Quantum gas engineering for atomic clocks
byLindsay I. Sonderhouse
B.A., Reed College, 2013
M.S., University of Colorado Boulder, 2020

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
2021

Committee Members:
Jun Ye, Chair
Ana Maria Rey
James K Thompson
Scott A Diddams
Vanja Dukic

ii
Sonderhouse, Lindsay I. (Ph.D., Physics)
Quantum gas engineering for atomic clocks
Thesis directed by Prof. Jun Ye

Optical atomic clocks with high precision allow minute frequency shifts to be measured
in short timescales. Low instability requires an atomic frequency reference with a high quality
factor that is prepared in the minimal amount of time. The advent of the Fermi-degenerate threedimensional (3D) optical lattice clock demonstrated that significant advances in precision metrology
can be made by incorporating ultracold quantum gases. This thesis focuses on improving the
precision of Fermi-degenerate clocks beyond their initial record demonstration by optimizing the
preparation, characterization, and scattering properties of degenerate Fermi gases (DFGs). We
first demonstrate that the SU(N)-symmetric nuclear spin degree of freedom in 87 Sr can be used
as a tool to enhance cooling to quantum degeneracy. This allows us to rapidly prepare a DFG,
evaporating to temperatures of 0.2 times the Fermi temperature in only 600 ms. With this anda new spin-polarizing method, we can prepare a spin-polarized degenerate gas in under 3 s. We
also systematically study the thermodynamics of the gas, measuring up to a 20% reduction of thecompressibility due to repulsive SU(N) interactions. We next explore how the natural lifetime ofan atomic state can be increased by embedding excitations within a degenerate Fermi gas. Witha light scattering experiment, we angularly resolve the suppression of the photon scattering rate
and measure up to a factor of two reduction near the absorption direction, where the momentum
transfer from the light is reduced. Our results agree with semiclassical calculations across a rangeof temperatures and Fermi energies. Finally, we discuss an experiment that attempts to directly
measure an increase in the natural lifetime of a narrow-linewidth transition. This Pauli blocking
mechanism provides a means to quantum engineer devices with decoherence rates below whatnature would provide alone.

Dedication

To all my mentors that have shown me how to think like a successful physicist and that havedemonstrated unbounded patience and kindness.

iv

Acknowledgements

I cannot begin to thank people enough for making me the scientist I am today. Graduate
school can be a rough time as you learn to be a competent scientist, and tough experiments
require a great team to not only look at problems from many angles, but to provide support andencouragement along the way. I am beyond lucky to get the opportunity to work with such an
excellent team, one that works hard, thinks critically, and takes considerable time out of their day
to help and provide feedback to others. There is a certain special bond you make with people
that you struggle very hard with on an important and fun problem. While you experience great
frustration together, you also experience childlike wonder, a truly special feeling that tends to put
all other things in life into perspective. For the science family I have developed here I am eternallygrateful.
I first want to thank my advisor, Jun Ye. Jun is a one-of-a-kind scientist, perhaps the most
optimistic and passionate scientist I have ever met. Jun has taught me to always be engaged
with what other scientists are telling you, and to ask questions about the details as they are whatreally matter. He has also taught me how special it is to create a sense of urgency in others throughenthusiasm. Getting others excited about what you are doing is certainly the best way to encourage
hard work. He has spent so much time with each of his students, which is truly outstanding given
how busy he is. I am incredibly grateful for how much feedback he has given me over the years,
and how much he advocates for his students.
I was lucky enough to work closely with Christian Sanner for the majority of the work reported
in this thesis. Christian is incredibly patient, kind, and knowledgeable about technical things such

v
as optics, electronics, and imaging, but also about a range of fundamental physics topics. I havelearned so much from him throughout my time here. I greatly admire his methodical attitude tosolving problems, his communication skills, and his creativity. I will never forget the roller coasterof discoveries we experienced while working on the Pauli blocking project, and how we supported
one another when the other was feeling pessimistic on progress (and, of course, when we wereboth down we could always count on Jun to lift up our spirits). I always looked forward to ourdaily ‘brainstorming’ sessions before going down to lab, something I hope to replicate in my future
projects as it was always a beneficial alignment for the day and a chance to think critically andbroadly about our project.
I worked with such a great number of other people. Ross Hutson always seems to know every
answer, and has spent a lot of time helping others with FPGA programming among other things.
He will succeed at whatever he decides to devote his time to. Aki Goban really helped me begin
to step into the arena. Will Milner is an incredibly hard worker and is constantly looking for new
things to do and problems to solve. With his dedication, he will continue to grow as a scientist
throughout the rest of his life. Lingfeng Yan has remarkable fortitude. Ed Marti spent many
helpful hours with me as a new student and always had a lot of insight into whatever problem I was
facing. I am very grateful for the time I spent with Eric Oelker, a kind, thoughtful scientist who
knows infinite things about feedback loops and electronics. I admire Sara Campbell’s attitude ofself-determinism, and her immense technical skills. Colin Kennedy is a quick thinker and physics
problem solver, as well as having his eye more broadly on interesting scientific problems. Toby
Bothwell is a very practical physicist who likes to work on problems that will immediately improvethe devices we work with. Alex Aeppli got up to speed very quickly on the Sr1 project and will
be a great person to take over Sr1 in the future. Sarah Bromley’s outcome-oriented attitude is aninspiration. Shimon Kolkowitz gave me some of the best advice on giving talks I have ever received
when I was a new PhD student. John Robinson is admirably technically competent. Maya Miklos
is a fast learner, asks on-point questions, and knows an incredible amount about AMO. Ming Tso
is a kind and a careful thinker. I’m positive Maya and him will do more than just squeeze their way

vi
towards success. Dhruv Kedar has incredible perseverance in addition to a wealth of knowledge
on technical details and AMO physics. Alex Staron is incredibly positive and kind, and is a do’er.
Craig Benko selflessly helped me at a hard stage of my graduate career. Wei Zhang is a very hard
worker and is willing to spend a lot of his free time helping others. Thank you to Jan who always
inspired an ego free curiosity in all aspects of science.
I am also grateful for Ana Maria Rey, who has always been very supportive of me and mydevelopment, and has provided a very fruitful experiment-theory collaboration. Thomas Bilitewski
has been a great theory partner and has put up with my many questions, as has Asier Orioli.
The machine and electronics shop members have been outstanding; I particularly admire Hans
Green’s optimism and happiness. Making others around you happy is one of the most significant
ways you can positively affect this planet, and Hans always made my day a little brighter afterI ran into him in the hallway. Thank you to all those in the computing department and the HR
department. Thank you to the ultrastable laser folks at PTB who we have a great collaboration
with and who always welcomed us warmly, including Thomas Legero, Dan Matei, Uwe Sterr, andFritz Riehle. Thank you to Michele Giunta for being a great collaborator. Thank you to all theother defense committee members. Thank you to the DoD and the NDSEG fellowship program forproviding my funding for my first few years. I have benefited from interaction with so many otherpeople, including Luigi DeMarco, Lee Liu, Kyle Matsuda, Will Tobias, Giacomo Valtolina, Junru
Li, Bryce Bjork, Tanya Roussy, Julia Cline, and many others. Thank you to my undergraduate
physics professors who really fostered my growth and interest in physics.
The lessons I learned in my climate activism work bled over into my PhD work, and I would
be remiss not to mention those people here. My climate work really taught me how to work withpeople, particularly those you disagree with. It taught me how to feel incredible fear (I’m leading
how many meetings with Members of Congress today??) without letting it affect my actions.
Kelsey Grant and Dan Palken made two phenomenal climate buddies. Ian Harrison is an incredible
mentor. He tells you your strengths and your weaknesses in a frank and yet incredibly supportive
way, and encourages you to stand up for yourself at all costs. Many other people contributed tovii
my growth, including David Kline, Grant Couch, Susan Secord, and many others.
Kelsey Grant is one of my most cherished friends, and her passion, standards, work ethic,
and values-oriented behavior inspire me daily. I can’t wait until she is President as the world will
be a better place. Steffi Couey is well-spoken, smart, thoughtful and uniquely empathetic. She
inspires me daily to be a better, more understanding person. These two ladies really hold me up at
every point in my life, and I hope to keep their friendships for the rest of my life. Numerous otherfriends have made my time here special - Christina Porter, Shaun Burd, Daniel Palken, Graham
Greve, Adrian Merritt, Cole Perkinson, Zach Ulibari, Max Urmey, Val Phillips, James Greenberg,Peter Johnsen, Rachel Steinberg, Haley Piske, Liz Strong.
My family has been very supportive of my dedication to physics, and for that I am incredibly
appreciative. Thank you to my dad, my mom, Dianne (my second mom), Amy, Aidan, and Ian.
Finally, thank you to Jeffrey Moore, who has supported me immensely throughout my PhD. Thankyou for respecting me as an independent individual, and for listening to my opinions and thoughts
and taking them seriously, even if we disagree. Your introspection, lifelong learner attitude anddesire to constantly improve inspire me daily.

viiiContents

Chapter
1 Introduction to atomic clocks
1.1

1.2

1.3

1

Noise sources in atomic clocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.1.1

Quantum projection noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.1.2

Dick effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6Next-generation systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.2.1

Ultrastable lasers with Si reference cavities . . . . . . . . . . . . . . . . . . .

8

1.2.2

Three-dimensional optical lattice clocks . . . . . . . . . . . . . . . . . . . . . 13

Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Degenerate Fermi gases

19

2.1

Colliding fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2

Noninteracting Fermi gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1

Statistics of trapped Fermi gases . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.2

Time of flight expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.3

Static structure factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.4

Pauli blocking of collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.5

Pauli blocking of light scattering . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.6

Density-density correlation function . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.7

Density fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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2.3

Interacting Fermi gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.1

Interaction Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3.2

Expansion dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3.3

Compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Making, measuring and probing Fermi-degenerate clocks
3.1

3.2

3.3

Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.1

Summary of experimental sequence . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1.2

Experimental apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1.3

Overview of cooling and trapping 87 Sr . . . . . . . . . . . . . . . . . . . . . . 50

3.1.4

Measuring trap frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.1.5

Transparency beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.1.6

TenS4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.1.7

Lattice loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Imaging techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2.1

Imaging expanded gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.2.2

In-situ imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.2.3

Imaging paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.2.4

Measuring camera parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.2.5

Measuring imaging parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Stability transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.3.1

Frequency comb technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.3.2

Stability transfer frequency chain . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.3.3

Differential noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.3.4

Optical path length stabilization . . . . . . . . . . . . . . . . . . . . . . . . . 91

4 Thermodynamics of a deeply degenerate SU(N )-symmetric Fermi gas
4.1

44

101

Cooling fermions to degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

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4.2

4.1.1

Common techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.1.2

Exploiting SU(N ) symmetry for cooling . . . . . . . . . . . . . . . . . . . . . 104

Rapid cooling of fermions to degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2.1

Transparency beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.2.2

The SU(N ) edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.3

Spin Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.4

Characterization of SU(N )-enhanced interactions . . . . . . . . . . . . . . . . . . . . 112

4.5

4.4.1

Isothermal compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.4.2

Expansion dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.4.3

Anisotropic density features . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.4.4

SU(N ) interacting model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5 Pauli blocking of atom-light scattering

126

5.1

Pauli blocking in a large Fermi sea . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.2

Light scattering experimental strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.3

Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.4

5.3.1

Temperature and Fermi energy dependence . . . . . . . . . . . . . . . . . . . 134

5.3.2

Pre-pulse application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.3.3

Spatially resolved scattering rate . . . . . . . . . . . . . . . . . . . . . . . . . 137

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6 Attempts to measure Pauli blocking of the natural lifetime using coherent control on a
narrow-linewidth transition

141

6.1

Pauli blocking with imbalanced Fermi gases . . . . . . . . . . . . . . . . . . . . . . . 142

6.2

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.3

Optical depth considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.4

Measurements with a cold sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

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6.5

Measurements in a 1D optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.6

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7 Conclusion

162

7.1

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

7.2

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Bibliography

167

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TablesTable
1.1

124 K Si cavity resonance frequencies, amplitudes, and linewidths . . . . . . . . . . . 11

xiii

FiguresFigure1.1(a) Schematic of a modern atomic clock. A local oscillator that has good short term
stability is referenced at long times to a narrow linewidth energy transition in an
atom. Here, the frequency reference is represented by neutral atoms trapped in a 1D
optical lattice clock, and the laser probes the 1 mHz linewidth 1 S0 − 3 P0 clock transition in 87 Sr. The number of optical cycles is then counted with traditional countersby converting the optical signal to a microwave one using a frequency comb. (b) Example of the fractional frequency stability of an atomic clock. The stability at shorttimes is determined by the local oscillator. At long times, the laser stability degrades
and is corrected by the atomic reference, which acts as a frequency discriminator thatprovides feedback to the local oscillator. . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

3(a) Contributions of the substrate, spacer, and dielectric SiO2 /Ta2 O5 coatings to
the thermal noise for different spacer and substrate materials. The thermal noise
is minimized when all-crystalline components are used. Here the frequency noise of
the Si/Si cavity is calculated at 124 K. (b) Coefficient of thermal expansion (CTE)for Si. The CTE crosses zero at 124 K and 17 K, and trends towards zero as T → 0.

1.3

9(a) Fractional frequency stability of the JILA ultrastable laser locked to a 124 K Si
resonator (b) Frequency noise of the JILA 124 K ultrastable laser. . . . . . . . . . . 10

xiv
1.4

(a) Dick effect (×10−17 ) at 1 s averaging time for different pulse times Tp and deadtimes Td , using Rabi spectroscopy. (b) Dick effect (×10−17 ) at 1 s averaging time
for different dark times Tdark and dead times Td , using Ramsey spectroscopy. . . . . 12

1.5

Schematic of a 3D optical lattice clock. Using retro-reflected laser beams along all
three directions, single atoms are trapped on individual lattice sites, guarding thesystem against particle interactions. The large arrows represent the propagation
direction, while the double arrows denote the polarization. The narrow line clock
laser is phase stabilized to the x̂ lattice beam. The oblique clock laser is used tomeasure sideband frequencies in along all three directions. . . . . . . . . . . . . . . . 14

1.6(a) Ramsey spectroscopy data, demonstrating a 6 s atom-light coherence time. (b)Fractional frequency stability of the frequency difference between two regions of the√
3D optical lattice, P1 and P2 . The blue solid line shows an instability of 3×10−17 / τ ,which when extracting the QPN of the total ensemble with twice as many atoms gives√
an instability of 1.5 × 10−17 / τ . When the number of atoms is reduced by a factor√of three in each region, the instability increases by 3, confirming the stability isgoverned by QPN. (inset) Spatially resolved image of the excitation fraction. The
frequency difference is extracted between two regions with excitation probability P1and P2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

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2.1

(a) In a T = 0 Fermi gas, all available harmonic oscillator states up to the Fermi
energy EF are filled with unity occupation. An excited atom (orange) acquires a recoil energy ER when returning to its internal ground state (blue). (B) In momentum
space, the atoms form a Fermi sea occupying all states up to the Fermi momentum~kF . Spontaneous decay of an excited atom with emission along α and total momentum transfer ~k can happen only if the final momentum state is not occupied
by another ground state fermion. If kF > 2kR , all final states are occupied and theatom cannot decay. The decay in both (a) and (b) is blocked since the final state is
occupied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2

Suppression of the scattering rate for a Fermi versus classical gas for various angles
α between the absorbed and emitted photons, and Fermi versus recoil momentum.
When k = 2kR sin(α/2) > 2kF , no suppression occurs. . . . . . . . . . . . . . . . . . 31

2.3

Pair distribution function versus scaled distance between the two particles. . . . . . 33

2.4

FQ , an important parameter for the collisional term in the Boltzmann equation . . . 41

3.1

Optical layout of the lower optical breadboard. Optical beams are brought to themain mezzanine through 45 degree mirrors. The 45 degree mirrors in the center ofthe diagram are directed onto the atoms in the center of the vacuum chamber on themain mezzanine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2

Optical layout of the middle optical breadboard. Atoms are trapped in the center ofthe vacuum chamber shown in the middle of the diagram. . . . . . . . . . . . . . . . 48

3.3

Optical layout of the top optical breadboard. . . . . . . . . . . . . . . . . . . . . . . 49

3.4

One side of the Al enclosure that surrounds the main experiment. . . . . . . . . . . . 50

3.5

Simplified energy level diagram of 87 Sr.

. . . . . . . . . . . . . . . . . . . . . . . . . 51

xvi
3.6

Left column: Parametric modulation of the ODT intensity leads to heating resonances at twice the trap frequency. The top figure shows the vertical trap frequency
and the bottom the horizontal frequencies. Right column: Trap frequencies are
extracted through dipole oscillations of the atomic cloud. . . . . . . . . . . . . . . . 54

3.7

Trap frequency measurements for various intensity setpoints extracted using bothparametric heating and dipole oscillations. The left shows the vertical trap frequency
and the right the horizontal trap frequencies. . . . . . . . . . . . . . . . . . . . . . . 55

3.8

Potential energy of the 1 S0 ground state atoms and 3 P1 excited state atoms in the
ODT. The transparency beam shifts the cooling light out of resonance for atoms inthe dimple. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.9

Calculated Stark shift of the excited 3 P1 stirring/trapping states for different nuclear
spin states. The linewidth of the 3 P1 state is Γ = 2π × 7.5 kHz. . . . . . . . . . . . . 56

3.10 Schematic of the reservoir and dimple dipole traps, as well as the propagation direction (single arrows) of transparency beam and the TenS4 laser, and their polarization(double arrows). The magnetic field while the transparency beam is on is denotedby Btsp and the magnetic field while the TenS4 laser is on is represented by BT enS4 .
The polarization of the TenS4 laser is along the magnetic field, so that the ellipticity
is zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.11 Atoms remaining in the xODT after applying a cleanup pulse to the xODT (leftimage) with the transparency beam on. Only atoms that don’t absorb the coolinglight remain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.12 (a) 3 P1 hyperfine structure and TenS4 detuning. The TenS4 detuning is referenced to
the zero magnetic field F=11/2 transition. (b) Clebsch-Gordan coefficients squared
for transitions from the 1 S0 ground state to the 3 P1 , F = 11/2 and F = 9/2 transitions. The F = 11/2 and F = 9/2 states have magnetic field sensitivities of 380 kHz
mF /G and 85 kHz mF /G, respectively.

