A Fermi-degenerate three-dimensional optical lattice clockbySara L. Campbell
B.S., Massachusetts Institute of Technology, 2010

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
2017This thesis entitled:
A Fermi-degenerate three-dimensional optical lattice clock
written by Sara L. Campbell
has been approved for the Department of Physics

Prof. Jun Ye

Prof. James Thompson

Date

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

iii
Campbell, Sara L. (Ph.D., Physics)
A Fermi-degenerate three-dimensional optical lattice clock
Thesis directed by Prof. Jun Ye

Strontium optical lattice clocks have the potential to simultaneously interrogate millions ofatoms with a spectroscopic quality factor Q ≈ 4 × 1017 . Previously, atomic interactions have forced
a compromise between clock stability, which beneﬁts from a large atom number, and accuracy, which
suﬀers from density-dependent frequency shifts. Here, we demonstrate a scalable solution whichtakes advantage of the high, correlated density of a degenerate Fermi gas in a three-dimensional
optical lattice to guard against on-site interaction shifts. Using a state-of-the-art ultra-stable laser,
we achieve an unprecedented level of atom-light coherence, reaching Q = 5.2 × 1015 with 1 × 104
atoms. We investigate clock systematics unique to this design; in particular, we show that contact
interactions are resolved so that their contribution to clock shifts is orders of magnitude lower thanin previous experiments, and we measure the combined scalar and tensor magic wavelengths forstate-independent trapping along all three lattice axes.

Dedication

To Bradv

Acknowledgements

First of all, I thank Jun for being the best advisor I can possibly imagine. Once I stopped
being so intimidated by him, Jun became a great friend who I could rely on for advice about
everything from the noise properties of mundane circuit components to life’s biggest questions.
What I most admire about Jun is his fearlessness. From years of working with him, I have come tobelieve that there is nothing we cannot gradually tackle via persistence and improved measurement
capabilities. Jan Hall surely played a key role in fostering Jun’s playful enthusiasm in the lab. I
thank Jan for his warm company in the electronics shop, and for setting an example to appreciate
every little bit involved in getting an experiment working. Indeed, even the smallest electronic
challenges can be “great fun!”
When I ﬁrst arrived at JILA, I felt a bit like a deer in the headlights. I am incredibly indebted
to Ben Bloom for taking me under his wing, teaching me how to debug and how to manage a project.
Travis Nicholson’s passion for our experiment convinced me to join. Jason Williams’ quiet wisdom
always came just when we needed it. Other strontium alumni I enjoyed working with include Mike
Martin, Mike Bishof, Xibo Zhang, and Matt Swallows.
I feel especially lucky to have had a great few years building the new system with the SrQ
Dream Team. Everyone’s unique strengths were essential to getting our new experiment working.
While technically we were supposed to be supervising our undergraduate student Rees McNally,
it was not long before he was the one teaching us! Among many other important projects, Rees
designed, built, and tested the high-current magnetic coil control electronics. Masters student
Nelson Darkwah Oppong built the 1064 laser system, ﬁgured out how to use the high-power ﬁbers,

vi
and made us some beautiful German-engineered low-noise photodiodes, complete with instructions
from Easy-E on how to set the gain. We still miss his laugh and exclamations of ”niiiiice!!” alongwith all of his insight and technical tips. We also enjoyed the company and curiosity of visiting
student Dan Reed who always made us laugh.
Ross Hutson is insanely good at everything from FPGA programming to theoretical threebody calculations, and he manages to do it all with quiet gooﬁness, modesty and thoughtfulness
towards others. Lab is rad, LabRad is rad, and Ross is the raddest. Ed Marti (my human Mendeley
library) is an AMO renaissance man, whose broad knowledge and insights both led us to new ideas
and prevented several disasters. During his time in the Ye group, Ed has also assumed the role of”Team Mom,” taking special care of all us little strontium ducklings, and being our go-to person
for life and physics advice. Aki Goban is a conscientious experimentalist and a deep thinker who
is always willing to jump in and get things done, as well as one of the kindest people I know. I likeit when Aki breaks character and gets sassy, because it means you deserved it. I feel privileged tohave been a a part of this amazing team, and I can’t wait to see what they do next.
Thank you to all of our collaborators. The Rey group regularly attended group meetings,
adding insights and excitement. Tom Loftus was meticulous, honest, and available to help – justthe person you need when having strange problems with a new prototype. Dylan Cotta from
Stephan Kuhr’s group graciously shared all of their knowledge on using photonic crystal ﬁbers
for high power and was patient with my incessant emails. David Tracy of TraTech Fiberoptics
did a beautiful custom job connectorizing these ﬁbers for us and was similarly patient with myincessant emails. Tim Darby of the UK Atomic Energy Authority helped with the design andconstruction of our recessed viewports, was accommodating when we had some unusual problems,
and was also patient with my incessant emails. Our team really enjoyed visits from both Darrick
Chang and Helmut Ritsch which helped us get a handle on the magnitude and mechanisms ofdipolar interactions in our present system, as well as gave us inspiration to work towards future
possibilities. While we play with new ideas for clocks, Judah Levine carries the responsibility ofactually telling the time, and kindly maintains a Cs-referenced 10 MHz signal so that all of Jun

vii
and Jan’s experiments - from clocks to combs to molecules - have a reliable RF reference. Every
few years, when we need to actually measure a number, it has been a real treat to take ﬁeld trips tohis Time Lord fortress to hear about “ﬁberology,” “jiggly-wigglies” and how one actually measures
time. In earlier accuracy studies, we worked with Marianna Safronova, Wes Tew, and Gregoroy
Strouse to further nail down the BBR shift. Insights on the state of our ﬁeld from Murray Hollandand Misha Lukin were greatly appreciated when I emerged from the basement to ponder suchthings.
We shared much of the strontium experience with the Sr1 gang. I admire Sarah Bromley’s
determination and level of focus in the lab. We have enjoyed the ideas and thoughtful companyof Shimon Kolkowitz (not to mention the donuts and breakfast burritos). Toby Bothwell seems to
get more excited about strontium every day. With a curiosity for the truth and an eye towards the
big picture, he will do a great job at steering Sr1’s future experimental direction. Dhruv Kedar
recently joined the team as well, adding his talent and positive attitude.
Finally, thank you to the stable laser crew, who provide the heartbeat and the gears of the
optical lattice clock. In the lab, I “grew up” taking Mike Martin’s 40 cm ULE cavity for granted.
The ﬁrst demonstration of the better stability of a many-particle frequency reference would nothave been possible without it. Wei Zhang has the superhuman capability of working two postdocs
at once and selﬂessly does whatever needs to get done. Lindsay Sonderhouse persevered in getting
the comb working again, and maintains a thoughtfulness towards the broader context of sciencein society. John Robinson was the next to join the team; I have never met a young student more
excited about Allan deviations. Yo dawg, I heard you are fond of the meta, so I put inﬁnities
inside your inﬁnities so you can self-similarity forever. The most recent addition was postdoc Erik
Oelker, who is applying LIGO-style transfer-function-ology to elucidate some of the ultrastablelaser voodoo.
A huge thanks to the JILA electronics and machine shops. Not only do they do beautiful,highly specialized work, they also teach us students to do these things ourselves. Terry Brown
always has time for impromptu lectures on feedback. Carl Sauer taught me some of his debugging

viii
and surface mount soldering skills. James Fung-a-fat and Chris Ho’s good-humored presence andtechnical tips were always appreciated.
Pushing the limits of metrology often requires pushing the limits of machinists’ patience, andso I am inﬁnitely grateful to everyone in the JILA machine shop. Kim Hagen had to put up withthe majority of the magnetic coil winding. Hans Green made our temperature sensors for bothaccuracy evaluations and somehow managed to keep a positive attitude. Blaine Horner’s wisdom
was crucial throughout the design process. Tracy Keep helped us design the water cooling manifold
and was the general magnetic ﬁeld guru. Todd Asnicar coordinated all of these jobs beautifully.
There are too many other JILA folks to really do justice to in these acknowledgements. Thankyou to the rest of the Ye group for their company and expertise, and also to the Thompson group
who graciously shared their knowledge and their lab space. Thank you to Brian Lynch, Dan Lewis,
and Jennifer Erickson in the supply oﬃce; Xu the custodian for her friendly hellos; Amy Allison
for her help with everything; Dave Alchenberger for all things tiny and/or coated; JR, Cory, andothers in computing for their help and company; Chris Purtell and Dave Errickson for assistancewith our FACMAN questions and problems; Beth Kroger for being there when things really needed
to get done. Finally, none of the work presented in this thesis would have been possible without
Debbie Jin, who laid the experimental groundwork for degenerate Fermi gases and continues to be
an inspiration.
I ﬁrst discovered how much fun AMO can be in Martin Zwierlein’s group at MIT (“djyeahhhhhhhh!”). Thanks to Martin for all of the excitement, Peyman Ahmadi for his kind leadershipand team spirit, Cheng-Hsun Wu for the laughter, and Ibon Santiago for the camaraderie. From
countless optics from BECII (aka ”Wal-Mart”) to his Solidworks ﬁles, Aviv Keshet has been graciously letting me steal his things for my entire AMO career. Outside of lab, Nergis Mavalvala
was the best junior lab instructor, Javier Duarte was the best junior lab partner, and my academic
advisor Gabriella Sciolla always fought for my best interest.
Outside of lab, my roommates (aka the men and women of Beefcake Manor) were my second
family. Thank you to Bob Peterson, Ben Pollard, AJ Johnson, Scott Johnson, Liz Shanblatt,ix
and Andy Missert. My friends and climbing partners including Brian O’Callahan, Kuyler andMadrone Coopwood, and Cathy Klauss kept me sane. Thank you to my parents for encouraging
and supporting my interests and my brother Jonathan for always being on my team. Thank you
as well to the Callahans, Auntie Jen and Uncle Mo, Julie Grandma, and Fred Grandma for all of
their help along the way.
Lastly I thank my husband Brad Johnson, the smartest person I know, for his love, patience,
and advice.

xContents

Chapter
1 Introduction

2

1.1

Historical perspective. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

The current frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.3

Coupled bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4

3D Lattice Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 New Apparatus
2.1

21

UHV System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.1

Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.2

Viewports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1.3

Bake-out procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2

Chamber mounts, custom breadboards, and other mechanical structures . . . . . . . 31

2.3

Magnetic ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.1

Compensation/bias coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3.2

Anti-Helmholtz coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3.3

Quadrant coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3.4

Single layer selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.3.5

Control electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.3.6

Mounting and water cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

xi
2.4

2.5

Laser systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.4.1

Optics layout around the main experiment. . . . . . . . . . . . . . . . . . . 63

2.4.2

Blue system upgrades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Imaging system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.5.1

Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.5.2

Measuring and testing resolution . . . . . . . . . . . . . . . . . . . . . . . . . 77

3 Preparation

80

3.1

Initial laser cooling stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.2

Crossed optical dipole trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.3

3.2.1

Initial loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.2.2

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.2.3

Measuring trap frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Red absorption imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.3.1

Magnetic ﬁeld cancellation and calibration . . . . . . . . . . . . . . . . . . . . 88

3.3.2

Optical pumping and spin state detection . . . . . . . . . . . . . . . . . . . . 94

3.4

Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.5

Kapitza-Dirac scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.6

Lattice loading and characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4 Experiments
4.1

4.2

4.3

108

Lattice AC Stark Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.1.1

Experimental conﬁguration and measurement . . . . . . . . . . . . . . . . . . 109

4.1.2

Data analysis and statistical methods . . . . . . . . . . . . . . . . . . . . . . 112

Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.2.1

On-site contact interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.2.2

Dipolar interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Narrow line spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

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4.4

Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5 Future prospects and conclusion
5.0.1

131

Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.1

Dark matter searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.2

The future of quantum metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Bibliography

140

Appendix
A Anti-reﬂection coating fused silica viewports: A cautionary tale

152

xiii

TablesTable
1.1

Typical operating parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2

Requirements for clock operation in a Mott-insulating regime with one
atom per site. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1

Speciﬁcations for important lasers used in the experiment. . . . . . . . . . . . . . . . 62

2.2

AOSense requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

xiv

FiguresFigure1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

Coupled bands for diﬀerent dressing laser Rabi frequencies . . . . . . . . . . . . . . . 13

1.4

Coupled bands for diﬀerent clock laser detunings . . . . . . . . . . . . . . . . . . . . 14

1.5

Coupled band lineshapes for diﬀerent tunneling rates . . . . . . . . . . . . . . . . . . 16

1.6

Coupled band lineshapes for diﬀerent phase shifts . . . . . . . . . . . . . . . . . . . . 17

2.1

CAD drawings of the UHV chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2

TAKOS AR coatings spectrophotometer measurements . . . . . . . . . . . . . . . . . 25

2.3

REO AR coating spectrophotometer measurement . . . . . . . . . . . . . . . . . . . 29

2.4

Which viewports have what AR coatings . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5

Chamber mounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.6

CAD drawings of the full system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.7

Mezzanine posts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.8

Optical table enclosure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.9

Compensation coils CAD model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.10 Teﬂon spacers in the compensation coil mounts . . . . . . . . . . . . . . . . . . . . . 38
2.11 Compensation coil design ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.12 Anti-Helmholtz coil design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

xv
2.13 Anti-Helmholtz coil ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.14 CAD model of quadrant coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.15 Quadrant coil schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.16 Constructing the quadrant coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.17 3D plots of ﬁelds generated by actual quadrant coils . . . . . . . . . . . . . . . . . . 46
2.18 Magnetic ﬁelds of actual quadrant coils . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.19 x lattice layer selection schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.20 Applying a magnetic ﬁeld to select a layer along x . . . . . . . . . . . . . . . . . . . 49
2.21 Range of transition frequencies due to magnetic ﬁeld curvature . . . . . . . . . . . . 50
2.22 Coil control electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.23 Coil switching logic logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.24 Coil switching explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.25 Anti-Helmholtz and quadrant coil current servo . . . . . . . . . . . . . . . . . . . . . 56
2.26 Anti-Helmholtz coil design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.27 Photo of all bucket coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.28 Bottom coil water and electrical hookups . . . . . . . . . . . . . . . . . . . . . . . . 58
2.29 Top coil water and electrical hookups . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.30 Interlock schemaitc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.31 Interlock photo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.32 Mezannine optics layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.33 Bottom optics layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.34 Top mezzanine layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.35 Artist’s depiction of the optics layout process. . . . . . . . . . . . . . . . . . . . . . . 67
2.36 Blue system schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.37 Imaging system section view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.38 Illustration of aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.39 Meniscus coating curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

xvi
2.40 Imaging system design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.41 Image of a two-pinhole test pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.42 Measuring resolution via the Rayleigh criterion . . . . . . . . . . . . . . . . . . . . . 78
3.1

Sr-87 level diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.2

3P

3.3

Timing diagram for the red MOT and ODT loading . . . . . . . . . . . . . . . . . . 86

3.4

Attempt to measure XODT frequencies via parametric heating . . . . . . . . . . . . 89

3.5

Measuring ODT trap frequencies via dipole oscillations . . . . . . . . . . . . . . . . . 90

3.6

HODT vertical calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.7

3P

3.8

Compensation coil ﬁeld calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.9

Polarization conﬁguration for measuring spin populations with 3 P1 absorption imaging 95

2 trap lifetime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

1 scans for compensation coil calibration. . . . . . . . . . . . . . . . . . . . . . . 92

3.10 3 P1 scans of nuclear spin states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.11 Time-of-ﬂight data of a degenerate gas in the optical dipole trap . . . . . . . . . . . 96
3.12 Dipole dipole trap lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.13 Kapitza-Dirac scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.14 Lattice loading timing diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.15 Experiment schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.16 Motional sidebands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.17 Doublon suppression for T = 50 nK. . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.18 Doublon suppression for T = 15 nK. . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.1

AC Stark cartoon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.2

AC Stark data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.3

AC Stark data autocorrelation function . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.4

Vector Stark shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.5

Level diagram for two-atom interactions . . . . . . . . . . . . . . . . . . . . . . . . . 118

xvii
4.6

Interaction spectroscopy data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.7

Spectrum of all transitions on doubly-occupied sites . . . . . . . . . . . . . . . . . . 124

4.8Line pulling calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.9

Narrow-line Ramsey spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.10 Narrow-line Rabi spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.11 Measured and calculated stability for diﬀerent clock conﬁgurations. . . . . . . . . . . 130
A.1 APC Quadband coating spectrophotometer measurement . . . . . . . . . . . . . . . 154
A.2 Never have your coatings done by American Photonics or Blue Ridge Optics . . . . . 155
A.3 TAKOS coating delamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

1

Chapter 1
Introduction

Of the four fundamental forces, the electromagnetic force is the one most intricately tied to
how we perceive and manipulate matter. Atoms, the building blocks of matter, are comprised ofmostly empty space. Our eyes cannot resolve the electron clouds around atoms, tiny particles ﬂying
around in a huge void. We see the world through photons, the gauge bosons that mediate electromagnetic interactions, scattered into the eye where they are focused on the retina and converted toelectric impulses that travel to the brain. The reason why we can touch and pick up a cup of coﬀee
is because of the force of a billion virtual photons a second being exchanged between the atomsof our ﬁngertips and the atoms of the cup, enforcing the Pauli exclusion principle that identicalelectrons cannot spatially overlap, a consequence of the unintuitive statistics of their minuscule
realm. It is only natural then that spectroscopy, the study of the interaction between matter andelectromagnetic radiation, was historically central to understanding the fundamentals of nature,and presently gives the physical quantities that can be the most accurately measured. SI base units
such as the second and the meter, the labels we assign to the magnitudes of human experience, are
all derived from spectroscopic measurements.

1.1

Historical perspective
Spectroscopy was our ﬁrst window into the quantum world. When Isaac Newton used a

prism to refract white light into its constituent colors in the mid-1600s, it appeared as a continuous
rainbow (see Fig. 1.1) [146]. Joseph von Fraunhofer’s inventions of the spectrometer and the3wire diﬀraction grating in the early 1800s both increased resolution, revealing discrete dark lines
within the rainbow, and enabled absolute frequency measurements [27]. Scientists in the mid-1800s
discovered that each element has its own unique spectrum [88]. The dark lines in the solar spectrum
were the shadows cast by absorption at discrete energy levels of atmospheric atoms. Atoms wereshown to emit at these discrete wavelengths, producing a negative of their absorption spectrum. In
1885, Johann Balmer found a simple formula that could predict the hydrogen spectrum in termsof integers [79]. Meanwhile, other experimental harbingers of quantum mechanics cropped up,
including the photoelectric eﬀect, the discovery of the electron, and the discovery of the discretized
nature of electric charge [69, 78, 116].To resolve the ultraviolet catastrophe, models by Planck andhis contemporaries began to quantize the energy emitted and absorbed by matter. Then, in 1905,
Einstein proposed a model where light came in discrete quanta of energy, in order to explain thephotoelectric eﬀect [54].
In 1917, Einstein laid the foundation for the modern laser when he described the three
types of light-matter interactions [55]. Absorption and spontaneous emission were evident in the
spectral lines ﬁrst observed a century ago. Additionally, to satisfy Plancks law for the distribution ofradiation at thermal equilibrium, Einstein introduced a third process: stimulated emission, whereby
a photon causes an atom to transition from the excited state to the ground state, emitting a second
photon identical to the ﬁrst. The fact that nature already held the key to making perfect copies
of quantized electromagnetic radiation was an early hint at the utility of this newly-discovered
non-classical world. In 1953, Charles Townes demonstrated the ﬁrst maser, a microwave ampliﬁer
based on stimulated emission on a 24 GHz transition in ammonia molecules [60]. Nikolay Basovand Aleksandr Prokhorov realized that these ampliﬁers could achieve continuous output by using
two transitions: one for incoherent pumping to maintain a population inversion, and another for
stimulated emission for coherent ampliﬁcation [120]. Thus began an evolution towards quantum
engineering, where the newly-discovered underpinnings of light and matter began to be exploitedin clever ways.
Another breakthrough in spectroscopy came in 1938 when, with the matrix and wave me-4

Figure 1.1: Album art from Pink Floyd’s Dark Side of the Moon [155].

Figure 1.2: Frauhofer lines in the spectrum of the sun [2].

5chanics formulations of quantum mechanics recently established, I.I. Rabi demonstrated the ﬁrst
coherent manipulation of internal states of atoms in his seminal work on nuclear magnetic resonance [130]. His student Norman Ramsey improved on this protocol, introducing the method of“separated oscillatory ﬁelds” in 1949 [131]. Armed with the technology to coherently control bothlight and matter, we could ﬁnally begin working towards the vision ﬁrst laid out in the Magna Carta
in 1215, that, “There is to be a single measure ... throughout our realm.” [159] Agreements in the
Magna Carta were not upheld, resulting in war within a year of the signing of the document, and as
recently as 1795, abuse of units of measures was one of the motives of the French Revolution [113],
thus highlighting the fallibility of man, and the superiority of the immutable laws of nature to enforce the equality of metrology standards. One of the new doctrines resulting from quantum theory
was that, if two people each had a Cs atom that they were each using as a frequency reference, andthey put their two atoms close enough together and in the same quantum state, the two quantum
particles would be fundamentally indistinguishable. This then led to the cesium beam clock where
the periodic “ticking” is provided by radio-frequency (RF) electromagnetic radiation and the absolute frequency reference is the ground hyperﬁne transition of the Cs-133 atoms, a constant of
nature that is identical everywheree [3].
Spectroscopists continued to devise new methods to improve energy resolution, putting Newtons rainbow under an ever-ﬁner microscope, looking for any anomalies that could force us tooverhaul our foundational understanding of the natural world. One of the main hindrances toatomic spectroscopy was Doppler-broadening due to atomic motion; control of external degrees of
freedom in precision measurement experiments remains important today. In 1947, Willis Lamb andRobert Retherford employed a clever use of microwave transitions to reduce Doppler eﬀects andobserve an unpredicted splitting between the 2 S1/2 and 2 P1/2 levels of the hydrogen atom, due toa one-loop correction to account for the zero-point energy of the vacuum causing absorption andemission of virtual photons [96]. This is another example where new measurement techniques mandated new understanding, and it spurred the development of modern quantum electrodynamics,
introducing the general concept of renormalization in quantum ﬁeld theory. Theoretical devel-6
opments in quantum ﬁeld theory, along with new particles discovered by generations of particle
accelerators with ever-increasing collision energies, led to the formulation of the Standard Modelof particle physics [129].
Meanwhile, with the foundations of quantum mechanics well established, atomic, molecular,
and optical (AMO) physicists spent the latter half of the 20th century developing new techniques
to control both light and matter. Beginning with the ﬁrst continuous-wave helium-neon (HeNe)
gas laser, the 1960s and 1970s saw the invention of several new kinds of lasers, frequency control,
and ﬁnally frequency stabilization techniques[65]. Laser technology enabled the meter to be thenext SI base unit to be fully deﬁned in terms of fundamental constants: ﬁrst in 1960 in terms of anumber of wavelengths of an atomic transition, and then again in 1983 in terms of the length of thepath traveled by light in 1/299792458 of a second [1]. Frequency-stabilized lasers were a necessary
tool in laser cooling and trapping of neutral atoms [150, 161, 36], which could then be loaded intoa magnetic trap for evaporative cooling, culminating in 1995 with the synthesization of a new stateof matter, the Bose-Einstein condensate (BEC), in which the majority of a gas of identical bosons
occupies the lowest motional state of the trap [40, 87]. Finally, all degrees of freedom of individual
quantum particles could be controlled. Just as the discovery of quantum mechanics led to thecesium beam clock, developments in laser cooling and trapping led to the cesium fountain clock,where the use of cold atoms allowed for longer interrogation times, improved stability, and a tenfold
improvement in accuracy [77].

