owing to the degeneracy of nρ . We start by expandingin powers of x and y.

hn| ρ̂4 |ni = hn| x̂4 + ŷ 4 + 2x̂2 ŷ 2 |ni

2~2
1
1
3~2
2
2
n
+
n
+
n
+
n
+
1
+
n
+
n
+
x
y
x
y
x
y
2m2 ωρ2m2 ωρ222
~2=2n2x − 2nx nρ + 3n2ρ + 5nρ + 4
2m2 ωρ2

=

(A.8)

160
Notice on the last line we have rewritten ny = nρ − nx . To resolve the aforementioned degeneracyof nρ , we take an average over the possible values of nx .

hn| ρ̂4 |ni = hn| x̂4 + ŷ 4 + 2x̂2 ŷ 2 |ni

nρ 
X
~2
1
2
2
=
2nx − 2nx nρ + 3nρ + 5nρ + 4
2m2 ωρ2 (nρ + 1)
nx =0


~2
4 2 8
= 2 2
n + nρ + 2
m ωρ 3 ρ 3(A.9)

This is key for reproducing the harmonic model in [28], but is only detailed in [7].

A.3

Expansions
For small x.

x8
x4 2x6
−
+
3
45
315
4
6
5x
34x
13x8
cos4 (x) = 1 − 2x2 +
−
+
3
45
63
4
6
8
x
2x
x
sin2 (x) = x2 −
+
−
3
45
315
2
3
x
x
ex = 1 + x +
+
2
6
4 x4
a
a6 x6
2
2
e−a x = 1 − a2 x2 +−26
4a6 x622e−2a x = 1 − 2a2 x2 + 2a4 x4 −3

cos2 (x) = 1 − x2 +(A.10)