. . . . . . . . . . . . . . . . . . . . . . . . . 60

xvii
3.13 (a) Height of the TenS4 potential versus detuning. At 266 MHz, the TenS4 potential
is zero for mF = +9/2. (b) Potential of the TenS4 beam as a function of position at
a detuning of 266 MHz. The beam is offset from the atoms by roughly 1/3 the trap
waist (dashed line) to create a varying potential across the cloud. Plots are shown
for a 5 G magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.14 Combined potential due to the ODT, the TenS4 beam, and gravity. Dashed curves
represent mF = −9/2 and mF = −7/2, which are removed through optical pumping. 62
3.15 Photoassociation spectra. The blue data points were taken with 45 µW of power,
and the red with 15 µW of power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.16 Optics setup for the TenS4 beam. Light from an ECDL is locked to an ultrastablelaser with a roughly 1 Hz linewidth at 689 nm (red master). A Fabry-Perot cavity
is used to filter the TenS4 light before being sent through a fiber to the experiment(to Sr2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.17 (a) Total atom number versus TenS4 application time. The majority of atoms are
removed after 100 µs. (b) Spin-resolved measurements of the mF = +/ − 7/2 andmF = +/−9/2 states. After 1 ms, single-particle scattering is visible. Measurementsin (a) and (b) were taken on different days with slightly different total atom numbers. 66
3.18 T /TF after switching on the TenS4 laser for variable amounts of time. After a longenough application time, the sample spin polarizes with minimal heating. . . . . . . 67
3.19 Fit of the atomic cloud to a Fermi-Dirac and a thermal distribution after the Fermi
gas is released from the crossed optical dipole trap. The Fermi-Dirac distribution
minimizes the residuals of the fit. The Fermi-Dirac fit extracts a T /TF = 0.08. Inset:
Absorption image of an expanded Fermi gas with a 10 ms time-of-flight. . . . . . . . 70

xviii3.20 Measured optical depth of an atomic cloud with densities greater than 1014 atoms/cm3 ,captured both in situ (top row) and after 500 µs time-of-flight (second row). Absorption imaging (left column) incorrectly extracts the optical depth due to collective
effects such as superradiance. The optical depth is conversely accurately extractedusing high-intensity fluorescence imaging (right column). The images are extractedusing a high resolution side imaging setup with an NA = 0.1 . . . . . . . . . . . . . 73
2
3.21 Variance of the difference image ∆Sdif
f versus the mean of the bright image S̄.

Fitting the data to a line allows one to extract the camera gain through the slope,and the read noise through the offset. Measurements are taken with both incoherent
and coherent light to confirm the absence of excess noise when using coherent light.

79

3.22 Time and frequency representation of a frequency comb. In the time domain, a seriesof pulses regularly spaced by time T are generated from the output of a mode-locked
laser. These pulses are the result of the coherent addition of many optical modesof a mode-locked laser. The carrier offset-frequency, defined as zero here, is due toa phase slip between the carrier and envelope each pulse. frep can be locked to a
microwave reference or a laser. The carrier-envelope offset f0 must also be stabilized.
Figure adapted from Ref. [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

xix
3.23 Outline of the Menlo frequency comb we use for stability transfer of the Si3 cavity
to other wavelengths relevant for Sr. Separate branches are used for each desired
frequency output. We have branches that stabilize f0 , our 813 nm lattice lasers, and acustom branch with minimal differential noise to transfer the stability of Si3 (the 124
K cavity) to 698 nm, the 1 S0 → 3 P0 clock transition. Our frequency comb is locked
to Si3, and the beat between the frequency comb and MJM (a pre-stabilized cavity
at 698 nm) provides the error signal to phase-lock MJM to the comb. Heterodyne
beats between Si3 and Si4 (the 4 K Si cavity), and between the frequency comb
and Si4, also occur inside the Menlo comb box. Acronyms are: erbium-doped fiber
amplifier (EDFA), highly nonlinear fiber (HNLF), periodically-poled lithium niobate(PPLN), bandpass filter (BPF), beat-detection unit (BDU), pm fiber splitter (FS),pm fiber splitter with fast axis blocked (FSB), fiber bragg grating (FBG), faraday
mirror (FM), acousto-optic modulator (AOM). All fibers and fiber components arepolarization-maintaining (PM). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.24 Stability transfer of the Si3 cavity to the 1D and 3D optical lattice clocks. A frequency comb is utilized to do the transfer. The comb is locked to Si3, and a prestabilized ULE cavity at 698 nm is locked to the comb. The stabilized light of the
ULE cavity seeds two injection locked lasers (not shown) before being distributed
to the Sr experiments and the frequency comb. The fiber links between the comb
and the Si3/ULE laser, and the Sr experiments and the distribution center, arephase-stabilized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.25 Electronics demonstrating how we lock MJM to the frequency comb. The MJM-comb
beat provides an error signal which we use to steer the demodulation frequency of
the MJM fiber noise servo. Both the fiber noise and the MJM-comb servo are closed
feedback loops. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

xx
3.26 (a) Picture of the frequency comb, including the outer Al box, the 2 inch breadboard,and the inner box that goes around a smaller Al breadboard that is water-cooled andclamped to the middle-sized breadboard. (b) Optical components of the frequency
comb, inside the inner box. Out-of-loop fiber components are housed in Al boxes
that are thermo-electrically stabilized to within 10 mK. . . . . . . . . . . . . . . . . 88
3.27 Differential noise between two identical frequency combs. Each comb is locked to acommon reference at 1542 nm, and a heterodyne beat is formed at 698 nm that isthen counted on a frequency counter. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.28 Instability of the out-of-loop fiber in the home-built BDU boxes. The measurements
include fiber noise from the phase-stabilized links.

. . . . . . . . . . . . . . . . . . . 92

3.29 PSD of the out-of-loop fiber noise measurements. When using the 0th order as thefiber noise reflection but the 1st order to perform spectroscopy, the differential noise
between the paths is above the Si3 noise floor. When using the 1st order to provide
the phase reference and perform spectroscopy, the noise is much smaller. . . . . . . . 93
3.30 Schematic of the updated fiber noise stabilization setup. Both a 0th order and a
1st order servo are utilized. The 0th order servo, which provides actuation on the
RF of AOM1, is continuously on, and provides phase-stabilization for the majorityof the fiber link. The 1st order servo removes any differential noise between the 0th
and 1st order paths. This servo is turned off (by shutting off AOM3) right before
spectroscopy, while the shutter is opened. . . . . . . . . . . . . . . . . . . . . . . . . 94
3.31 1st order fiber noise servo electronics, which reduces phase excursions when the servo
is re-engaged. While AOM3 is on, the 1st order diffracted beam provides a reference
for the fiber noise beat, which actuates on the RF to AOM3. AOM3 is shut off before
spectroscopy using a TTL. During this time, a synthetic beat signal is fed into thefiber noise input. To get the right phase and frequency input, a servo is used on thesynthetic beat signal before AOM3 is extinguished to properly reference it to fref . . 96

xxi
3.32 Phase transients of the 1st order fiber noise servo once AOM3 is powered and thefiber noise beat is connected to the servo. 3V corresponds to 90 degrees. Here, thepurple traces show the TTL that turns AOM3 on, and simultaneously switches theservo input to the fiber noise beat. The blue traces show the in-loop error of thefiber noise servo. The top and bottom traces show the phase deviation without the
feedback loop on the synthetic beat’s phase. The center trace shows the reduced
phase deviation when the loop is engaged. The magnitude of the phase transientwith the synthetic beat phase stabilization enabled corresponds to an AOM phase
chirp frequency shift of below 10−19 for a 450 ms Ramsey dark time and 15 ms π/2pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.33 Full electronics for the 1st order fiber noise servo, that reduces phase transients whenreengaging the servo and provides maximal frequency agility. The electronics feed ina synthetic beat signal with the right phase and frequency when AOM3 is shut off,right before spectroscopy. The electronics also include the ability to scan the beatfrequency over a wide range by feeding forward on fref and the beat signal. . . . . . 99
3.34 Fiber noise box. The first mirror of the box contains the 1st order FN reference, andthe last mount contains the lattice retro mirror (a dielectric that reflects 813 nm,
transmits 698 nm) that provides a common phase reference for the lattice light andthe clock light. ND filters can be inserted inside the box with a motorized actuator,
and an SRS shutter is also mounted to the top of the box. . . . . . . . . . . . . . . . 100
4.1

Unlike bosons which can occupy the same state, indistinguishable fermions must
separate into different energy levels. SU(N ) fermions, on the other hand, can have
N particles per state. In a given level, each particle has (N − 1) distinct partners,
as shown in the top right, and interactions are correspondingly enhanced. . . . . . . 105

xxii
4.2

a, Timing diagram showing stages of the experimental sequence. The times are
optimized for clock operation in the band insulating regime. Single spin samples
with 20,000 atoms at T /TF = 0.2 are prepared in a 3D optical lattice in under 3 s.
b, Atoms are confined in a crossed optical dipole trap consisting of a reservoir anddimple trap, and accumulate in a dimple after a “transparency” laser is applied. The
transparency and TenS4 lasers overlap with the dimple, and form a 10◦ angle withrespect to gravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.3a, Level diagram depicting the energy transitions for the transparency beam andTensor Stark Shift Spin Selector (TenS4). b, Time to reach T /TF = 0.12 for different numbers of nuclear spin states N participating in evaporation. Each trajectory is
prepared with 9.6(8) × 105 atoms per spin state and T /TF = 2.0(1). The gray band
denotes a 1/(N − 1) scaling based on the number of initial collisional partners, given
a ±20% change in the atom number. The error bars are upper bounds, reflecting theresolution of the sampling grid. c, Combined optical and gravitational potential after application of the TenS4 laser. The beam creates a spin-dependent modificationof the potential, and only atoms in spin state mF = +9/2 (purple line) are supported
against gravity. Atoms in mF = −9/2 and −7/2 are removed through optical pumping. d, Spin purity after application of the TenS4 laser for 10 ms. Atoms are loaded
into a 3D optical lattice and the spin state population is determined using selective
excitation on the ultranarrow 1 S0 – 3 P0 transition. Gray triangles correspond tospin states that are removed through optical pumping. The error bars denote thestandard error of the mean (s.e.m.) and are smaller than the marker size. . . . . . . 107

4.4

Three-body loss measurement, which is used to extract k3 . . . . . . . . . . . . . . . . 109

4.5

Evaporation trajectories for three different total times, 0.6 s (orange), 1.2 s (blue),and 2.4 s (green). The time constant of the ramps are adjusted to increase the total
evaporation length while maintaining the same final trap depth. The top plot shows
T /TF as a function of time, and the bottom T /TF versus atom number per spin state. 110

xxiii4.6

a, Density profile of a Fermi-degenerate gas after 10 ms time-of-flight with 10 spinstates and T /TF = 0.07. b, Residuals after fitting the cloud to a Maxwell-Boltzmann
distribution. Color bars in each subplot correspond to the number of atoms per
1.37 µm2 . c, Radially averaged fit of the density profile given in a to a MaxwellBoltzmann distribution and a Fermi-Dirac distribution. The thermal distribution
has larger residuals than the Fermi-Dirac fit. . . . . . . . . . . . . . . . . . . . . . . 111

4.7

Density fluctuations after 11.5 ms time-of-flight for a degenerate cloud with N = 10
nuclear spin states (red circles). The data is fit using an SU(N ) interacting model
to extract T /TF = 0.16 (red solid line), with shading representing a 2σ uncertainty of ±0.02 T /TF . Fitting the data instead to a non-interacting ideal Fermi
gas gives T /TF = 0.13 (blue dashed line), showing an interaction-induced suppression of ∼20%. The difference between the interacting and non-interacting fits ismuch less than the scatter in the data, highlighting the indistinguishability between
interacting and non-interacting systems by measuring density fluctuations alone at
the given signal-to-noise ratio. The total density fluctuations are 25% of that of thethermal gas. A thermal cloud (purple triangles) reproduces Poisson statistics with∆Ñ 2 /Ñ = 1 (purple dot-dashed line). Each data point is obtained by looking at theatom number variation and mean in a subregion of the cloud for a series of images.

114

xxiv
4.8

a, b, c, Line-of-sight integrated atomic density for 0.5 ms, 3.5 ms, and 8.5 ms time-offlight (ttof ) expansion times. The colorbar’s unit corresponds to the number of atoms
per 1.37 µm2 . d, Aspect ratio of a cloud of cold atoms with N = 10 nuclear spin states
released from an optical dipole trap for variable expansion times (red circles). After∼3 ms, the aspect ratio passes through unity (black dashed line), a clear signatureof interactions in the gas. At long times, the sample approaches an aspect ratio of
1.12 (grey line). The sample has an initial trap asymmetry of νz /νr = 6.4. In panels
d - f, the expected aspect ratio is shown for N = 10 (red dashed line), N = 7 (purple
dotted line), N = 4 (green dash-dotted line), and N = 1 (blue solid line) nuclear spinstates. e, Aspect ratio versus initial trap asymmetry of a degenerate gas for N = 10
(red circles) and N = 1 (blue squares) spin states after time-of-flight expansion for
15.5 ms. f, Aspect ratio versus temperature. Data is shown with roughly the sameatom number per shot. The data is fit using an interacting model that includes botha mean-field interaction and an additional collisional term. Neglecting the collisionalterm fails to explain the results (gray dashed line). All error bars represent the s.e.m.
and are smaller than the marker size. . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

xxv
4.9

a,b,c, Line-of-sight integrated atomic density n(x, z) after time-of-flight for a single
shot of the experimental data, non-interacting theory, and interacting theory, respectively. The colorbar’s units represent the number of atoms per 1.37 µm2 . Images
are shown with an initial trap asymmetry of νz /νr = 1.8 and T /TF = 0.17. d,e,f,
The anisotropy of the cloud, defined as n(x, z) − n(z, x), is shown in the second row,where lobes are clearly visible for clouds with interactions. To improve the signal-tonoise ratio the experimental image is first symmetrized by reflection along the x andz-axes. g,h,i, If the anisotropy is integrated along one direction, peaks symmetricto the center of the gas appear for the interacting distribution that are sensitive totemperature, while the non-interacting signal shows a different signature that displays only weak temperature dependence. Here, the red lines show the integrated
anisotropy of the images in (d-f), while the blue dashed lines in (h) and (i) show the
integrated profile for a 50% higher temperature.

. . . . . . . . . . . . . . . . . . . . 120

xxvi
5.1

Light scattering on atoms embedded inside a Fermi sea.
(A) Indistinguishable fermions obey the Pauli exclusion principle. If the thermalenergy kB T is sufficiently low they fill almost all available harmonic oscillator states
up to the Fermi energy EF with near-unity occupation. An excited atom (orange)acquires a recoil energy ER when returning to its internal ground state (blue). (B) In
momentum space, the atoms form a Fermi sea occupying most states up to the Fermi
momentum ~kF . Light scattering with re-emission along α and total momentum
transfer ~k can happen only if the final momentum state is not occupied by anotherground state fermion. A detector covering a solid angle dα registers the emitted
photon. (C) Pauli blocking leads to a characteristic angular distribution of scatteredphotons in the deeply degenerate regime (here T /TF = 0.1). For EF ∼ ER (leftsphere) scattering is preferentially suppressed in a small cone around the forward
direction, while EF >> ER (right sphere) causes strong suppression for all scattering
angles α. The suppression factor specifies the scattering rate relative to a non-Pauliblocked sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

xxvii
5.2

Experimental Setup.
(A) Off-resonant probe light excites 87 Sr atoms inside a Fermi sea. Spontaneously reemitted photons are collected simultaneously along two imaging axes under anglesof 24◦ and 72◦ , with their numerical apertures (N A) shown respectively. Small
scattering angles correspond to a small momentum transfer with k/kF < 1, whereas
the transversal observation detects photons from scattering events with k/kF > 1.
The circularly polarized probe beam has an intensity of 5Isat where the resonant
saturation intensity is Isat = 41 mW/cm2 . (B) On resonance the atomic cloud is
optically thick for the probe beam and the image formed on Camera 1 displays a holein the cloud center due to multiple scattering. At a detuning of ∆ = 40Γ the atom
cloud is optically thin and the corresponding image resembles the atomic density
distribution. The detuned frequency is used in the Pauli blocking experiment. A
magnetic bias field of 3 Gauss applied in the horizontal plane along the y-direction
defines the atomic quantization axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

xxviii
5.3

Suppression of light scattering in a 87 Sr Fermi gas over a range of temperatures and Fermi momentums.
All measurements are performed with a 10-component Fermi gas containing 18 000
atoms per spin state. The scattering behavior is distinctly different for the two observation angles of 24◦ (blue circles and squares) and 72◦ (red circles). Raw photon
counts are normalized with respect to measurements on non-Pauli-blocked reference
samples (see main text). Each circle data point is derived from 150 iterations of theexperiment, while each square point results from 50 experimental runs. Solid theory
curves are calculated with no free parameters. The widths of the theory lines reflect
the experimental uncertainties of Fermi energy and temperature. The error bars are
purely statistical and indicate one-standard-deviation confidence intervals. (A) At
a constant Fermi wavevector of kF /kR = 0.93 (EF /ER = 0.86), the atom ensemble’s
scattering cross section decreases as the gas approaches deep quantum degeneracy.
The suppression observed under 24◦ is pronounced and reaches 50% at T /TF = 0.13.
In contrast, under 72◦ , the suppression is negligible. (B) At constant T /TF = 0.13,
kF is varied by adiabatically changing the confinement. A larger kF results in astronger suppression. (C) The data reported in A and B are measured along 4
trajectories (dotted lines) through the parameter space spanned by k/kF and T /TF .
Depending on the scattering angle k varies between 0 and 2kR . Light collected under
an off-axis angle of 24◦ corresponds to a momentum transfer ~k < ~kF for the given
Fermi gas, leading to substantial reduction of the density of available final states.
On the contrary, for the 72◦ collection angle, the corresponding momentum transfer~k > ~kF . Thus most final states are not blocked and scattering is not suppressed.