While the reductionist approach that dominated 20th century physics was widely successful
in predicting a wide range of phenomena in terms of mean-ﬁeld theory and perturbative expansions,
it left a broad swath of physical phenomena unexplained, namely strongly-interacting many-bodysystems. Some outstanding problems falling in this category include the fractional quantum Hall effect, high-Tc superconductivity, and the quantum chromodynamics governing the physics of atomic
nuclei, neutron stars, and quark matter. These phenomena are all physical manifestations of themore general numerical sign problem, in which the antisymmetrization required by Fermi statistics

7
requires summing over a large number of wavefunctions with opposite sign, resulting in convergenceissues which become prohibitive in the thermodynamic limit [103]. While quantum entanglement
was proposed in 1935 and demonstrated in the 1980s, Hilbert space grows exponentially with thenumber of particles, and so our understanding was limited to systems which either only have a few
particles, or have an underlying symmetry.
“...it does seem to be true that all the various ﬁeld theories have the same kind of behavior, andcan be simulated in every way, apparently with little latticeworks of spins and other things.”- R. Feynman [57]
In 1982, Richard Feynman considered that, while the complexity of quantum many-bodysystems quickly overwhelms classical computers, one can build a universal quantum simulator to
predict their behavior [57]. At the turn of the millennium, some key tools and ideas were unearthedthat enabled the realization of Feynmans vision.
As many of the outstanding problems in condensed matter and particle physics follow Fermi
statistics, one challenge was to realize a controllable quantum gas of fermions, which are morediﬃcult to evaporatively cool than their bosonic counterparts. A second challenge was in relatingthese highly-controllable synthetic ultracold atom systems to unexplained physical phenomena thatare orders of magnitude away in temperature and density.
In 1999, Debbie Jin and student Brian deMarco produced the ﬁrst degenerate Fermi gasof ultracold atoms [46]. Then, not long after, along came the idea of operating these quantum
simulators in regimes giving access to so-called universal thermodynamics. At a Feshbach resonance [74], when the energy of two atoms in the open scattering channel is brought into resonancewith the energy of a bound state in the closed channel, the scattering length a diverges. In theunitarity limit of r  n−1/3  a, thermodynamic quantities depend only on the Fermi energy
EF (with corresponding Fermi temperature kB TF = EF ) and the temperature kB T . The internalenergy E can then be described in terms of a universal function f of a dimensionless parameter:
E = N EF f (T /TF ) [132, 10]. All unitary Fermi gases, from ultracold quantum gases at a Feshbach

8resonance (≈ 10 nK), to hot dense objects such as neutron stars (≈ 106 K) and quark-gluon plasmas(≈ 1012 K), can be described via a single equation of state spanning over 20 orders of magnitude.
An early triumph in the quantum simulation of natural phenomena was the measurement of theequation of state for the unitary Fermi gas in 2012 [94].
Additionally, quantum phase transitions, which occur at zero temperature by tuning therelative strengths of competing terms in the Hamiltonian, oﬀer another link to the universal physics
of quantum many-body systems. Just as in classical thermodynamics, the behavior of a system neara phase transition is governed by universal critical exponents which only depend on dimensionality
and the power law of interactions. Inspired by crystal lattices governing the behavior of electrons
in solid state materials, optical lattices made from standing waves of laser light were used to realize
Hubbard models for an atomic gas. This led to the observation of the superﬂuid to Mott-insulator
phase transition in a BEC [63]. Ultracold atomic systems operating at quantum criticality sincehave measured universal thermodynamic behavior.
While Debbie Jin was upstairs at JILA cooling potassium atoms, down in the basement, amid
towering piles of papers, precariously air-wired circuits, and tangled webs of RF, Jan Hall and his
team demonstrated the ﬁrst octave-spanning frequency comb in 1999 [51].1

This provided themissing step in converting between optical and microwave frequencies, as it allowed for measurement
and stabilization of the carrier oﬀset frequency f0 , and thus opened the possibility of using opticalfrequency standards. The two fundamental limitations to clock stability are local oscillator noise
which is aliased by running with a ﬁnite duty cycle via the Dick eﬀect [76] and quantum projection
noise (QPN), which depends on both the spectroscopic quality factor and the number of atoms
being interrogated. Optical frequency standards achieve a much higher spectroscopic quality factor
Q = ν/Δν than their RF counterparts by operating at a frequency ν that is over 4 orders ofmagnitude higher. The NIST mercury-ion and aluminum-ion clocks went on to break records inclock stability and accuracy [164, 66, 73].
1

Little did Debbie and Jun know, they would soon combine these powers to cool potassium-rubidium molecules
to their rovibrational ground state!

9
Also in 1999, work done at JILA, Caltech and in Japan [64, 83, 168] developing stateindependent trapping potentials planted the seeds for optical lattice clocks, based on thousandsof neutral atoms trapped in a one-dimensional (1D) optical lattice. These optical lattice clocks
went on to gain another order order of magnitude in stability due to the lower quantum projection noise (QPN) limit [122, 70] and demonstrate record-breaking accuracies [121, 23]. With
optical frequency transfer technology rapidly approaching the performance of the best ultra-stablelasers [49, 101], we are ﬁnally poised to upgrade our international time standard and frequency
distribution infrastructure to an optical-frequency clock.

1.2

The current frontier“The longing for a frontier seems to lie deep in the human soul... While there are clearly many
nonscientiﬁc sources of adventure left, science is the unique place where genuine wildness may
still be found.”- Robert Laughlin [133]
Try as we might, we have little intuition for worlds we have not yet experienced. Human
exploration encompasses both the outer limits of the cosmos and the tiniest energy shifts resolvable.
Every time we measure something new, we gain new understanding. This new understanding
extends the kinds of things we can make our tools out of and extends the limits of human perception.
To quote Jun’s old website, “Every time you peel oﬀ another layer of nature and look in a little bit
further, it gives you the most fantastic feeling.”2
In metrology, better stability ultimately leads to better accuracy, as systematic shifts can beevaluated to lower uncertainty during the ﬁnite number of hours in a day and years in a graduate
student’s career [23, 121] . The two fundamental limitations to clock stability are local oscillator
noise which is aliased by running with a ﬁnite duty cycle via the Dick eﬀect and quantum projection
noise (QPN), which depends on both the spectroscopic quality factor and the number of atoms being
2

I would like to note that thus far, every time I have looked in a little bit further, all I’ve found is insidioustechnical noise. But I’m still totally holding out for dark matter.

10interrogated. The QPN limit for Ramsey spectroscopy can be given as,
1
σQPN (τ ) =2πνTT + Td,Nτ(1.1)

where ν is the clock frequency, T is the free-evolution time, Td is the dead time, and τ is thetotal averaging time. Typically, OLCs operate at a stability above this limit due to the Dickeﬀect [76]; however, operation at or near the QPN limit has been demonstrated in systems throughsynchronous interrogation of two clocks [122, 154] or interleaved interrogation of two clocks withzero dead time [140]. For the future generation of optical lattice clocks, extended coherence time√and more atoms will lead directly to smaller QPN. However, reaching the next goal of 10−18 / τstability is extremely challenging for 1D OLCs as collisional eﬀects force a compromise between
interrogation time and the number of atoms that can be simultaneously interrogated [109, 121].
Throughout history, so-called fundamental limitations have merely been a consequence ofthe prevailing scientiﬁc paradigm. As science begins to tackle emergent behavior, we move beyond
quantum phenomena that can be described in terms of its constituent components, to entangled
many-body states, which can be engineered to overcome classical noise and be robust against perturbations. Here we take an initial step towards metrologically useful quantum correlated matter,
presenting the ﬁrst scalable solution to the central limitation of collisional eﬀects in optical latticeclocks. We load a two-spin degenerate Fermi gas into the ground band of a 3D optical lattice in
the Mott-insulating regime where the number of doubly occupied sites is suppressed [63, 80, 142].
In this conﬁguration, the atom number can be scaled by orders of magnitude, while strong interactions prevent both systematic errors and decoherence associated with high atomic density. This
work demonstrates the utility of Mott insulator physics for precision metrology, opening up new
possibilities for novel schemes in which quantum gas technology overcomes limits of atom-lightcoherence.
More is diﬀerent. - P. W. Anderson
More is a reduction in fundamental noise. - J. Ye3
3

At least half of the quotes in this thesis are fabricated.

11

1.3

Coupled bands“Bandz a make her dance.” - Juicy J (the rapper, not Jun Ye)
Now we consider the eﬀect of tunneling on the clock transition lineshape. While historically

we have operated clocks deep in the Lamb-Dicke regime where the tunneling rate J is negligible,
tunneling opens the door to spin-orbit-coupling studies [91] and other exciting quantum simulation
prospects [61]. Furthermore, eﬀorts to increase the atom-light coherence time beyond what has
been demonstrated in this thesis may require contending with the upper clock state decay due to
Raman scattering from the magic wavelength lattice. Studies of this eﬀect in 1D systems positthat this may limit the excited state lifetime to ≈ 10 s [115]. Eﬀorts to conﬁrm this eﬀect in our
3D system without the complication of lossy collisions between excited state atoms are currently
underway.
As is a theme for other systematic eﬀects and broadening mechanisms, there are two complimentary facets to spin-orbit-coupling in an optical lattice clock. First, lineshape distortions
provide an exciting measurement tool, allowing us to access spin-orbit-coupling physics using precision spectroscopy and uncover exciting many-body physics. Second, once we understand thisphysics, we can cleverly engineer our lattice geometry to eliminate these eﬀects and enable clock
operation at heretofore impossibly low trap depths, thus overcoming an imminent roadblock on thepath towards the full potential of the strontium atom’s ≈ 160 s natural lifetime.
The following Hamiltonian describes atoms on 1D lattice sites with index m, spacing a anda tunneling rate J, in two diﬀerent electronic orbitals e and g, coupled with Rabi frequency Ω witha laser with detuning δ, and a wavenumber kc that results in a site-to-site phase shift kc a, using
the dressed-atom picture and the rotating wave approximation:
H=


 
J a†g ag,m+1 + c.c. + a†e,m ae,m+1 + c.c.
m

 δ


+ Ω e−ikc am a†g,m ae,m + c.c. +a†e,m ae,m − a†g,m ag,m
2

(1.2)
(1.3)

To solve for the two coupled bands of this Hamiltonian, we use the following ansatz for the energy

12eigenstates φk which have quasimomentum k:
ψk =


m


c1 a†g,m e−ikam + c2 a†e,m e−i((k−kc )a)m |0(1.4)

Plugging the ansatz into the time-independent Schrödinger equation Hψk = E± (k)ψk gives thefollowing system of equations for the coeﬃcients c1 and c2 :⎞⎛ ⎞ ⎛ ⎞⎛Ω⎟ ⎜c1 ⎟ ⎜c1 ⎟⎜2J cos ka − δ/2⎠⎝ ⎠ = ⎝ ⎠⎝Ω
2J cos((k − kc )a) + δ/2
c2
c2

(1.5)

We then solve the characteristic equation to determine the energy eigenvalues E± (k),2J cos ka − δ/2 − E± (k)ΩΩ

2J cos((k − kc )a) + δ/2 − E± (k)

= 0,

(1.6)

ﬁnding that,
E± (k) = J(cos ka + cos((k − kc )a)
±

(1.7)

(J(cos ka − cos((k − kc )a)))2 − 2J(cos ka − cos((k − kc )a))

δ
2


+

δ
2

2

+ Ω2 . (1.8)

These two bands are plotted for diﬀerent Rabi frequencies in Figure 1.3, for the case ofstrontium in a magic wavelength lattice with spacing 813/2 nm, and a copropagating 698 nm clock
laser, which gives a site-to-site phase shift of kc a = 1.16π. Turning on the clock laser with Rabi
frequency Ω opens up a gap of ≈ Ω between the two bands.
To calculate the eﬀect of spin-orbit-coupling on the clock transition lineshape, we assume
that Ω  J, δ, which simpliﬁes the calculations but preserves the relevant physics. In this limit,
the bands become,E− (k) = 2J cos ka −δ
2

E+ (k) = 2J cos((k − kc )a) +

(1.9)
δ
2

(1.10)

Figure 1.4 shows the intersecting bands for diﬀerent detunings δ. At a given detuning δ,
atoms that are initially in the lower band are driven to the excited band at the quasimomenta k13Bands at different Rabi frequencies for δ = 0, kc a = 1.16π
E+ (k), Ω/J = 0.1

E+ (k), Ω/J = 0.5

E+ (k), Ω/J = 1

E+ (k), Ω/J = 0.1

E+ (k), Ω/J = 0.5

E+ (k), Ω/J = 1

3
2

E(k)/J

1

Ω
0
−1
−2
−3
−4

−3

−2

−1

0
ka

1

2

3

4

Figure 1.3: Plot showing how turning on the dressing clock laser with Rabi frequency Ω opens up
a gap of ≈ Ω between the two bands.

14

Bands at different detunings for Ω ≈ 0, kc a = 1.16π
E+ (k), δ/J = 0

E+ (k), δ/J = 1

E+ (k), δ/J = −1

E+ (k), δ/J = 0

E+ (k), δ/J = 1

E+ (k), δ/J = −1

Bands intersect when:δ = 2J[cos ka − cos((k − kc )a)]3
2

E(k)/J

1
0
−1
−2
−3
−4

−3

−2

−1

0
ka

1

2

3

4

Figure 1.4: Plot showing how the energies of the two coupled bands varies with detuning of the
dressing (clock) laser, in the limit of small Rabi frequency Ω compared to the detuning δ andtunneling rate J (Ω  δ, J). Atoms are initially in the bottom band, and at a given detuning, they
are driven to the excited band at the quasimomenta where the two bands cross.

15
where the two bands cross according to,δ = 2J[cos ka − cos((k − kc )a)]




kc a
kc a
sin k −
= −4J sin22

(1.11)(1.12)

By inverting Equation 1.12, we can solve for the two quasimomenta k1 and k2 that are driven to
the excited band at a given δ:1
k1 (δ) = sin−1aδ
4J sin(kc a/2)

+ kc /2k2 (δ) = π/a + kc − k1

(1.13)(1.14)

To ﬁnally calculate the clock transition lineshape, our remaining task is to sum the density of states
as a function of energy Dn (E) at these two quasimomenta. We assume an evenly-ﬁlled ground band
such that Dn (k)dk is a constant.

Dn (E)dEdkdE= Dn (k)dk(1.15)

dE.
dk(1.16)

so therefore,
Dn (E)dE ∝As explained in Equations 1.13 and 1.14 and illustrated in Figure 1.4, for a given δ, there are two
k-values for which the bands cross. So, to ﬁnd the total density of states D(δ) at a particular δ,
we add the density of states at k1 and k2 to ﬁnd,dE− (k) dE− (k) D(δ) =+,
dk k=k1 (δ)
dk k=k2 (δ)(1.17)

where k1 (δ) and k2 (δ) are given by Equations 1.13 and 1.14. Approximate clock transition lineshapes for Rabi spectroscopy can then be determined by convolving the Rabi lineshape with D(δ).
Figure 1.5 plots the lineshapes for the standard 1D strontium conﬁguration with φ = 1.16π
as the tunneling rate J is increased, showing a splitting of the line of approximately 8J sin φ. Figure
1.6 plots the lineshapes for J = 2 Hz for diﬀerent φ, illustrating that phase matching of the two16Lineshapes for different tunneling rates J with φ = 1.16π
J = 0.0 Hz

J = 0.3 Hz

J = 2.0 Hz

ka

Excitation fraction

1.0
0.8

8J sin(φ/2)

0.6
0.4
0.2
0.0
2
1
0
−1
−2
−3
−10

Detuning (Hz)E− (k)
E+ (k)ΔE(k)−50Detuning (Hz)5

10

Figure 1.5: Plot showing splitting of the clock transition as the tunneling rate J is increased. Thisis shown for the standard conﬁguration with co-propagating 698 nm clock laser and 813 nm lattice(with 813/2 nm spacing), giving a site-to-site clock laser phase shift of φ = 1.16π.

17Lineshapes for different phase shifts φ with J = 2 Hz
φ = 0.22π

φ = 1.16π

φ = 2.00πkaExcitation fraction

1.0
0.8

8J sin(φ/2)

0.6
0.4
0.2
0.0
2
1
0
−1
−2
−3
−10

Detuning (Hz)E− (k)
E+ (k)ΔE(k)−50Detuning (Hz)5

10

Figure 1.6: Plot showing splitting of the clock transition as a function of the clock laser site-to-site
phase shift φ at a ﬁxed tunneling rate J = 2 Hz. While the line can split by as much as 16 Hz,
by carefully phase-matching the ground and excited bands, the momentum kick imparted by theclock laser can be canceled, and we could operate the clock at much lower lattice depths.

18bands such that φ = 2π preserves the original lineshape, even at a tunneling rate that would
dramatically split line in the standard φ = 1.16 conﬁguration.
However, while phase-matched bands allow for Doppler-free, recoil-free spectroscopy even atshallow lattice depths, this technique does not prevent the atoms from tunneling. In 1D opticallattice clocks, inelastic collisions between excited state atoms have been shown to result in atom
loss and excitation fraction suppression [109, 121]. As we will describe in this thesis, we havealready shown that we can overcome atomic interactions by trapping one atom per site in a 3D
optical lattice. The use of a quantum gas in the Mott-insulating regime, where atomic interactions
are responsible for doublon suppression, oﬀers a way to continue to suppress atomic interactions
and block tunneling even at relatively low trap depths, thus providing an option to overcome theupcoming limitation of Raman scattering. With each barrier to optical lattice clock performancethat we solve, there are always more lurking at the next decimal place. We hope that the use ofquantum correlated matter for precision metrology experiments will allow us to continue to engineer
many-body states to push against future limitations.

1.4

3D Lattice Design
So, a Fermi-degenerate 3D optical lattice clock seems like a good idea. What kinds of lasers,

optics, etc. should we spend all of Jun’s money on?4 Here we review the basic technical requirements we considered when designing our new experiment.
First, we consider both the requirements for loading into the ground band of the lattice and forreaching the Mott-insulating regime. Loading the ground band requires that EF,ODT , kB Tlattice 
Erec , where EF,ODT is the Fermi energy in the ODT, kB is the Boltzmann constant, TODT is
the temperature in the ODT, and Erec is the recoil energy from a lattice photon. Competition
between tunneling (J) and repulsive interactions (U ) initializes the spatial distribution of theatoms. As the lattice depth increases, multiple occupancies are suppressed when 12J  U andkB Tlattice  EF  U , where Tlattice is the temperature in the lattice [80, 142].
4

That Jun Ye gravy train. Choo choooo.

Parameter
Fermi energy in XODT
Temperature in XODT
Lattice recoil energy
Clock laser recoil energy
Contact interaction energy
Bloch bandwidth
Tunneling rate along the clock laser axis
Spectroscopy time
Lattice trap depth
Lattice trap frequency

SymbolEF,XODT
TXODT
Erec
Erec,clock
U
4J
Jx /h
τ
U0,x , U0,y , U0,z
ν x , νy , νz

Typical Value
75 nK·kB
15 nK
167 nK·kB
226 nK·kB
2 kHz·h
6 × 10−7 Erec
0.5 mHz
6s(100, 70, 50)Erec,lattice(65, 55, 45) kHz

19

Table 1.1: Typical operating parameters.

Next, we consider how ﬁnite tunneling rates aﬀect clock spectroscopy. We require 1/Jx  τ ,where Jx is the tunneling rate along the clock laser propagation direction and τ is the spectroscopy
time, as the ﬁnite Bloch bandwidth of the lattice potential causes a ﬁrst-order Doppler broadeningof 8Jx . This requirement is satisﬁed for our longest spectroscopy times τ = 6 s by using lattice
depths above 80Erec . To achieve a suﬃciently deep trap as well as mode-match with the XODT,
we use elliptical beams for the x̂ and ŷ lattice axes with horizontal and vertical waists of 120 μm
and 35 μm, respectively. The ẑ lattice beam is round with a 90 μm waist. Typical operating
parameters in our experiment are summarized in Table 1.1, and a list of the requirements on theseexperimental parameters is summarized in Table 1.2.
OH YEAH! -Kool Aid Man

20

Requirement
Ground band loading
Double occupancy suppression
Resolved contact interactions
Lamb-Dicke
Doppler suppression

Inequality
EF,XODT , kB Tlattice  ErecJ  U, kB Tlattice  EF,lattice  U
ΔU  1/τhν  Erec,clock
8Jx /h  1/τ

Table 1.2: Requirements for clock operation in a Mott-insulating regime with one atom
per site.

Chapter 2
New Apparatus

2.1

UHV System
One of the requirements for both evaporating to Fermi degeneracy and exploring the limits of

atom-light coherence is a long vacuum lifetime. This ultrahigh vacuum (UHV) system was designed
to reach a pressure ≤ 1×10−11 Torr for a vacuum lifetime ≥ 60 seconds for optically trapped atoms.
2.1.1

DesignThe strontium source is a commercial system from the company AOSense, including an oven,
Zeeman slower, and two 2D MOTs which both cool the atoms and direct them into the mainchamber, allowing for no line-of-sight between the oven and the main experimental chamber. This
design is advantageous, as it prevents blackbody radiation (BBR) photons from the hot oven fromreaching the main experimental chamber, which would result in the atoms experiencing an athermal
BBR spectrum, and more complicated clock systematics.
The strontium source is connected to the main system via a 4 inch long, 0.24 inch diameter
diﬀerential pumping tube, corresponding to a suﬃciently-low 1 L/s conduction of hydrogen to themain chamber. The main chamber design is based oﬀ of similar designs from MIT (cite Aviv
thesis, Cheng thesis), and featuring recessed top and bottom viewports (“bucket windows”), whichallow for both high-resolution (≈ 1.2μm) imaging and the placement of magnetic coils close to
the center of the chamber for high magnetic ﬁeld gradients (> 1 Gauss/(Amp·cm)). The mainchamber was custom-made by the company Sharon Vacuum, from 304 stainless steel. They also

22oﬀered 316 stainless steel, but the company said that since these parts can get magnetized during
the machining process, it is not worth the extra cost. The bucket windows were custom madeby the UK Atomic Energy Authority (UKAEA), using 304 stainless steel for the ﬂanges, and 316
stainless steel for the recessed portion, which would sit closer to the magnetic coils.
The science section of the main chamber connects to the pumping section via a wide squaretube, so that the pumping on the main chamber is not conduction-limited. For pumps, we use
a Varian titanium sublimation pump (TSP), which is contained in a large cylinder for maximal
pumping surface area, a NEXTorr 300 L/s (for H2 ) Non-evaporable Getter (NEG) and 6 L/2 (forAr) ion combination pump, and we ultimately had to add a Varian 150 L/s StarCell ion pumpbecause of an unexpectedly-high noble gas load. Additionally, we shipped the entire chamber
to Jeﬀerson Laboratories to have them coat the interior with a sputtered Titanium-ZirconiumVanadium (TiZrVn) NEG coating. This coating reactivates at a temperature of 200o C (actually it
can reactivate at lower temperatures, it just takes longer).
2.1.2

Viewports

The closer the laser wavelength is to the UV, the more one must worry about the bubble
content of the viewport glass, as the glass can continue to ﬂuoresce long after the laser beams
are shut oﬀ, resulting in stray photons that can be detrimental to low-entropy Fermi gases andprecision clock experiments. Because several steps of our experimental sequence depend on using
a 461 nm transition, we took care to use grade A (maximum index variation ≤ 1 ppm) fused silica
with an inclusion class of 0 (≤ 0.03 mm2 total inclusion cross section per 100 cm3 of glass witha maximum size of 0.10 mm). We custom ordered our viewports from Larson Electronic Glasswith the best speciﬁcations available for a reasonable price: 20/10 scratch-dig and λ/4 ﬂatness.
(They do not specify better than λ/4 ﬂatness after warping that results from making the glassto-metal seal without re-grinding and polishing.) We had some diﬃculties with our anti-reﬂection
coatings delaminating from the fused silica (see Appendix for the full saga), but we ultimately
settled on a combination of viewports with the AR coating curves shown in Figures 2.2 and 2.3,

23

TSP

Top bucket
window
150 L/s ion pump
AOSense system

Gate valve

Bottombucketwindow

Ion gauge

NEXTorr NEG + ion pump

150 L/s ion pump

All-metal valves

TSP

Top bucket
window
NEXTorr NEG + ion pump

Ion gauge
AOSense system

Gate valve

Figure 2.1: CAD drawings of the UHV chamber.