135

xxix
5.4

Photon scattering from a Fermi gas after exposure to a pre-pulse of variable duration.
For the chosen pre-pulse detuning and intensity, the Pauli blockade is destroyed after exposure for a few µs. The scattering signal from the deeply degenerate sample(green squares) increases by almost a factor of 2 while the barely degenerate sample (purple squares) shows only minimal increase, as expected for a Fermi sea withkF /kR = 0.93 under an observation angle of 24◦ . Data in the plot is normalized
relative to the mean counts detected for 4 and 5 µs pre-pulse durations. . . . . . . . 137

5.5

Spatially resolved light scattering from a trapped Fermi gas at T/TF = 0.12.
Radially averaging the spatially resolved mean signal (inset) from 1100 individual
images obtained along the z-axis yields a radial light scattering profile (blue data
points). In-situ column density images, separately obtained using a high intensity
fluorescent imaging technique, are used to predict the scattering signal for a nondegenerate gas (purple curve). The spatial profile of light scattering calculated for
the T /TF = 0.12 ensemble (blue curve) agrees well with the measured data. . . . . . 139

6.1(a) Pauli blocking is maximized for atoms in the center of the Fermi sea. Atoms get a
directional momentum kick ~kabs from the absorbed light, and a diffusive momentum
kick ~kemi from the emitted light, giving a net momentum kick ~k in the center of
the Fermi sea. An atom in the center has the maximal suppression since it is least
likely to find an unoccupied final state. (b) Exciting atoms in the center of the
Fermi sea can be approximated using a three-level scheme with imbalanced Fermi
gases. Two ground states with an imbalanced atom number are created and theminor Fermi gas is excited so that it predominantly decays into the major Fermi gas.
Under the optimal conditions this decay channel is blocked. . . . . . . . . . . . . . . 143

xxx
6.2

3 P level scheme. Imbalanced Fermi gases with a 10:1 atom number ratio are created1

using two nuclear spin states of the ground 1 S0 state. Atoms in mF = 9/2 are thenexcited to the 3 P1 , F = 11/2, mF = 9/2 excited state using a π pulse with π polarized
light. The excited atoms are Pauli blocked from decaying to the mF = 7/2 groundstate. The numbers in boxes refer to the Clebsch-Gordan coefficients squared. . . . . 146
6.3

Rabi scan as the frequency of the excitation laser is changed. Zero corresponds tothe mF = 9/2 to 3 P1 , F = 11/2, mF = 9/2 transition. Given the magnetic field of 1
G, the mF = 11/2 transition is on resonance at -.5 MHz. . . . . . . . . . . . . . . . . 147

6.4

Prediction of Pauli blocking suppression for a three-level scheme where the atom
number ratio between the two ground state Fermi gases is 10:1 and the ClebschGordan coefficients for the decay are 0.82:0.18. Suppression is shown for various
temperatures T/TF and Fermi energies EF /ER , quoted for the majority component. 148

6.5

Total suppression for a three-level scheme where the Clebsch-Gordan coefficientsfor the decay are 0.82:0.18, and T /TF = 0.1 with variable EF /ER for the major
component. Here the suppression is shown for different atom number imbalances
for the major component and minor component (EF is quoted for the majority
component in the legend). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.6(a) Strategy to measure the excited state lifetime and the branching ratio from the
mF = 9/2 excited state. Atoms are excited on the narrow linewidth 3 P1 transition
using a π pulse and the excited atoms are read out using the broad 1 P1 transition.
The upper and lower right panels describe how we measure the lifetime of excitedatoms and the branching ratio of excited atoms back to mF = 9/2, respectively, andare discussed in the main text. (b) Experimental setup. Excitation light probes
a Fermi gas on-resonance with the 1 S0 , mF = 9/2 to 3 P1 , F = 11/2, mF = 9/2
transition. The ground state population is read out by taking a fluorescence image
using the 1 P1 state, where the fluorescence is measured using a high NA imaging
system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

xxxi
6.7

Fraction of atoms that return to the mF = 9/2 ground state after the mF = 9/2 atoms
are excited to the 3 P1 , F = 11/2, mF = 9/2 state. Measurements are performed witha thermal sample where T /TF = 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.8

Rabi flopping on 1 S0 , mF = 9/2 → 3 P1 , mF = 9/2 transition, measured on a thermal
Fermi gas. The red line represents a fit using no free parameters and assuming the21.28 µs natural lifetime for the excited state. . . . . . . . . . . . . . . . . . . . . . . 153

6.9

Lifetime versus mF = 7/2 OD. Measurements are performed with a thermal sample
where T /TF = 0.75 at a 7/2 OD of 2.7. The 9/2 OD is fixed at 0.7 for all measurements.154

6.10 Measurements of the excited state lifetime for variable temperatures (here quoted
for the majority component) while the OD is kept relatively constant. The 7/2 OD
varies from 2.0 to 1.5 as T /TF is increased. Here the 9/2 OD is 0.7, and T /TF isquoted for the 7/2 component. Measurements of the lifetime performed for a thermal(T /TF = 0.75) gas at equivalent ODs yield shorter lifetimes. . . . . . . . . . . . . . . 155
6.11 Full lifetime curve for a Fermi gas with a 7/2 OD of 2, a 9/2 OD of 0.7, andT /TF = 0.2 for the major component. The blue line is an exponential fit to thedata, which gives a 1/e lifetime of 25.7 ± 0.3 µs. Data points reflect the mean, anderror bars are 1 standard deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.12 Fraction of atoms returning to mF = 9/2 while the 7/2 and 9/2 optical depth are kept
fixed. The temperature is quoted for the 7/2 component. As the sample is heated,
the ratio decreases towards the non-Pauli blocked ratio. Here T /TF is quoted for the
major component. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.13 (a) Experimental setup to measure the lifetime in a 1D optical lattice. The opticallattice is formed along gravity, g, and the excitation probe polarization (ê is along
the magnetic field, which points into the page. (b) Lifetime of excited atoms in
the 1D optical lattice versus EF /ER . The 9/2 OD is 0.8, and the temperature is
T /TF = 0.6 in the XODT when the 7/2 OD is 5.5. . . . . . . . . . . . . . . . . . . . 158

xxxii6.14 (a) Lifetime as the excitation pulse area is varied for a fixed 7/2 OD of 5 and a fixed
9/2 OD of 0.8 (b) Lifetime as the excitation pulse area is varied for a 7/2 OD of 3
and a 9/2 OD of 0.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Chapter 1

Introduction to atomic clocks

Time is a physical quantity that has had practical relevance for the majority of human history.
In our modern capitalist society, we often view time in terms of its monetary value, realizing thattime, like energy, is a vital yet finite currency that is necessary to get anything done. Early
agrarian cultures associated time with sustenance — Egyptians observed the annual flooding of the
Nile River, and the Mayans were cognizant of the reproducibility of the optimal harvest time.
Time is unidirectional, irreversibly succeeding from past through present to the future. Aninterval separating two points on this timeline can be measured by a device that is cyclical innature. Time is measured by counting the number of cycles that have passed during an interval.
Numerous phenomena are cyclical and could serve as a standard for time. One example is Earth’s
revolution around the sun, which defines a year. The sun is a remarkably good timekeeping device
due to its accessibility and reasonable stability (if the sun is high in the sky, it’s around noon), but
it has limitations. First, its definition of time is not stable as its location in the sky changes with theseasons, and Earth’s rotation is gradually decreasing over time. Additionally, it is challenging toaccurately measure time intervals below tens of minutes. Another example is a grandfather clock.
Each half cycle of a grandfather clock takes a second, which allows time to be defined well overthe course of seconds but poorly for much less than a second. Over many days, the error per cycle
accumulates, meaning it does not work well over long timescales. Additionally, since the oscillation
frequency depends on the length of the pendulum, it is difficult to create two grandfather clocks
with the same definition of time.

2

To create an optimal definition of time, we need a system that works over many timescales
and is accurate, stable, and consistent. High accuracy implies the time can be told with few
systematic errors, while low instability means the answer doesn’t change significantly with eachmeasurement. A consistent definition implies that we can create two clocks at two different places,such as a clock in Paris and a clock in Boulder, and be able to arrive at exactly the same definitionof time. Furthermore, how can we maximize the societal use of measuring this quantity?
To satisfy all these requirements, a clock consists of multiple components. A local oscillator(LO) keeps track of time over short timescales, and a frequency reference creates an absolutedefinition of time while also providing a stable standard over long timescales. Finally, clocks requirea device that counts the number of oscillations of the clock. For example, since a grandfather clock
works well over the course of seconds, it makes a decent LO. However, because of its accumulated
error per cycle and the variance in the oscillation frequency with each clock, it makes a lousy
frequency reference. The sun, on the other hand, is a better frequency standard, since it works wellover the course of days and two people in the same neighborhood observe the same sky so that acommon definition of time can be implemented. The sun and the grandfather clock can work intandem, with the sun periodically updating the local oscillator to create a time standard that works
over many timescales. This system works well enough for those trying to be home by dinnertime,
but needs improvements for people who want modern GPS accuracy or precise navigation in deep
space 1 .
The SI unit of time, the second, is currently defined as 9 192 631 770 periods between two
hyperfine transitions in the unperturbed ground state of a 133 Cs atom [2]. The hyperfine transition
is probed with a microwave oscillator, which acts as the LO. Microwave electromagnetic radiation
can be thought of as a grandfather clock except with a 1010 times higher oscillation frequency. This
allows the Cs clock to achieve a much higher quality factor Q = ν/∆ν than a grandfather clock.
Energy transitions in atoms provide a stable frequency reference and a universal definition of time,1

See for example NASA’s deep space atomic clock.

3
(a)
3

P0

~1 mHzlinewidth
1

S0

(b)

(Frequency
standard)(Local oscillator)(Counter)Figure 1.1: (a) Schematic of a modern atomic clock. A local oscillator that has good short term
stability is referenced at long times to a narrow linewidth energy transition in an atom. Here, thefrequency reference is represented by neutral atoms trapped in a 1D optical lattice clock, and thelaser probes the 1 mHz linewidth 1 S0 − 3 P0 clock transition in 87 Sr. The number of optical cycles isthen counted with traditional counters by converting the optical signal to a microwave one using afrequency comb. (b) Example of the fractional frequency stability of an atomic clock. The stability
at short times is determined by the local oscillator. At long times, the laser stability degrades andis corrected by the atomic reference, which acts as a frequency discriminator that provides feedback
to the local oscillator.

since a transition frequency in a given element is identical for all atoms of the same isotope, as longas they are in the same environment.
Optical clocks capitalize on this idea and probe electronic transitions that operate at opticalfrequencies 4 orders of magnitude higher than a hyperfine transition. Fig. 1.1 outlines a modern
atomic clock. The LO is a laser, where an oscillation cycle is defined by the laser wavelength andthe speed of light. With the advent of ultrastable laser technology, the oscillation frequency can be
defined with sub-Hz resolution, making modern ultrastable lasers incredibly pristine oscillators. At
long times, the laser frequency drifts owing to environmental perturbations so that a long-term frequency standard is required. The LO excites a narrow linewidth transition in an atom, which whenprobing the slope of the atomic lineshape (red dots in Fig. 1.1) acts as a frequency discriminator

4that corrects the laser frequency and keeps the laser on resonance with the atomic transition. Finally, to count the rapid optical cycles of the clock, a frequency comb is utilized. A frequency comb
allows a low noise conversion of optical frequencies into microwave, where traditional electronic
counters can count the number of cycles.
As mentioned previously, accuracy and stability are two important variables for atomic clocks.
Accuracy is how far away the measured clock transition frequency is from the true unperturbed
value. A low uncertainty implies that the offset is very well known. In contrast, precision andstability define the noise of the measurement. A worse stability means that the standard deviationof frequency measurements is larger. It takes less time to average away the noise if the clock has ahigher stability, allowing sub-Hz frequency shifts to be measured faster.
Both ion clocks and neutral atom optical clocks have surpassed Cs. Compared to ion clocks
that generally trap single ions, neutral atom optical clocks interrogate thousands of atoms simultaneously. This grants a lower instability, and neutral atom optical clocks have demonstrated low
10−17 instability after averaging for 1 s [3, 4]. It currently takes Cs standards 106 s to achieve a
10−17 level of instability [5]. Optical clocks have demonstrated systematic uncertainty over twoorders of magnitude lower than a Cs clock, with accuracies in the low 10−18 [6, 7, 8, 9]. To achievea low uncertainty, potential systematics that can shift the clock transition frequency have to becontrolled and understood. Nuclear clocks are an exciting new research direction since the nucleus
is expected to be less sensitive to external perturbations such as electromagnetic fields [10]. A
prominent example is the 229 Th clock, which features a long-lived few eV nuclear transition. Given
the ability to connect ultrastable lasers at telecom wavelengths to the ∼150 nm nuclear transition frequency using an XUV frequency comb [11], this avenue of research is bound to have many
exciting discoveries over the upcoming decades.
The exceptional accuracy and stability of optical atomic clocks drives the consideration ofa redefinition of the SI second [12]. Moreover, besides timekeeping there are many exciting applications of atomic clocks. Because clocks are relatively insensitive to their external environments,
they can be also used to detect small frequency shifts of the clock transition that are normally5invisible to other experiments. As a result, precision metrology affects a range of applications inquantum sensing and fundamental science. Atomic clocks can enable relativistic geodesy [13, 14],which motivates physicists to develop transportable clocks that can be used outside the laboratory[15]. In addition, clocks can be used to detect possible drifts in fundamental constants [16, 17, 18],
to test fundamental symmetries [19, 20], to look for the existence of ultralight dark matter candidates [21, 22, 23], and potentially to measure gravitational waves [24]. Optical clocks are also
ideal many-body platforms to study light-matter and matter-matter interactions and to further our
understanding of quantum science [25, 26, 27, 28]. Such fundamental physics applications are connected to the rapidly advancing frontier of quantum technology, and they also provide a necessarymeans to further improve atomic clocks. Physics that is uncovered at a given level of performancemust be accounted for or circumvented for further improvement. This relation between time andphysics can never fully be disentangled and is one of the drivers of our discoveries and passions.
In Section 1.1 we detail the various noise sources in atomic clocks that can lead to an increase
in instability. In Section 1.2 we then outline next generation atomic clocks that attempt to solve
the instability problems that have plagued past systems.

1.1

Noise sources in atomic clocks
Precision is essential to measuring small frequency shifts in short timescales. Our state-of-

the-art clock uses an ultrastable laser referenced to a mHz-wide optical transition in trapped 87 Sr
atoms. Sources of instability are thus rooted in both the local oscillator and the matter-basedfrequency reference. Here we discuss limitations to the intrinsic stability of the frequency reference,quantum projection noise, and noise due to the local oscillator, called the Dick Effect.

1.1.1

Quantum projection noise

The intrinsic instability of an atomic clock is determined by quantum projection noise (QPN).
When an atom is in a superposition of ground and excited states |ψi = a |gi + b |ei, measuring an
atom in the ground or excited state is not deterministic but instead follows a binomial distribution,

6
where the probability finding the atom in |gi (|ei) is given by pa = |a|2 (pb = |b|2 ), assuming|a|2 + |b|2 = 1. This noise that is inherent to quantum mechanics is called quantum projection
noise, since atoms are projected into a given state upon measurement [29]. If the atoms areuncorrelated, they can be regarded as independent so that the excited state variance is
σe2 = N pb (1 − pb ) ,(1.1)

where N is the total number of atoms. Converting this to an excitation fraction Pee = Ne /N ,where Ne is the number of excited state atoms, gives us σP2 ee = pb (1 − pb )/N . Generally atomic
clocks operate with pb = 0.5, where the slope between the frequency and the excitation fraction is
maximized. The excitation fraction noise can then be converted to fractional frequency noise δν/νfrom the analytic shapes of the Rabi and Ramsey spectroscopy lineshapes. This givesr

Td + TpNτr
Td + Tp
1
σyRamsey =,2πTdark νNτ
0.264
σyRabi =
Tp ν

(1.2)
(1.3)

where Tp is the pulse time, Tdark is the Ramsey dark time, τ is the averaging time, and Td is the deadtime, or preparation time, of the atomic state. The dead time is typically not zero, since atomic
clocks so far operate in a destructive fashion - the atomic sample is destroyed upon measurement
so that the atoms have to be continually re-prepared, which takes time Td . More details on this in
the upcoming subsection.
In order to achieve a better intrinsic stability, it thus helps to 1.) operate a clock with more
atoms N , 2.) maximize the coherence time Tp (or Tdark ), and 3.) reduce the preparation time
Td . With 1000 atoms, Tp = 5 s and Td = 1 s, our QPN-limited stability for Rabi spectroscopy is√
10−17 / τ , and sub-10−18 frequency shifts can be resolved in 100 s.

1.1.2

Dick effectComparisons between independent clocks generally have an instability above the QPN limit
due to noise from the local oscillator called the Dick Effect [30]. The clock transition typically7cannot be measured continuously, since most techniques that probe the excited state fraction are
destructive and heat atoms out of the trap. As a result, an atomic clock operates in cycles ofpreparing the atomic sample, probing the clock transition, and reading out the quantum state.
During state preparation and detection, also called the dead time, the laser frequency is not measured and thus acts like a flywheel. This periodic sample of the laser phase leads to aliasing of highfrequency laser noise down to lower frequencies, and causes a random false frequency correction of
the laser phase that can degrade the performance of a clock. The Dick effect stability is given by[30]
1σy2 (τ ) =τ∞X|G(n/Tc )|2|G(0)|2

m=1

Sy (n/Tc ) ,(1.4)

where Tc is the cycle time, Sy (f ) is the power spectral density of fractional frequency fluctuationsof the laser at frequency f , and G(f ) is the Fourier transform of the response function, g(t), definedby1δPee =
2

Z Tp
g(t)δω(t)dt .

(1.5)

0

That is, the sensitivity function defines how changes in the laser frequency δω(t) over the coherent
interrogation pulse changes the excitation fraction Pee . To reduce the aliased signal, it is advantageous to have a high duty cycle, where the majority of time is spent coherently probing the atomic
sample. Lower laser noise will also reduce the Dick effect.
Typically, independent clock stability of neutral atom clocks is limited by the Dick effect[6, 9, 31, 4]. However with the advent of ultrastable lasers based on crystalline materials, which are
discussed in the next section, the Dick effect can be on par with state-of-the-art QPN stability [3].
In addition, interleaved interrogation of two atomic clocks in a zero-dead-time configuration candrastically reduce the Dick effect [4]. Synchronous measurements, performed between two atomic
regions of a single clock [32] or between two separate systems [3, 33], can cancel the Dick effect andare useful if one is not using a clock as a time standard but instead to measure frequency shifts
brought on by physical phenomena, such as the gravitational redshift.

8

1.2

Next-generation systems
Here we discuss next-generation systems whose goal is to increase the precision of atomic

clocks. In subsection 1.2.1 we introduce a local oscillator that is composed of a laser locked to a
Si reference cavity. The high quality factor and low temperatures afforded by this system permit
a low thermal noise, and allow demonstration of record-breaking laser stability [34]. Next in
subsection 1.2.2 we give an overview of a 3D optical lattice clock. Confinement of atoms on
individual 3D lattice sites guards against interactions, and allows large numbers of atoms to beprobed coherently for over 10 s, reducing QPN.

1.2.1

Ultrastable lasers with Si reference cavities

A state-of-the-art optical oscillator is composed of a laser that is locked to a stable passive
resonator. Fundamentally, the frequency stability of the oscillator is limited by the length stabilityof the resonator. Barring technical noise, the length stability is given by the composite thermal
noise of the substrates, spacer, and coatings that make up the resonator. For the majority ofstate-of-the-art cavities, the noise of the substrates and coatings are much larger than that of theTspacer, so that we can write that the fractional frequency stability as σT2 N ∝ QEL
2 , where T is thetemperature, Q is the mechanical quality factor, E is the Young’s modulus, and L is the cavity
length. While the total thermal noise of the spacer depends on the length of the resonator, thethermal noise of the substrates and coatings do not. This is why the fractional length stabilityof the substrates and coatings is ∝ L−2 , and the fractional length noise can thus be reduced byincreasing the length of the resonator. However, the resulting increase in the vibration noise on thecavity and the temperature non-uniformity of the spacer create technical noise that is challenging
to mitigate. These technical noise sources were overcome in a cavity that was 40 cm long in ourgroup in 2013 [35, 36]. Thermal-noise-limited instability has been demonstrated in cavities aslong as 48 cm [37]. However, increasing L significantly beyond this length remains difficult. A
different strategy is thus warranted for significant advances in ultrastable laser technology. Up

9
(a)

(b)

Figure 1.2: (a) Contributions of the substrate, spacer, and dielectric SiO2 /Ta2 O5 coatings to thethermal noise for different spacer and substrate materials. The thermal noise is minimized when
all-crystalline components are used. Here the frequency noise of the Si/Si cavity is calculated at
124 K. (b) Coefficient of thermal expansion (CTE) for Si. The CTE crosses zero at 124 K and 17
K, and trends towards zero as T → 0.

until 2012, ultra-low expansion (ULE) glass was predominantly used as a spacer material in stateof-the-art cavities due to its low coefficient of thermal expansion (CTE) at room temperature. A
significant advance was made when crystalline materials with more optimal material properties wereconsidered. Crystalline materials such as Si have many orders of magnitude higher Q comparedwith conventional glass materials, and a large Young’s modulus [38]. A crystalline cavity can thus
demonstrate significantly lower thermal noise than one composed of amorphous materials, as shownin Fig. 1.2 (a). The stiffness of Si provides the additional benefit that the cavity is more robust
against vibrations [38].
Thermal noise is also reduced when the cavity is operated at a lower temperature. However,technical limitations prevent ULE and other materials from working at cryogenic temperatures.
Critical to demonstrating a thermal-noise limited cavity is the reduction of technical noise sourcesthat can change the cavity length. Temperature fluctuations and vibrations constitute two major
technical contributions that must be mitigated. ULE and other materials have an appreciable
coefficient of thermal expansion (CTE) at cryogenic temperatures (the CTE instead passes zero

10
(a)

(b)

Figure 1.3: (a) Fractional frequency stability of the JILA ultrastable laser locked to a 124 K Si
resonator (b) Frequency noise of the JILA 124 K ultrastable laser.

around room temperature). As such, operating these cavities at low T can enhance the coupling
between thermal fluctuations and the cavity length. In contrast, as shown in Fig. 1.2 (b), Si has
a CTE zero crossing at 124 K and 17 K. In addition, as is true of all materials, the CTE trends
towards zero as T approaches zero. Si cavities can thus operate at a lower T with a reduced couplingof thermal fluctuations to the cavity length.
Thus far, lasers locked to Si resonators at 124 K and 4 K have been developed at JILA andPhysikalisch-Technische Bundesanstalt (PTB). Fig. 1.3 (a) shows the fractional frequency stabilityof a laser locked to a 124 K Si cavity, extracted using a three-cornered hat comparison [39] with a
4 K Si cavity and our second clock laser based on a 40 cm ULE cavity. The laser with a 124 K Si
reference cavity demonstrated a thermal noise limited instability of 4 × 10−17 , corresponding to acoherence time up to 55 s at 1550 nm [34]. This was roughly a factor of two improvement in thethermal noise over our previous clock laser [35].
The JILA setup has an improved long-term stability thanks to a few upgrades. First, theintensity of the laser light is stabilized in the transmission port of the cavity, minimizing the cavity’s
thermal fluctuations due to changes in the intracavity power. Second, superpolished optics andbetter thermal control of the environment were added, both which reduce slowly drifting parasitic

Index
1
2
3
4
5
6
7

fi (Hz)
5.7
12.7
20.0
30.0
40.0
50.0
55.0

ai (1/Hz3 )7.0e-34
1.5e-34
4.0e-34
5.0e-34
5.0e-34
1.0e-34
4.0e-34

Γi (Hz)
1.0
1.5
0.1
0.1
0.1
4
1.2

11

Table 1.1: 124 K Si cavity resonance frequencies, amplitudes, and linewidths

etalons. These improvements lead to a thermal-noise-limited stability from 0.2 − 104 s and a drift
rate of the Si cavity below 60 µHz/s [40], a more than two orders of magnitude improvement in
the drift rate over our previous clock laser. An interesting question is if the drift rate of this cavity
decreases over time. We are continuining to monitor the drift rate of this cavity, and as of thewriting of this thesis, the lowest drift rate we have seen over the course of days is 30 µHz/s. Thisultralow drift makes Si cavities useful as stand-alone oscillators [40].
Fig. 1.3 (b) shows the frequency noise of the 124 K Si cavity, extracted using a crosscorrelation measurement [41] between a 124 K Si cavity, a 4 K Si cavity, and our old 40 cm
ULE clock laser [3]. From this measurement, we can extract a frequency noise model of the Si
laser, given by
N

Slaser (f ) =X
h−1
ai f 2+ h0 + h2 f 2 +.
f
1 + ( f −fi )2
i=1

(1.6)

Γi /2

This model includes a white frequency noise term (h0 = 4 × 10−34 Hz−1 ), a flicker frequency noise
term (h−1 = 1.5×10−33 ), a white phase noise term (h2 = 3×10−36 Hz−3 ), and a series of resonancesthat arise from for e.g. vibrations that are listed in Table 1.1.
Using the Si noise model quoted above, we can predict what the Dick effect-limited instabilityof the clock will be for different dead times and pulse times as shown in Fig. 1.4. The Dick effectstability quoted in this model was confirmed by transferring the stability of the 124 K Si cavity to
the 698 nm clock transition using a frequency comb (see Chapter 3), and subsequently performing

12
(a)

(b)

Figure 1.4: (a) Dick effect (×10−17 ) at 1 s averaging time for different pulse times Tp and deadtimes Td , using Rabi spectroscopy. (b) Dick effect (×10−17 ) at 1 s averaging time for different darktimes Tdark and dead times Td , using Ramsey spectroscopy.

intercomparisons between two Sr atomic clocks [3] 2 . Prior to 2019, the Dick effect limited clock√stability was around 10−16 / τ [31, 4, 42, 9]. With the implementation of the 124 K Si cavity, an√
independent clock stability of 4.8 × 10−17 / τ was reached with a 570 ms pulse time and a 600 ms
dead time [3]. Further improving the stability is possible with improvements in the coherence timeand additional reduction of the dead time.
At 4 K, vibrations from the closed-cycle cryostat limited previous results [43]. With theaddition of a custom-designed cryostat with many layers of thermal isolation and minimal coupling
to the cooling mechanism [44], vibrations and thermal fluctuations were suppressed below thethermal noise floor to demonstrate a 4 K cavity with thermal-noise-limited instability of 6.5 × 10−17
and a drift rate as low as 30 µHz/s, the latter of which depends on the intracavity power [45].
Decreasing the Dick noise contribution will of course benefit from an improvement in the laser
noise. As shown in Fig. 1.2 (a), the predominant noise contribution in a Si cavity is the mirror
coating noise. Crystalline semiconductor coatings made from AlGaAs/GaAs epitaxial layers havedemonstrated an order of magnitude reduction in Brownian noise at room temperature compared
to commonly used dielectric mirror coatings [46]. These coatings also have high optical quality in
2

More details on these measurements will be contained in upcoming theses.

13

the telecom band, producing a finesse up to 300,000 at 1550 nm with scattering and absorption
losses of a few ppm [47]. Implementation of these coatings in a 4 K Si cavity could result in athermal-noise floor in the lower half of the 10−18 range, roughly an order of magnitude better thanthe current state-of-the-art [34].

1.2.2

Three-dimensional optical lattice clocks

Neutral atom clocks probe thousands of atoms simultaneously, resulting in a favorable QPN.
Typically optical lattice clocks consist of thousands of neutral atoms that are trapped, cooled toµK temperatures, and loaded into a 1D optical lattice so that atoms are confined to individual
pancakes. The optical lattice reduces the effect of atomic motion on the transition probabilityof the atom. In particular, if the probe laser is along the direction of the optical lattice in theLamb-Dicke regime, the recoil caused by the probe light is absorbed by the lattice so that theatom’s motional state is unaffected [48]. Furthermore, the wavelength of the lattice laser can be set
such that the polarizability of the ground state equals the polarizability of the excited clock state,creating equivalent trapping conditions for the two states. 1D optical lattice clocks (OLCs) havereached record accuracies of low 10−18 [6, 9], but since they only have strong confinement along
one direction, they suffer from collisions between atoms that dephase and shift the clock frequency.
These atomic interactions place a limit on the number of atoms per lattice pancake and maximum
coherence time, and thus limit the achievable QPN.
One possible solution to this problem is to create a 1D optical lattice with a large beam
waist [4], which increases the distance between atoms and thus reduces the interaction strength
throughout the interrogation pulse. This solution is accompanied by various technical challenges
that scale with the size of the atomic sample. For example, correctly modeling the temperatureenvironment over the extended sample will be critical to correctly account for blackbody radiation,a leading systematic for many optical lattice clocks [6, 9] and ion clocks [8, 7].
An alternative scalable solution is to create a 3D optical lattice clock [49, 50], which is the

14

Figure 1.5: Schematic of a 3D optical lattice clock. Using retro-reflected laser beams along all three
directions, single atoms are trapped on individual lattice sites, guarding the system against particle
interactions. The large arrows represent the propagation direction, while the double arrows denotethe polarization. The narrow line clock laser is phase stabilized to the x̂ lattice beam. The oblique
clock laser is used to measure sideband frequencies in along all three directions.

subject of this thesis. A spin-polarized, unity filled 3D lattice clock contains one atom per lattice
site, as shown in Fig. 1.5, which removes contact interactions and thus reduces density shifts.
3D optical lattice clocks can thus support more atoms per unit volume than a traditional 1D
optical lattice clock. Additionally, residual atoms in other spin states that create doubly occupied
lattice sites have an interaction energy that is much larger than in a 1D lattice clock due to thesignificant spatial overlap between the wavefunction of the two atoms. These kHz interaction
shifts are spectroscopically resolvable from singly-occupied sites given our sub-Hz resolution of theclock transition. The clock laser can consequently singularly address solely-occupied sites, guarding
against contact interactions that affect the clock transition frequency. Interactions present in a 1D

15

lattice clock can also dephase the atoms during the clock interrogation, reducing the atom-atomcoherence time and thus the maximum allowable probe duration during spectroscopy. As discussed
earlier, increasing the probe duration reduces not just QPN but also the Dick Effect, and is thus
beneficial to obtain better clock stability.
To create a near-unity filled 3D optical lattice, it is necessary to load a Fermi-degenerategas. Fermi-degenerate gases have entropy on the order of a few kB per particle, where kB is theBoltzmann constant. Adiabatically loading this low-entropy gas into the optical lattice allows only
the lowest energy states to be occupied so that atoms are loaded into the ground band of the opticallattice with a minimal number of holes. 3D optical lattice clocks consequently require extra stagesof cooling to prepare a near-T = 0 gas, as is discussed more in Chapter 3. We have demonstrated
temperatures as low as 0.07 times the Fermi temperature [51], corresponding to only a few tens ofparts per billion above absolute zero.
In the first implementation of this new technology, we demonstrated a record QPN of 1.5 ×√
10−17 / τ [32, 49]. This was achieved using 6000 atoms with a coherent interrogation time of 4s and a dead time around 16 s. We also observed atom-light coherence times up to 6 s, limitedby the coherence time of the clock laser [35]. A long-standing question was whether a 3D opticallattice could support the small AC Stark shifts required for highly accurate optical clocks. To
avoid interference effects, each lattice beam needs either orthogonal polarization or a different
lattice frequency. Despite these challenges, we demonstrated state-independent trapping based ona proposal in Ref. [52] and measured the magic frequencies of each lattice beam to around 10 ppb[49]. Higher order Stark shifts such as hyperpolarizability, magnetic dipole, and electric quadrupole
interactions [53], will be a fruitful subject for future study.
To extract the QPN and cancel the Dick effect, we take advantage of a high-resolution
imaging system with an NA = 0.23 to spatially map the frequency shifts of our atomic sample[32]. The clock transition frequency can be compared in different parts of the atomic sample to

16
(a)

Excitation fraction(b)

Detuning (Hz)Figure 1.6: (a) Ramsey spectroscopy data, demonstrating a 6 s atom-light coherence time. (b)Fractional frequency stability of the frequency difference between two regions of the 3D optical√lattice, P1 and P2 . The blue solid line shows an instability of 3 × 10−17 / τ , which when extracting√
the QPN of the total ensemble with twice as many atoms gives an instability of 1.5 × 10−17 / τ .
Whenthe number of atoms is reduced by a factor of three in each region, the instability increases by√
3, confirming the stability is governed by QPN. (inset) Spatially resolved image of the excitation
fraction. The frequency difference is extracted between two regions with excitation probability P1and P2 .

perform a spatially resolved synchronous comparison. This technique rejects the Dick effect since
all atoms in the sample see the same laser noise. The stability of this frequency comparison isthus limited by QPN. Spatially resolving frequency shifts can identify important clock systematics
such as AC Stark shifts [32], dipolar interactions, and few-to-many-body physics [54]. In addition,
since synchronous comparisons improve a measurement’s precision, they open up the possibility ofmeasuring gravitational redshifts at 10−19 stability within a single experiment.
Further improvements in the stability of 3D optical lattice clocks requires increasing theatom number and the coherence time, and decreasing the dead time. To further scale the atom
number beyond what was demonstrated in Ref. [49], larger lattice waists can be used. A bigger
challenge for 3D optical lattice clocks is the extended preparation time associated with preparing a
Fermi degenerate gas. The workhorse technique for preparing a Fermi degenerate gas is evaporative
cooling, which typically takes 10s of seconds. This prolonged dead time is significantly longer than

17
the hundreds of ms dead time demonstrated in 1D optical lattice clocks. Decreasing the dead timeto prepare a Fermi degenerate gas requires novel cooling techniques and is the subject of Chapter 4.
Coherence time limitations in 3D optical lattice clocks are an active field of study [55, 56, 57].
In particular, we found that our atom-atom coherence time was limited to 12 s [55], despite the
118 s lifetime [58] of the 3 P0 clock transition. This coherence time was found to be limited by
Raman scattering of lattice photons from the deep 3D optical lattice. Raman scattering is reduced
when the optical lattice is operated at lower lattice depths. However, an additional dephasing
mechanism due to site-to-site tunneling is then introduced. The interplay between tunneling andRaman scattering leads to a maximum coherence time in our system of 12 s. Further improving
the coherence time may be achieved by operating in the band-insulating regime, where tunnelingis suppressed, or by operating with a larger lattice spacing. The latter requires different latticegeometries to maintain the magic wavelength for each optical beam.
Eventually, the coherence time of atomic clocks will be limited by the lifetime of the excitedstate. Such a limitation would be an exciting development. However, it is prudent to think oftechniques that can engineer the coherence time beyond this seemingly fundamental limit. One
mechanism to achieve this is by altering the density of available final atomic motional states.
Measurements to observe this intriguing idea, predicted back in the 1980’s, will be discussed in
Chapters 5 and 6.

1.3

Outline of this thesis
The work reported in this thesis aims to increase the stability of quantum degenerate clocks bydecreasing the preparation time of the system and increasing the coherent interrogation time. This
thesis in particular highlights the essential interplay between improvements in quantum metrology,
and understanding and exploiting advances in quantum physics. The outline of this thesis is asfollows. First we will discuss the theory behind Fermi degenerate gases in Chapter 2. The consequences of Fermi statistics and the Pauli exclusion principle are profound and lead to striking
physics, which will be detailed. The theory introduced in this chapter is the basis of many measure-18
ments we make throughout the rest of this thesis. Then, in Chapter 3 we discuss the experimental
setup of the 3D optical lattice clock, with a particular focus on the spin polarizing sequence, imaging techniques, and stability transfer of the ultrastable laser referenced to a 124 K Si cavity, which
operates at 1542 nm, to the narrow linewidth atomic transition at 698 nm.
In Chapter 4 we discuss new cooling techniques to rapidly create Fermi degenerate systems.
By taking advantage of the many collisional partners a multi-spin component gas contains and thelack of inelastic loss for atoms with SU(N ) symmetry, we demonstrate a sub-3 s preparation time
to reach deep degeneracy that includes only 600 ms of evaporation. This is roughly a factor of
5 decrease in the preparation time over our prior work. We also perform a detailed study of theenhanced SU(N ) interactions in the Fermi gas.
Chapters 5 and 6 outline efforts to quantum engineer the natural lifetime of the atomic stateusing Pauli blocking, which exploits the Pauli exclusion principle to prohibit decay of an excited
atom to states already occupied by an identical fermion. In Chapter 5 we discuss experiments thatsuccessfully observe Pauli blocking on the 1 P1 transition in a light scattering experiment that angularly resolves the suppression of the spontaneous emission rate. Chapter 6 details measurements
made on the 3 P1 transition that attempt to directly measure the lifetime of the atomic state as
opposed to an angularly-resolved scattering rate. Finally, we conclude and provide an outlook for
future research in Chapter 7.

Chapter 2

Degenerate Fermi gases

Studying and controlling the quantum many-body system that comprises neutral atom atomic
clocks is essential for accurate and precise quantum metrology, and in tandem creates unique opportunities to advance our understanding of quantum science. Since atomic clocks use cold matter,the quantum statistics of the constituent atoms governs their behavior. The statistical properties of bosons and fermions dictates how particles interact. For example, the required asymmetricwavefunction of fermions implies that spin-polarized ground state fermions cannot interact at low
temperatures. Since particle interactions cause frequency shifts and dephasing, this was an essential
original argument for using fermions in neutral atom clocks.
The rich physics of Fermi-degenerate systems near T = 0 creates intriguing reasons to use such
low temperatures in atomic clocks. Three-dimensional optical lattice clocks rely on loading a gas
into the ground band of a 3D optical lattice. In order to load the lattice at high filling, it is necessary
to load a Fermi-degenerate gas. Near T = 0 and without particle interactions, the textbook example
of an ideal Fermi gas can be realized. Even though the system is theoretically simple, it provides
many opportunities to build our physical understanding and to use new discoveries to advancequantum science. We will see an example of this when we discuss changing the decoherence rateof an atom by exploiting the well-known phenomena that is Pauli exclusion. Adding particle
interactions creates a complex system that can describe for example the liquid state of electrons ina solid. These normal Fermi liquids are an active field of study, and we will see later on in this thesisthat such interactions can be utilized for example for novel cooling mechanisms. The intriguing

20
capabilities that are granted by the use of Fermi-degenerate systems requires a deep understandingof the quantum physics underlying them. In this chapter, we thus review the physics of Fermi gases.
First we theoretically outline particle interactions and how they are simplified at low temperatures.
We next detail the statistics and thermodynamics of noninteracting Fermi gases. We focus on
observables such as the shape of the gas after releasing it from its confining potential, as well as the
structure factor and corresponding density correlations of the gas. The quantum statistics produce
striking results in both cases. The former gives rise to Pauli blocking of collisions and spontaneous
decay, while the latter changes the compressibility and thus particle number fluctuations of thesample. Subsequently, we review the corresponding thermodynamics of interacting Fermi gases.
We describe the expansion dynamics of an interacting sample and how interactions modify thesystem’s compressibility.

2.1

Colliding fermions
In this section, we discuss scattering between fermions at low temperatures. The theory

explains when interactions need to be considered to properly describe the system. Further details
on non-interacting and interacting Fermi gases are the subject of the rest of the chapter.
In dilute atomic gases, interactions between particles are predominantly governed by pairwise
collisions. Here, we will discuss elastic collisions1 . Scattering theory is discussed in detail in many
textbooks [59, 60]. For a spherically symmetric scattering potential V (r), where r is the distance
between the particles, the wavefunction after scattering can be represented for r → ∞ byψ(r) = A e

iki ·r
eikf r+ f (ki , kf )
|si ,
r

(2.1)

where ki and kf are the incoming and outgoing wave vectors, A is a normalization constant, and|si is the two-particle spin wavefunction. This equation consists of an incoming plane wave and an
outgoing spherical wave with scattering amplitude f (ki , kf ), which can be expanded using spherical1

We neglect inelastic collisions, which matter at high densities.