24
in the locations detailed in Figure 2.4. The coating scheme was determined by some compromise
between technical requirements, availability, and time constraints. Apparently ion beam sputtering
is better than electron beam sputtering (which both of our companies used). In the future, whenasking companies for quotes, I would be more emphatic about the diﬀerent thermal expansion
coeﬃcients of fused silica vs. stainless steel, and have any company assure me that they will docareful temperature control to minimize any mechanical stress.
2.1.3

Bake-out procedure

Due to complications introduced by using several getter pumps in conjunction, we had many
steps to our bakeout procedure.
(1) Remove ion pump magnets, make sure all cables for ion pumps, TSP, NEG pump, ion
gauge, and residual gas analyzer (RGA) are plugged in.
(2) Pump down with roughing pump and turbo pump, leak check using helium gas and theRGA.
(3) Heat the whole system to 150-160o C, put the NEXTorr NEG into conditioning mode, run
25 A through each TSP ﬁlament to heat and clean (but not to deposit any titanium yet).
We couldn’t ﬁnd a commercial TSP controller which allowed the current to go that low, so
we borrowed a regular power supply from the electronics shop, hacked apart a TSP cable,and put all TSP ﬁlaments in parallel because they had similar resistance (to a few percent).
(4) When the ion gauge reads 2 × 10−8 Torr, clean/degas/ﬂash all components. Ion gauge:degas for the full 15 minutes as the manual suggests. RGA: Degas 3 times via the control
software. Ion pump: Flashed twice by turning on the high voltage (saw nothing on the
RGA). TSP ﬁlaments: Ramp them all at the same time to 37-42 A each and leave for 21
minute before ramping down to 25 A. Do this until the pressure rise from the gases otherthan hydrogen reads in the low 10−9 range on the RGA.

25

TAKOS Coating Comparison on Larson Electronic Glass 2.75 CF
100

99

X: 1064
Y: 99.41

X: 698
Y: 99.2
X: 461
Y: 98.81

98
X: 1064
Y: 98.01

X: 813
Y: 97.97

97%T

X: 698
Y: 96.95

96

95

94

93
400

Quad−Band
New Broad−Band

X: 461
Y: 94.61

500

600

700

800

900

1000

1100

Wavelength (nm)Figure 2.2: Spectrophotometer measurements of both coating runs from TAKOS used on the
viewports.

26(5) Heat the whole system to 180 − 190o C for several weeks. We realized that with the NEG
in conditioning mode at 250o C, the pressure was not going to go below the low 10−8 level,
so we turned oﬀ the conditioning mode to check the base pressure of the system, and foundit to be below 10−8 Torr.
(6) Clean the TSP ﬁlaments again. On the RGA, we saw mostly hydrogen, a little nitrogen,and a huge argon spike. Throughout most of the vacuum work, we saw odd spurious argon
peaks on the RGA. After this, we had to wait overnight for the hydrogen pressure to return
to normal. The next morning, we “ﬂashed” (turned on the high voltage to) the ion pumpagain. We never saw any change to the RGA or ion gauge readings when ﬂashing any ion
pump in this manner.
(7) Activate the NEXTorr NEG. Before activation, the ion gauge read a pressure of 1.2 × 10−8
Torr. First we tried regular activation, but we got scared when the pressure was a few 10−7 .
Then we repeatedly turned on activation mode and turned it oﬀ every time the pressure
reached 5 × 10−6 . Finally, we gave up and let the pressure go up as high as it wanted to,to a maximum of 3.5 × 10−5 . We activated the NEXTorr NEG at 550o C for 3 hours, andthe pressure dropped to 3.6 × 10−6 Torr, corresponding to a capacity of 1.6 Torr·liter.
(8) Cool down the system to room temperature.
(9) Re-attach magnets, turn on ion Andrs D Herrerapumps, then turn them oﬀ.
(10) Subsequently ﬁre all TSP ﬁlaments to clean them (while the turbo pump is still pumping)in preparation for re-ﬁring them a year or more later. We rotated ﬁring one ﬁlament at thetime, stopping when the pressure rose to the low 10−6 Torr level. After a few iterations,the pressure only rose to the 1 × 10−7 level and we deemed the ﬁlaments suﬃciently clean.
After all this was done, we found that the pressure measured by the ion gauge was 2−3×10−11
when the all-metal valve connecting the main chamber to the turbo pump was open, which isincredibly good for something connected to a turbo; from looking at the hydrogen partial pressure

27
on the RGA, we concluded that the main chamber was actually pumping hydrogen away from thefront of the turbo pump. However, the pressure curiously rose to 5 × 10−11 when we closed the allmetal valve. With the valve closed, we turned the 6 L/s ion pump portion of the NEXTorr oﬀ andmeasured the pressure rise over time to verify that it was indeed pumping as speciﬁed. Therefore,
we concluded that we had an unexpectedly-high noble gas load in the main chamber, and so wewarn that for other groups with similarly large vacuum chambers, the 6 L/s in the NEXTorr pump
may not be suﬃcient, and a regular large ion pump may be required. Additionally, we observedthat the pressure rise in the main chamber from opening the gate valve with the oven on was
initially much higher than what we would expect from the 300 L/s hydrogen pumping speed fromthe NEXTorr NEG and the 1 L/s conduction for hydrogen, but after several days of running theoven, the pressure rise was as expected. We concluded that there was likely some residual noble gasload, perhaps due to oven chamber being backﬁlled with argon for strontium loading, that quickly
baked oﬀ within several days of operation.
In any case, we ordered a custom tee so that we could add a 150 L/s ion pump to the chamber.
(RTFM: The pumping speed decreases with pressure and depends on what is being pumped. The
labeled pumping rates correspond to diﬀerent gases for diﬀerent companies; the Varian StarCell
pump was 150 L/s for hydrogen at higher pressures, but only 30 L/s for argon at 1 × 10−11 Torr.
Still, this was 5 times better than the NEXTorr ion pump.) Also, Varian put the ion pump magnet
on backwards when they refurbished it. So, if anybody else has trouble getting their refurbishedion pump to start, this may be a useful thing to check, before you follow tech support’s advice toaggressively ”hit the pump with a rubber mallet to try and shake some electrons loose.”
We were also able to do some tests to estimate the pumping speeds of the various pumps. The
turbo pumping speed for hydrogen was 45 L/s (and we calculated that it should not be conduction
limited). After the next round of ﬁring all TSP ﬁlaments to clean them during the bake and before
activating any getters, the hydrogen partial pressure on the RGA decreased by a factor of 9, leading
us to estimate that the conduction-limited TSP pumping rate on the RGA was approximately 360
L/s, and so the TSP pumping rate on the main chamber was at least that high.

28
As we were cooling down and after activating the NEXTorr NEG pump, we closed the allmetal valve to test our hydrogen pumping speeds (the titanium ﬁred by the TSPs was saturated
and not pumping at this point). At our pressure of 6 × 10−11 Torr, the StarCell hydrogen pumping
speed was speciﬁed to be 40% of its nominal 150 L/s, or 60 L/s. Turning the ion pump oﬀ caused
the pressure to rise to 8 × 10−11 Torr, which is consistent with the hydrogen being predominantly
pumped by those two pumps. Therefore, we concluded that the Ti-Zr-V coating on the chamber
was actually not doing any pumping, so perhaps it was not worth all the eﬀort we went throughto make sure that it was activated. The folks at Jeﬀerson Labs quoted 0.1 L/ft2 for the pumping
speed, very low indeed. However, it may be an eﬀective outgassing barrier, but without an identical
uncoated chamber to compare it to, it is hard to know. Once we ﬁnished cooling down and closed
the gate valve, the ion gauge measured a base pressure of approximately 5 × 10−12 Torr. We used
the UHV-24P ion gauge (and corresponding controller) from Agilent (formerly Varian), as it seemed
to be the only hot ion gauge that could measure pressures that low.
Unfortunately, we when we unwrapped the chamber, we discovered to our horror that thebottom bucket window was completely covered with metal dust from the Ti-Zr-V chamber coating.
Somehow something has been going wrong lately with Jeﬀerson Laboratory’s coating process; wehave not heard about this happening to anyone eles. The dust is ﬁne (maybe even good) for
vacuum, but obviously not ideal for optical lattices or high-resolution imaging. We vented thechamber, blew the dust out, and re-baked at 180o C, after the experts at Jeﬀerson Lab assured
us that baking would not produce any more dust. Jun thought (for no good reason) that 150o C
would be a better temperature, but he let us go to 180o C. Sure enough, more dust ﬂaked oﬀ thewalls and fell on the bottom viewport. So, we vented yet again, blew dust oﬀ the bottom bucket
window, baked at 150o C, and henceforth believe that Jun Ye is a magical wizard who should not
be doubted. Since there was a compromise between pressure and metal dust on the viewport, wesettled for 1 × 10−11 Torr and a handful of particles on the bottom viewport.

29

REO Tri−Band AR Coating on Sr2 Viewport
105

100

95

%T

90

85

80

75

70
400

500

600

700

800

900

1000

1100

Wavelength (nm)Figure 2.3: Spectrophotometer measurements of the tri-band coating by REO on a viewport fromthe old Sr2 chamber, which we had to vigorously clean via the Jun Ye method of hanging the
viewport above a beaker of boiling solvents (yummmmm). Note the high transmission and how
beautifully they achieved the target wavelengths of 461, 698, and 813 nm. Again, they don’t makethings like they used to. Note: NEVER put fused silica viewports in a sonicator. It might be ﬁne
for 7056 glass or other materials which have a diﬀerent kind of glass-to-metal seal (an anonymous
postdoc and senior graduate student on this experiment had sonicated many a viewport on theirpast experiments), but it is NOT ﬁne for fused silica, where the glass is soldered to the metal.
Something about sonicating it eats away the solder. Ask us how we know.

30

Red + Blue MOTTAKOS Broad-band
TAKOS Quad-band
REO Tri-band

813 lattice +clock laser
813 lattice

HODT

Abs.
Imaging

Figure 2.4: Illustration of the diﬀerent AR coatings on the diﬀerent viewports

31

2.2

Chamber mounts, custom breadboards, and other mechanical structures
The chamber was baked on a temporary 80-20 mounting structure. Once the bake wasﬁnished, we transported the chamber into the main lab and placed it on permanent mounts madeof solid aluminum, shown in ﬁgure 2.5. Because the mounting tabs on the chamber were not weldedon perfectly, and also following the customary wisdom that any mechanical structure referenced toa plane (i.e. the optical table) by more than three points must be done so with compliant material,
we put “blue stuﬀ”1 between the chamber and the mounts, between the mounts and the optical
table, and between the mounts and the clamps to the optical table. Since the magnetic coils would
be clamped directly to the bucket windows (and therefore transfer vibrations to the chamber), the“blue stuﬀ” has the added beneﬁt of damping vibrations. The mounts were designed with crossbraces and gussets to reduce any potential rocking or swaying vibrational modes. A slot was milled
out of the bottom of the mounts to accommodate clamping to the optical table.
Two custom breadboards from the Technical Manufacturing Corporation (TMC) were ordered to ﬁt snugly around the optical table as a “mezzanine” level, as shown in Figure 2.6. Weordered the 77 Series honeycomb core stainless steel breadboard in a 2 inch thickness, with holes
on both the top and bottom, to help with future mounting structures and cable management. The
breadboards were mounted on 3” diameter aluminum cylinders ﬁlled with lead ball bearings forvibration damping and welded shut. As with the chamber mounts, blue stuﬀ was placed aboveand below the mezzanine mounts both to have a compliant material to minimize mechanical strain
and for additional vibration damping. To make sure that both mezzanine breadboards were rigidly
referenced to one another in order to minimize drifts in optical alignment, we connected them with
two T-beams and one right angle brace. This also served to stiﬀen the board longitudinally to
avoid “ﬂapping” vibrational modes between the two halves of the breadboard.
Additional hardware upgrades included an optical table enclosure, as since we were planningon using a 50 Watt ﬁber ampliﬁer for evaporation, we decided that it would be a good idea in
1

ISODAMP C-1002-06 made by EAR, available from Raithbun Associates

32

Figure 2.5: Top: CAD drawing showing the chamber mounts in blue. Bottom: Clamping themounts to the optical table.

33

Figure 2.6: CAD drawings showing the full system design, including custom breadboards surrounding the chamber.

34

Figure 2.7: Top: Photo of a mezzanine post. Bottom: Photo of the full assembled system.

35
our dusty lab to run a HEPA ﬁlter so that dust wouldn’t get burned onto all of the optics. The
HEPA ﬁlter air intake is connected to a water-cooled heat exchanger to allow for the possibilityof temperature stabilizing the optical table (the lab temperature has been stable enough that wehave not been motivated to implement this yet).

2.3

Magnetic ﬁelds
The magnetic ﬁeld control requirements for our experiment include bias/compensation coils

for both canceling ambient magnetic ﬁelds and applying bias ﬁelds for atom manipulation; antiHelmholtz coils for magneto-optical trapping; and new “quadrant coils” capable of applying strong
bias ﬁelds along any axis, which, when used in conjunction with the gradients from the antiHelmholtz coils, allow spectroscopic selection via the clock laser of a single lattice layer along any
lattice plane.
2.3.1

Compensation/bias coils

The compensation coils were designed to be as compact as possible so as not to impede
optical access. It is also desirable to be able to move the narrow-line red MOT, which operates ata magnetic ﬁeld gradient of ≈ 3 G/cm by at least 2 mm to aid alignment the next stages of laser
cooling and trapping, thus requiring a bias ﬁeld capability of at least 2 G. The compensation coils
in our previous generation experiment were very large (a total of ≈ 500 feet of 20 gauge wire) andso became hot to the touch when run at the full current of 3 A. Since the magnetic ﬁeld scalesas I 2 N , where I is the current and N is the number of turns, while the power scales as IN , it isgenerally advantageous to have as many turns of wire as will ﬁt.
A CAD drawing of the compensation coil mounts is shown in Figure 2.9. They were designed
to ﬁt snugly together around the vacuum chamber. The mount for one of the x-coils (shown in red)
was machined from two separate pieces that ﬁt together around the chamber, and, for mechanicalclearance reasons, was wound before the viewports. As in [14], we used Duralco NM25 epoxy (500o F
maximum temperature and free of magnetic particles and ﬁllers), along with Kapton-insulated wire

36

Figure 2.8: Top: Photo of the optical table enclosure, including sliding doors, top and bottom tabletrays and optical ﬁber enclosures. The black Plexiglass doors double as a whiteboard!

37(400o F maximum temperature), so that the coil could be baked along with the rest of the vacuum
chamber. The mounts were machined out of aluminum, so in order to achieve reasonably fastswitching times of ≈ 1 ms, we had to add Teﬂon spacers (shown in Figure 2.10 to break the loopof aluminum and reduce Eddy currents. In retrospect, it might be better to just make the entire
mounts out of some kind of high-temperature plastic.
The calculated magnetic ﬁelds from these coils are plotted in Figure 2.11, and achieve ourdesign goals, reaching a ﬁelds > 4 G in all directions when operated at 3 A, with less than 1/4 ofthe power consumption of the previous design.
2.3.2

Anti-Helmholtz coils

The anti-Helmholtz coils (Fig. 2.12) serve the dual purpose of providing magnetic ﬁeld
gradients of 50 G/cm and 3 G/cm for the blue and red MOTs, respectively, as well as providing
the capability to apply strong gradients of 0.62 G/(A·cm) for maximum gradients of 310 G/cm at
500 A, as shown in Figure 2.13. One can use a combination of magnetic ﬁeld and high resolution
spectroscopy to select a particular lattice layer in the z direction by running the top anti-Helmholtz
coil only, which simultaneously provides a bias ﬁeld and a gradient. Given an 813 nm magicwavelength optical lattice with 406.5 nm spacing between sites, a 310 G/cm gradient gives a 12.6
mG magnetic ﬁeld diﬀerence between adjacent lattice planes. From the 109·mF μB Hz/G diﬀerential
g-factor between the ground 1 S0 and clock 3 P0 states, where μB is the Bohr magneton, there would
be a 6 Hz clock shift between π-polarized lattice sites on adjacent lattice planes, which is easily
resolvable with, for example, 1 Hz Fourier-limited clock spectroscopy.
To provide suﬃcient water cooling, each set of coils is wound with 3 sets of two layers, with
the inner layer wound top to bottom, and the outer layer wound bottom to top. A CAD modelshowing the layers and dimensions is shown in ﬁgure 2.12. This way, the inner and outer layers
spiral up and down in opposite directions with respect to the horizontal plane, thus largely cancelingstray transverse ﬁelds resulting from that tilt. Additionally, each set of 2 layers is designed withdecreasing height, so that each set has approximately the same length, and thus heat load. The 3

38

Figure 2.9: CAD model of the compensation coils.

Figure 2.10: Teﬂon spacers in the aluminum compensation coil mounts, added to reduce eddy
currents.

39

X field (Gauss/Amp)
3.0
2.5
2.0
1.5
1.0
0.5

-10

Distance from center (cm)

-5

0

5

10

5

10

Y field (Gauss/Amp)
3.0

2.5

2.0

1.5

-10

Distance from center (cm)-50

Z field (Gauss/Amp)
3.00

2.95

2.90

2.85

-4

-2

Distance from center (cm)02

4Figure 2.11: Calculated magnetic ﬁelds for the compensation coils.

40
sets of coils are connected electrically in series and connected to the water cooling in parallel.

Figure 2.12: Anti-Helmholtz coil design.

2.3.3

Quadrant coils

The anti-Helmholtz coils apply a gradient dBAH /dx in the x̂ and ŷ directions, and a gradient
2 · dBAH /dx in the ẑ direction. In order to eﬀectively select a lattice layer along, for example, thex̂ direction, we must ensure that the magnitude of the ﬁeld |B| varies with x̂ more quickly thanit varies with ŷ or ẑ. Then, if the atom cloud size is suﬃciently small, all of the atoms along theplane with a certain x̂ value will have the same transition frequency, while all subsequent planes
along the x̂ direction will be spectroscopically resolved. This can be accomplished by applying alarge bias ﬁeld Bbias along the x̂ direction so that,






dBAH 2dBAH 2dBAH 2
Bbias +
|B(x, y, z)| =
x +
y + 2
z .
dx
dy
dx

(2.1)

For small excursions from the center,
dBAH
Bbias
x 
dx



dBAH
y
dx

2 

dBAH 2
, 2
y ,
dx

(2.2)

41

On-axis field from one set of AH coils
Magnetic field (Gauss/Amp)

4.0xx xx xx  Measured
xx 
xx 
xx 
x Theory
3.0
xx 
xx 
xx 
xxx
2.5
xxx
xxx xxx xxx 2.0
xxx 
xxx
3.5

1.5
0

5

10

15

20

25

Distance from can (mm)Anti-Helmholtz coils
Axial field (Gauss/Amp)

642
1.24 Gauss/(Amp·cm)
0
-2
-4
-6
-10

-5

0

5

10

Distance from atoms (cm)Top coil only
Axial field (Gauss/Amp)

64

1.93 Gauss/Amp
0.62 Gauss/(Amp·cm)

2
0
-2
-4
-6
-10

-5

0

5

10

Distance from atoms (cm)Figure 2.13: Anti-Helmholtz coil comparison of measurement and theory, gradient from both coils,
and bias ﬁeld and gradient from running the top coil only.

42
and so for suﬃciently small y and z, |B| varies linearly with x and does not change with y and z:|B(x, y, z)| ≈ Bbias +

dBAH
x
dx

(2.3)

Here, we introduce “quadrant coils”: a scheme for applying large bias ﬁelds in any direction,using only coils in the top and bottom bucket windows. A CAD model of the coil shape is shownin Figure 2.14 and a schematic depicting the principle of operation is shown in Figure 2.15. To
visualize how these coils work, we assume that the ﬁelds produced by wires farther away from theatoms are much less signiﬁcant than the ﬁelds produced by wires closer to the atoms (i.e. the wires
closest to the bucket window viewports) and can be ignored. Then, we can group these wires into
a pair of eﬀective Helmholtz coils, where each eﬀective coil in the Helmholtz pair is shown by theblue and green arrows in Figure 2.15, which also indicate the direction of current ﬂow. Therefore,by changing the direction of current ﬂowing through these 8 quadrant coils, one can apply bias
ﬁelds in the x̂, ŷ and ẑ directions. Since a horizontal layer (a plane of constant z) can be selectedby running the top anti-Helmholtz coil only, for now, we have only implemented the x̂ bias ﬁeld andŷ bias ﬁeld conﬁgurations. One must simply reverse the direction of current in 4 of the 8 quadrant
coils to change between these two orthogonal conﬁgurations.
In order to construct the quadrant coils, the wires are wound around square Teﬂon jigs, thenpressed into an arc of the correct radius of curvature to ﬁt inside the bucket window, as shown in
Figure 2.16. We tested both the anti-Helmholtz and quadrant coils using ﬂowmeters and foundsimilar ﬂow rates, and so conclude that this method of winding then clamping does not appreciably
occlude the hollow core used for water cooling.
The eﬃcacy of this strategy for producing strong bias ﬁelds at the atoms depends on thevalidity of the assumption that the inner wires are much closer to the atoms than the outer wires,
so it is desirable to place the quadrant coils as close to the atoms as possible.
In order to simulate the magnetic ﬁelds produced by these oddly-shaped coils, a CAD model
was imported into the COMSOL simulation software. To simplify the computation, all 5 turns ofcoils were lumped into one solid object with a constant current ﬂux density of 5 times the current

43

Figure 2.14: CAD model showing one set of quadrant coils. The two sets of quadrant coils areplaced inside the bucket windows, on opposite sides of the atoms.

z
y

x

z

z

z

y

y

y

B

B
x

Z bias field:

Y bias field:

X bias field:x

Bx

Figure 2.15: Schematic showing how the quadrant coils can be used to generate a bias ﬁeld in any
direction.

44

Figure 2.16: Clamps used to bend the quadrant coils into the correct shapes.

45
in one turn of wire, as shown in Figure 2.17. We had some engineering compromises. Originally,
I had assumed an optimistically-small radius of curvature for winding the coils around the squareedges, and I assumed that the coils would be touching. However, just some slight adjustments to
the CAD model to account for reality had a signiﬁcant eﬀect on the maximum achievable bias ﬁeld,
reducing it by a factor of 3. Figure 2.18 plots the COMSOL results for the ﬁelds as a function ofarc length plotted along short lines through the center of the chamber along the x and z latticeaxes. The coils still produce a bias ﬁeld of 0.08 G/(A·cm), corresponding to a bias ﬁeld of 40 G at
500 A, which will be suﬃcient for our needs.
2.3.4

Single layer selection

Now we consider how the anti-Helmholtz and quadrant coils may be used together to selecta single layer of the 3D optical lattice in any direction. Figure 2.19 shows the atom cloud, viewed
from the side, where a 310 G/cm (620 G/cm) gradient is applied in the x and y (z) directions,and a 40 G bias ﬁeld is applied in the x direction to select a single layer along that direction. As
one can see, there is still some curvature along the y and z directions of in the surfaces giving a
constant |B|, but the 40 G bias ﬁeld largely ﬂattens this out so that the magnetic ﬁeld value isgenerally constant over a lattice layer. In Figure 2.20, we plot the magnetic ﬁeld as well as contour
lines along vertical and horizontal slices through the atom cloud. Finally, Figure 2.21 shows thatwhen the lattice layers in the x-direction are resolved by ≈ 6 Hz, the clock frequency varies by ≈ 2over a lattice layer for a cloud size of ±50 lattice sites (40 μm) in the x and y directions, and ±25
lattice sites) (20 μm) in the z direction.
2.3.5

Control electronics

The coil switching and current servo electronics for both the quadrant and the anti-Helmholtz
coils are shown in Figures 2.22 and 2.25. First we explain how the switching circuit works. Figure
2.23 shows truth tables for both the quadrant coils and anti-Helmholtz coils explaining how thetwo logic signals L1 and L2 determine the behavior of each switching circuit, as well as the logic

46

Figure 2.17: 3D plot of the ﬁeld produced by the actual quadrant coils.

47

Figure 2.18: Magnetic ﬁeld generated by the actual quadrant coils (after some engineering compromises) when conﬁgured to give a bias ﬁeld in the y-direction. Plots were generated using a
COMSOL simulation based on the coil geometry imported from a CAD model.

Figure 2.19: A to-scale schematic showing the selection of lattice layers in the x-direction. The white circles indicate lattice layers in thex-direction, and the multicolored surfaces are 3D contour surfaces of constant magnetic ﬁeld. The atoms in one particular spectroscopicallyselected layer are shown in red.
48

49

Figure 2.20: A gradient along all directions from the anti-Helmholtz coils, plus a strong bias ﬁeld
along the direction causes the magnitude of the magnetic ﬁeld to vary predominantly along the x
direction.

50

Figure 2.21: A large bias ﬁeld in the direction of layer selection minimizes the gradient in theabsolute value of the magnetic ﬁeld along the two orthogonal directions, with the goal of achieving
a constant clock transition frequency within the entire lattice layer that is to be selected. However,for lattice layers in the x direction that are resolved by 6 Hz, there is still a ≈ 2 Hz spread intransition frequencies along the y direction, and a ≈ 2 Hz spread in transition frequencies along thez direction. Therefore, when applying these techniques, the cloud size must be kept to within ±50
and ±20 lattice sites along the x/y directions and the z direction, respectively. This corresponds
to atom cloud dimensions of 40 μm in the x and y directions, and 20 μm in the z direction.

51
inverters and MOSFET drivers that distribute the logic signals to the MOSFETs. Figure 2.24
provides a schematic description of how the two circuits work for each logic conﬁguration.
The MOSFETs essentially act as shorts when given a logic high (5V) control signal and as
open circuits when given a logic low (5V) control signal, so to help visualize how the circuit works,
we represent them as normal switches that are closed for logic high and open for logic low. Bothcircuits use logic L1 to control the overall on/oﬀ state. For the quadrant coils, an H-bridge circuit
controlled by logic L2 is used to reverse the current through half of the coils (as shown for switching
between x̂ and ŷ bias ﬁelds in Figure 2.15) in order to switch between providing a bias ﬁeld in thex̂ direction and the ŷ direction. For the anti-Helmholtz circuit, logic L2 is used to run with either
the top coil only (for magnetic ﬁeld gradient selection) or with both the top and bottom coils.
The MOSFETs that we are presently using for current switching are the Inﬁneon IPT007N06N,
which are rated up to 300 A. (These will need to be upgraded if we choose to use a higher-current
power supply.) The MOSFETs are driven using an an LT1010 buﬀer chip driving a IXYS FDA217
photovoltaic MOSFET driver. The IT 400-s Ultrastab Hall probe is used to measure the current.
It outputs a sense current of 1/2000 of the measured current, which is then run across a stable 20Ω
resistor to give a sense voltage of -1/100 V/A. The sense voltage then goes into the current servo
shown in Figure 2.25 where I changed R6 so that the overall transfer function is -3/100 A/V. The
actuator is the SKM900GA12E4 IGBT by Semitrans, which can handle 2kW of power dissipation.
It is recommended to put a diode between the gate and the source, as word on the streets of JILA
is that this prevents the IGBT from blowing up. The power supply used is the Sorensen DHP60330. It is important to set the current limit high enough so that the power supply does not switch
between constant current mode and constant voltage mode, which can result in very large (4000amp) current spikes. The power supply should remain in constant voltage mode at all times.
Due to the large inductive load L of the coils, suddenly switching them oﬀ produces a backEMF much larger than the maximum speciﬁcation of the MOSFETS (which was accidentallyexperimentally veriﬁed!), so it is necessary to include varistors as shown in Figure 2.22 which will
limit the voltage to the clamping voltage Vvar of the resistor. Then the rate of change of current

52
through the coils will be limited to dI/dt = −Vvar /L so the current decreases linearly from an
initial current I0 with a switching time Toﬀ = I0 L/Vvar and the MOSFETs do not explode.
2.3.6

Mounting and water cooling

As shown in Figures 2.26 and 2.27, the anti-Helmholtz and quadrant coils were potted inside
Teﬂon cans using Duralco NM25 epoxy. This way, if the main chamber needs to be re-baked at all,the coils can withstand the heat and do not need to be removed from the bucket windows. Because
the coils can vibrate when switched on and oﬀ, it is important to include damping material in themounting structure. The Teﬂon cans are clamped inside the bucket windows with set screws thatcause the Teﬂon can to push down on a layer of Viton foam placed between the can and the insideof the bucket window.
The coils are made from hollow wires which allow them to be water cooled from the inside.
The square-shaped magnet wire is soldered inside a round adapter that allows for the connection
to regular 1/4” copper tubing that can be connected using normal Swagelok. As shown in Figures
2.28 and 2.29, the wires are connected to manifolds that provide both electrical and water coolingconnections.
Because any kind of failure in the water cooling system could cause a catastrophic disasterin the lab, we made sure to include a thorough interlock system, detailed in Figure 2.30. Eachportion of water-cooled coil has an attached thermocouple, which then is read by a Keithley 2701
digital multimeter (via the Keithley 7710 diﬀerential multiplexer). If any of the coil temperatures
exceeds a threshold of 22o C, it causes a TTL output signal to drop low. Our two high-power
water-cooled 1064 nm ﬁbers also have thermocouples going to the same interlock, as they can burnif their temperature becomes too high (due to misalignment, etc). Originally, I attempted to makemy own hardware-based interlock based on the AD595 thermocouple reader chip, comparators,
and a series of digital logic chips. However, since thermocouples sense such small voltages andare thus susceptible to electrical pickup and grounding issues when traveling across the lab, this is
not such a trivial task: the interlock tripped every time the magnetic coils switched or somebody

53

Quadrant coils

AnƟ-Helmholtz coils

V+

V+

Quad coils(in series)
Top Q1, Q3
BoƩom Q1, Q3

D
L2

D

G

L2
S

25V varistor
V39ZA6PDMOSFET
IPT007N06N

G

L2

S
Quad coils (in series)
Top Q2, Q4; BoƩom Q2, Q4

DL2Top AH coil

25V varistor
V39ZA6P

D

G

L2
S

S

ŽƩŽŵ AH coil

D

G

L2
S

G
S

D
L1

D

G

L1

G

S

S
C

LT1010bufferCurrent
servo

MOSFET
IPT007N06N

G

Hall
probe
IT 400-S

C
IGBT
SKM900GA12E4G

IGBT
SKM900GA12E4G

E

E
Current
servo

Hall
probe
IT 400-S

Figure 2.22: Schematic of coil switching and current servo electronics. The Anti-Helmholtz electronics switch between the anti-Helmholtz and top coil only conﬁgurations. The quadrant coilelectronics switch between conﬁgurations that give a bias ﬁeld in the x and y directions.

54

Quadrant coils:

Anti-Helmholtz coils:

L1 L2 Behavior
0 Off
1 0 x bias field
1 1 y bias field

L1 L2 Behavior
0 Off
1 0 Top AH only
1 1 Top + Bottom AH
LT1010buffers

LT1010buffers
L1

L1

L1

L1

L2

L2

To MOSFETs

L2
L2

L2

L2

L2

To MOSFETs
L2
L2
L2

Figure 2.23: Truth table, inverters, and buﬀers for driving the MOSFETs for both the quadrant
coils and the anti-Helmholtz coils.

55

Quadrant Coils:ŶƟͲ,ĞůŵŚŽůƚǌĐŽŝůƐ͗V+

V+

In series:
Top Q1,Q3; bottom Q1,Q3

L2
OffIn series:
Top Q2,Q4; bottom Q2,Q4

Top AH

L2

L2

L2

Off
Bottom AH

L2

L2
L1

L1

V+

V+

In series:
Top Q1,Q3; bottom Q1,Q3

L2

In series:
Top Q2,Q4; bottom Q2,Q4

Top AHL2

X bias field

L2

L2
Top AH only

L2

Bottom AH

L2
L1

L1

V+

V+

In series:
Top Q1,Q3; bottom Q1,Q3

L2
Y bias field

In series:
Top Q2,Q4; bottom Q2,Q4

L2

Top AH

L2
Top + Bottom AH

L2

L2ŽƩŽŵ,

L2
L1

L1

Figure 2.24: Schematic showing the principle of operation of both the quadrant coils and the antiHelmholtz coils. Here, all the MOSFETs are depicted as switches controlled by the corresponding
logic signals. The color red is used to indicate components that have current ﬂowing through them;the color black is used to indicate components with no current ﬂowing through them. Arrows
indicate the direction of current ﬂow. The logic signal L1 is used on both coils to control theoverall on/oﬀ state. For the quadrant coils, L2 is used to control the H-bridge which can reverse
the direction of current through half of the coils, thereby switching the bias ﬁeld from the x direction
to the y direction. For the anti-Helmholtz coils, L2 is used to either turn on the bottom coil or
short it out so that no current ﬂows.

Error inControl in+/- 10V

R6/R5: Sets error gain vs. control signal gain
R7/R6: Proportional gain
R7/C8: Roll off (not used)
R7/C7: PI corner

Control out

56

Figure 2.25: Circuit diagram for the servo used to control the current in the anti-Helmholtz andquadrant coils. For now, I have changed R6 to 3 kΩ in order to use more of the ±10V range of the
DAC output, giving -3/100 Amp/Volt.

57

Figure 2.26: CAD model showing all magnetic coils inside the two Teﬂon cans that are clamped
inside the top and bottom bucket windows.

Figure 2.27: All bucket coils potted with epoxy in the Teﬂon can.

58

Figure 2.28: Top: CAD model of the water and electrical hookups for the bottom bucket coils.
Bottom: Photo of the connections under the main chamber.

59

Figure 2.29: Left: CAD model of the water and electrical hookups for the top bucket coils. Right:
Photo of the connections on the mounting tabs on main chamber.

60with some static electricity touched the optical table and got shocked. It is rather diﬃcult to getanalog comparators to work when the lab ground sometimes swings by 10 V (as measured with an
oscilloscope when delivering a shock to the lab ground... yes I kid you not). Of course it should havebeen possible to get this to work, but it would have been quite an ordeal. In the interest of time, we
opted for the proven Keithley solution. In retrospect, if one is trying to make homemade electronics,
it may be better to sense temperatures using a larger signal from thermistors, rather than spending
great eﬀort trying to shield and ﬁlter the tiny voltages from thermocouples. Keithley’s oﬀ-the-shelfproduct does a ﬁne job at this, however.
In addition to the thermocouples, the return lines of pairs of water coils are combined together
so that there is a ﬂowmeter for every two water-cooled coils, in order to detect a leak or any plumbing
obstruction, as shown in Figure 2.31. We used the Proteus Industries 0804BN1 ﬂowmeters whichhave an internal relay so that if any ﬂow rate drops below a set value, the ”normally open” channel
shorts to ground and the signal which is attached with a pull-up resistor to logic high drops low.
The quadrant coil was measured to have a nominal ﬂow value of 1.6 L/minute and the middle
anti-Helmholtz coil was measured to have a nominal ﬂow rate of 1.25 L/m. The threshold values
for tripping the interlock were adjusted in situ for the two coils going to each ﬂowmeter. The twologic signals from the ﬂowmeters and the Keithley are then fed through an AND gate that travels
to both the power supply and through a second set of AND gates that turn oﬀ the AOMs to thehigh power ﬁbers if either the interlock signal or the computer control drops low. After hearing
some horror stories, we were sure to make it so that logic high means that things are ok, and we also
added appropriate pulldown resistors, so that if anything is accidentally turned oﬀ or unplugged,
the interlock will shut things oﬀ.

61

TTL

Keithley 2701
digital multimeter

Keithley 7710
differential
multiplexer

1064 highpower fibers

AOM

AND

AOM

AND

Computer
control TTL
Computer
control TTL

thermocouples

AND

Power supply

Magnetic coils
Water cooling return lines

COM

NO

Proteus Industries
0804BN1 flowmeters+5V

Figure 2.30: Schematic of safety interlock for both the magnetic coils and the 1064 high powerﬁbers.

Figure 2.31: Photo of part of the water cooling manifold, with ﬂowmeters shown.

BeamBlue MOT
RepumpRepump
Red MOT trapping
Red MOT stirring
Optical pumping
HODT
VODT
Horizontal Lattice
Vertical Lattice
Clock laser

Transition
1S − 1P0
1
3P − 3S
01
1S − 1P0
1
1 S (J = 9/2) −3 P (J = 11/2)
0
1
1 S (J = 9/2) − 3 P (J = 9/2)
0
1
1 S (J = 9/2) − 3 P (J = 9/2)
0
1
Far red-detuned
α1064 = −1.3 Hz/(mW/μm2 )Far red-detuned
α1064 = −1.3 Hz/(mW/μm2 )Far red-detuned
α813 = −1.1 Hz/(mW/μm2 )Far red-detuned
α813 = −1.1 Hz/(mW/μm2 )1 S −3 P
01

λ (nm)
461
679
707
689
689
689
1064

Γ
30 MHz
1.4 MHz
7.3 MHz
7.4 kHz
7.4 kHz
7.4 kHz-Isat
40 mW/cm2
0.58 mW/cm2
2.7 mw/cm2
3μW/cm2
3μW/cm2
3μW/cm2-

Typical power range
30 mW total
5 mW total
5 mW total
1 mW to < 1μW total
1 mW to < 1μW total
100 μW
10 W to 200 mW

-

ω0
2 cm
1 cm
1 cm
1 cm
1 cm
1 cm
ω0,x = 340μm
ω0,z = 17μm
25 μm
ω0,z = 17μm
ω0,x = 120μm
ω0,z = 35μm
90μm

1064

-

-

813

-

-

813

-

698

1 mHz

0.4 pW/cm2

0.36 mm

10 mW

2 W to 100 mW
1 W each axis
1W

Table 2.1: Speciﬁcations for important lasers used in the experiment.

62

63

2.4

Laser systems

2.4.1

Optics layout around the main experiment

The laser beams required by our experiment are: blue Zeeman slower, blue 2D MOTs, blue
3D MOTs, two blue MOT repumps, red MOT trapping laser, red MOT stirring laser, verticaland horizontal optical dipole traps (ODTs), blue absorption imaging (horizontal, vertical highresolution, vertical low-resolution), red absorption imaging, optical pumping, 3 lattice beams toform a 3D lattice, horizontal and oblique clock lasers, and a blue probe for PMT ﬂuoresence
detection during clock readout. We eliminated the need for the blue probe and enhanced oursignal-to-noise by re-trapping the atoms in the blue MOT for clock readout, leaving only 20 morebeams to ﬁgure out how to arrange around our experiment (see Figure 2.35)! The speciﬁcations
for each of these beams are summarized in Table 2.1.
The optics layout for the mezzanine which surrounds the vacuum chamber is shown in Figure
2.32. It was helpful to import the Solidworks drawing into Adobe Illustrator and make the wholeto-scale schematic beforehand, as it is easier to drag images around the screen than to realize
something won’t ﬁt and then re-do half of the optics. We were able to save some space by putting
high-resolution blue absorption imaging, red absorption imaging, and optical pumping beams allalong the same path. We also added a low-resolution blue absorption imaging beam path with alarger ﬁeld-of-view to aid in alignment.
The layout underneath the vacuum chamber on the main optical table is shown in Figure
2.33. To save space, we combined all red-colored beams (both blue MOT repumps and both red
MOT lasers) on this level, and then launched the 4 combined beams up to the mezzanine witha periscope. The red MOT, blue MOT, high resolution absorption imaging, and vertical lattice
beams are all launched from the bottom and combined with polarizers and dichroics as shown in
Figure 2.33. They arrive on the top mezzanine breadboard, shown in Figure 2.34. After passing
through the high-resolution imaging objective, the vertical MOT beams must be re-focused before
they are retro-reﬂected. The high-resolution absorption imaging beam has the opposite helicity as

ff
ko
Pic

64

2D
MOT

BS

Spin Pol. +Red Imaging
PD

PBS
Red +
RepumpParasc.

ro

r2

"

BS
man
Zee er
Slow

PD
QPD

Pic
k

off

PBS

is

m

Po
la
r

irr
or

2”

PBS

BS

PBS

f = 150mm
cylindrical

λ/2

f = -22mm
cylindrical

Pickoff

λ/2
f = -30mm
cylindrical

Abs.
Probe

MOT

mir

PD

roic
ich

f

of

k
Pic

PBS

PD

Td
MO

EOT Faraday
Rotator

Razor
Stack

Lattice
PBS

λ/2
λ/4
LC WP
λ/4
λ/2

BS

Razor
Stack

Pic
k

of
f

Pola

ris

mirr

or

1"

f = 80mm
cylindrical

PD

f=500

Lens

dic 30 cc
hro
ic
Beam
Dump

PBS

λ/4

r
High power shuttet)(taken out of moun

T
MO

PD
Lens

MO
T dic

hro

ris
Pola

is mi

λ/4

1"

mirror 1”

rror

2”

λ/4
λ/4

λ/4

Lens

PD

Polar

mirror

n

mirr

or

1"

ic

Gla

mir

ror

Polar

2”

is m

irror

1"

Lens
un

mir

ror

Clock
Laser

Len

s

λ/2

Po

shu

PD

lari

sm

irro

r 1"

ow
er

rvo

JILA

hp

Se

f = 50 mm
cylindrical

f=500

Hig

ff
Picko

Lens

HR
8
30 13,cc HT
dic 69
hro 8
ic Cu

n
Gla

m

n

λ/4

f=200 m

Gla

sto

m

mo

PMT

t

λ/4

Lens
Lens

PBS

tte

f = 80mm
cylindricalrzo
Ra ack
St

AOM

PBS

f = -22mm
cylindricalf = 1000mm
cylindrical

m

Pi
ck

or
irr

of

f
EOT Faraday
RotatorLens
λ/2

Lattice
Razor
Stack

Lens

1”

PBS

PBS

Camera

f=100

PD

ODT

r

Figure 2.32: A to-scale drawing showing the layout of the optics on the mezzanine breadboard thatsurrounds the chamber.

r1

"

65

JI

Red +
RepumpParascope

LA

PBS

m

irro

Red MOT
β
r 1"
irro

r1

"

JI

LA

m

irro

r

la
Po

ris

JILA

m

m

irro

1"

m

r

LA

irro

JI

1"

off
m
Po

la

ris

PBS

λ/2

High Res.
Abs. Imaging

irro

r1

"

Pic
k

la
Po
ris
r1

50/50 BS

irro
"

ro

r

2"

Blue MOT

m
ris
Po
la

Servo PDir

679 + 707
Repumps

irro

r1

"

PBS

m

PBS

Servo PD

m

off

PBS
la
Po

ris

m

irro

r1

k
Pic

"

JI

LA

m

irro

r

1"

JI

LA

m

irro

r

1"

m

PBS

λ/4
or
irr

ris

2”

la
Po
m
irro

r

Red MOTα

1"

To
45 p O
de DT
g
m
irr
or

T: 461, 689
R: 813
Semrock FF725-SDi01

r

Mirro

m

irro

r2

"@

45

de

g

Glan

am
Be mp
Du

Po
la
ris
m
irro
r

99/1

1"

Po
la

ris

m

irro

r

1"

Low Res.
Abs. Imaging
PBS

f = 1000

JI

LA

m

Hig

irro

r

1"

hp

ow
er

s

Len

shu

tte
r

D

oP
Serv

Le
ns

e
lad
rB
zo ck
Ra Sta

off

k
Pic

rv
o

PD

Se

f = -250

f = -100

λ/2

PBS

irr
or

EOT Faraday
Rotator

M

PBS
Lattice

Figure 2.33: A to-scale drawing showing the layout of the optics on the top mezzanine, which sitson the mezzanine level above the chamber.

66
the MOT beams and so it is separated via a PBS so it can then be imaged on the Andor camera.
The low-resolution vertical absorption imaging beam travels at a slight angle to miss the otherbeam paths and is imaged on a cheaper Allied Vision Guppy camera. Additionally, the vertical
ODT beam (VODT) and the oblique clock laser are launched from the top.
2.4.2

Blue system upgrades

As shown in Figure 2.36, we made some minor changes to our blue laser system in order
to meet the speciﬁcations required by the commercial AOSense source. The master laser is acommercial external-cavity diode laser (ECDL) at 461 nm. We originally used a New Focus Vortex
laser, which was the ﬁrst available of its kind. The power output degraded over time (possibly
to be expected with any blue diode), and also the laser controller seemed to have some noiseand grounding issues, so we had to add a buﬀer and a low-pass ﬁlter between the controller andthe servo that actuates on the PZT to stabilize the laser frequency to the atomic transition. We
realized that much of our shot-to-shot atom number ﬂuctuations from absorption imaging could
actually be attributed to frequency noise on the blue master diode laser due to noise and grounding
issues involved with the laser controller and the lock. In particular, though the servo has a highoutput-impedance, high frequency noise on the laser controller external PZT control input leaked
backwards through the whole servo circuit, as the high output-impedance only applies for signals
within the bandwidth of the circuit. In addition to this high-frequency noise, we also saw 60 Hz
noise, and we struggled to ﬁnd a grounding conﬁguration that would minimize this. Eventually, we
upgraded to a new blue master laser and controller from AOSense, which so far has not had anyof the aforementioned noise and grounding problems.
The light from the blue master (¿30 mW output) is then appropriately frequency-shifted,
then travels to 3 injection-locked slave lasers. The light from the 2D MOT slave is split equally
between both of the 2D MOTs in the AOSense system. Previously we had used a high-frequency
AOM from Brimrose for the Zeeman slower, with a quoted 40% eﬃciency. However, we realizedthat we could get up to twice as much power double-passing a 90% eﬃciency AOM. A 3-way ﬁber

67

AOM

λ/4

f = 300

Mi
rro
r

Lens

ror
Mir

f = 75Collimating

λ/4

PBS

PBS

VerticalClock

Low res. abs. imaging
MOT
beams

VerticalODTCameraCameraAndor iKon

High res.
abs. imaging

f = 300

Figure 2.34: A to-scale drawing showing the layout of the optics on the optical table below thechamber.