21
harmonics into∞

1X
f (ki , kf ) =(2l + 1)eiδl sin δl Pl (cos θ) ,
k

(2.2)

l=0

where θ is the angle between ki and kf , l is the angular momentum of the partial wave, δl is theacquired phase shift that depends on the interaction potential, Pl is the Legendre polynomial, and|ki | = |kf | = k. The two-body wavefunction can then be written as [60],"#X
Pl (cos(θ)) 2iδl eikr
e−i(kr−lπ)
1
e
|si .
(2l + 1)−ψ(r, θ) =2ikrr(2π)3/2(2.3)

l

To reflect the quantum mechanical nature of the colliders, the total spin and spatial wavefunction ψ(r, θ) must be antisymmetric. If two fermions interchange, θ → θ − π. Under this exchange,
the spatial part of the wavefunction is symmetric (antisymmetric) for even (odd) l. This implies
that two fermions in the s = 0 spin singlet state can only interact through even partial waves, while
two fermions in one of the s = 1 spin triplet states can only scatter through odd partial waves.
Thus, elastic collisions between two fermions in the same spin state can occur through odd l, whilefermions in different spin states can interact through even and odd l, based on whether the spinstate is symmetric or antisymmetric. Two ground state atoms with low energy k → 0 have a phaseshift δl ∝ k 4 for l > 1, and δl ∝ k 2l+1 for l = 0 or 1 [61]. The phase shift is thus smaller as l increases
at ultralow temperatures where k is small. Thus, the scattering amplitude given in Equation 2.2 islargely governed by low l partial waves at ultralow temperatures. As a result, two cold fermions inthe same spin state to a good approximation only undergo p-wave (l = 1) scattering. In contrast,
two fermions in different spin states can undergo s-wave (l = 0) or p-wave scattering depending on
if the spin is in a singlet or a triplet state. The proportionality factors of the phase shifts are called
the scattering length for s-wave scattering, a, and the scattering volume for p-wave scattering, b3 .
The ratio δ1 /δ0 = k 2 b3 /a is proportional to T in the classical limit. P-wave scattering is thus
suppressed at ultralow temperatures below the p-wave barrier. The threshold energy Eth at whichcollisions occur can be approximated as
Eth (l) =~2 l(l + 1) C6
− 6 ,
2mb2
b

(2.4)

22
where l is the corresponding partial wave, m is the mass of the atom, and
s
6C6 m
2
b =
.
~2 l(l + 1)

(2.5)

The C6 coefficient for the 1 S0 state is 3107(30) a.u and for the 3 P0 state is 5360(200) a.u. [62],where 1 a.u. = 1 Eh a60 , Eh is the Hartree energy and a0 is the Bohr radius. For l = 1, thecentrifugal barrier is thus 35 µK in 1 S0 and 26 µK in 3 P0 . Using narrow-line laser cooling we
can reach temperatures of 1 µK. P-wave scattering, although small at µK temperatures, has beenmeasured in both 87 Sr and 171 Yb optical lattice clocks in the excited clock states [63, 64, 25]. In
the upcoming sections we will discuss the Fermi-degenerate regime, where T is 100s of nK. P-wave
scattering is negligible at these temperatures and only s-wave scattering is allowed.
R
The elastic scattering cross section is σ = |f (θ)|2 dΩ, which can be simplified to σ =P
4π/k 2 l (2l + 1) sin2 δl . For fermions in different spin states that undergo s-wave scattering, thisfurther simplifies to (k → 0)
σ = 4πa2 ,(2.6)

where a is the scattering length. However, for fermions in the same spin state, the s-wave scatteringcross section is zero.

2.2

Noninteracting Fermi gases
S-wave collisions, the predominant collisions at ultralow temperatures, are forbidden for

spin-polarized fermions. As a result, a spin-polarized ultracold Fermi gas can be described as
an ideal Fermi gas consisting of noninteracting particles. Noninteracting theory is also a goodfirst approximation for multi-component Fermi gases with small scattering lengths. This theory
highlights the unique statistics of fermions and its consequences.
In this section, we first describe trapped Fermi gases in subsection 2.2.1. We subsequently
describe the expansion of noninteracting Fermi gases after being released from the confining potential in subsection 2.2.2. Next we describe the structure factor, the response of the system to an
external wavevector, in subsection 2.2.3, and the consequences of the Pauli exclusion principle on

23
the structure factor, including Pauli blocking of collisions in subsection 2.2.4 and Pauli blocking oflight scattering in subsection 2.2.5. We conclude with a discussion of a Fermi gas’s correlation function, which is its response to spatial perturbations, in subsection 2.2.6 and the resulting suppressionof density fluctuations in subsection 2.2.7.

2.2.1

Statistics of trapped Fermi gases

The thermodynamics of non-interacting Fermi gases can be described through statistical
relations. The average number of fermions in a given spin state with energy  is given by the
Fermi-Dirac distribution [65]F() =1,eβ(−µ) + 1(2.7)

where β = 1/kb T , kb is the Boltzmann constant, and µ is the chemical potential. Non-interacting
trapped Fermi gases can be described by the HamiltonianH=

p2+ U (r) ,2m

where p is the momentum, m is the mass, U (r) = 12 m

(2.8)
2 2
i ωi xi is the trapping potential, and ωiPare the trap frequencies due to the confinement along direction xi . For a symmetric trap whereωx = ωy = ωr , we can rewrite the potential as U (r) = 12 mωr2 (x2 + y 2 + λ2 z 2 ) with λ = ωz /ωr . The
total atom number isZ ∞g()F()dωr kb T 3=−
Li3 (−ζ) ,ωz ~ωr

N=

0

(2.9)

where ζ = eβµ is the fugacity, Lin is the generalized polylogarithm of order n and g() is the densityof states as a function of energy , which in 3D is equal to [66]g() =
where ω̄ =

Q

j ωj

1/3

2
,
2~ω̄ 3(2.10). At T = 0, the occupation is one for each distinct energy state below EF
REand zero above EF . The above equation can then be simplified to N = 0 F g()d and integrated

24
to give EF = ~ω̄(6N )1/3 . The total energy of the system isZ ∞g()F()

E=
0

=−

3 (kb T )4
Li4 (−ξ) .
λ (~ωr )3(2.11)At T = 0 the mean energy per particle is 43 EF . The fugacity is related to the reduced temperature
through
Li3 (−ζ) = −1.
6(T /TF )3(2.12)For large numbers of atoms, a semiclassical phase space distribution can be used to describe
the many-body wavefunction, where f (r, p) = (2π~)−3 F(H(r, p)). A semiclassical transport equation can be derived by assuming the phase space density is conserved, that is df (r, p, t)/dt = 0.
Expanding this derivative gives us ∂f /∂t + ∇r f · ṙ + ∇p f · ṗ = 0. This simplifies to


∂t +


p
· ∇r − ∇r U (r) · ∇p f = 0 .
m(2.13)

The above equation is called the Vlasov, or the collisionless Boltzmann, equation. We can see that
the Fermi phase space distribution function, given by
f0 (r, p) =1(2π~)3

1
e

β



p2
+U (r)−µ2m,

(2.14)+1solves Equation 2.13 in equilibrium, when ∂t f = 0. The real space density n(r) =R

d3 pf (r, p) isthen given byn(r) = −(kb mT )3/2
2
2
Li3/2 (−ζe−mωr βρ /2 ) ,3/2
3(2π) ~(2.15)

where ρ2 = x2 + y 2 + λ2 z 2 .

2.2.2

Time of flight expansion

As is further discussed in Section 3.2, high-SNR images of the atomic cloud at high densities
are best performed after the atoms are released from the confining potential and measured after
long expansion times. The expanded gas can then be appropriately fit to extract T /TF and the25atom number of the gas. For a noninteracting gas, atoms expand ballistically and thus the phasespace density f (t, r, p) = f0 (t, r − pt/m, p). This can be rewritten as [67]f (r, p, t) = f0

ri
, λi (t)pi − mλ̇i (t)ri
λi (t)

,(2.16)

where f0 is the equilibrium distribution function given in Equation 2.14. The parameters λi arescaling parameters that represent the dilution of the gas along the ith direction and can be solved
to give the distribution function for different expansion times. We assume that before t = 0, thegas is in equilibrium with trap frequency ωi where i denotes the cardinal direction. The dynamics
are then described by a sudden trap frequency change at t = 0 from ωi to ωf [67]. Multiplying
Equation 2.13 by ri pi and integrating in phase space, we findλ̈i + ωf2 λi −

ωi2
= 0,
λ3i

(2.17)To study the expansion after switching off the trap, ωf in Equation 2.17 is set to 0. Theqdifferential equation is then solved for the scaling parameters, which yields λi (t) = 1 + ωi2 t2 . This
simple rescaling is unique for a harmonic trap. The scaling parameters can then be plugged into
the scaling ansatz to yield the phase-space distribution after time-of-flight. We get that the density
evolves according ton(r, 0)n(r, t) = Q q
.
2 t2
1
+
ω
i
i

(2.18)

That is, the 3D density is simply rescaled after expansion out of the trap.
The 3D density is hard to extract experimentally. Instead, the line-of-site integrated density
is imaged along a particular direction (here integrated along y) which gives
m(kb T )2
1
2
2
2
2
p
n(x, z) = − p
Li2 (−ζe−x /2σx −z /2σz ) ,3
2 1 + (ωr t)2 1 + (ωz t)2 π~ ωr(2.19)

where
σi2 (t) = x2i 0 (t) =kb T 1 + ωi2 t2.
m
ωi2(2.20)

26
Fitting the expanded gas cloud to the above equation after time of flight allows us to extract the
fugacity. As the fugacity depends on the shape of the atomic cloud and not on other extracted
parameters, it allows an accurate determination of T /TF . The aspect ratio of the line-of-site
integrated density is
σz
ωx
=
σx
ωz

s

1 + ωz2 t2.
1 + ωx2 t2(2.21)

At long expansion times (ωi t >> 1), the aspect ratio approaches one. This reflects the fact that
expansion occurs ballistically. After long time-of-flight, the cloud reflects the isotropic momentum
distribution even if the confining potential is anisotropic.

2.2.3

Static structure factor

Fermi statistics has profound effects on the scattering behavior of particles. The static structure factor Sk describes the response of a system to a perturbation with wave vector k. Perturbations can come from, for example, light, or from collisions with other non-identical particles. The
consequences of the Pauli exclusion principle are evident in the static structure factor, given by
Sk =

XX
u



ni (v) 1 − nf (u) | hu| eik·r̂ |vi |2 ,(2.22)v

where the sums are over all energy eigenstates and r̂ represents the position operator. This equation states that the probability a fermion scatters from energy state |vi into energy state |ui is
proportional to the occupation number of the initial state ni (v), times the probability that the
final state nf (u) is empty. The product must be weighted by the probability that the fermion will
scatter into state |ui given a momentum transfer of k. In a harmonic trap with many harmonic
oscillator levels filled, the structure factor can be approximated with a semiclassical equation as
S(k) = h−3

Z

d3 r

where f (p, r) is given in Equation 2.14.

Z
d3 pfi (p, r) 1 − ff (p + ~k, r) ,(2.23)

27
2.2.4

Pauli blocking of collisions

The Pauli exclusion principle describes a fundamental phenomenon that gives rise to properties such as Fermi pressure, which causes the size of a Fermi gas to saturate as the temperature
is lowered [68]. The above discussion demonstrates a consequence of the Pauli exclusion principle,
that scattering is not allowed if the final momentum state is occupied by another identical fermion.
The momentum transfer k during this process can be imparted for example through collision withanother particle. This has a measurable consequence on the equilibrium thermodynamic properties and collisional dynamics of a multiple component Fermi gas. In the degenerate regime, Pauli
exclusion reduces a gas’s ability to perform binary collisions, which can be illustrated by noting
the following: Imagine that the system consists of two equal T = 0 Fermi gases, so that all levels
below EF are occupied, and where each Fermi sea has a different spin so that scattering between
the seas is allowed. The initial energies of any scattered particles E1 and E2 must be less than EF .
In order for two particles to scatter, the final states with energies E3 and E4 must be greater thanEF , since all states below EF are occupied. That is, our conditions for scattering are E1 , E2 < EF
and E3 , E4 > EF . However, energy conservation requires that E1 + E2 = E3 + E4 . This is in
direct contradiction with the energy bounds we set. Thus, scattering between two Fermi seas iscompletely blocked at T = 0.
The elastic cross-section for a two-component Fermi gas goes from the classical value of σel =4πa2 to 0 at T = 0 [69]. The suppression of collisions causes a reduction in the evaporation efficiency
at degenerate temperatures [70] and in the energy per atom in an imbalanced two-component Fermi
gas [71]. In the latter case, different Fermi energies cause a disparate mean energy per particle
between the two components. In a classical system, collisions between components would equalize
the mean energy per particle, but those collisions are blocked for a Fermi gas. The more degeneratecomponent has a larger Fermi sea, and collisions that cause a reduction in energy are blocked due
to occupied final states.

28
2.2.5

Pauli blocking of light scattering

The momentum k imparted to an atom can additionally be caused by light. When an atom
absorbs a photon, it receives a recoil of energy Er = ~2 k 2 /2m, where k is the light’s wavevector.
For a light scattering experiment where an atom undergoes a directed absorption followed by adispersive spontaneous emission event, the total momentum transfer k can range anywhere from√0 to 2kr , where kr = 2mEr is the recoil momentum. This has the striking consequence that if
T = 0, k is much less than the Fermi momentum radius kF , and the Fermi sea is weakly excited
such that the ground state Fermi sea remains unperturbed after absorption of the probe light, most
excited atoms are prohibited from decaying. Only atoms near the edge of the Fermi sea, which
can be excited to unoccupied states outside of the Fermi sea, will be able to decay. Experiments
to observe this Pauli suppression of the spontaneous emission rate of an atom will be discussed in
more detail in Chapters 5 and 6. Here we review the theory of Pauli suppression in light scatteringexperiments, focusing on the intuition gained from analyzing the case of a T = 0 spin-polarizedFermi gas.
Equation 2.23 can be compared to a Maxwell-Boltzmann distribution to give the relative
scattering rate [72]SF M (k)
=
SM B

h−3

R

d3 r



d3 pfi (p, r) 1 − ff (p + ~k, r)RR.
h−3 d3 r d3 pfi (p, q)

RThis integral can be simplified by making the variable substitutions r̃j = rjppj β/2m to give
SF M (k)6=1− 3 3 3
SM B
π EF β

Z

3

d r̃

Z

d3 p̃

1

1

(2.24)qβmωj2 /2, p̃j =√,1 + ζ −1 e(p̃2 +r̃2 ) 1 + ζ −1 e((p̃+ βEF k/kF )2 +r̃2 )(2.25)

where ω̄ = (ωx ωy ωz )1/3 is the average trap frequency, and we used the fact that ~kF = (2mEF )1/2
and EF = ~ω̄(6N )1/3 . This makes the integral spherically symmetric. We can align our coordinate
system such that the momentum transfer k is along the z-direction, that is k = kẑ. The position
and two other momentum directions can be expressed in 5-dimensional spherical coordinates with29

radius s to give
SF M (k)
16
=1−
SM B
πEF3 β 3

Z
dss

4

Z
dpz

1

1

1 + ζ −1 es2 +p2z 1 + ζ −1 es2 +(pz +kscal )2,(2.26)

where we have used the fact that the volume element, integrated over the 4 angular directions, is√equal to 8π 2 /3s4 ds and kscal = βEF k/kF . Finally, we let a = s2 to arrive at the formula
8
SF M (k)=1−
SM B
πEF3 β 3Zdaa3/2Zdpz

1

1
1 + ζ −1 ea+p2z

1 + ζ −1 ea+(pz +kscal )2.

(2.27)

This can be numerically solved for finite T /TF to extract the reduction factor of the scattering
rate. Equation 2.27 can be explicitly solved at T = 0 to obtain [72]SF M (k)
32
k
=1−
χ(
)Θ(2 − k/kF ) ,SM B
5π 2kF(2.28)

where Θ is the Heaviside step function and√x 1 − x2
15
χ(x) =(−8x4 + 26x2 − 33) +cos−1 (x) .
48
48(2.29)For a light scattering experiment, an atom first receives a momentum kick kabs along a defined
direction, and a second randomly oriented momentum kick kemi upon spontaneous decay, where|kabs | = |kemi | = kR . The total momentum transfer k is then given by k = kabs + kemi wherek = 2kR sin(α/2) and α is the angle between the absorbed and emitted photon. This is illustrated
in Fig. 2.1 for the specific case of a single excited atom in the center of the Fermi sea in a T = 0
gas. Here, if kF > 2kR , complete suppression is achieved and SF M (k) = 0.
Equation 2.28 is plotted in Fig. 2.2 as a function of α and kF /kR . Decay can occur if the
momentum transfer kicks the atom outside the Fermi sea. No suppression occurs when k > 2kF ,as is reflected in the Heaviside step function in Equation 2.28. This is because all atoms, regardlessof their position within the Fermi sea, will get kicked outside it. For k < 2kF , suppression isdominant in a cone near the absorption direction, where the total momentum transfer is small.
Larger suppression can be achieved at all angles α when EF >> ER . However, even at large Fermi

30

a

EF

ER

} ħω
b
kemi
k

α
kabs

incoming
photonkF
emittedphotonFigure 2.1: (a) In a T = 0 Fermi gas, all available harmonic oscillator states up to the Fermi energy
EF are filled with unity occupation. An excited atom (orange) acquires a recoil energy ER whenreturning to its internal ground state (blue). (B) In momentum space, the atoms form a Fermi sea
occupying all states up to the Fermi momentum ~kF . Spontaneous decay of an excited atom withemission along α and total momentum transfer ~k can happen only if the final momentum state is
not occupied by another ground state fermion. If kF > 2kR , all final states are occupied and theatom cannot decay. The decay in both (a) and (b) is blocked since the final state is occupied.

energies, atoms near the edge of the Fermi sea can be excited to available decay channels outside

31

1.0

SFM (k)/SMB

0.8
kF = 20kR
0.6

kF = 2kR
kF = kR

0.4

kF = 0.5 kR
0.2
0.0
0

Angle α (degrees)

50

100

150

Figure 2.2: Suppression of the scattering rate for a Fermi versus classical gas for various angles
α between the absorbed and emitted photons, and Fermi versus recoil momentum. When k =2kR sin(α/2) > 2kF , no suppression occurs.

the Fermi sea. Complete suppression thus only occurs for EF >> ER and when only the atoms
near the center of the Fermi sea are excited. Note that Equation 2.28 only depends on the ratio
k/kF = 2kR /kF sin(α/2). Thus, changing the ratio of kR /kF or the angle α are equivalent. At
finite temperature, holes within the Fermi sea provide additional paths for atoms to decay, andeven with EF >> ER only finite suppression is achieved at all angles.

2.2.6

Density-density correlation function

If Fermi statistics modify the response of a Fermi gas to a perturbation of wavevector k,then it must similarly modify the response to perturbations at a distance scale of r = 1/k. This is32

reflected in the correlation function ν(r), which is the Fourier transform of Sk [73]:1
ν(r) =(2π)3Z

d3 kSk eik·r ,(2.30)

where r = r1 − r2 is the distance between two particles at positions r1 and r2 . The idea to lookat density-density correlations of ultracold atoms was originally proposed in Ref. [74], and was
first measured between different spin states of a Feshbach molecule [75]. To further analyze ν(r),we take advantage of the local density approximation (LDA) [76] and absorb the spatial variationof the potential into a locally changing chemical potential µLDA ≡ µ − U (r). This allows us tolocally treat the trapped Fermi gas as a homogeneous gas. The density-density correlation function
C(r) = nν(r), where n is the density, can then be written as [77]
"
#2
p
Z ∞
u sin(u kB T /EF kF r)
kB T kF2
C(r) = nδ(r) −du.
EF 2π 2 r 0
e−βµLDA eu2 + 1(2.31)For a T = 0 gas, the correlation function further simplifies to [78]
cos(kF r) sin(kF r) 2
9−C(r) = nδ(r) − n(kF r)2
kF r(kF r)22= nδ(r) − n2 (g 2 (r) − 1) ,(2.32)

Equation 2.32 is divided into “auto-correlations”, which associate the correlation of the particle
with itself, and pair-wise correlations given by g 2 (r), which is the conditional probability of finding
two particles separated by distance r. g 2 (r) is plotted in Fig. 2.3 versus kF r. From this figure, one
can see for distances r ∼ 1/kF that g 2 (r) < 1 and approaches 0 as r → 0. This reflects the reduced
probability of finding two fermions closer than ∼ 1/kF due to Pauli exclusion.
Combined with the autocorrelations, C(r) is negative for fermions. This fermionic ‘antibunching’ was originally observed in a band insulator of 40 K atoms released from an optical lattice[79], and in a gas of 3 He [80]. Higher order correlations beyond second order lead to the formationof Pauli Crystals, geometrical structures that are formed in few-body systems [81, 82].