Figure 2.35: Artist’s depiction of the optics layout process.

68
splitter carries the light to the 3 arms of the ﬁnal blue MOT.

Table 2.2: List of power, shape, and polarization requirements for the new AOSense strontium
source.

2.5

Imaging system
A high-resolution imaging system helps us to characterize our atom cloud, eliminating someof the guesswork and indirect measurements in previous work. Combining the energy resolutionof clock spectroscopy with the spatial resolution of a relatively high numerical aperture (NA)imaging system will both provide new tools for quantum simulation as well as the capability todiagnose spatially-dependent inhomogeneous broadening due to systematic shifts. As this system
was built with both next-generation clocks and quantum simulation in mind, we decided to makethe horizontal lattice beams parallel to the ground and make the vertical lattice beams parallel togravity. Whereas tunneling is generally not desirable during clock operation, we still wanted to allow
for tunneling for experiments that could investigate synthetic gauge ﬁelds [39, 151, 172, 15, 106]and simulate charge currents [104, 6]. Therefore, as the interesting physics in the lattice would
occur along 2D sheets in the xy plane, we wanted the high-resolution imaging path to run parallelto the vertical lattice beam, as well as the vertical MOT beams.
The simplest design was to retroreﬂect the vertical lattice oﬀ of the ﬁrst optical surface of theimaging system and pass both the red and blue MOT beams through the entire imaging system. A

69

f0 = 88Sr resonance frequency

SrEOM

f0 + 30
MasterLaser

Pre-Spectrometer-1st
85 (87Sr) MHz
or 137 (88Sr) MHz

f0 + 30
2D MOT AOM-1st Single Pass
122 MHzf0 + 30Inj
Lock

2D MOTf0 - 52 - 40 = -92

Inj
Lock

f0 - 552

ZS Tuning AOM-1st double pass582/2 = 291 MHz

Spectrometer+1st
110 MHzf0 - 55
ZS to Atoms
f0 - 52 - 500 = -552

f0 + 30Inj
Lock

MOT to Atoms
f0 - 52 - 40 = -92

MOT AOM-1st
122 MHz

Fluorescence AOM-1st , 79 MHz

Probe Absorption
to Atoms
Probe Fluorescence
f0 - 38
to Atoms
f0 - 49

Absorption AOM-1st , 68 MHz

Figure 2.36: Schematic showing updates to the blue system that were made in order to satisfy the
speciﬁcations required by the AOSense system. All frequency values are shown in units of MHz.

70
polarizing beam splitter (PBS) that works at both 461 and 489 nm then separates the MOT beams
from the absorption imaging beam (these travel through the atom cloud with opposite helicities)
so that the MOT beams can be re-collimated and retroreﬂected, while the imaging beam is sent to
the CCD camera, as shown in Figure 2.34. The eﬀective focal length f of the imaging system isapproximately equal to the distance from the objective to the atoms. We wanted to use ≤ 50 mm
optics ≈ 250 mm away from the objective to re-collimate and retroreﬂect the MOT beams. For an
initial blue MOT beam diameter of 20 mm, this therefore requires that
(20mm) · (250mm)≤ 50mm,f(2.4)

so f ≥ 100 mm. This therefore constrains the ﬁrst lens to be ≈ 90 mm away from the atoms. A
section view showing the imaging system’s location in the man experiment is shown in Figure 2.37.
We also had to allow room for the vertical optical dipole trap (VODT), as we were planningto evaporate to quantum degeneracy in a crossed optical dipole trap (XODT) conﬁguration. Since
during the design phase we were unsure of our evaporation protocol and wished to maintain ﬂexibility to reﬁne this detail of our experiment, we allowed room for the VODT to pass at an obliqueangle outside of the imaging objective. This also allowed room for an oblique clock laser for 3Dband detection and manipulation, as well as a low-resolution vertical imaging beam path which,due to its larger ﬁeld of view (FOV) is a convenient tool for alignment. Given those constraints,
the maximum NA we can attain is 0.23.
2.5.1

DesignSo how do you design a high-resolution microscope?2

Because of the diﬀraction limit, you

generally want to capture as much of the solid angle from the source as possible (given the geometricconstraints of your system). This is characterized by the numerical aperture NA = n sin θ, where
n is the index of refraction of the material surrounding the lens (air in our case), and θ is thehalf-angle of the cone of light traveling to the lens.
2

Many groups seem to work with optics companies to do this. Our postdoc Ed Marti was chieﬂy responsible forthis. This section aims to summarize and de-mystify the design process, as Ed was right that it’s not too bad to just
do it yourself!

71

Figure 2.37: Section view of the high-resolution imaging system, in its position inside the top bucket
window.

72
However, the diﬀraction limit is not the only thing that can limit microscope performance.
The preceding discussion relied on the paraxial approximation, which assumes that a small angle
of incidence between the all rays of light and the normal to the surface. So, whereas Snell’s law
describes refraction of a ray of light as it travels from a medium with index of refraction n1 at anangle θ1 to the surface normal, to a medium with index of refraction n2 at an angle θ2 to the surface
normal according to n1 sin θ1 = n2 sin θ2 . In the paraxial approximation where θ1,2 ≈ 0, this justbecomes n1 θ1 = n2 θ2 . The Taylor series expansion for sin θ is,sin θ = θ −θ3 θ5 θ7
+
−
+ ···
3!
5!
7!

(2.5)

So in other words, the paraxial approximation amounts to just taking the ﬁrst term in thisexpansion. According to paraxial theory, all rays focus at the same point in space, with a spot size
determined by Gaussian beam propagation. When the assumption that sin θ ≈ θ begins to break
down, this is no longer the case, leading to image distortion referred to as aberration. Sphericalaberration refers to the way that diﬀerent rays focus at diﬀerent points in space, as shown in Figure
2.38 [41].
The ﬁrst step in designing a new imaging system is to pick the ﬁrst lens (closest to theobject), which will set the NA of the system and thus the diﬀraction-limited resolution. In order toensure that our imaging system is diﬀraction limited, we must take care to minimize aberrations.
One option is an asphere where the lens surface is designed so that, to a certain polynomial order,
all rays are incident at the same angle. Whereas small aspheres can be molded, custom-grinding alarger asphere is a bit too costly and diﬃcult for our application (though it is done for telescopes,etc). A second option is an achromatic doublet, which combines two spherical lenses: a positive
focal length lens made from low-index glass with a negative focal length lens made from high-index
glass so that the θ3 spherical aberrations cancel but the focusing powers do not [41]. This also has
the eﬀect of mitigating chromatic aberrations (hence the name).
Importantly, aberration cancellation must be done with particular angles of incidence and73

Figure 2.38: Illustration of aberrations from a plano-convex lens. Image from [41].

74
refraction in mind. In other words, aspheres and achromatic doublets only minimize aberrations
for a particular magniﬁcation (also referred to as “conjugate ratio”). Commercial achromats are
generally designed for an inﬁnite conjugate ratio; that is, they are designed to collimate the lightfrom a point source. A third option is to use a custom meniscus formed by two surfaces with
diﬀerent radii of curvature. Similar to an achromatic doublet, this gives us enough degrees of
freedom to cancel third-order spherical aberrations.
In general, there is a bound on how short one can make the focal length before higher-orderaberrations begin to dominate. Since achromats are designed for inﬁnite conjugate ratio, they
only minimize spherical aberrations when placed a focal length away from the source. Due tohigher-order aberrations, they typically only work well for NA ≤ 0.1. A custom meniscus has theadvantage of being easier to manufacture than an asphere, and it also can achieve a higher NA,
because we can design it to minimize aberrations for the conjugate ratio of our choosing. A custom
meniscus can therefore achieve a higher NA than a commercial achromat, as it need not collimate
the image in order to eﬀectively minimize spherical aberrations, and so it can be placed closer tothe image. We therefore choose a custom meniscus for the ﬁrst lens of our system.
The four-lens imaging system is shown in Figure 2.40. Our design process began with theconstraint of a 90 mm radius of curvature for the ﬁrst surface of the meniscus, which is coatedwith the AR coating shown in Figure 2.39 to serve the dual purpose of retroreﬂecting the verticallattice. The radius of curvature of the second surface was then chosen to minimize aberrations.
The ﬁrst lens brings the image location farther away so that a commercial achromat is then able tocollimate it. One can try several diﬀerent options for this achromat and then go back and modify
the meniscus so that the combined two lens system minimizes aberrations. If this combination does
not allow enough degrees of freedom to satisfactorily cancel aberrations, one can add a third lensbetween the two, such as the chosen plano-concave lens shown in Figure 2.40. Finally, a planoconvex lens is used as the last piece of the objective, and it focuses the collimated beam onto the
CCD camera.
Sudden switching of the magnetic ﬁeld coils can induce Eddy currents that could potentially75

Figure 2.39: Coating curve for the meniscus, which transmits the 461 and 689 nm MOT beams
and retroreﬂects the 813 nm lattice.

76

Newport KPX208
f = 400 mm plano-convex
Newport PAC088
f = 250 mm achromatic doublet
Newport KPC064
f = -500 mm plano-concave
53.17 mm radius of curvature
Custom meniscus, SF11 glass, f = 165.6 mm
90 mm radius of curvature

Figure 2.40: Section view of the high-resolution imaging system design, showing all part numbers
and speciﬁcations.

77
shake the optical system. We avoid this by using a plastic (Delrin) lens holder and spacers. The
only metal in the mount is a SM2RR Thorlabs retaining ring, which can be replaced with a plastic
part as well.
2.5.2

Measuring and testing resolution

One way to test imaging system resolution is to image a pinhole. For our test pattern, we
image two 350 nm pinholes,3 as shown in Figure 2.41. The Airy disk describes the diﬀractionlimited pattern formed by focusing a point source with a perfect lens with a spherical aperture,and the angular intensity I is given by,I(θ) = I0

2J1 (ka sin θ)
ka sin θ

2

,

(2.6)

where θ is the angle of observation, k = 2π/λ is the wavenumber of the light from the point source,
a is the radius of the aperture, and J1 is the order-one Bessel function of the ﬁrst kind. Figure 2.42
plots an azimuthal average of a 2D ﬁt of the Airy function to the data.
There are many ways to deﬁne imaging system resolution. The Rayleigh criterion statesthat the minimum resolvable distance between two point sources occurs when the ﬁrst diﬀraction
minimum of one source overlaps with the maximum of the other. Theoretically, the minimum
resolvable radius from an Airy disk rAiry is,rAiry = 0.5 ·1.22λ= 1.2μm
NA(2.7)

It is generally diﬃcult to get data where the wings match well with the theoretical ﬁt function.
The ﬁt in Figure 2.42 shows a ﬁrst diﬀraction minimum at 1.5μm, while the data itself shows akink at 1.2μm.
Another option is to use the 1/e2 radius of the ﬁt,
ωGaussian = 0.68rAiry = 0.41
3

λ= 0.82μm,
NA

Made by and borrowed from the Kapteyn-Murnane group. Thank you!

(2.8)

78

Figure 2.41: Image of two 350 nm pinholes taken with our lens system in order to test the resolution.

Figure 2.42:

79
which isn’t too far oﬀ from the measured 1.0μm. This corresponds to a depth of focus zRayleigh ofzRayleigh =and a ﬁeld of view of 150 μm.

πω02
λ= 4.6μm= 0.53
λ
NA2

(2.9)

Chapter 3
Preparation

The strontium gas leaves a hot eﬀusive oven at a temperature of 500o C and then undergoes aseries of laser cooling and trapping stages to realize the goal of a degenerate Fermi gas in the groundband of a 3D optical lattice. In this section we summarize the steps necessary to bridge over 10orders of magnitude in temperature in order to ﬁnally realize complete control over all internal andexternal degrees of freedom for atoms conﬁned in a 3D lattice. All of the experiments describedin this thesis use the fermionic 87 Sr isotope, which is the preferred choice for clock operation, andalso has a large nuclear spin of I = 9/2, which is both advantageous for evaporative cooling andprovides a sizable basis in a spin synthetic dimension which can be used for quantum simulation
experiments.

3.1

Initial laser cooling stages
A simpliﬁed level diagram for strontium is shown in Figure 3.1. The ﬁrst stages of laser

cooling occur on the 32 MHz linewidth 1 S0 − 1 P1 transition at 461 nm (”blue transition”). We
use a commercial source from the company AOSense, which includes a Zeeman slower and two 2D
MOTs, which slow the atoms and direct them into the main chamber. We just input the speciﬁed
beam powers, shapes, and polarizations (see Table 2.2), and Tom Loftus produces a magical cold
strontium beam. Then, once in the main chamber, the atoms are trapped in the ”blue MOT” onthis transition, with a magnetic ﬁeld gradient of 30 G/cm. Up to 107 atoms are trapped in ourblue MOT, at an optical density high enough that the atoms actually cast a shadow on the vertical

81

3S
1P
1
1D
2

Blue MOTߣ ൌ Ͷ͸ͳ nmȞ ൌ ͵ʹ MHz

1

Blue MOT repumpsߣ ൌ ͸͹ͻǡ ͹Ͳ͹ nm
3D
3
3D
2
3D
1

2.9 μmRed MOTߣ ൌ ͸ͺͻ nm
Ȟ ൌ ͹Ǥͷ kHz

3P
2
3P
1

2.6 μm3P
0

Clock transitionߣ ൌ ͸ͻͺ nm
1S
0 Ȟ ̱ ͳ mHz

Figure 3.1: Level diagram of relevant states and transitions in 8 7Sr.

82
MOT beam (and so sometimes MOT performance is enhanced by some slight misalignment of theretroreﬂection so that the shadow avoids the atoms).
The blue transition is mostly-closed, with some leakage to the 1 D2 state, which can thendecay to the metastable 3 P2 state. Therefore, operation of the blue MOT beneﬁts from repumping
via the 707 nm 3 P2 − 3 S1 state, which can decay to the 3 P0 clock state, thus requiring anotherrepump to address the 679 nm 3 P0 − 3 S1 transition. Note that this is not the only repumping
option; other groups rempump via the 497 nm 3 P2 to 3 D2 transition[149]. The repumps are freerunning ECDLs (i.e. their wavelengths are not externally-referenced), and are modulated over a≈ 3 GHz range by modulating their PZTs at a frequency of ≈ 1 kHz in order to address all thehyperﬁne levels of the I = 9/2 87 Sr atom. In the absence of these repumps, atoms accumulate in themagnetically-trappable 3 P2 state and are trapped by the MOT ﬁelds [166]. Atoms in the magnetic
trap do not suﬀer the same loss due to light-assisted collisions as atoms in MOT, and so measuring
the lifetime of atoms in the magnetic trap oﬀers a ﬁrst lower-bound on the vacuum-limited lifetime.
To measure the lifetime, ﬁrst the MOT is operated without repumps, then all lasers are shut oﬀ
while atoms are held in the magnetic trap, then after some hold time, the repumps and MOT beams
are switched on to re-capture atoms in the MOT where they can be counted via side absorption
imaging, as shown in Figure 3.2. Here, the lifetime is limited by blackbody radiation (BBR) driving
the 2.9 μm 3 P2 − 3 D1 transition, which can decay to both the non-magnetically trappable 3 P 0 and(via decay through the 3 P1 state) 1 S0 states. The measurement in Figure 3.2 is consistent withan ≈ 40 s lifetime estimated from the intensity of room temperature BBR, and was thus the ﬁrst
conﬁrmation that our vacuum-limited lifetime was greater than 40 s.
After the atoms have been cooled to 2-3 mK on the blue transition, they are then loaded into
the second-stage ”red MOT” which operates on the 7.4 kHz 1 S0 − 3 P1 intercombination transition
at 689 nm. Nice descriptions of this so-called ”narrow-line MOT” are given in Refs. [110] and [9],so we just provide a quick summary here. One typically thinks of the Doppler limit to laser cooling
TDoppler on a transition of linewidth Γ as being the ultimate limit of MOT temperature, where
TDoppler = hΓ/2kB results from competition between the rates of cooling forces and spontaneous

83

Figure 3.2: Lifetime of the 3 P2 atoms in the magnetic trap, limited by blackbody radiation photons
that drive transitions which then decay to states that are not magnetically-trappable.

84
emission. However, because the intercombination transition has such a small Doppler limit of ≈ 180
nK, one must also consider the recoil temperature Trecoil ≈ 230 nK, when the transition is so narrowthat a single momentum kick from spontaneous emission of a cooling photon can Doppler-shift theatom out of resonance with the cooling laser. The theoretical limit on laser cooling due to thiseﬀect is Trecoil /2, and so narrow-line MOT dynamics are complicated by the interplay of Doppler
and recoil eﬀects, which are of similar order. Trapping and cooling forces come from our trapping
laser “α” which drives the 1 S0 , F = 9/2 − 3 P1 , F = 11/2 transition.
However, as the ground state only has nuclear spin, whereas the excited state has electron spin, the ground and excited states have very diﬀerent g-factors. Because of this, the “lessstretched” states (with smaller |mF |) are too far detuned (and sometimes detuned so they canabsorb a photon from the wrong MOT beam), and so experience insuﬃcient cooling forces. Luckily, these transitions also have the smallest line strengths, and so by using a second “stirring” laser“β” on the 1 S0 , F = 9/2 − 3 P1 , F = 9/2 transition, we can quickly pump atoms back towards thestretched states, and thus achieve the desired trapping and cooling forces on average.
A timing diagram showing the red MOT loading, red MOT, and ODT loading is shownin ﬁgure 3.3. To span the temperature diﬀerence from the ≈ 2 mK blue MOT to the ≈ 1 μK
red MOT, the red MOT lasers are ﬁrst operated at a higher intensity and frequency-modulated
to maximally address all velocity classes from the blue MOT. Then, as shown in Figure 3.3, thefrequency modulation envelope is ramped down until the red MOT is operated at a single frequency,
while simultaneously the intensity is ramped down by a factor of 100. Note that midway throughthe broadband ramp, we switch the transimpedance gain for the in-loop photodiodes so that we
may eﬀectively servo the light intensity over 3 orders of magnitude.

85

3.2

Crossed optical dipole trap

3.2.1

Initial loading

For evaporative cooling, the atoms are next loaded into a 1064 nm crossed optical dipole
trap (XODT), which is formed by a horizontal and a vertical beam crossing at their respective
waists. The XODT is held on throughout both the broadband and narrow-line cooling stages
in the second-stage MOT. To ensure that narrow-line cooling works eﬃciently inside the XODT,
the spatially-dependent diﬀerential ac Stark shifts of the 1 S0 − 3 P 1 cooling transition must be
minimized. The horizontal beam has a sheetlike geometry with waists of 340 μm and 17 μm in thehorizontal and vertical directions, respectively. By focusing more tightly in the vertical direction,
the horizontal beam achieves the same conﬁnement against gravity for a lower light intensity. The
vertical beam, which is aligned at a small angle with respect to gravity, is a circular Gaussian
beam with a 25 μm beam waist. The HODT and VODT have initial powers of ≈ 10 W and ≈ 2
W, respectively. During XODT loading, the quadrupole magnetic ﬁeld from the red MOT has
cylindrical symmetry about the vertical axis, and so the following polarizations are listed withrespect to a vertical quantization axis. Similar to [149], the vertical beam is circularly polarized tominimize diﬀerential ac Stark shifts. We perform a further stage of cooling by blue-detuning thecooling light relative to the free space resonance to address atoms inside the XODT. After the red
MOT is switched oﬀ, 5 × 106 atoms at 1.5 μK are loaded into the XODT, in an equal mixture of
all 10 nuclear spin states.
3.2.2

TheoryTo calculate the vertical trap potential U (z), we must include the eﬀects of gravity, and ﬁnd,

1
2
2
U (z) = − αĒ 2 e−2z /ω0z + mgz,
2

(3.1)

86

AH Field(G/cm)

Load

Broadband

Cool
in ODTSF

Hold
in ODT

30
20
10
1

On

Blue MOT lasers and repumps
ODT lasersOffα PD Voltage

65Transimpedance
gain x100

1000 Isat432

0.5 Isat1

α freq offset (kHz)

0
200
0
−200
−400
−600
−800
−1000
−1200

+36 kHzβ PD Voltage

3.0Transimpedance
gain x100

2.5
2.01000 Isat

1.5
1.0

0.5 Isat

0.5

β freq offset (kHz)0.0+60 kHz

100
0
−100
−200
−300
−400
−500
0

100

200

300

400

500

600

700

800

Figure 3.3: Timing diagram for the red MOT loading, red MOT, and ODT loading, showing
the intensities and frequencies throughout the red MOT loading, broadband, single frequency andin-ODT cooling stages.

87
where Ē is the average electric ﬁeld,

Ē =

2P
,
π0 cω0z ω0x(3.2)

ω0x,z are the beam waists along the x and z axes, respectively, P is the average beam power, α isthe atomic polarizability which depends on the frequency of the trapping light.
The trap depth ΔU is given by the diﬀerence between the local minimum and local maximum,

ΔU = U (zmax ) − U (zmin ),(3.3)

and the vertical trap frequency νz is given by the curvature of the potential at the local minimum,
d2 U (z) 2m(2πνz ) =(3.4)
dz 2 z=zmin
3.2.3

Measuring trap frequencies

With the atoms now loaded into the XODT, we still must do a few measurements before wecan determine the Fermi temperature and optimize the evaporation trajectory. Here we discuss themeasurement of trapping frequencies in the XODT.
Initially, it seemed like it would be convenient to measure the trapping frequency via parametric heating, a process by which intensity noise of the trapping beam at twice the trap frequency
causes heating (and therefore atom loss) by exciting harmonic oscillator modes |n → |n + 2. The
2 S (2ν ) [137], S (ν) is the power spectral
heating rate is given by Ė = ΓE, where Γ = πνtrtr
density of the laser intensity noise, and νtr is the trapping frequency. However, attempts to measure the trapping frequencies in this way resulted in spectra such as those in Figure 3.4. Since thetemperature of the atoms was too close to the trap depth, parametric heating was inﬂuenced bytrap anharmonicities and coupling of the vibrational modes along all three axes. This diﬃculty was
further compounded by the fact that at the time of initial loading, we try and get as much power
to the optical table as possible, so we are close to saturating our mixers, RF ampliﬁers, and AOMs,
thus operating all of the intensity control electronics in a very nonlinear regime where it is diﬃcult
to modulate the intensity at a particular frequency without also introducing Fourier components at

88
harmonics and sub-harmonics. In retrospect, it makes sense why this should work for cold atoms
in a deep optical lattice, but not for atoms in the ODT, where the temperature is generally ﬁxed
via the trap depth during the loading and evaporation process.
We had much better luck measuring trap frequencies via dipole oscillations, where, at aparticular trap depth, we pulse on a second dipole trap with a slight oﬀset from the atom cloud inorder to “kick” the atoms along a particular axis, thus causing “sloshing” along that axis. Initially,
this was accomplished using a second 1064 trap; now that we have our 3D optical lattice beams
installed, we can conveniently ﬂash them on one at a time. We preform 2D Gaussian ﬁts to ﬁnd
the center of the atom cloud in absorption images taken at diﬀerent times after the kick, resulting
in measurements of the trapping frequencies like those shown in Figure 3.6.