33

1.0

g2(r)

0.8
0.6
0.4
0.2
0.0
0

2

4

6

8

10

kF r
Figure 2.3: Pair distribution function versus scaled distance between the two particles.

2.2.7

Density fluctuations

The correlation function C(r) = hn(r1 )n(r2 )i − hn(r1 )i hn(r2 )i is related to density fluctuations through2(δn(r))

Z=3Z

d3 r2 C(r)δ(r1 − r2 − r) ,d r1

(2.33)

where the notation h·i represents the statistical mean, which is the result of a large number of
independent realizations. In an experimental context, this implies the result of a large numberof repeated measurements. The particle number fluctuations in a subvolume V = 4/3πR3 of the
Fermi gas can be expressed as
(δN )

2

Z R
=
0

1
C(r)h(r) 4πr dr =
(2π)3
2

Z

d3 knS(k)H(k) ,(2.34)

where h(r) is a geometrical overlap factor, given byZ
h(r) =
V

d3 r1

Z

d3 r2 δ(r1 − r2 − r) .

(2.35)

V

Physically, this represents the overlap region between a subvolume V and that subvolume shiftedby r, while H(k) is its Fourier transform. To gain intuition about Equation 2.34, we first look at

34
a T = 0 gas. In this limit, the number fluctuations can be rewritten for large kF R as [76, 83](δN )2 =i(kF R)2 h γ−1/2
ln
4e
k
R
+ O(1/kF R) ,F
2π 2

(2.36)

where 4eγ−1/2 = 4.32. In a T = 0 gas, there are no thermal fluctuations since every state isalways occupied below EF . However, Equation 2.36 reveals an additional source of atom number
noise. Upon measurement, atoms are projected into the subvolume with a probability given by
the overlap of the particle’s wavefunction inside V . This finite probability for small enough box
sizes causes atom number fluctuations, similar in concept to quantum projection noise. For a box
with many atoms, quantum projection noise is a particular problem on the edges of the subvolume,
since projection noise puts these atoms either inside or outside the box 2 . Hence, the fluctuations
depend on the surface area of the box which goes like R2 . Written in terms of the atom number,
Equation 2.36 can be written as(δN )2
=3
N



3
32π 4

1/3

i
h
N −1/3 ln 10.45N 1/3 ,(2.37)and thus the effect disappears in the thermodynamic limit, when a subvolume contains many
particles and the projection noise tends towards zero.
In the thermodynamic limit at finite temperature, H(k) → V δ(k) so that Equation 2.34
simplifies to (δN )2 = N S(0), yielding the result of the fluctuation-response theorem [84] withgeneralized force µ, which is the chemical potential:(δN )2 = N kB T nκT ,(2.38)

where κT = −1/V (∂V /∂P )T,N is the isothermal compressibility. This result can also be derived
using the Grand partition function
Z=

YX
p

2

e−βnp (p −µ) ,(2.39)

np

Technically, the same underlying source of projection noise exists for smaller boxes where there are only a few
atoms, but it is difficult to accurately model the fluctuations in this case as it depends on the precise shape of thesingle-particle wavefunctions.

35
where the sum is over states p with occupation per state np . For identical fermions, the occupation
can be either zero or one so thatZ=

Y

(1 + e−β(p −µ) ) .

(2.40)

p

The average atom number hN i is related to the Grand Potential Ω = −kB T ln(Z) through N =− (∂Ω/∂µ)V,T , which after making the sum continuous gives the familiar equation
V
N = 4π 3hwhere fn (ζ) = −Lin (−ζ) and λ =

Z ∞
0

1
V
dpp2 −1 βp= 3 f3/2 (ζ) ,λζ e +1(2.41)p2π~2 /mkB T is the thermal deBroglie wavelength. The varianceis then [85]
2

(δN )



∂N
∂ 2
log(Z) = kB T.
= kB T∂µ∂µ(2.42)

κT can be expressed using Maxwell’s relations as κT = V /N 2 (∂N/∂µ)T,V . Equation 2.42 thus
reproduces Equation 2.38. Given the relation ∂fn (ζ)/∂ζ = fn−1 (ζ)/ζ, we find(δN )2 =V
f (ζ) ,
λ3 1/2(2.43)and thus(δN )2 /N =f1/2 (ζ)= nkB T κT .
f3/2 (ζ)(2.44)Atom number fluctuations in the above equation are of purely thermal origin and can be understoodas follows. Thermal fluctuations manifest on the Fermi surface, where the occupation number is
not unity and the majority of holes lie. In real space, this corresponds to fluctuations delocalized
equally over the whole homogeneous Fermi gas. As T /TF decreases, the width of the Fermi surface
narrows and the number of holes decreases so that the thermal fluctuations correspondingly decline.
Taken in light of the LDA, this implies that fluctuations on the edge of a trapped Fermi gas, where
the local T /TF is larger, will experience increased fluctuations compared to the center of the Fermi
gas where µ is maximum. To first order in T /TF , Equation 2.44 becomes3(δN )2 /N = T /TF ,2(2.45)

36
3
implying κT = 2nE. In contrast, for a classical gas, (δN )2 /N = 1, reflecting the Poisson statistics
Fof the gas 3 .
In the actual experiment, we do not measure the 3D atomic density but instead the column
density, which is the 3D density integrated along the line-of-site (LOS) of the imaging camera.
We can obtain the expected fluctuations by integrating the fugacity in Equation 2.44 through theimaging direction x3 4 :R
f1/2 (ζ(x1 , x2 , x3 ))dx3
f1 (e−β(µ−U (x1 ,x2 )) )2
(δN ) /N LOS = R=.
f3/2 (ζ(x1 , x2 , x3 ))dx3
f2 (e−β(µ−U (x1 ,x2 ) )(2.46)

To obtain T /TF from the experimental data, we look at (δN )2 versus N in the same subvolume
across a series of images. We perform the noise analysis after time-of-flight to avoid any collective
scattering processes that can occur when imaging an atomic cloud at high density. Because thedensity is simply rescaled after time-of-flight, the noise properties (δN )2 /N = kB T /n ∂n/∂µ are
not altered. To account for shot-to-shot atom fluctuations due to imperfections in the experimental
preparation, we high pass each image by fitting each atom cloud to a gaussian and looking at theresiduals. In comparison to scaling each image, this also minimizes the effect of center-of-massfluctuations on the results. Repeating this protocol for subvolumes in the center vs edge of the
Fermi gas gives us T /TF as a function of the harmonic confinement, which we then fit with theory.
The subvolume size is set by looking at (δN )2 /N and observing when the fluctuations saturate as
we vary the box size. Finally, photon shot noise must be subtracted to get an accurate calibrationof the thermal fluctuations. This can easily be obtained for each subvolume by looking at the brightimage in the absorption image analysis. See Chapter 3 for more details on our imaging methods.
Sub-Poissonian fluctuations were first measured by the groups of Ketterle [86] and Esslinger[87] in noninteracting trapped Fermi gases, showing up to around 50% of suppression compared to aclassical gas. Reduced fluctuations have also been seen in the Mott-insulating and band-insulating
regimes in an optical lattice, see for eg [88, 89, 90], and have been used as a complementary3

The number fluctuations for a classical gas follow a binomial distribution. For large atom numbers, this isapproximately Poissonian and the probability of having atom number N in subvolume V is pN = hN iN e−hN i /N !.
The atom number variance is thus (δN )2 = hN i.
R∞√2
4
We use the identity −∞ fn (ζe−x )dx = πfn+1/2 (ζ)

37technique to confirm the degenerate temperatures that were extracted through global profile fits in
KRb [91].

2.3

Interacting Fermi gases
When more than one spin component exists in a Fermi gas, s-wave collisions allow interactions

between components with a strength that depends on the scattering length a. In this section,
we outline the physics of interacting two-component Fermi gases, which are the simplest multicomponent system to analyze and gain intuition. This will provide a theoretical basis which will be
expanded on in Chapter 4 to describe interacting SU(N ) systems with N = 10 spin components. In
subsection 2.3.1 we derive the Hamiltonian for an interacting Fermi gas, and in subsection 2.3.2 wediscuss the expansion dynamics of interacting systems. We conclude with a discussion of density
fluctuations in a repulsively interacting trapped Fermi gas in subsection 2.3.3.

2.3.1

Interaction Hamiltonian

The scattering potential between particles is complex at short-range, but can be simplified
at long range into a Van-der Waals potential in the ground state, V (r) ∼ C6 /r6 . For excitedstate atoms, dipole-dipole interactions are also important, with V (r) ∼ C3 /r3 . Fermi discoveredthat a simpler pseudopotential can describe many properties correctly, including the energies andscattering phase shifts. The pseudopotential for s-wave scattering is given by [92]V (r) =

4πa~2
δ(r) .
m(2.47)The use of a pseudopotential greatly simplifies the interaction potential between particles, and isaccurate as long as the details of the short-range physics are not important.
Given the simplification granted by the Fermi pseudopotential, we will now derive the interaction potential in an ultracold two-component Fermi gas. As derived in Section 2.1, s-wave collisions
are predominant at ultralow temperatures. Interactions thus largely occur between particles with

38
different spins. The Fermi pseudotential allows us to write the total interaction energy as
ZHint = d3 rd3 r0 ψ̂↑† (r)ψ̂↓† (r0 )V (r − r0 )ψ̂↓ (r0 )ψ̂↑ (r)Z= g d3 rψ̂↑† (r)ψ̂↓† (r)ψ̂↓ (r)ψ̂↑ (r) ,(2.48)

where we have identified the interaction strength g = 4π~2 a/m, and ψ̂σ (r) (ψ̂σ† (r)) annihilates(creates) a fermion of spin σ at point r. We have labeled the two spin states as |↑i and |↓i. The
fermionic operators satisfy {ψ̂σ (r), ψ̂σ† 0 (r0 )} = δσ,σ0 δ(r − r0 ). Given the density operator n̂σ (r) =ψ̂σ† (r)ψ̂σ (r), and assuming the density for both components is equal, that is n̂↑ (r) = n̂↓ (r) = n̂(r),the interaction energy can be further rewritten as
Z
Hint = g d3 rn̂↑ (r)n̂↓ (r)Z= g d3 rn2 (r) .
The total Hamiltonian for a single component can be written as


Z
Z


∇2
3
†
Hσ = d r ψ̂σ (r) −+ U (r) ψ̂σ (r) + d3 rψ̂σ† (r) gn(r) ψ̂σ (r) ,2m(2.49)(2.50)

where the first term describes the kinetic and potential energy of the fermions, which are confinedPby a harmonic trapping potential U = m/2 i ωi2 x2i . The above equation makes it apparent thatthe interaction energy looks like an additional effective potential with a strength given by the mean
field energy:
VM F = gn(r) .

(2.51)The derived interaction Hamiltonian is accurate for a two-component Fermi gas. In general,interactions can exist between many spin components, and the interaction potential must take
into account all possible two-body interactions, which in general have different scattering lengths
a. Under SU(N ) symmetry, the nuclear spin degree of freedom is decoupled from the electronic
angular momentum so that all scattering lengths a are equal for atoms in different nuclear spinstates, and the interaction Hamiltonian is greatly simplified. This allows enhanced interactions
without inelastic spin loss that normally result when a gas has many-components. We will discuss
SU(N ) interactions in more detail in Chapter 4.

39
2.3.2

Expansion dynamics

To solve for the expansion dynamics of the gas after switching off the harmonic confinement,
we can use the same method as that used for noninteracting Fermi gases in subsections 2.2.1 and2.2.2, but replace U (r) with the effective potential U (r) + gn(r). The Boltzmann equation becomes



∂t +


p
· ∇r − ∇r [U (r) + gn(r)] · ∇p f = 0 .
m(2.52)

Solving the equation in equilibrium (∂t f = 0) gives us the interacting equilibrium distributionfunctionf0 (r, p) =1(2π~)3

1
β



.

p2
+U (r)+gn(r)−µ2m



(2.53)+1e
Due to the dependence of the distribution function on n(r), the equilibrium distribution must be
solved self-consistently [59]. First a zeroth order distribution function is calculated by assumingRthere are no interactions. The density is then calculated through n(r) = d3 pf (r, p), and thenusing the calculated density, the next order distribution function is derived. This process is repeatedseveral times until the distribution function converges. The time-of-flight expansion dynamics canthen be solved for using the same scaling method as introduced for the non-interacting gas. The
scaling ansatz isf (r, p, t) = f0

ri
, λi (t)pi − mλ̇i (t)ri
λi (t)

,(2.54)

where f0 is the equilibrium distribution function. The scaling parameters λi satisfy the differential
equation [93]ω2λ̈i + ωf2 λi − 3i + ξi ωi2
λi

1
1
− Q
3
λi j λjλi!
= 0,(2.55)

where ξi = g hni /(g hni + p2i /2m). For a zero temperature Fermi gas, ξ ∼ kF a, where kF a
is commonly referred to as the interaction parameter. It can be understood as the ratio of thescattering length to the interparticle distance, and characterizes the strength of interactions.
If we switch off the harmonic trap, ωf is set to zero and we can numerically solve for λi . Theqaspect ratio of the gas after time t is then hrz (t)i2 / hrr (t)i2 ≡ σz /σr = λz (t)ωr /λr (t)ωz . For a40
repulsively interacting Fermi gas (a > 0), the cloud aspect ratio will invert for long time-of-flight.
This can be understood as follows. Interactions provide an additional force during expansion that is
proportional to dn(r)/dr. A trap that is initially tightly confined in the z-direction will thus expand
more after time-of-flight, since interactions will preferentially push atoms along the direction of thelargest density gradient. This is in contrast to the non-interacting case, where the aspect ratio
approaches one after long time-of-flight, as dictated by the initial homogeneous momentum.

2.3.2.1

Collisions during expansion

An implicit assumption made thus far is that collisions do not occur during the expansion.
Collisions change the momentum distribution and as a result can alter the expanded density distribution. While the mean-field force is non-dissipative, collisions are dissipative and the availability of
the final momentum states as governed by Fermi statistics needs to be considered [94]. The Boltzmann equation, which can be derived by assuming unimpeded semiclassical motion of particles,
needs to be modified. With collisions the Boltzmann equation becomes [95]


∂t +


p
· ∇r − ∇r [U (r) + gn(r)] · ∇p f = Ic [f ] ,m(2.56)

where Ic [f ] is the collisional term. In general the collisional term describes two-body collisions andis given by [94]a2
Ic [f ] = 3
h m

Z



d3 p2 dΩ|p1 − p2 | (1 − fσ (p1 ))(1 − fσ0 (p2 ))fσ (p01 )fσ0 (p02 ) − (σ ↔ σ 0 ) ,(2.57)

where the last term refers to the first term with σ and σ 0 swapped. This equation can be simplified
by using the relaxation time approximation [95, 94, 96]. Here, we assume that the distributionfunction relaxes to equilibrium over some timescale τ :Ic [f ] =f − fle,τ(2.58)

where fle is the local equilibrium distribution function, which defines the distribution at a given
point in time where collisions no longer change the temperature of the gas. The relaxation time τ

41

2.0

FQ

1.5
1.0
0.5
0.0
0.0

0.5

1.0

1.5

2.0

T/TF
Figure 2.4: FQ , an important parameter for the collisional term in the Boltzmann equation

can be defined as [94](ω̄τ )−1 =4
5 31/3 π(N 1/3 a/aho )2 FQ (T /TF ) ,where N is the number of atoms per species, aho =(2.59)

p~/mω̄ is the harmonic oscillator length, and

FQ is a function that is only dependent on T /TF , and whose precise form is given in Ref. [94].
Fig. 2.4 plots FQ and displays the important behavior. At very low temperatures, Fermi statistics
suppresses the number of collisions between atoms and one finds that the gas is in the collisionlessregime. At high temperatures, collisions are again less likely because for a fixed atom number,a higher temperature causes the density to decrease. However, at intermediate temperatures,
collisions during expansion can be an important element of the dynamics.
To properly describe the expansion dynamics with collisions, the scaling parameters have to
be modified. Collisions occur when the local temperature is not constant. As such, in addition to
the parameters λi that describe the spatial expansion of the gas, we introduce additional scaling
parameters θi to describe when the temperature along all three axes are not equal. The distribution

42function can now be scaled as [96]f (t, ri , vi ) = Q

1
1/2

j (λj θj )

f0

ri 1
,
λi θ1/2

λ̇ivi − ri
λi

and the scaling parameters to describe the expansion are solved using
!
θ
θ
1
i
i
λ̈i + ωi2 + ξi ωi2
− Q
=0
λi
λi λi j λj


λ̇i
1
1X 
θ̇i + 2 θi = −
θi −
θj ,
λi
τ
3

!!
,

(2.60)(2.61)(2.62)jwhere we have already set the term that describes the harmonic confinement to zero. One canP
see from these equations that when the effective temperatures are equal (θi = θ̄ = 1/3 j θj ),collisions are effectively shut off and we get the corresponding scaling solutions without interactions
with θi = λ−2
i . This is equivalent to sending τ → ∞. The aspect ratio can then be solved usingσz /σr = λz (t)ωr /λr (t)ωz . Interestingly, while the mean-field interaction depends on the sign of thescattering length, a, the collisional term depends on a2 , and thus is not sensitive to whether theinteractions are repulsive or attractive.
The first observation of anisotropic expansion was reported in Ref. [97]. Here, the twocomponent Fermi gas was prepared with large attractive interactions (a ≈ −104 a0 ) so that the gas
was a superfluid. The mean-field treatment derived above is incorrect at these large interaction
strengths, and instead the hydrodynamic equations of superfluids must be used as in Ref. [93].

2.3.3

Compressibility

Interactions also cause a change in the compressibility of a gas. Repulsive interactions, much
like Pauli exclusion, create a pressure that resists a change in volume. The compressibility κT =1/n2 ∂n/∂µ can be derived numerically using the equilibrium distribution function for an interacting
gas, as given in Equation 2.53. To gain intuition about the results, we can look perturbatively at
the compressibility for low temperatures and small interaction parameters. At finite temperature,
we can take use of the Sommerfield expansion and arrive at for the chemical potential [98, 73]"#
 
π2 T 2
4µ(T, a) = EF 1 −kF a + O((kF a)2 ) .
+(2.63)
12 TF
3π

43
Given TF and kF ’s dependence on the density, we can then derive that the number fluctuations are(δN )2
3 T /TF=,
N
2 1 + π2 kF a
showing that the fluctuations decrease for both lower temperature and larger interactions.

(2.64)

Chapter 3

Making, measuring and probing Fermi-degenerate clocks

This chapter discusses the experimental details of creating and probing a highly precise Fermidegenerate optical lattice clock. In section 3.1, we describe the process to create a Fermi-degenerate
gas of 87 Sr atoms and load it into a 3D optical lattice. We then discuss the imaging techniquesthat we use to spatially resolve our sample in section 3.2. We finish the chapter in section 3.3 witha discussion of how we transfer the stability of our ultrastable laser based on a Si resonator to the
87 Sr atoms.