3.3

Red absorption imaging
Red absorption imaging is performed with the same laser on the same 1 S0 , F = 9/2 to

3P , F1= 9/2 transition, along the same beam path used for blue absorption imaging. 50 μs

seemed to be a good pulse duration. By scanning the frequency of the absorption imaging laserand counting the atom number as a function of frequency, we are able to measure the relative spin
populations in the optical dipole trap, as shown in Figure 3.10.
3.3.1

Magnetic ﬁeld cancellation and calibration

We ﬁrst perform red absorption imaging to zero the background magnetic ﬁelds and calibrate
the compensation coils. By measuring the frequency splitting between diﬀerent nuclear spin states
as shown in Figure 3.7 for diﬀerent bias coil control voltage settings, we were able to both zero
the magnetic ﬁeld and calibrate the control voltage to ﬁeld conversion factors for all three axes asshown in Figure 3.8

894Modulate HODT

10 5

HODT = 1.4 V, VODT = 1.6 V
HODT = 1.4 V, VODT = 1.6 V
HODT = 1.4 V, VODT = 0.8 V
HODT = 1 V, VODT = 1.6 V

3.53

Atom Number

2.5

2

1.5

1

0.5

0
0

100

200

300

400

500

600

700

800

900

700

800

900

Modulation Frequency (Hz)4

Modulate HODT

10 5

HODT = 1.4 V, VODT = 1.6 V
HODT = 1.4 V, VODT = 1.6 V
HODT = 1.4 V, VODT = 0.8 V
HODT = 1 V, VODT = 1.6 V

3.53

Atom Number

2.5

2

1.5

1

0.5

0
0

100

200

300

400

500

600

Modulation Frequency (Hz)Figure 3.4: Attempt to measure XODT trap frequencies via parametric heating. I do not recommend trying to measure trap frequencies in this way. The vibrational modes along all threeaxes were coupled, and also some nonlinearities in the RF electronics made a confusing forest of
sub-harmonics, super-harmonics, and god-knows-what.

90

Figure 3.5: Measuring HODT trap frequencies as a function of photodiode voltage (calibration:
5W/1V) by pulsing on the the lattice beams and causing dipole oscillations.

91

Vertical trap freq (Hz)600

HODT vertical trapping calibration
Trap depth = -42.4 ± 3.4 μK/V
Beam waist = 25.6 ± 1.1 μm

500
400
300
200
100
0

Trap depth (uK)

101

100

10−1
−0.5

−0.4

−0.3
−0.2
−0.1
Final HODT setpoint (V)0.0

Figure 3.6: A two-parameter ﬁt to the data shown in Fig. 3.6 determines both the voltage (calibration: 5W/1V) to trap depth conversion factor and the beam waist at the location of the atoms.

Atom Number

Atom Number

Atom Number

Atom Number

92

2

X Coil: Zero Field10 4

5V dataFit|B| = 2.43(0.01) G
0
-2
19.7
5

10

19.8

19.9

20

20.1

20.2

20.3

20.4

4

-5V data
Fit|B| = 2.02(0.02) G
0
-5
19.7
5

19.8

19.9

20

20.1

20.2

20.3

20.4

10 4

-7V data
Fit|B| = 2.77(0.02) G
0
-5
20
5

10

20.05

20.1

20.15

20.2

20.25

20.3

20.35

20.4

4

-10V data
Fit|B| = 3.69(0.03) G
0
-5
20

20.05

20.1

20.15

20.2

20.25

20.3

20.35

20.4

20.45

20.5

SRS Frequency (MHz)Figure 3.7: Measuring the frequency splitting between diﬀerent nuclear spin states in order to zero
the magnetic ﬁeld for atoms in the XODT.

93

Zero Fields: X Coil4

A = 0.372(0.01) G/V
Background = 1.244(0.1) G
Minimum at -0.586 V

3.5

Measured |B|

3

2.5

2

1.5

1
-12

-10

-8

-6

-4

-2

0

2

4

6

Comp Coil Control Voltage (V)Zero Fields: Y Coil4

Zero Fields: Z Coil

6

3.5
5

Measured |B|

2.5

Measured |B| (G)A = 0.309 +/− 0.00 G/V
Background = 0.489 +/− 0.06 G
Minimum at 0.032 V

3

2

1.5

A = 0.53(0.00) G/V
Background = 0.60(0.04) G
Minimum at −0.02 V

4

3

2

1
1
0.5

0
−12

−10

−8

−6

−4

−2

0

2

Comp Coil Control Voltage (V)

4

6

0
−12

−10

−8

−6

−4

−2

0

2

4

6

Comp Coil Control Voltage (V)Figure 3.8: Zeroing the magnetic ﬁeld and calibrating the compensation coils using absorption
imaging on the 3 P1 transition. (Each plot is formatted slightly diﬀerently, just to annoy you.)

94
3.3.2

Optical pumping and spin state detection

To prepare an equal mixture of the stretched mF = ±9/2 states, we perform optical pumping
via the 1 S0 , F = 9/2 ↔ 3 P1 , F = 9/2 transition in a 3 G magnetic bias ﬁeld, which splits neighboringmF states by 260 kHz. First, a σ − -polarized (relative to the bias ﬁeld axis) optical pumping beam isfrequency-chirped from the mF = −1/2 → −3/2 transition to the mF = −7/2 → −9/2 transition,
which pumps all mF < 0 atoms into the mF = −9/2 state. In the second step, a liquid crystal
waveplate is used to switch the laser polarization to σ + , and similarly all the mF > 0 atoms are
pumped into the mF = 9/2 state.
In order to get an accurate comparison of the relative number of spin states, we must ensure
equal scattering cross sections for each of the spin states. We therefore operate with the absorption
imaging laser polarization at a 54.6o to the quantization axis. This is the “magic angle” known√
from NMR, where θ = cos− 1(1/ 3). This gives equal amplitudes of π, σ − , and σ + transitions,which when added together, give the same scattering cross section for each spin state, as shown in
Figure 3.9.

3.4

Evaporation
Laser cooling followed by optical pumping to the mF = ±9/2 stretched nuclear spin states

produces two separate Fermi gases with mF = +9/2 and mF = −9/2, each with an initial phasespace density of ≈ 0.1 in the crossed optical dipole trap. (This estimate ignores atoms that arein the HODT but not in the VODT as well, and so the actual phase space density is lower.)
Evaporative cooling to degeneracy proceeds by exponentially decreasing the trap depth in a 7 s,
two-stage ramp [48, 149]. For diﬀerent measurement goals, we optimize particular ﬁnal parameterssuch as temperature (10 to 60 nK) and atom number (104 to 105 ). Figure 3.11 shows a freely
expanding degenerate Fermi gas of 4 × 104 atoms. The temperature T and Fermi temperature TF
are determined from a ﬁt to the Fermi-Dirac distribution, giving T = 60 nK and T /TF = 0.2.
After evaporation, we are also able to perform a better measurement of our vacuum-limited

95

Atom number (arb.)

1000
π transitions
σ − transitions
σ + transitions

800
600
400
200

Atom number (arb.)

0
1200
1000

Polarization at 54.7 degree angle to quantization axis

800
600
400
200
0
−1000

−500

0
Frequency (kHz)

500

1000

Figure 3.9: Illustration of the transition amplitudes for diﬀerent polarization conﬁgurations withrespect to the quantization axis for the 1 S0 , F = 9/2 to 3 P1 , F = 9/2 transition.

Figure 3.10: Measurement of the mixture of 10 nuclear spin states in the crossed optical dipole
trap via absorption imaging on the 1 S0 , F = 9/2 to 3 P1 , F = 9/2 transition.

96

Figure 3.11: Momentum distribution data of a two-spin Fermi gas after being released from thecrossed optical dipole trap. The inset shows an absorption image of the freely expanding gas. The
azimuthally averaged column density deviates from a Maxwell-Boltzmann ﬁt, but agrees well witha Fermi-Dirac ﬁt giving T /TF = 0.2.

97
lifetime by ramping the ODT depth back up (so that heating is not mistaken for atom loss) andmeasuring the number of atoms as a function of time, as shown in Figure 3.12. We measure a 1/elifetime of ≈ 90 s, so we know that our vacuum-limited lifetime is at least that long. In the processof making these measurements, we realized that the atoms are very sensitive to any stray light,
especially from our blue laser system, which is on the same optical table. The blue laser system is
now housed inside a black plastic box to seal out any stray photons.

3.5

Kapitza-Dirac scattering
Another measurement that can be performed after evaporation is Kapitza-Diract scattering[82] in order to calibrate the high-resolution top imaging system. To do this, we quickly pulse on
one axis of the optical lattice for 4 μs which is less than 1/ωrec , where ωrec is the recoil frequency ofthe particle, given by ωrec = 2 k 2 /2m, where k is the wavevector of the 813 nm lattice photons,
and m is the mass of the strontium atom. In this case, we can disregard the motion of the atoms
with respect to the light ﬁeld (Raman-Nath approximation).
The potential produced by the lattice can be written as U (z, t) = U0 (t) sin2 (kz) where U0 (t)is the time-dependent envelope for the potential. Then, starting with an initial wavefunction ψ0 ,the wavefunction |ψ immediately after the lattice is pulsed on and oﬀ can be written as,−i
dtU0 (t) sin2 kz|ψ = e|ψ0 = e−iA eiA cos 2kz |ψ0 ,
where,
A1=
2

dtU0 (t).

(3.5)

in Jn (α)einβ ,(3.6)

Using the identity,∞eiα cos β =n=−∞where Jn are Bessel functions of the ﬁrst kind, we ﬁnd,
ψ = e−iA∞n=−∞in Jn (A)ein(2kz) |ψ0 

(3.7)

98

12

Lifetime in Dimple

10 4

11

1/e lifetime = 90.8(5) s

10Atom Number

9

8

7

6

5

4

3

2
0

20

40

60

80

100

120

140

Time (s)Figure 3.12: Lifetime of ground state atoms in the optical dipole trap, demonstrating that thevacuum lifetime is at least 90 seconds.

99
Therefore, pulsing the lattice on quickly causes states with momentum 2nk to be populated withprobability Jn (A). For our purpose of calibrating the top imaging system, it is suﬃcient to justknow that the momenta will be multiples of 2k, where k is the lattice laser wavevector, and so
the velocity packets will be multiples of 2h/mλ8 13 = 11.28 mm/s. Then for a time-of-ﬂight tau,
we precisely know the distance between the momentum packets will be (11.28 mm/s)τ and that is
how we calibrate the magniﬁcation of the imaging system.

3.6

Lattice loading and characterization
The atoms are then adiabatically loaded from the crossed optical dipole trap into the groundband of a 3D optical lattice. We use the high resolution vertical imaging system for horizontalalignment of the lattice beams to the atom cloud at the end of evaporation. Vertical alignment,
however, is trickier, as all dimensions in the vertical direction are smaller and also we do nothave high-resolution imaging in the horizontal direction. Initially, we overlapped the location of
the atoms at the end of evaporation with the location of the atoms in the lattice loaded directlyfrom the red MOT. However, this was not a very precise method for vertical alignment, as, due
to gravitational sag, the position of the thermal atoms in the lattice beam depends on variables
such as the beam intensity and atom temperature. A better method of vertical alignment is to
use the same technique used to measure the XODT trapping frequency: pulse the lattice on for400 μs and measure dipole oscillations. Measuring the oscillation amplitude for diﬀerent latticesteering mirror positions enables us to ﬁnd the center position which causes minimal oscillations in
the plane transverse to the vertical beam.
A timing diagram for the loading procedure is shown in Figure 3.14. Whereas in previous
stages of the experiment, the atoms were hot enough that we did not have to worry about transients
when initially engaging any servos; here, we had to be very careful to make sure that integrator
windup (from having the lattice beam intensity servos disengaged until now) and other transients
did not add any entropy to the gas. Therefore, we added analog switches across the capacitors for
all integrators in the intensity servo circuit so that the integrators are all shorted out until it is100

Figure 3.13: Kapitza-Dirac scattering for calibration of the vertical imaging system.

101

On

ODT AOMsOffOn

ODT Shuts.
OffOn

Lat. AOMs,
Int. Servo
Integrators Off

On

Lat. Shuts.
OffODT Power

On

Off

Lattice setpoint (V)

0
−200
−400
−600

x
y
z

−800
80

Trap depth (Erec )

70
60
50
40
30

x
y
z

20
10
0

0

5

10

15

20

25
30
Time (ms)

35

40

45

50

55

Figure 3.14: Timing diagram for the lattice loading. Importantly, the mechanical shutters for thelattice are close to the experiment, after the in-loop photodiodes used for the intensity servos.
Therefore, we can turn on the lattice AOMs and servo integrators before opening the shutters, sothat the atoms are protected from any transients resulting from engaging the servos.

102
time to turn the lattice on. Then the AOMs and integrators are turned on to a very low setpoint
voltage while the shutter remains closed so that the atoms are still shielded from any transients
while the lattice beams are still being ramped up. Finally, after 5 ms, the shutters are opened.
Loading the ground band requires that EF  Erec , where EF is the Fermi energy, kB is theBoltzmann constant, and T is the temperature. As the lattice depth rises, the increasing role ofinteractions relative to tunneling suppresses multiple occupancies in the Mott-insulating regime.
At the ﬁnal lattice depths of 40 to 100 Erec , where Erec is the lattice photon recoil energy, theLamb-Dicke requirement is satisﬁed for clock light along all directions [100].
With a low-entropy Fermi gas loaded into a 3D lattice, clock spectroscopy is then performedon the 698 nm 1 S0 (|g; mF ) ↔ 3 P0 (|e; mF ) clock transition. As shown in Fig. 3.15, the clock
laser propagating along the x̂ lattice beam is used for precision spectroscopy, while an oblique clock
laser enables a systematic characterization of the lattice via motional sideband spectroscopy (seeFig. 3.16). Adiabatic loading of the lattice with respect to the lattice band spacing is veriﬁed by
the lack of visible red sidebands in motional sideband spectrum, indicating nearly perfect loading of
the ground motional band. For data presented in this thesis, we detected no population in excited
bands of the lattice.
However, there are much stricter requirements for adiabaticity when considering interactions
between atoms. We quantify the amount of entropy added to the system through a “round-trip”
loading from the XODT (crossed optical dipole trap) to the lattice and subsequent unloading of
the lattice back into the XODT. We verify lattice loading adiabaticity by measuring the T /TF both
before ramping up the lattice and after a subsequent reversed ramp back into the XODT. For theconditions of N = 105 and T = 50 nK, we observe no increase for T /TF = 0.30 ± 0.05, whichallows us to put a conservative upper bound on the bulk gas entropy for the data in Figs. 3.11
and 4.6. Our lowest-temperature data was taken under diﬀerent conditions: N = 104 , T = 15 nK
and T /TF = 0.2. We have veriﬁed, again through a round-trip measurement of T /TF , that loadingof such a sample to the lattice is nearly adiabatic. We detected less than a few nK increase intemperature and a 30% loss in atom number, corresponding to a increase in T /TF of 10%.

103

Figure 3.15: Schematic showing propagation direction (large arrows) and polarization (doublearrows) of the 3D lattice and clock laser beams. The quantization axis is deﬁned by the magnetic
ﬁeld B. The narrow line clock laser used for precision spectroscopy is phase-stabilized to lattice x̂.
The oblique clock laser is used drive motional sidebands along all three lattice axes.

104

0.30

Excitation fraction

0.25
0.20

3P0

xˆ blue SByˆ blue SBzˆ blue SB
Red SBs
Carrier

0.15
0.10

1S0

0.05
0.00
−60 −40 −20 0
20 40
Detuning (kHz)60

Figure 3.16: Motional sideband spectroscopy using the oblique clock laser shows no observable red
sidebands, illustrating that atoms are predominantly in the ground band of the lattice.

105
Since the interactions between ground state strontium atoms cannot be “turned-oﬀ,” weestimate the eﬀects of interactions as compared to a non-interacting system through numericalcalculations based on the measured trap frequencies, temperature, and entropy of the bulk Fermi
gas before applying the lattice potential.
To estimate suppression of doubly-occupied sites (doublons) in the lattice due to interactions,
we use a model which assumes conservation of the total entropy and atom number measured in
the XODT to compare the doublon fraction for the interacting and non-interacting cases [80, 142].
For our coldest samples, a large suppression of doublons, as well as a central region of vanishingcompressibility, is expected (see Fig. 3.18). The model predicts a dublon fraction more than 100
times smaller than that of the non-interacting case. Clock spectroscopy comﬁrms the suppressionof doublons. The next step is to use a high-resolution imaging objective to verify the vanishingcompressibility in the center of the trap and the existence of a low-entropy Mott-insulator [80, 142].
(see Fig. 3.17 below). From this investigation, we conclude that we operate our clock in the
Mott-insulating regime where atomic interactions suppress the number of doubly-occupied sites(doublons) [80, 142]. We have also explored lower-temperature conditions in the XODT: N = 104 ,T = 15 nK and T /TF = 0.2 in the XODT. Here, a much greater suppression of doublons, as well
as a central region of vanishing compressibility, is expected (see Fig. 3.18 inset). For our coldest
samples (N = 1 × 104 , T = 15 nK, TF = 0.2) used in spectroscopy, a non-interacting model would
predict a doublon fraction greater than 60%. In contrast, both the interacting model and our clock
spectroscopy demonstrate a vanishing number of doublons. This level of suppression of double
occupancies is characteristic of a unit-ﬁlled Mott-insulator.
The next step in our experiment is to use a high-resolution imaging objective to verify the vanishingcompressibility in the center of the trap for a low-entropy Mott-insulator [80, 142].
Accurate temperature measurement in the lattice is experimentally challenging and represents an active area of research. We estimate the temperature in the lattice through numericalcalculations based on an interacting model. In the case of TODT = 15 nK and N = 1 × 104 in the
XODT, the temperature drops in the lattice yielding Tlattice = 4 nK.

106

Doublon fraction

107

0.4
0.3
Non- interacting

0.2

Interacting

0.1
0.0

2

4

6

8

10

Atom number (104)
Figure 3.17: Suppression of doubly occupied sites (doublons) in the Mott-insulating regime. A
comparison of calculated doublon fraction between interacting and non-interacting cases is plottedfor T = 50 nK and T /TF = 0.2 − 0.4 in the XODT, depending on atom number. The inset shows
the calculated density proﬁle in the lattice for N = 4 × 104 and T /TF = 0.25.

Doublon fraction

0.8

0.6

0.4

Non- interacting
Interacting

0.2

0.0
2

4

6

8

10

Atom number (10 )4Figure 3.18: Suppression of doubly occupied sites (doublons) in the Mott-insulating regime. A
comparison of calculated doublon fraction between interacting and non-interacting cases is plottedfor T = 15 nK and T /TF = 0.1 − 0.2 in the XODT. The inset shows the calculated density proﬁlein the lattice for N = 1 × 104 and T /TF = 0.2. For our coldest samples at T = 15 nK and
N = 1 × 104 used for narrow-line spectroscopy in Fig. 4, the number of doubly occupied sites isgreatly suppressed as compared to calculations for the non-interacting case.

Chapter 4
Experiments

Now that we have produced a degenerate Fermi gas, loaded it into the ground band of a 3D
optical lattice, and veriﬁed doublon suppression, we can ﬁnally answer the question: Is it possible
to employ correlated quantum matter to enhance both stability and accuracy in a state-of-the-art
atomic clock?
We still had to do a lot of technical groundwork to answer this question. First, there has
been a long-standing question as to whether or not state-independent trapping in a 3D lattice
could be achieved at a level to accommodate state-of-the-art narrow line spectroscopy and atomic
clock. The challenge arises from complications involving vector and tensor ac Stark shifts, as wellas interference between diﬀerent lattice beams. Next, we had to measure the interaction energies
and characterize the clock shifts due to clock transitiosn occuring for any residual doublons.
With these questions answered, we demonstrated a record quality factor in the ﬁrst implementation of this new technology. We conclude this chapter by discussing future clock protocols
for enhancing stability, and discuss the implications of our work for the stability of future clocks.

4.1

Lattice AC Stark Shifts
There has been a long-standing question as to whether the overall ac Stark shift in a 3D

lattice can be managed to allow state-of-the-art narrow line clock spectroscopy. We implement asolution to this challenge, inspired by the proposal in [160]. The diﬀerential ac Stark shift from thelattice trapping beams at a particular trap depth U0 can be expressed in terms of its scalar, vector,

109
and tensor components as [160, 25],
Δν = (Δκs + Δκv mF ξêk · êB + Δκt β)U0 ,(4.1)

where Δκs,v,t are the scalar, vector, and tensor shift coeﬃcients, respectively, ξ is the lattice lightellipticity, and êk and êB are unit vectors along the lattice beam wave vector and magnetic ﬁeld
quantization axis, respectively. The parameter β can be expressed as β = (3 cos2 θ − 1)[3m2F −F (F + 1)], where θ is the angle between the nearly-linear lattice polarization and êB .
4.1.1

Experimental conﬁguration and measurement

We achieve state-independent trapping by operating the lattice at the combined scalar andtensor magic frequency and ensuring that the vector shift is zero [56, 121]. Linearly polarized
lattice light (ξ = 0) suppresses the vector shift, and the tensor shift is minimally sensitive to driftsin θ when the polarization either parallel (θ = 0◦ ) or perpendicular (θ = 90◦ ) to the quantization
axis. The frequency of the trapping light is then tuned to adjust the scalar shift so that it precisely
cancels the tensor component. The θ = 0◦ conﬁguration has been thoroughly studied in 1D latticeclocks [121, 56]. For the 3D lattice, we set the horizontal (x̂, ŷ) and vertical (ẑ) lattice polarizations
to be parallel and perpendicular to êB , respectively, as shown in Fig. 3.15. The two polarization
conﬁgurations have distinct magic frequencies due to their diﬀerent tensor shifts.
We measure the magic frequencies for the vertical and horizontal lattice beams. A continuouswave Ti:Sapphire laser is used for the lattice light because of its low incoherent background [144].
The absolute frequency of the lattice laser is traceable to the UTC NIST timescale through an
optical frequency comb. For a given lattice laser frequency, we measure the diﬀerential ac Stark
shift using four interleaved digital servos that lock the clock laser frequency to the atomic resonancefor alternating high and low lattice intensities and mF = ±9/2 spin states [23, 121, 56]. From
the data shown in Figure 4.2, we measure that the vertical (θ = 90◦ ) and horizontal (θ = 0◦ )magic frequencies are 368.554839(5) THz and 368.554499(8) THz, respectively, in agreement with
[121, 56, 144]. From these two magic frequencies, we ﬁnd that the scalar magic frequency is

z/ Erec )
H
(m
t
if
h
s
k
r
a
St

110

ˆ ŷ
x,
ẑ

De
ma t uni
gic ngfre fromq.
(M scaHz lar)mF

Figure 4.1: Depiction of the diﬀerent lattice AC stark shifts for the two diﬀerent beam orientations,
showing the quadratic dependence of the tensor component on the nuclear spin state.

111

20

ac Stark shift (mHz/Erec )15
10x̂: θ = 0◦ŷ : θ = 0◦ẑ: θ = 90◦

5
0
−5
−10
−15
−20
−1.5

−1.0
−0.5
0.0
0.5
1.0
Detuning from scalar magic freq. (GHz)

1.5

Figure 4.2: Determination of the magic wavelengths for the horizontal (x̂, ŷ) and vertical (ẑ) lattices,with |mF | = 9/2. The vertical and horizontal lattices have distinct magic wavelengths due to theirdiﬀerent tensor shifts arising from diﬀerent angles θ between the lattice polarization and magnetic
ﬁeld quantization axes. For presentation, the measured frequency shift is scaled by the diﬀerencein peak trap depths at the high and low lattice intensities. The diﬀerence in slopes is caused bytrapping potential inhomogeneities and does not aﬀect the determination of the magic frequencies.

112
368.554726(4) THz, in agreement with [144].
Since there are only two θ conﬁgurations with a stable tensor shift, in a 3D lattice two of thethree lattice beams will necessarily have the same magic frequency. However, the frequencies of
all lattice beams must be oﬀset to avoid interference, which is known to cause heating in ultracold
quantum gas experiments [107]. We choose our two horizontal beams to have the same polarizationand operate them with equal and opposite detunings (±2.5 MHz) from their magic frequency, giving
equal and opposite ac Stark shifts from the two beams. From the slopes in Fig. 4.2, we determinethat for a 10% imbalance in trap depths, detuning the two horizontal beams ±2.5 MHz from theirmagic wavelength results in a < 1 × 10−18 systematic shift, the exact magnitude of which can bemeasured to much better accuracy.
Another issue resulting from our 3D geometry is that one lattice beam must operate withêk · êB = 1, which can give rise to a vector ac Stark shift due to residual circular polarization.
The vertical (ẑ) beam has this conﬁguration, and we measure a 3 × 10−18 × mF /Erec vector shift,corresponding to an ellipticity of ξ = 0.007 [160]. In contrast, the horizontal beams (x̂, ŷ) withêk · êB = 0 enjoy an additional level of vector shift suppression. Nevertheless, the clock operates
by locking to alternating opposite mF = ±9/2 spin states, so the net vector shift is removed byaveraging the two spin states, the same as for the 1st order Zeeman eﬀect.
4.1.2

Data analysis and statistical methods“Yo dawg, I heard you like precision measurement. So I put some systematics on yoursystematics.” -Xzibit
Elimination of systematic shifts and accurate error estimation are the two complementary
cornerstones of precision measurement; in this section, we discuss our techniques for handlingboth. We evaluate the ac Stark shift using four interleaved but independent servos locked to themF = ±9/2 transitions for each of the high and low lattice intensity conﬁgurations, with high andlow lattice depths diﬀering by approximately 30Erec . Each servo tracks the detuning between one

113
transition of the strontium atoms and the TEM00 mode of an ultra-stable cavity [109].
Laser frequency drift gives a nonuniform shift to the interleaved locks that can lead to an erroneous systematic oﬀset in the measured ac Stark shift. Every four experimental cycles, we measureii
ii
independent frequencies locked to the four conﬁgurations {fhigh,+9/2, flow,+9/2, fhigh,−9/2, flow,−9/2},

where high and low refer to lattice depths, ±9/2 refer to the mF states, and i is the iteration
number of the experiment. A four-point string analysis [53, 112] removes linear and quadratic laser
drift using the following linear combination of eight consecutive measurements:



5  i
3  i
i
i
−
fhigh,+9/2 + fhigh,−9/2
flow,+9/2 + flow,−9/2
16
16


5  i+1
3  i+1
i+1
i+1
+
−
,
fhigh,+9/2 + fhigh,−9/2
flow,+9/2 + flow,−9/2
16
16


3  i
5  i
i
i
i
−
=
fhigh,+9/2 − fhigh,−9/2
flow,+9/2 − flow,−9/2
Δfvector
1616

5  i+1
3  i+1
i+1
i+1
+
−
.
fhigh,+9/2 − fhigh,−9/2
flow,+9/2 − flow,−9/2
16
16iΔfscalar+tensor
=

(4.2)

(4.3)

In addition to the dominant eﬀect of clock laser reference cavity drift, this technique of point
string analysis also removes systematic oﬀsets due to linear and quadratic drifts in uncontrolled
systematics, including shifts from room temperature BBR, a drifting quadratic Zeeman shift due
to background magnetic ﬁeld ﬂuctuations, and potential DC Stark eﬀects. It is crucial to performthe analysis in this way so that we extract frequency changes corresponding to the one systematic
we are modulating: the AC Stark eﬀect.
With systematic shifts mitigated, we now turn to proper estimation of statistical error. In
previous clock experiments with an ≈ 1 s duty cycle, the interleaved servos were able to adequately
track the fundamental“1/f ” noise from the local oscillator. However, thus far, our new degenerate
gas experiment has not been optimized for quick evaporation and thus runs on an ≈ 20 s duty cycle
where 1/f noise plays a role. The measured frequencies are then correlated in time because theservos low-pass ﬁlter the system response. We correct the servo frequency using the error signal,

114
which contains higher-frequency components of the system response, according to,

fcorrected = funcorrected +

Γ
× e,
2A

(4.4)

where Γ is the FWHM linewidth, e is the error signal (the diﬀerence between the excitation fraction
from probing the left and right sides of the line), and A is the maximum peak height of the spectroscopic feature. Fig. 4.3 demonstrates that this procedure ﬂattens the autocorrelation functionof the corrected data (blue) compared to the uncorrected data (green).
To estimate the autocorrelation function from the data, we use the formula [13],
n−j


ρj =

i=1

n


[Xi − X̄(n)][Xi+j − X̄(n)],(n − j)S 2 (n)2S (n) =

i=1

[Xi − X̄(n)]2n(n − 1).

(4.5)

The standard deviation of the mean is only a good measure of error when the noise is Gaussian
and the data points that are averaged together are statistically independent; it underestimates theerror when the data is correlated and overestimates the error when the data is anti-correlated. We
must therefore use the autocorrelation function to calculate the unbiased error for correlated data,



n−1

j
σ 

√
ρj ,
1−
1+2
n
n

(4.6)

j=1

where σ is the standard deviation of the point-string diﬀerence frequencies Δf i , ρj is the autocorrelation function for data separated by j points, and n is the number of measured frequencies [13].
This empirical formula for the autocorrelation function is known to be unreliable unless one
uses hundreds of data points (we only had tens of data points for each measurement), so to verify
the validity of this technique, all of these steps are validated on simulated data modeled by clock
laser noise similar to the actual laser [19], with a power spectral density S ∝ 1/f , where f is thefrequency, and a linear frequency drift from imperfectly canceled material creep of the reference
cavity between -2 and 2 mHz/s. We use weighted least-squares ﬁtting to determine the magic
wavelength for each lattice direction. From the diﬀerence in the splittings between the ±9/2 states
for the high and low lattice intensities, we also extract the vector ac Stark shift, as shown in Fig. 4.4.

115

1.0

Corrected
Uncorrected

0.8

Autocorrelation

0.6
0.4
0.2
0.0
−0.2
−0.4
0

1

2
3
Number of points

4

5

Figure 4.3: Autocorrelation functions of the measured ac Stark shifts for clock laser servo frequencies
uncorrected (green) and corrected by the measured error signal (blue).

Vector ac Stark shift (mHz×mF /Erec )

116

1.5

1.0

0.5

x̂: θ = 0◦ŷ : θ = 0◦ẑ: θ = 90◦

0.0

−1.5

−1.0
−0.5
0.0
0.5
1.0
Detuning from scalar magic freq. (GHz)

1.5

Figure 4.4: Measuring the vector ac Stark shift for the horizontal (x̂, ŷ) and vertical (ẑ) conﬁgurations. The vector shift from the vertical lattice beam is sensitive to a small ellipticity in thepolarization because êk · êB = 1, while the horizontal beams with êk · êB = 0 beneﬁt from additional
suppression of the vector shift.

117
The reduced chi-squared χ2red provides some estimate of whether or not we modeled the noiseproperly. Note that according to [8], it is bad practice to rescale the error bars such that χ2red = 1,
as this assumes that the noise is Gaussian (it is not: it is dominantly 1/f noise), the model is linear
in all parameters (it is not: the frequency lists come from a digital servo which includes a double
integrator), and it already assumes that the noise model is correct (it might not be: that is thepoint of using χ2red to test whether it is!).

4.2

Interactions

4.2.1

On-site contact interactions

In the ground band of the lattice, each site can be occupied by either one atom, or by
two atoms with opposite nuclear spin. Tight conﬁnement in the 3D lattice gives rise to strong
interactions on doubly occupied sites. We label the two-particle eigenstates of the two-orbital√
interaction Hamiltonian as |gg; mF , mF  = |gg⊗|s, |eg + ; mF , mF  = (|eg + |ge)/ 2 ⊗ |s,√|eg − ; mF , mF  = (|eg − |ge)/ 2 ⊗ |t, and |ee; mF , mF  = |ee⊗|t with corresponding ener+ , U − , and U
gies Ugg , Uegee [61, 171, 138, 31]. Here, |s and |t represent the singlet and triplet
egwavefunctions for the two spins, mF and mF . A schematic showing the relative energy levels
and symmetries of these eigenstates is shown in Figure 4.5. An applied magnetic bias ﬁeld mixes
the |eg + ; mF , mF  and |eg − ; mF , mF  states due to a diﬀerential Landé g-factor between the twoorbitals. We label the new eigenstates of the combined interaction and Zeeman Hamiltonian as|eg u ; mF , mF  and |eg d ; mF , mF .
Clock light resonantly couples the ground state |gg; mF , mF  only to the states |eg u,d ; mF , mF .
+ and U − diﬀer from U
The energies Ueg
gg on the h·kHz scale, where h is the Planck constant, reeg

sulting in transitions on doubly occupied sites that are well-resolved from the single-atom clocktransitions and strong suppression of two-photon transitions to the |ee; mF , mF  state. In 1D optical lattice clocks, interaction shifts are less than clock transition Rabi frequencies; in 2D clocks,
interaction shifts and Rabi frequencies are comparable [152]. In contrast, the interaction shifts in

118

g, e
↑,↓
atom 1 atom 2,: symmetric: anti-symmetric

eg+
+

Ueg
Ueg

ege
gg

Ugg

g

Figure 4.5: Level diagram showing transitions for lattice sites with one atoms and two atoms atzero magnetic ﬁeld.

119
a 3D lattice are 103 times greater than the clock transition Rabi frequency. Although for normal
clock operation we load at most one atom per site, to measure contact interactions we increase theﬁnal Fermi temperature so that a small fraction of lattice sites are ﬁlled with two atoms in the|gg; +9/2, −9/2 state. Fig. 4.6A shows clock spectroscopy for a magnetic ﬁeld B = 500 mG. At
this ﬁeld, π-polarized clock light gives a negligible transition amplitude to the |eg u ; +9/2, −9/2
state due to destructive interference between the two oppositely signed transition dipole moments
for the stretched states. Therefore, we observe excitation to the |eg d ; +9/2, −9/2 state, highlighted
in red, and no excitation to the |eg u ; +9/2, −9/2 state, highlighted in blue.
Clock operation predominantly probes atoms in singly occupied sites, with only a negligible
systematic shift due to line pulling caused by transitions from the few atoms in doubly occupied
sites. For a comprehensive study of line pulling from doubly occupied sites, we use our mea− −U+surements of Ueg
gg and Ueg − Ugg at B = 0 to calculate the spectrum of all transitions on

doubly occupied sites as a function of B, as shown in Fig. 4.6 B. We account for imperfect clock
laser polarization and residual spin populations by considering π, σ + , and σ − transitions to all|eg u,d ; mF , mF  states, as indicated in the shaded regions in Fig. 4.6B. At intermediate magnetic
ﬁelds, transitions from doubly occupied sites can cross clock transitions and potentially cause signiﬁcant frequency shifts. However, this is avoided if B is kept suﬃciently low. At our chosen bias
ﬁeld of 500 mG [23, 121], transitions on doubly occupied sites are at least 500 Hz away. Froma 1% upper bound on residual transition amplitudes, we estimate the fractional frequency shift
due to line pulling eﬀects from doubly occupied sites to be below 1 × 10−24 for 1 Hz linewidths.
We also calculate the fractional frequency shifts due to superexchange interactions [156] between
neighboring sites at typical lattice depths and ﬁnd them to be below 1 × 10−22 .
While on-site interaction shifts can be eliminated in 3D lattice clocks, atoms may also interact
via long-range electric dipole forces, which can lead to many-body eﬀects such as collective frequency
shifts, superradiance, and subradiance [28]. At unit ﬁlling, clock shifts from dipolar interactions
could reach the 10−18 level [32, 93], though employing diﬀerent 3D lattice geometries is a promising
approach for canceling these shifts. Exploring diﬀerent strategies for accurately measuring and120
eliminating dipolar frequency shifts will be a fruitful avenue of future study.
We calculate the detunings of transitions from doubly occupied sites, relative to the singleatom clock transition |g; mF  → |e; mF  at zero magnetic ﬁeld. Since the clock operates under amagnetic bias ﬁeld B, we consider how competing Zeeman and interaction energies determine theenergy eigenstates for atoms on doubly occupied sites with arbitrary m1 and m2 . The Zeeman
shifts for |g; mF  and |e; mF  are gI μB BmF and (gI + δg)μB BmF , where gI is the nuclear g-factor(gI μB = h · 184.4 Hz/Gauss) and δg is the diﬀerential g-factor between the ground and clock states(δgμB = h · 108.4 Hz/Gauss). The diﬀerential Zeeman shift between the two clock states introducesa coupling between the |eg + ; m1 , m2  and |eg − ; m1 , m2  states. Thus, the combined interaction anddiﬀerential Zeeman Hamiltonian in the |eg ± ; m1 , m2  basis can be expressed as [171, 138]:⎛

Δ1 +Δ2+⎜Ueg + 2

Δ1 −Δ22

Δ1 −Δ22− + Δ1 +Δ2
Ueg
2

Ĥ = ⎝

⎞
⎟
⎠,

(4.7)

where Δ1,2 = δgμB Bm1,2 are the diﬀerential Zeeman shifts. The two eigenstates |eg u ; m1 , m2  and|eg d ; m1 , m2  have eigenenergies given by,E{u,d} (m1 , m2 ) = V +
Δ1 + Δ 2
±
2

2 +
Vex

Δ1 − Δ2
2

2

,

(4.8)

+ + U − /2 and V+−
where V = Ueg
ex = Ueg − Ueg /2 are the direct and exchange interaction enereg

gies, respectively. The detunings of the |gg; m1 , m2  → |eg {u,d} ; m1 , m2  transitions relative to the
B = 0 single-atom clock transition are,ΔE{u,d} (m1 , m2 ; m1 , m2 ) = E{u,d} (m1 , m2 ) − Ugg + gI μB B(m1 + m2 − m1 − m2 ),(4.9)

when the two-particle Rabi couplings Ω{u,d} (m1 , m2 ; m1 , m2 ; q) with π(q = 0) or σ ± (q = ∓1)polarized clock light are non-zero, as discussed below.
The atom-light Hamiltonian for a single particle driven by π(σ ± )-polarized clock light is givenby,Ĥq(1) =(ΩmF ,q |e; mF − qg; mF | + h.c.) ,2m
F(4.10)

121
Here, the single-particle coupling ΩmF ,q is expressed using the Wigner-Eckart theorem as,ΩmF ,q = −Fg , mF ; 1, q|Fe , mF − qFg ||d||Fe  ·Eq,

(4.11)

where Fg ||d||Fe  is the reduced matrix element and Eq is the electric ﬁeld amplitude. Then, thetwo-particle atom-light Hamiltonian is written as,Ĥq(2) = Ĥq(1) ⊗ (1)2 + (1)1 ⊗ Ĥq(1) .
1

2(4.12)

We ﬁnd the two-particle couplings Ω± (m1 , m2 ; m1 , m2 ; q) between the |gg; m1 , m2  and |eg ± ; m1 , m2 transitions, driven by π(q = 0)- or σ ± (q = ∓1)-polarized clock beams to be, ±Ω (m1 , m2 ; m1 , m2 ; q) = eg ± ; m1 , m2 |Ĥq(2) |gg; m1 , m2 
2


 
√ Ωm1 ,q δm1 ,m1 −q δm2 ,m2 ∓ δm1 ,m2 δm2 ,m1 −q
=
2 2


+Ωm2 ,q ±δm1 ,m1 δm2 ,m2 −q − δm1 ,m2 −q δm2 ,m1 .

(4.13)

Thus, the two-particle couplings Ω{u,d} (m1 , m2 ; m1 , m2 ; q) between the |gg; m1 , m2  and |eg {u,d} ; m1 , m2 states at a given B are obtained from linear combinations of Eq. 4.13. Fig. 4.7 shows all the detunings ΔE{u,d} (m1 , m2 ; m1 , m2 ) from Eq. 4.9 for non-zero Ω{u,d} (m1 , m2 ; m1 , m2 ; q) with π (blacklines), σ + (red lines), and σ − (blue lines) clock laser polarizations, as well as all single-atomtransitions. In the case of an equal mixture of mF = ±9/2 stretched states, the π-polarizedclock light does not drive the |gg; 9/2, −9/2 → |eg + ; 9/2, −9/2 transition, since Ωm1 =9/2,q=0 =−Ωm2 =−9/2,q=0 gives zero coupling strength as seen in Eq. 4.13. At our operating B, |eg u  ≈ |eg + ,
so π-polarized clock light only drives the |gg; 9/2, −9/2 → |eg d ; 9/2, −9/2 transition, as shown in
Fig. 4.6 A.
Next, we investigate line pulling eﬀects due to the additional lines shown in Fig. 4.7. Singleparticle line pulling eﬀects have already been discussed in [56]; here we focus on line pulling fromtransitions on doubly occupied sites. We approximate the lineshapes as Lorentzians with full widthat half maximum Γ. For typical clock operation, the error signal ε for the digital PID that steers theclock laser center frequency f0 to the atomic resonance is generated by measuring the normalized

122
excitation fraction Nexc at f0 + Γ/2 and at f0 − Γ/2 [23, 121]. The clock frequency is locked
to the atomic reference such that ε = Nexc (f0 − Γ/2) − Nexc (f0 + Γ/2) = 0. In Fig. 4.8, westudy how an additional line with a given amplitude relative to the clock transition modiﬁes thecenter frequency f0 for which ε = 0. Transitions from atoms on doubly occupied sites begin to− )/[(9δg + g )μ ]. At our chosen bias ﬁeld
overlap with the clock transitions when B > (Ugg − Ueg
I B

of 500 mG [121, 23], the transitions on doubly occupied sites are separated from the main clocktransitions by at least 500 Hz. Using a 1% upper bound on residual transition amplitudes andassuming 1 Hz transition linewidths, we estimate the fractional frequency shift due to line pullingeﬀects from doubly occupied sites to be below 1 × 10−24 . If one were to operate their clock using
Ramsey spectroscopy (instead of Rabi), we recommend ﬁrst performing a narrow Rabi “clean-up”
pulse to the excited state, removing all ground state atoms (including residual doublons), and thenperforming “top-down” Ramsey spectroscopy. Then the preceding discussion of systematic eﬀects
due to doublons is still valid.
4.2.2

Dipolar interactions

With shifts from on-site interactions now eliminated in 3D lattice clocks, we must now consider long-range electric dipole interactions, which can lead to many-body eﬀects such as collective
frequency shifts, superradiance, and subradiance [28]. At unity ﬁlling, clock shifts from dipolarinteractions could reach the 10−18 level [32, 93], but there are strategies for accurately measuring
and eliminating these shifts. Exploring diﬀerent lattice conﬁgurations is a promising avenue forcanceling dipolar frequency shifts. Also, since the dipolar shift scales with the ﬁlling fraction ρas ρ2/3 , clocks can measure these systematic eﬀects with Ramsey spectroscopy by modulating ρ[109]. Future studies of dipolar interactions in quantum degenerate lattice clocks will not only benecessary for clock accuracy but will also provide powerful connections to other physical systems
based on polar molecules [167, 118], Rydberg gases [139], and magnetic atoms [4, 45, 81, 29].

123

Figure 4.6: (A) Clock spectroscopy data of a two-spin Fermi gas in the mF = ±9/2 stretched states
for a 500 mG magnetic ﬁeld, where a small fraction of the lattice sites contain both spin states.
All transitions are saturated. The |gg; −9/2, 9/2 → |eg u ; −9/2, 9/2 transition is absent due to its
vanishing dipole matrix element at small magnetic ﬁelds. Inset: Level diagram at zero magnetic
ﬁeld. (B) Calculated detunings for transitions on singly and doubly occupied sites. The solid lines
correspond to transitions on singly occupied (orange) and doubly occupied (red, blue) sites withmF = ±9/2. Transitions on doubly occupied sites for arbitrary mF and clock laser polarization lie
within the shaded regions. At our operating magnetic ﬁeld of 500 mG, all resonances for doubly
occupied sites are well-resolved from the clock transitions.

124

6

Detuning (kHz)4

Clock transitions
π transitions

σ + transitions
σ − transitions|eg u 2
Single atom0d−2 |eg 

Operating ﬁeld
−4
0.0

0.5

1.0

1.5
Bias ﬁeld (G)

2.0

2.5

3.0

Figure 4.7: Detunings (relative to the unperturbed clock transition) for all transitions that can be
driven on singly and doubly occupied lattice sites, shown for typical trap depths. Fig. 3B in themain text is a simpliﬁed version of this ﬁgure, with the shaded regions covering the range of alllines, and the solid lines showing the transition frequencies for only the states with mF = ±9/2.

125Figure 4.8: Systematic shifts due to line pulling from a residual line. Though the linewidth of theresidual line will vary depending on what transition is being driven, we perform this calculation for
1 Hz linewidths to give a reasonable upper bound. The amplitude ratio indicates the amplitude of
the residual line relative to that of the clock transition.

126

4.3

Narrow line spectroscopyWith atomic interactions and lattice ac Stark shifts controlled, we demonstrate a new record

for narrow-line clock spectroscopy. Fig. 4.9 shows a progression of Ramsey fringes with freeevolution times from 100 ms to 6 s, beyond what has been demonstrated in 1D lattice clocks [109,121, 140]. Fig 4.10 shows 4 s Rabi spectroscopy at the Fourier limit, giving a linewidth of 190(20)mHz with full contrast. The x̂ lattice beam is operated at a depth suﬃcient (>80Erec ) to prevent
atoms from tunneling along the clock laser axis during the 6 s free evolution period. Spectroscopy
is performed on a spin-polarized sample, prepared by ﬁrst exciting |g; −9/2 → |e; −9/2, thenremoving all ground state atoms via resonant 1 S0 − 1 P1 light. Our longest observed coherencetime approaches the limit of our clock laser based on its noise model [19] and the 12 s dead time
between measurements. For the demonstration of Ramsey fringes at longer free-evolution times,
maintaining atom-light phase coherence will require a signiﬁcant reduction in fundamental thermal
noise from the optical local oscillator. Additionally, the observation of narrower lines will require
magnetic ﬁeld control below the 100 μG level. The contrast of the observed Ramsey fringes is likely
limited by lattice light causing both dephasing over the atomic sample and excited state population
decay. Inhomogeneities are minimized for our smallest samples with an atom number of 1 × 104
and temperature of 15 nK.
Using Rabi spectroscopy with a 4 s pulse time, we measure a full-contrast Fourier-limited
linewidth of 190(20) mHz, as shown in Fig. 4.10. This is diﬀerent than spectroscopy in a 1D lattice,where clock scans at the longest probe times show substantially reduced contrast due to atomic
contact interactions [109, 121].