3.1

Preparation
Here we review the experimental steps necessary to prepare a Fermi-degenerate 3D opticallattice. Many of these steps have been detailed in other theses1 , and thus will only be discussed in
brief here. Focus is put on the stages that are new as of this thesis, such as the transparency and theTenS4 lasers. First in subsection 3.1.1 we summarize our experimental sequence. In subsection 3.1.2
we describe the main experimental apparatus, including the optical layout for each laser used in
the experiment. In subsection 3.1.3 we provide an overview of cooling 87 Sr and trapping the atoms
in an optical dipole trap, and describe how we measure trap frequencies in subsection 3.1.4. We
then detail the transparency beam technique in subsection 3.1.5, which allows us to overcome lightassisted loss and increase the density of atoms before evaporation. In subsection 3.1.6 we discuss the
TenS4 laser that allows us to spin-polarize a gas after evaporation with minimal heating. Finally,1

See mainly Ref. [99].

45
we detail loading the spin-polarized gas into the optical lattice in subsection 3.1.7.

3.1.1

Summary of experimental sequence

To create a Fermi-degenerate clock, we perform the following steps.
(1) A high flux of hot Sr atoms is created with an atomic beam oven that runs at a temperatureof 480◦ C, which are then slowed using a Zeeman slower and directed into the main vacuum
chamber with two 2D MOTS.
(2) Atoms are cooled to mK temperatures in a ‘blue’ MOT, which addresses the 1 S0 → 1 P1transition.
(3) Atoms are then transferred to a second-stage MOT on the 1 S0 → 3 P1 transition that cools
the atoms to µK temperatures.
(4) For evaporative cooling, the atoms are loaded into a crossed optical dipole trap (XODT).
(5) The transparency laser is applied to the dimple region of the XODT and a further stageof cooling with the single frequency red MOT occurs to cool atoms in the reservoir of the
ODT while the density of atoms in the dimple is increased.
(6) Nuclear spin states mF = −9/2 and mF = −7/2 are optically pumped to other spin states.
(7) Evaporation with the remaining 8 spin states then proceeds to reach temperatures less than
0.1 times the Fermi temperature with about 50,000 atoms per spin state.
(8) The Tens4 laser is applied to spin-polarize the sample with minimal heating.
(9) Atoms are adiabatically loaded into the 3D optical lattice to form a band insulator.
(10) Clock spectroscopy is performed on the atoms to measure the atomic transition frequency.
Advances in atomic clocks hinge on optimizing, understanding, and harnessing the physics ateach stage of the preparation process. In particular advances in quantum technology benefit from46

controlling and manipulating quantum properties of lab-made quantum platforms. Inspired by thisidea, Chapters 4, 5, and 6 focus on studying, understanding, and manipulating Fermi gases. In
these chapters, the above procedure is stopped after step (7). Frequently all 10 spin states are also
used in these experiments.

3.1.2

Experimental apparatus

Numerous lasers are required to implement all the steps detailed above. Our experimentconsists of three optics breadboards stacked vertically to create optical beam paths that can enterthe vacuum chamber from one of the 15 different available viewports. Beam paths for the bottom,
middle, and top optical breadboards are shown in Figs. 3.1, 3.2, and 3.3, respectively. Further
details on the various beam paths in these drawings will be referred to in their corresponding
sections later in the chapter.
Our experiment is surrounded by an Al enclosure to remove air currents and improve temperature stability, as shown in Fig. 3.4. The box has two ‘garage doors’ that open to allow opticalaccess to the main parts of the experiment. Small holes on the sides of the box, far away fromimportant optical beams, allow cables to escape easily. We found that adding this box significantly
reduced fringe issues in our imaging system. The air currents caused by leaving panels of the box
open were found to cause residual parasitic etalons to fluctuate. Dust is also reduced inside thebox, which decreases the number of stationary fringes. The box is equipped with foam inside eachof the panels to damp acoustic vibrations. We additionally temperature control the main optical
table using water cooling that is hooked up to a 100 mK stable chiller. The chiller temperature isroughly 22 C so that when we open the garage doors there is no significant temperature perturbation. We also temperature control any significant heat sources inside such as AOMs, the oven,

47

off

Blue MOTPD

679 + 707
Repumps

PBS

PBS

PD

λ/2

Pic
k

PBS

BS

High Res.
Abs. Imag.

Red +RepumpParasc.

PBS

Red MOT
α

off

k
Pic

PBS

Red MOT
β

PBS

λ/4

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

Mirro

r

de Top
g
m OD
irr T
or

45 degmirror45

45 degmirror

am
Be mp
Du

High Res.
Abs. Imag.

Glan

PBS

Fluor.
Imaging

99/1

Flippermirror

hp

ow
er

PBS

s

t.

Len
Bandpass

Pickoff

Bandpass

Le
n

s

PD

r
zo
Ra ackSt

shuLR
Im Abs
ag
ing

Hig

λ/2

f = 1000PD

f = -250

f = -100
λ/2
PBS

EOT Faraday
Rotator

irr
or

PBS

M

Lattice

Figure 3.1: Optical layout of the lower optical breadboard. Optical beams are brought to the main
mezzanine through 45 degree mirrors. The 45 degree mirrors in the center of the diagram are
directed onto the atoms in the center of the vacuum chamber on the main mezzanine.

and beam dumps. With these implementations, the temperature stability inside the box is +/100 mK throughout the day and is relatively immune to larger temperature drifts in the main lab.
We find that we have to realign our optical beams significantly less with this control. For ease, we
use picomotor mirror mounts on optics mirrors that are frequently adjusted such as for the lattice

r1

"

48
irro

JIL

m

ror

2”

irro

r

2
or
irr

mir

A
JIL

m

A

m

1"

"
r1

”

”

LA
JI

2
ror
mir

2D
MOT

BS

Red +
RepumpParascope

λ/2

λ/2

f = -30mm
cylindrical

r2

roic

ro

"

PBS

ich

λ/4
λ/2

2”

λ/2

PD

Po
la

man
Zee er
Slow

Pola

mirr

or 1"

m
irro

1"

irr
or
2”

irro
r

m

PBS

ris

PBS

1"

Polar
is mi

Po
la

BS

Lens

or
mirr

1"

30
c
dic c
hro
ic

ris
Pola

Pic
ko
ff

1"

ror

r

2"

mir
s
lari
Po

2”
ror

Razor Blade
Stack

Beam
Dump

mir

irro

m

r
High powershutte)(taken out of mount

λ/4

ris

la
Po

f=500

99/1

PBS

Lens

T
MO

Servo PD

Servo PDf = 80mm
cylindricalmirror
2”

Po
la
ris

m

f = -22mm
cylindricalBS

r1

Dichroic
R 461 nm
T 689 nm

BS

rror

PBS

"

1"

off

r

Pic
k

irro

m

ris

ris

QPD

f = 150mm
cylindrical

mir

λ/4
LC WP

1"

ror

Td
MO

mir

mirr
or

Razor Blade
Stack

PD

JILA

EOT Faraday
RotatorAb
Pro sorp
be tio
n

f

MOT

PBS

Lattice

PBS

of

1"

irro

PD

ck
Pi

m

ff
ko
Pic

irro
m

r

ris

Po
la

Spin Pol. +Red Imagingmirro

Polaris mirror

1"

"

r1

MOT dic
hroic

Pola

λ/4

mirr

1"

Pola
ris

or
mirr

mir
ro

1"

ris

rror

2”

λ/4
λ/4

λ/4

mirror 1”

is mi

Lens

MOT
Fl. PD

Polarnmirror

r 2”

or

1"

Gla

mir

ror

2”

ris

or
mirr

1"

Pola

mo

un

t

30 cc dichroic

HR 813, HT 698
dichroic
813 ﬁlter

Gla
n

Po

sm

r1
irro

"

ra
Ca
me

2"

o

irr

m

m
o

tte

irro
"

PBS

Flipper mirror
1"
rror

f = -22mm
cylindrical

0th order
FN servo

mi
JILA

f=100

PBS

Flipper mirror

r1

Mako 2
cameram

f = 200 mm

LA

JI

Clock
Laser

r

f = 80mm
cylindrical

e
lad
rB
zo ck
Ra Sta

ODT

irr

m

shu

Serv
o
PD

irr

mirro

r2
"

ow
er

"
r2

Iris

or

f = 50 mm
cylindrical

r 1"

JILA

hp

0th
FN ordeser rvo

To topmezzaninef = 1000mm
cylindrical

AOM3

Oblique
clock

lari

f=500

Hig

f
PickofObjective

BS

2"

1st
FN orde
ser r
vo

Ch
fan risti
cy an’s
FN
bo
x

λ/2

or

ins
e
ND rtabﬁlt le
ers

λ/4

Len
s

irr

f=200 mm

Gla
n

m

Cu
s

tom

λ/4

f = 250 mm

AOM2

irr

m
or

Iris

2"
A

JIL
r

JI

irro

m

r

1"

LA

LA

1"

irro
m

"
r1

irro

m

JI

Pi
ck

or
irr

of

f

m

r 2"

1”

λ/2

Razor BladeStack

EOT Faraday
Rotator

PBS

Servo PD

Lens

PBS
Lattice

CameraAndor iKon

mirro

Figure 3.2: Optical layout of the middle optical breadboard. Atoms are trapped in the center ofthe vacuum chamber shown in the middle of the diagram.

PBS

PBS

rro

r

Lens

MirrorLow res. imaging

Telescope

Mirror mounton motorized rail

Mi

Lens

High res.
imaging

Lens
MOT
beams

NPBS

Lens

λ/4

BS
f = 300

M

Focusing
Dichroic
lens

or
irr

Shutterλ/4Obliqueclock

f = 300

Top
mirrorPD

Figure 3.3: Optical layout of the top optical breadboard.

beams.

f = 75PD

CameraKuro Princeton

Vertical
ODT

TenS4

Mako1
CameraTransp.
beam

49

PBS

BS

50

Figure 3.4: One side of the Al enclosure that surrounds the main experiment.

3.1.3

Overview of cooling and trapping 87 Sr

Preparing a Fermi degenerate gas starts with cooling and trapping 87 Sr. For a more detailed
account of cooling 87 Sr to µK temperatures using a two-stage MOT, see Refs. [100, 101]. Refs. [99,102] discuss trapping Sr in a crossed optical dipole trap (XODT); the latter reference is not the
JILA experiment but the preparation has many overlaps.
A simplified level diagram of 87 Sr is shown in Fig. 3.5. The first stage of laser cooling begins
on the 30.4 MHz 1 S0 → 1 P1 transition. A high flux of hot atoms first leave a commercial AO Sense
oven, and are slowed in an AO Sense Zeeman slower and transversely cooled in two 2D MOTs
before entering the main chamber. We have had to replace the AO Sense oven multiple times over

51

3

1

Blue MOT
Γ/2π = 30.4 MHz
461 nmP1

Repumpers
707 nm,679 nm~1:20 000
1

D2

S1

Transparency
beam
688 nmΓ/2π = 4.4ΜHz

1:2

Red MOT, TenS4
Γ/2π = 7.4kHz
689 nm

3

PJ

2
1
0

Clock transition
Γ/2π = 1 mHz
698 nm1

S0
Figure 3.5: Simplified energy level diagram of 87 Sr.

the course of my PhD. During replacement, we have frequently seen a ‘volcano’ of Sr on the exit
aperture of the oven, where Sr sticks and accumulates. Eventually the Sr creates a bridge thatshorts two disconnected pieces of the chamber, which degrades the temperature stability of theoven and significantly reduces the oven’s clear aperture. We recently replaced our oven with themore modern AO Sense oven that has improved temperature control and are hoping this fixes ourintermittent issues.
After the atomic beam enters the chamber, atoms are trapped in a ‘blue’ MOT that has a
30 G/cm gradient. The 1 P1 transition is almost closed, with a 1:20,000 chance of decaying to the
1D

2 state [103], which then either decays to the

3 P state or the 3 P state with 1:2 odds.
2
1

The

52
atoms that decay to 3 P2 are then trapped in that state due to the 40 s lifetime of the state, which
limits the lifetime of the MOT to a few 10s of ms. Multiple repumping transitions can be used; we
historically pump atoms to the 3 S1 state, which requires two repumping lasers at 679 nm and 707
nm to pump atoms out of the 3 P2 and 3 P0 states; the latter becomes populated from decay out of
the 3 S1 state.
Atoms are then loaded into a second stage ‘red’ MOT on the 7.4 kHz 1 S0 → 3 P1 transition.
Cooling on this narrow line transition with a J = 0 ground state requires two lasers to effectively
cool the atoms and pump them out of nuclear spin states that are not trapped [104]. In the firstbroadband red MOT stage, the frequency of the red MOT cooling lasers is modulated to capture
and cool more atoms from the blue MOT. The modulation is then gradually ramped down, as is
the intensity of the lasers, until the red MOT is operated at a single frequency with an intensity
below the saturation intensity. This allows us to achieve µK temperatures with roughly 107 atoms.
We perform optical evaporation of the atoms which requires the atoms to be trapped inside
an ODT. In particular, we employ two ODTs, one ‘reservoir’ horizontal ODT (HODT) that hasa sheet-like geometry with waists of 340 µm and 20 µm in the horizontal and vertical directions,
respectively. The second ODT is a ‘dimple‘ vertical ODT (VODT) that is aligned at roughly
10◦ with respect to gravity and is circular with a 30 µm waist. While the reservoir ODT traps
many atoms and sets the trap depth, the dimple ODT sets the density of atoms, which can becorrespondingly higher than in the reservoir due to the small waist. The ODT is on during thebroadband and single frequency red MOT, and we adjust the frequency of the MOT to cool directly
into the ODT.

3.1.4

Measuring trap frequencies

Trap frequencies are measured using two different methods. In the first technique, we pulse ona second dipole trap that is slightly offset from the cloud and look at the center of mass oscillation
frequency inside the dipole trap. In particular, we use a lattice laser with the retro-reflected beam
blocked to form the extra dipole trap. The dipole trap is then extinguished at variable times and53

the center of mass position of the atoms is measured after a fixed time of flight.
Second, we parametrically heat the atoms by modulating the intensity of the dipole trap.
This is achieved by modulating the RF power of an AOM. Since we have servos that stabilize our
ODT intensity, we add the modulation either at the output of the servo or at the servo reference,
depending on if the modulation frequency is outside or inside the servo bandwidth, respectively.
The heating rate is maximized at twice the trap frequency [105]. We do not measure atom loss,
which requires high modulation intensites that can lead to nonlinearities in the drive signal andthe excitation of atoms beyond the minimum of the trap where the harmonic approximation holds.
Instead we look at the width of the cloud after time of flight, which is sensitive to the temperatureof the atoms in the trap. Minimal heating can be easily measured with this technique and small,clean modulation frequencies can be applied that result in high SNR heating resonances.
Examples of the dipole oscillations and parametric heating resonances are given in Fig. 3.6.
Dipole oscillations are performed by exciting the gas along a particular direction and thus cannot
separately measure trap frequencies along x and y. In contrast, we clearly observe the asymmetryin our VODT through parametric heating measurements. The trap frequencies extracted using thetwo methods agree well, as shown in Fig. 3.7.

3.1.5

Transparency beam

A large density in the dimple of our ODT is advantageous for efficient evaporative cooling.
For a ground state with SU(N ) symmetry, the evaporation rate is given by Γ = (N − 1)nσvrel ,where N is the number of spin components, n is the single-component density, σ = 4πa2 is thes-wave collisional cross-section, and vrel is the relative velocity (see Chapter 4). Inelastic lightassisted collisions [106] and reabsorption of the cooling light, which leads to an effective repulsion
between atoms [107], scale with the density, making it challenging to achieve high density samples.

54

Figure 3.6: Left column: Parametric modulation of the ODT intensity leads to heating resonancesat twice the trap frequency. The top figure shows the vertical trap frequency and the bottom thehorizontal frequencies. Right column: Trap frequencies are extracted through dipole oscillations of
the atomic cloud.

To overcome light-assisted loss, we render atoms inside the dimple transparent to the cooling light.
This is achieved by applying an additional laser to atoms in the dimple that shifts the cooling lightout of resonance, see Fig. 3.8. The large density of atoms that accumulate inside the dimple thenthermalize with atoms in the surrounding reservoir.
Using a transparency beam to overcome light-assisted loss has been demonstrated in a low
phase-space density MOT [108], and was used in bosonic 84 Sr to reach the critical phase-space

55
Parametric heating
Sloshing

160
Horizontal trapping frequency (Hz)Vertical trapping frequency (Hz)

700
600
500
400
300

140
120
100
80
60

0.1

0.2

0.3
0.4
0.5
HODT*coef (V)

0.6

0.7

0.2

0.4

0.6
VODT (V)0.8

1.0

Figure 3.7: Trap frequency measurements for various intensity setpoints extracted using bothparametric heating and dipole oscillations. The left shows the vertical trap frequency and the rightthe horizontal trap frequencies.

density for Bose-Einstein condensation without any evaporation [109]. The critical phase spacedensity is reached when nλ3dB > 2.612, where λdB is the thermal de-Broglie wavelength and nλ3dB ∝nT −3/2 . In comparison, Fermi-degeneracy does not have a critical point and the transition is instead
continuous. While the effects of Fermi degeneracy begin when nλ3dB > 1 (T /TF < 0.6), if one wants
to reach deep degeneracy, say so that one can load a band-insulator state into an optical lattice withover 90% filling, the requirements are much more strict and one then needs nλ3dB > 20 (T /TF < 0.1).
This makes using the transparency beam alone as a replacement for evaporation in fermions morechallenging. In addition, hyperfine structure creates additional complication, since a sufficient AC
Stark shift needs to be created for each nuclear sublevel. Transparency from both the trapping andstirring second stage MOT lasers is also required [110].
Accordingly, the transparency beam is blue-detuned from the 3 P1 to 3 S1 transition by 25
GHz. The laser spatially overlaps with the VODT (see Fig. 3.3) but has a slightly larger waist
to ensure all atoms in the dimple do not absorb cooling light, and delivers 5 mW of power to the56

Figure 3.8: Potential energy of the 1 S0 ground state atoms and 3 P1 excited state atoms in the
ODT. The transparency beam shifts the cooling light out of resonance for atoms in the dimple.
(a)

(b)

1400

1100

1200
1000

Ushift [-1]

Ushift [-1]

900

800

800

600

700

400
200
0

F=11/2

1000

-4

-2

600

F=9/2
0

mF

2

4

500

-4

-2

0

2

4

mF

Figure 3.9: Calculated Stark shift of the excited 3 P1 stirring/trapping states for different nuclear
spin states. The linewidth of the 3 P1 state is Γ = 2π × 7.5 kHz.

experiment. The beam is linearly polarized along x, while a small magnetic field is applied along z(see Fig. 3.10). In this configuration, the smallest Stark shift still detunes the excited state by ∼200

57

z
x

B Tsp

TenS4 laseryg

B TenS4
Transparency
beam

Reservoir

DimpleFigure 3.10: Schematic of the reservoir and dimple dipole traps, as well as the propagation direction(single arrows) of transparency beam and the TenS4 laser, and their polarization (double arrows).
The magnetic field while the transparency beam is on is denoted by Btsp and the magnetic field
while the TenS4 laser is on is represented by BT enS4 . The polarization of the TenS4 laser is along
the magnetic field, so that the ellipticity is zero.

linewidths, see Fig. 3.9, and keeps spontaneous scattering events beyond relevant experimental timescales. In general the Stark shifts are sensitive to the transparency beam’s polarization, and we
calculate that circular light would not give a sufficient shift to all spin states for both cooling lasers.
The effect of the transparency beam on the sample is shown in Fig. 3.11. Here, we apply thetransparency beam to the dimple part of the XODT and apply 3 P1 light to remove atoms from thereservoir. The atoms in the dimple do not absorb the 3 P1 light and thus remain in the trap after
the pulse, as shown in the image on the right.
The transparency light is derived from an extended cavity diode laser (ECDL) that is filteredby a volume Bragg grating (VBG). Without the VBG, residual amplified spontaneous emission

58

Figure 3.11: Atoms remaining in the xODT after applying a cleanup pulse to the xODT (left image)with the transparency beam on. Only atoms that don’t absorb the cooling light remain.