4.4

Stability
The combination of large atom numbers with a long atom-light coherence time in this systemopens the possibility for improving the quantum projection noise (QPN) limited clock stability bymore than an order of magnitude over the current state-of-the-art [140]. The QPN limit for Ramsey

127

A

B

C

D

Figure 4.9: Ramsey spectroscopy data taken with 1 × 104 atoms at 15 nK for (A) 100 ms, (B) 1 s,
(C) 4 s, and (D) 6 s free-evolution times, using 10 ms π/2 pulse times. With contact interactions
and ac Stark shifts controlled in a 3D lattice, we are able to measure fringes at a record 6 s freeevolution time with a density of over 1013 atoms/cm3 . The data shown in (D) is an average of two
measurements, with 1σ error bars shown.

128

Figure 4.10: Rabi spectroscopy data for a 4 s π-pulse time, showing a 190(20) mHz Fourier-limited
linewidth, taken with mF = 9/2 and rescaled by the relative spin population.

129
spectroscopy can be given as,
1
σQPN (τ ) =2πνTT + Td,Nτ

(4.14)

where ν is the clock frequency, T is the free-evolution time, Td is the dead time, and τ is thetotal averaging time. Typically, OLCs operate at a stability above this limit due to the Dickeﬀect [76]; however, operation at or near the QPN limit has been demonstrated in systems throughsynchronous interrogation of two clocks [122, 154] or interleaved interrogation of two clocks with zerodead time [140]. A synchronous comparison between two copies of our system would achieve a QPN√
limited stability of σQPN (τ ) ≈ 3 × 10−18 / τ for T = 6 s and Td = 12.7 s, assuming perfect contrast.
√
The current experimental contrast limits this stability to 1 × 10−17 / τ (see Figure 4.9D). Our
design can accommodate even greater atom numbers which, combined with improved preparationof degenerate gases and the next generation of ultra-stable lasers, should enable operation of a zero√dead time clock below the 10−18 / τ level. Reaching such performance is extremely challenging for
1D OLCs as collisional eﬀects force a compromise between interrogation time and the number ofatoms that can be simultaneously interrogated [109, 121].

130

Figure 4.11: Measured and calculated stability√ for diﬀerent clock conﬁgurations. Top: Total
deviation of data (blue) from Fig. 3 divided by 2 to demonstrate single clock performance. The
black dashed line represents the predicted Dick eﬀect limit given the laser noise model [19] of ourcurrent cavity (ULE), a 12.7 s dead-time and a 0.4 s interrogation time, demonstrating that weunderstand the eﬀects of laser noise on our clock stability. Bottom: Calculated stability at 1 second
as a function of spectroscopy for various generations of clock lasers and dead times. Si represents
our next generation ultra-stable laser [111]. The 12.7 second dead-time corresponds to that of
our Fermi-degenerate 3D lattice clock while the 0.5 second dead-time corresponds to our previous
generation 1D lattice clock [121]. The black line shows the QPN limit of our 3D lattice clock for
104 atoms.

Chapter 5
Future prospects and conclusion

Optical lattice clocks have now entered the quantum degenerate regime. With atoms thatare frozen into a 3D cubic lattice, we have advanced the state-of-the-art in coherent atom-lightinterrogation times. Already, using synchronous comparisons, we have demonstrated an unprece√
dented stability of 3 × 10−18 / τ . Further improvements will be enabled by the next generations ofultra-stable optical reference cavities based on crystalline materials [86, 38]. The latest advancesin the frequency references and local oscillators that together constitute atomic clocks will lead toa new era for clock performance, resulting in new measurement capabilities [92].
Quantum degenerate clocks also provide a promising platform for studying many-body physics.
Future studies of dipolar interactions will not only be necessary for clock accuracy, but will also
provide insight into long-range quantum spin systems in a regime distinct from those explored bypolar molecules [167, 118], Rydberg gases [139, 95], and highly magnetic atoms [4, 45, 81, 29].
When clocks ultimately confront the natural linewidth of the atomic frequency reference, degenerate Fermi gases may be useful for engineering longer coherence times through Pauli blocking of
spontaneous emission [136] or collective radiative eﬀects [125, 93]. Ultracold quantum gases provide
new capabilities for precision metrology.
5.0.1

AccuracyMoving forward, we envision using high resolution imaging and spatially-pinned atoms in the
3D optical lattice to directly see systematic inhomogeneities. Whereas previously, we measured

132
systematic shifts via a self-comparison of alternating digital PID locks to alternating experimental
conditions (e.g. high and low lattice intensities for measurement of the AC Stark eﬀect), one canimagine combining high resolution imaging with long dark time Ramsey spectroscopy in order toperform a self-comparison within a single experimental cycle.
We could either trap each half of the cloud in a diﬀerent experimental condition, or we could
apply a linear gradient across the cloud. The systematic shift would then be determined from therelative phases inside atom cloud, and so the free evolution time could be much longer than thecoherence time of the laser. This technique would remove laser noise entirely from the measurementof systematics, as well as increase the measurement Q factor far beyond that of the clock laser.
In this conﬁguration, the statistical uncertainty for systematic measurements is only limitedby the atoms’ QPN which depends on the atom number N and atomic quality factor Q, thus
providing further motivation for improving control and coherence time of the atomic referencecomponent of the clock. This technique can improve clock accuracy whether or not laser noisefrom the Dick eﬀect limits stability during normal operation. Here we discuss some outstanding
systematics and how they might beneﬁt from the capabilities of a quantum degenerate gas.
The most outstanding source of systematic uncertainty in OLCs is Stark shifts due to ambient
blackbody radiation (BBR). Since the room-temperature BBR spectrum is largely in the infrared,
while most of the transitions in Sr are optical, to ﬁrst order, one can think of the BBR shift as a DC
Stark shift which only depends on the total magnitude of the electric ﬁeld. This is referred to as
the “static BBR shift,” and it can be written as Δνstatic T 4 . Additionally, we apply a second-ordercorrection known as the “dynamic BBR shift” (Δνdynamic T 6 ) which depends on the frequencydependent couplings of the various strontium transitions with the ambient BBR spectrum.
The strongest contributor to the dynamic BBR shift is the 2.6 μm 3 P0 to 3D1 transition. To
improve the uncertainty of the dynamic BBR strength in [121], we performed a better measurementof the 3 D1 lifetime to the 0.5% level in order to better constrain atomic structure calculationsof Δνdynamic , achieving a 1.4 × 10−18 fractional frequency uncertainty, limited by uncertainty invarious experimentally-measured line strengths. The uncertainty can be further improved by a133
better measurement of the 3 D1 lifetime, but only by a factor of 2 or so, before we would have togo and perform accurate measurements on all of the other coupling strengths.
A better strategy is to reduce the temperature T to reduce the contribution of uncertainty
in Δνdynamic . This can be accomplished using a cryogenic shield as in [157]. The diﬃculty withholding the atoms inside a shield with a diﬀerent temperature, away from equilibrium with theoutside world, is uncertainty introduced by both thermal gradients in the cryogenic shield and any
open aperture used to shuttle the atoms inside. Ray-tracing models can be used to estimate theseeﬀects, but they are never as convincing the direct clock measurements which are performed fornearly every other systematic. Cryogenic shielding will continue to be a challenging engineering
problem for any group pursuing a 10−19 -level clock.
For example, one could imagine shuttling the 3D lattice into a cryogenic shield1 , performinga Ramsey sequence with a long dark time, and then imaging the spatial distribution of relative
accumulated phases among the atomic references. Since an athermal BBR spectrum directly corresponds to a temperature gradient across the sample, this would allow us to directly measure theinﬂuence of, for example, room-temperature BBR leaking in through an aperture. These measurements would provide a direct check on the ray-tracing modeling, reducing uncertainty, and allowing
me to sleep better at night.
The lattice AC Stark shift can also be measured via this method. We are already considering
using digital micromirror arrays to engineer a 3D box potential for our lattice, it is not too muchof a stretch to imagine using this technology to trap, for example, the left half of the cloud at ahigh lattice intensity, and the low half of the cloud at a low lattice intensity. As discussed earlier,the use of a Mott insulator will also allow us to run the clock at much lower lattice depths, and theuncertainty in our extrapolated shift would scale down accordingly.
A ﬁnal example is that we could have half of our atoms be in the mF = −9/2 state and theother half of the atoms in the mF = +9/2 state. We could then apply a magnetic ﬁeld gradientalong a particular direction and for each magnetic ﬁeld value, add the phases from the opposite
1

Yes, this would be quite the engineering problem, but I would even say I could do it.

134spin states. This would directly extract the quadratic Zeeman shift as a function of magnetic ﬁeld.

5.1

Dark matter searches“Those years, when the Lamb shift was the central theme of physics, were golden years for all thephysicists of my generation. You were the ﬁrst to see that this tiny shift, so elusive and hard tomeasure, would clarify our thinking about particles and ﬁelds.”-Freeman Dyson, speaking to Willis Lamb on his 65th birthday[143]
Our innate desire to continue looking closer and closer is further fueled by the fact that weare far from ﬁguring everything out. The Standard Model (SM) of particle physics explains only
5% of the mass-energy content of the universe. Another 27% of the universe is comprised of dark
matter, which until recently was postulated to have a mass on the TeV/c2 scale and only couple to
the weak force. However, the Large Hadron Collider and direct detection experiments have so far
failed to detect these weakly interacting massive particles, thus inspiring more searches in diﬀerent
parameter regimes. Furthermore, the increasing costs and technical challenges with each decade ofparticle accelerator energy motivate the need for new approaches to explore beyond SM physics.
From astrophysical measurements, we know that DM in our Milky Way has an energy density ρDMof 0.3 GeV/cm3 and a mean virial velocity Vvir of 10−3 c with a variance of similar order. Structureformation in the early universe puts a 1 × 10−22 eV/c2 lower bound on the DM rest mass mDM [108].
Fermionic DM must have a rest mass greater than 1 keV/c2 , as Pauli exclusion prohibits lighter
particles from reaching the required density. In contrast, bosonic DM allows for mDM below 1
eV/c2 at high phase space density. In this case, the DM is a Bose condensate with amplitude andfrequency mDM c2 /h, where h is the Planck constant [12]. The wave follows the scalar equation (indimensionless units),
φ(t, x) = φ0 cos mDM t − kφ × x + · · · .

(5.1)

2 around the rest mass energy m
2
The spread in kinetic energy ≈ mDM vvir
DM c sets a quality factor

Q for dark matter oscillations of 106 , corresponding to a coherence time τcoh of 106 h/(mDM c2 ).

135Improvements in precise atomic tests can search for ultralight dark matter oscillations at the 1 mHz,
1 Hz, or 1 kHz scale, corresponding to mDM of 4 × 10−18 eV/c2 , 4 × 10−15 eV /c2 and 4 × 10−12 eV/c2 ,and coh of 30 years, 10 days, and 20 minutes, respectively. Candidates for ultralight DM particles
include the axion [126], motivated by QCD, and the dilaton, as motivated by string theory. The
dilaton is predicted by all versions of string theory as a scalar partner of the tensor Einstein
graviton [42, 52]. Light scalar ﬁelds from extra dimensions oﬀer a solution for the so-called hierarchy
problem, wherein the SM oﬀers no explanation for discrepancies in gauge coupling parameters forthe fundamental forces, the most outstanding of which is the puzzle as to why the weak force is 102 4
times stronger than gravity [11, 72]. The dilaton could be detected via possible scalar couplings di to
the four fundamental forces of the SM, which cause oscillatory perturbations to the eﬀective masses
and gauge couplings of fundamental particles [43]. In particular, the electromagnetic coupling term
de would lead the ﬁeld φ to cause coherent oscillations in the ﬁne structure constant α accordingto,∂ ln α= de ,
∂(κφ)

(5.2)

√
where the ﬁeld is normalized to the Planck mass MPl using κ = MPl / 4π. Additionally, thegluonic, quark mass, and electron mass coupling terms can also cause variations in the proton-toelectron mass ratio mp /me . The frequencies fA of atomic transitions scale with mp /me and α as,

fA ∝

me
mp

ξA

αξA +2 ,(5.3)

where ξA = 1 for hyperﬁne transitions and ξA = 0 for optical transitions [12]. The variable ξA
depends on the properties of the atom and is listed for common atomic frequency references in[98]. Until recently, equivalence principle (EP) tests such as lunar laser ranging [162, 163] andthe Eöt-Wash experiment [141] placed the best constraints on various dilaton couplings, including|de | ≤ 3.6 × 10−4 for all dilaton masses (or equivalently, oscillation frequencies). However, following
the proposal in [12], spectroscopy data for two isotopes of dysprosium was analyzed in [158] tofurther explore parameter space, improving limits on |de | to below the 1 × 10−7 level for masses in

136
the 1 × 10−22 eV/c2 range. The work in [67] used 6 years of comparison data between the hyperﬁneclock transitions in Rb and Cs, which, in addition to putting improved limits on |de | at low mass,
was also sensitive to variations in quark mass and the quantum chromodynamic mass scale. Thereis great room for improvement on searches for ultralight DM taking advantage of state-of-theart optical clocks and ultrastable reference cavities [148]. A direct frequency comparison of the
JILA strontium optical lattice clock with a laser referenced to a cryogenic silicon cavity is linearlysensitive to variations in α and therefore is sensitive to perturbations from ultralight DM. This
experiment will be sensitive to oscillations in the mHz to kHz frequency range (10−19 eV/c2 to10−12 eV/c2 mass scale), with the capability to measure Planck-scale coupling in one second andsurpass limits from EP tests after one week of averaging. Experiments using next-generation cavities
with crystalline spacers, substrates, and mirror coatings will be able to reach this limit within a
few hours. Synchronous clock comparisons between the JILA strontium clock and NIST ytterbium
clock can improve limits on |de | over a wide mass range, including lower masses approaching 10−22
eV/c2 .
The better we make our atomic clocks, the more sensitive they will be to measuring newparticles. Furthermore, better stability not only allows for shorter averaging times or lower uncertainties, it also enables the detection of transients.

137

5.2

The future of quantum metrology“The simplest prototype of emergent exactness, however, is the regularity of crystal lattices, theeﬀect ultimately responsible for solid rigidity. The atomic order of crystals can be perfect onbreathtakingly long scales in very good samples, as many as one hundred million atomic
spacings.”- Robert Laughlin, A Diﬀerent Universe: Reinventing Physics From the Bottom Down
We discover new physical laws and make deeper connections by traveling to new places andparameter regimes. Measurement precision is a dimension of exploration unto itself. For thejourney to continue, we must anticipate and overcome future limitations. The major impediments
to measurement precision are shot noise, decoherence, and uncontrolled systematics.
Great strides have already been made to tackle the ﬁrst two issues, as quantum correlations
can both move errors outside of the measurement basis and enhance the maintenance of an out-ofequilibrium state. Spin squeezing via the collective interaction of an atomic ensemble with a modeof an electromagnetic ﬁeld in a cavity has been used to redistribute the quantum noise between
diﬀerent degrees of freedom by entangling the atoms in the ensemble [117]. This has been used toovercome the Standard Quantum Limit, where measurement noise scales with the particle number√
N as 1/ N , to make progress towards the Heisenberg limit where noise scales as 1/N . However,to this day nobody has yet demonstrated spin squeezing at suﬃciently long time scales to truly
improve the state-of-the-art in frequency metrolgoy. This is a great and fun challenge to tackle in
the near future.
Many-body synchronization has been studied in the context of superradiance [50] and driven
time crystals that prevent thermalization [170, 35]. Superradiant lasers in particular can overcome
the noise from the optical cavity and may well be the future of ultrastable lasers [114, 24, 124, 123].
To protect against both decoherence and quantum noise, quantum error correction schemes store
the information from one qubit among several highly-entangled qubits.
While improvements in short-term stability increase both overall sensitivity and bandwidth

138
for the detection of transients, uncontrolled systematic shifts present a separate, albeit related,challenge in precision measurement. When developing tools such as atomic clocks, accuracy andprecision turn out to be two sides of the same coin. Nature provides fundamental constants thatare either not changing or changing so slowly that we cannot tell, and so we use them to reference
our measurement devices to minimize long-term drift, which is equivalent to low-frequency noise.
Something that does not drift over time has no limit to how well it can be known. More data
always improves the error bars. However, every real-world Allan deviation turns up at some point.
Atomic transition frequencies are, in principle, constants of nature, but in practice they ﬂuctuate
within the bounds of our ability to mitigate and account for environmental perturbations. Even
our “perfect” atomic reference is a source of noise which is not guaranteed to average down beyond
its accuracy.
We demonstrate ﬁrst successful use of a quantum many-body system (a Mott insulator) toessentially eliminate a systematic eﬀect (collisional shifts) in an atomic clock. We anticipate that
we will soon be able to make a band insulator and run at much lower trap depths which will also
reduce the lattice AC Stark shift. This opens up a new approach to clock accuracy. Instead of
measuring and stabilizing every systematic eﬀect, we can engineer quantum many-body states thatare immune to them.
This inspires us to take another look inside our quantum toolbox and see how far we can takethe ideas of quantum entanglement, emergent behavior, and topological matter towards protecting
our clock against systematic errors. The idea would be to engineer a synthetic atomic transition thatis not susceptible to the usual systematics. One could think of entangling two ensembles of diﬀerent
atoms with diﬀerent sensitivities to various environmental perturbations. Perhaps the synthetic
atomic transition would need to be entangled with and provide feedback to the environment. One
starting point could be systematics which couple to another observable, so we could extract anerror signal by other means than direct measurement on the clock transition. Inspired by thedecoherence-free subspaces employed in quantum error correction, I wonder if one can engineer asystem which has an “error-free subspace” which is immune to errors in DC values of clock transition

139
frequencies. Another idea is that we can engineer large energy gaps for systematics (similar to what
we have already shown for clock transitions for doublons), so that they are energetically forbiddenfrom befouling the clock transition.
For example, diﬀerential lattice AC stark shifts coming from the diﬀerential polarizability
between the ground and clock states cause the atoms position and motion to become entangledwith its electronic state. Could a cavity QED system provide some mechanism for detection andampliﬁcation of this diﬀerential atomic motion? Can shifts be canceled via quantum feedback?
A quantum many-body frequency reference that never decoheres and uses quantum feedback toeliminate all systematics may be a pipe dream, but then again, we do live in Boulder.

When logic and proportion
Have fallen sloppy deadAnd the White Knight is talking backwards
And the Red Queen’s oﬀ with her head
Remember what the dormouse said
Feed your head
Feed your head
- Jeﬀerson Airplane, White Rabbit

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Appendix A
Anti-reﬂection coating fused silica viewports: A cautionary tale

Here we highlight some things that can go wrong when you have a company that is accustomed
to coating 7056 glass viewports try and coat your fancy fused-silica viewports. Fused silica viewports
have a maximum bakeout temperature of 200o C, a maximum thermal gradient of 25o C/minute1 ,and the glass-to-metal seal is done via solder. Compare this to 7056 glass viewports (common in
many other AMO labs), which have a 400o C maximum bakeout temperature, a maximum thermal
gradient of 10o C/minute1 , with a glass-to-metal seal comprised of a “matched expansion seal.” The
7056 glass is more brittle and sensitive to thermal ﬂuctuations, and so the glass-to-metal seal ismade from the iron-nickel-cobalt alloy Kovar which has low thermal expansion similar to the glass.
In contrast, fused silica is more resistant to thermal shock and are brazed directly into the Conﬂatﬂange using PbAg solder or similar. However, with this improved resistance to thermal shock comes
more compilability, and the bending of the glass as it is rapidly heated (perhaps at diﬀerent rate
than the metal ﬂange if the coating company is not careful) can place too much mechanical strain
on the coating itself, causing it to fracture or peel oﬀ as the viewport bends during the coatingprocess as shown in A.2. One of our coatings even arrived intact, but, as shown in Figure A.3, oneside peeled oﬀ after the slightest disturbance of a careful methanol rinse. (In case the reader iswondering: Delamination occurred as soon as we poured the methanol on the viewport. It did not
happen because we thermally-shocked the viewport by via evaporative cooling from aggressively
blasting it with gas to dry it.) This possibly can also aﬀect the thickness of the thin ﬁlm layers,
1
Though, according to Kurt J. Lesker: “Opinions vary about maximum heating rate, but there is no penalty forbeing cautious and using the lowest quoted heating rate of ≈ 2o C/minute.”

153
which can then cause the ﬁnal coating to miss the target wavelength as shown in Figure A.1.
We recently learned of yet another AR coating failure mode from another group who used oneof the same companies that we did.2 The company did not mask the knife edge of the viewports.
It is my understanding that this is typically not an issue (our knife edges were coated as well, as
I believe is common), but for some reason their coating was permeable, so they saw leaks on the
RGA at the ≥ 1 × 10−9 Torr level. They were able to ﬁx some of the windows by attaching andreattaching them so that the copper gaskets would cause the coating to be somewhat scraped oﬀ,
and scrubbing the knife edge with acetone and then methanol in between. They were not conﬁdent
about the larger viewports, however. In any case, if for some strange reason the AR coating ispermeable (I wonder if it was somehow defective in a new strange way), it is crucial to have thecompany mask the knife edge. They subsequently had success with viewports from MPF. We also
think that the company FiveNine Optics sounds also promising (as REO always did a fantastic job
with AR coating viewport).
In conclusion, we recommend that other groups wishing to coat fused silica viewports makesure that their company understands the diﬀerence between the glass-to-metal seals for fused silica
and 7056 glass, and has had speciﬁc experience with successfully coating fused silica viewports. If
their coating is not UHV-compatible, they need to mask the knife edge as well. Additionally, we
advise checking that the company does ion beam sputtering, which is more robust than electron
beam sputtering.

2
Thank you to Paul Lauria of the Barreiro lab at UCSD for providing this information so that we may informthe community ASAP and prevent future headaches such as those that have occurred both of our labs.

154

Independent Measurement of Coated Viewports
100

98

96

%T

94

92

90

88

Spectrophotometer Measurement
461 nm target wavelength

86

84
400

500

600

700

800

900

1000

1100

Wavelength (nm)Figure A.1: Spectrophotometer measurement illustrating the decline of American manufacturing.
APC/Blue Ridge Optics completely missed the 461 nm target wavelength.

155

Figure A.2: Top: Shattered AR coating on the bucket window sub-assembly due to mechanicalstress caused by improper temperature control during the sputtering process. Bottom: Viewport
after Blue Ridge Optics etched away the faulty AR coating. We agreed that they would do one
test viewport and let us know how it goes. Instead, they damaged $10,000 of equipment, leaving
deep scratches and pockmarks on every viewport that we shipped back to them. Surprisingly, they
were willing to just ship everything back to us and pretend that there was no problem. One of
the conclusions of this thesis is that one should never, ever have their coatings done by American
Photonics or Blue Ridge Optics.

156

TAKOS Broadband AR Coating on Larson Electronic Glass 2.75 CF
100

98

96

%T

94

92
Before Cleaning
After Cleaning

90

88

86
400

500

600

700

800

900

1000

1100

Wavelength (nm)Figure A.3: A reminder that one should remind companies to use special care to temperaturecontrol fused silica viewports during coating. Spectrophotometer measurements before and after
one side of the viewport AR coating from TAKOS delaminated. Another set of viewports madewith 7056 glass for the Thompson group in the same coating run had no problems.