(ASE) on-resonance with the 1 S0 − 3 P1 transition limits the lifetime of atoms in the dimple to 250
ms. With the filter, the lifetime is extended to 5 s.
To separately extract the number of atoms in the dimple and reservoir, the HODT is extinguished so that atoms fall under gravity. Atoms in the dimple are guided by the VODT which has
a small angle with respect to gravity, while atoms in the reservoir undergo free expansion. This
spatially separates the atoms in the dimple from atoms in the reservoir, and respective atom numbers are then measured using absorption imaging. The number of atoms in the dimple saturates
at 5 × 106 , i.e. about 50% of the total atom number, after 400 ms of cooling with the transparency
beam. This leads to a total density of 2 × 1014 atoms/cm3 , corresponding to a three-body lifetimearound 10 s [54]. T /TF is then determined for atoms in the dimple by calculating TF and measuring

59
the temperature. The latter is determined by fitting all atoms to a Gaussian after time-of-flight, andgives a temperature of 2 µK with T/TF = 1.9. Lower temperatures can be achieved by adjusting
the MOT light, but without further improving the phase space density.
After the transparency beam is applied for 400 ms, we proceed with standard evaporation
by lowering the trap depth of the HODT but keeping the VODT high to maximize the density
throughout. We evaporate with all 10 spin states to maximize the evaporation efficiency, as discussed further in Chapter 4. For final trap depths of a few 100 nK and a vertical (radial) trap
frequencies around 200 Hz (100 Hz), we can reach temperatures as low as 0.07 times the Fermi
temperature, with roughly 50,000 atoms per spin state, after around 1 s of evaporation.

3.1.6

TenS4

After evaporation, the spin distribution is manipulated to subsequently load the optical lattice
into a band-insulating state. Previous attempts to manipulate the spin distribution through opticalpumping after evaporation resulted in light-induced heating of the atoms. Here we detail a method
to manipulate the spin distribution of the gas where the atoms remain confined to the ground state
so that there are no spontaneously emitted photons. This method can produce a spin-polarized gas
with >90% spin purity with minimal heating.

3.1.6.1

Stark shift

Our spin manipulation technique builds off of previous work that used optical beams tocreate differential spin forces in alkaline-earth atoms during time of flight [111, 112, 113]. In theseprevious experiments, atoms in different spin states could be separated after releasing them fromthe harmonic trap to image individual spin components. Our method, the Tensor Stark Shift
Spin Selector (TenS4), creates a spin-selective repulsive force on the atoms from the tensor Stark
shift of a laser while the atoms remain trapped in the XODT. This force can overcome the small
confinement that holds the atoms after evaporation and can subsequently push the atoms out of
the trap. In addition, our beam parameters have to be carefully chosen such that the spin state we

60
(a)

(b)
F=7/2

+1414.12

F=9/2+283.86

F=11/2-1179.29

TenS4

-9/2
3

-7/2

P1

0.18

266 MHz

-9/2
-11/2

-7/2

-9/2

S0 , F = 9/2

0.25
0.32

-7/2
0.02

1

-5/2

0.42

-5/2

0.33

-7/2

-3/2

-5/2

0.48

-1/2

-3/2

0.51

1/2

-1/2

0.48

3/2

1/2

0.42

3/2

0.32

7/2

5/2

0.18

7/2

P1 , F = 9/2

1

9/2
9/2

0.82

1

11/2 3S0 , F = 9/2

P1 , F = 11/2

0.18

0.33
0.05

5/2

3

0.82

7/2
0.65

0.44
0.11

9/2
0.18

0.49

5/2

0.51
0.51

0.18

7/2
0.32

0.25

3/2
0.38

0.55
0.27

5/2
0.42

0.09

1/2
0.27

0.55
0.38

3/2
0.48

0.01

-1/2
0.18

0.51
0.51

1/2
0.51

0.01

-3/2
0.11

0.44
0.65

-1/2
0.48

0.09

0.05

0.82

-9/2

-3/2
0.42

0.32
0.49

0.18

1

-5/2

0.18
0.82

0.02

1

9/2

S0 , F = 9/2

Figure 3.12: (a) 3 P1 hyperfine structure and TenS4 detuning. The TenS4 detuning is referenced to
the zero magnetic field F=11/2 transition. (b) Clebsch-Gordan coefficients squared for transitions
from the 1 S0 ground state to the 3 P1 , F = 11/2 and F = 9/2 transitions. The F = 11/2 andF = 9/2 states have magnetic field sensitivities of 380 kHz mF /G and 85 kHz mF /G, respectively.

desire to keep in the trap feels no force. Any residual force on this state will heat the atoms in the
trap. The ac Stark shift for a ground state atom with nuclear spin mF is given by [114]∆ν = ∆κs + ∆κv mF ξêk · êB + ∆κt β U0 ,(3.1)

where U0 is the trap depth, ∆κs , ∆κv , and ∆κt are the scalar, vector, and tensor shift coefficients,and ξ is the beam ellipticity. The light propagation direction and quantization axis are given
by êk and êB , respectively. Finally, β = (3|ˆ
 · êB |2 − 1) 3m2F − F (F + 1) , where ˆ is the beampolarization vector [52, 114]. The TenS4 laser is detuned from the 3 P1 , F = 11/2 state as shown in
Fig. 3.12, and is offset from the atom cloud to create a spin-selective repulsive force on the atoms.
The beam has a 30 µm waist with a peak intensity of 0.15 kW/cm2 , and is linearly polarized along
the magnetic field (ξ = 0), which has a strength of 5 G. Due to the slight tilt of the beam, themagnetic field mostly points along y but also has a small component along z (see Fig. 3.10). The
scalar shift provides an appreciable shift to all spin states, while the tensor shift, ∝ m2F , produces
a differential force between atoms with different |mF |. A small differential shift between positive
and negative mF states is further created with a magnetic field, which changes ∆κs and ∆κt due
to a slightly different detuning from the atomic resonances.
The Stark shift cancels for atoms in a particular spin state at the proper detuning. For nuclear

61
(a)

(b)

-9

TenS4 peak potential (uK)

50

2

-7
2
-5
2

-3

30

Out[ ]=Out[ ]=

1

10

2
3

0

2
5
2

100

Detuning (MHz)
200

300

400

500

7
2
9
2

2

-5

20

2

15

2

10

-1

20

-10

2

-7

25

TenS4 potential (uK)

40

-9

30

2

-3
2

-1
2

1
2

5

3
2

0

-5

5

-40

-20

2

z position (um)
0

20

40

60

7
2
9
2

Figure 3.13: (a) Height of the TenS4 potential versus detuning. At 266 MHz, the TenS4 potential
is zero for mF = +9/2. (b) Potential of the TenS4 beam as a function of position at a detuning of
266 MHz. The beam is offset from the atoms by roughly 1/3 the trap waist (dashed line) to createa varying potential across the cloud. Plots are shown for a 5 G magnetic field.

spin state mF = +9/2, this occurs at 266 MHz, where the polarizability from 3 P1 , F = 11/2 cancels
the polarizability from 3 P1 , F = 9/2. This is demonstrated in Fig. 3.13 (a). The spatial dependenceof the TenS4 potential for the different spin components is shown in Fig. 3.13 (b).

3.1.6.2

Experimental implementation

The TenS4 laser is overlapped with the VODT and the transparency beam, as shown in
Fig. 3.3, which facilitates initial alignment. The TenS4 and transparency beam also share the same
telescope to save space. The laser, which is at a 10◦ angle with respect to gravity, is then offset fromthe atomic cloud to provide a force both vertically and horizontally. The combined potential due tothe ODT, the TenS4 laser and gravity is shown in Fig. 3.14. With only a 5 G applied magnetic field,a residual attractive potential remains for mF = −9/2 and −7/2. We thus conventionally remove
these spins via optical pumping prior to evaporative cooling. Our optical pumping is performed
using σ + light that is selectively tuned to the 1 S0 → 3 P1 , F = 9/2 resonance. First we apply
a few ms pulse that is on resonance with the 1 S0 , mF = −9/2 to 3 P1 , mF = −7/2 transition.
We then step the frequency and apply a second pulse on resonance with the 1 S0 , mF = −7/2 to62

Potential (µK)

1
0
-1
-2
-20

0
20
40
Vertical position (µm)

Figure 3.14: Combined potential due to the ODT, the TenS4 beam, and gravity. Dashed curves
represent mF = −9/2 and mF = −7/2, which are removed through optical pumping.

3 P , m = −5/2 transition. This allows us to remove the m = −9/2 and m = −7/2 populations
1
F
F
F

by 90%, as measured after evaporation Fig. 3.17. Going to a larger magnetic field and aligning
the beam horizontally to provide maximum force along the vertical direction would remove theneed to optically pump. A magnetic field of 50 G would more than satisfy this requirement. The
spin-polarized gas can then be loaded into the lattice and coherently manipulated if one desires towork with a different spin state.
When we initially implemented the TenS4 laser, we measured an unexpected loss of atoms.
Loss due to the TenS4 laser can be described byṅ = −n/τ − 2Kn2 − k3 n3 ,(3.2)

where the first term represents single-particle photon scattering with a lifetime of τ , and the second

63

term represents a two-body loss term with a loss rate of K. The vacuum-limited lifetime of theatoms was measured to be 120 s, significantly longer than any relevant experimental timescales.
The last term represents three-body inelastic collisions with three-body loss coefficient k3 , which
for 87 Sr give us a lifetime around 10 s, much longer than the observed scattering lifetime [54].
Published photoassociation (PA) spectra on 87 Sr is limited to 1 P1 [115], where the spectra
is relatively simple due to the broad linewidth, which causes the hyperfine spectra to be unresolved. Narrow line photoassociation was measured on 3 P1 for bosonic 88 Sr [116], but no publishedmeasurements on 87 Sr exists. For 87 Sr, the narrow linewidth 3 P1 transition is predicted to have acomplex spectra due to the resolvable hyperfine structure and large nuclear spin [117]. To measure
photoassociation spectra, we evaporate the atoms to a 500 nK trap depth and then ramp up the
HODT to a few µK. The atoms are then exposed to the TenS4 laser at a power of 10s of µW for 100
ms - 1 s in a 5 G magnetic bias field. At this power and trap depth, the TenS4 does not modify the
optical potential. The gas is then released from the dipole trap and the atom number is counted
using absorption imaging. At a PA resonance, the atom number drops as hot molecules are formedthat cannot be confined to the trap. Fig 3.15 shows the atom loss spectra for a variety of detunings
near the 3 P1 , F = 11/2 resonance. Due to the F = 9/2 and F = 7/2 states, resonances exist aboveand below F = 11/2. The TenS4 laser is referenced to our red MOT trapping laser, which is itself
locked to a laser that has a ∼1 Hz linewidth. As such, the zero, which refers to the zero magnetic
field value of the F=11/2 resonance, can be trusted to within 10s of kHz. Clear PA resonances
exist around 80 MHz and 160 MHz and a broad loss feature consisting of many resonances is shownin the -400 MHz to -200 MHz region. The other peaks at -60 MHz, -100 MHz, -140 MHz, and320 MHz consist of a single data point and thus need to be remeasured to confirm the existenceof a resonance at these frequencies. The large intensities used for the measurements result in alarge single-particle scattering rate near the 3 P1 resonance, and mask any PA resonances in thisregion. A more detailed scan at the TenS4 detuning, taken at a variety of powers, did not reveal aNormalized atom number

1.0

64

TenS4

0.8
0.6
0.4
0.2
0.0

400

300

200

100

0

100

200

Detuning from 3P1, F = 11/2 (MHz)

300

400

Figure 3.15: Photoassociation spectra. The blue data points were taken with 45 µW of power, andthe red with 15 µW of power.

resonance.
Single-body loss is due to photon scattering with the TenS4 light. The predicted light scattering lifetime 266 MHz from resonance with the described beam parameters is 10s of ms, twoorders of magnitude above the measured lifetime. However, the lifetime can be limited by amplified spontaneous emission (ASE) from the laser diode at the 3 P1 resonance frequencies. ASE
can be suppressed through spectral filtering of the light using a Fabry-Perot cavity. In particular,
we implemented a cavity with a 10 MHz linewidth and a finesse of 200 to suppress on-resonancescattering by 3 orders of magnitude. After installation of the cavity, the lifetime of atoms became
consistent with the off-resonant scattering rate.
The optics of the TenS4 beam are shown in Fig. 3.16. The TenS4 laser is locked to an
ultrastable laser (‘red master’) at 689 nm. This ultrastable laser is also used to lock our 3 P1 MOT
beams. The light is then filtered with a Fabry-Perot cavity that has roughly 50% transmission
before being sent to the main experiment. The Fabry-Perot cavity is locked to the TenS4 laser

65

Figure 3.16: Optics setup for the TenS4 beam. Light from an ECDL is locked to an ultrastablelaser with a roughly 1 Hz linewidth at 689 nm (red master). A Fabry-Perot cavity is used to filter
the TenS4 light before being sent through a fiber to the experiment (to Sr2).

using PDH locking.

3.1.6.3

Spin purity measurements

To spin polarize the gas, we first evaporate down to a few 100 nK trap depth and then nonadiabatically turn on the TenS4 laser for a few ms. The TenS4 spin purity is measured by looking
at the total atom number using blue absorption imaging, and by performing spectroscopy on the
3P

0 transition after the atoms are loaded into a 3D optical lattice.

Fig. 3.17 (a) shows the total

66
(b)
(a)

Atom number

2 x 105

2 x 104
TenS4 application time (s)Figure 3.17: (a) Total atom number versus TenS4 application time. The majority of atoms are
removed after 100 µs. (b) Spin-resolved measurements of the mF = +/ − 7/2 and mF = +/ − 9/2
states. After 1 ms, single-particle scattering is visible. Measurements in (a) and (b) were taken on
different days with slightly different total atom numbers.

atom number after the TenS4 laser is applied for variable times. After 100 µs, most atoms in otherspin states are removed. Spin-resolved measurements are shown in Fig. 3.17 (b), giving > 90%
spin purity after 1 ms application time. For longer times, single particle scattering degrades the
mF = +9/2 population. At 5 ms, 96% spin purity is measured.
We also measure T /TF after applying the TenS4 laser for variable times, as shown in Fig. 3.18.
The TenS4 laser does not change the temperature for short application times, since the atoms do
not have time to change their momentum during the pulse. Around 10 µs, the pulse begins to heat
the atoms but atoms do not have time leave the trap. With a pulse duration greater than 1 ms, theheated atoms have enough time to leave the trap and again cause minimal heating to the sample.
Any single collision between an mF = 9/2 atom and another atom leaving the trap would cause
heating. However, this issue is avoided since the time for atoms to leave the trap is much less than

67

Reduced temperature T/TF

0.5
0.4
0.3
0.2
0.1
0.0

10 6

10 5

10 4
Pulse duration (s)10 3

Figure 3.18: T /TF after switching on the TenS4 laser for variable amounts of time. After a longenough application time, the sample spin polarizes with minimal heating.

the collisional time after evaporation. If we do not optically pump mF = −9/2 and mF = −7/2
prior to evaporation, we notice residual heating after applying the TenS4 laser. Since a residual
potential exists for these spin states, atoms undergo heating but not all atoms leave the trap; theatoms that remain in the trap heat the sample. With the TenS4 technique, we can demonstrate a> 90% spin polarized sample with minimal heating. The spin distillation technique enables us toload a single spin with T /TF = 0.1 into a 3D optical lattice to create a band insulator.

3.1.7

Lattice loading

The spin-polarized, deeply degenerate gas is then adiabatically loaded into the 3D opticallattice. To measure the entropy increase after loading, we perform a round-trip measurement,

68
first loading the atoms into the lattice and then back into the XODT. Afterwards, we extinguish
the XODT and measure T /TF by fitting the expanded gas cloud after time-of-flight. We measureapproximately 10% of heating throughout this process, confirming that the lattice is adiabaticallyloaded. We can reach trap depths above 100 Erec in each direction, sufficient to be in the LambDicke regime.
If the 3D lattice is loaded adiabatically from a low entropy Fermi gas, a band-insulator could
be formed with one atom per lattice site. To check this, we perform high-resolution fluorescenceimaging on the atoms while they are trapped in the optical lattice. We image atoms simultaneouslywith a high-resolution camera from the side (NA = 0.1) and from the top (NA=0.23). These images
allow us to extract the 3D optical density in the center of the trap, and the results are confirmed
using an inverse Abel transform from the image on the side [118]. Preliminary measurements show
over an 85% filling fraction in the center of the gas, consistent with a roughly adiabatic latticeloading given the finite temperature of the gas before loading the lattice. More information on thismeasurement will be contained in an upcoming publication.
It’s worth noting that the experiments discussed in Chapters 4, 5, and 6 study Fermi gases
and thus do not include this lattice loading step. However, this step is crucial for creating a
Fermi-degenerate 3D optical lattice clock.

3.2

Imaging techniques
High resolution imaging is a key tool in our experiment that allows us to look at the atomic

density distribution and at locally varying frequency shifts. In this section, we detail the techniques
we use to image the atomic density distribution in-trap and after time-of-flight. In subsection 3.2.1
we describe how we experimentally implement absorption imaging, which we use to image atomic
clouds that have expanded to a low density after time-of-flight. When imaging in-situ clouds,
we instead use high-intensity fluorescence imaging, which we discuss in subsection 3.2.2. We thendetail our various imaging paths in subsection 3.2.3. In subsection 3.2.4 we describe how to measurecamera parameters such as the read noise and the gain. Finally, in subsection 3.2.5 we discuss how

69

relevant imaging parameters such as the magnification and the imaging resolution are extracted.

3.2.1

Imaging expanded gases

To image expanded gas clouds, we use the standard method of absorption imaging. Absorption imaging looks at the transmission of an atom cloud after it is exposed to on-resonanceprobe light. For low probe intensities, the transmitted intensity I is given by the Beer Lambert
law I = I0 e−

R

σ0 ndz , where I is incident intensity, σ is the low-intensity scattering cross section,00

n is the atomic density, and z is the imaging probe direction. Taking into account saturation at
higher intensities and integrating along the atomic cloud gives us the more general formula for thenumber of atoms per pixel [119]N=A
2log(Sb /Sa ) +(Sb − Sa ) ,σ0
Γtp g q(3.3)

where A is the effective pixel area2 , Sb (Sa ) is the number of counts in area A without (with)atoms, tp is the imaging pulse time, Γ is the spontaneous emission rate, g is the camera gain, andq is the detector quantum efficiency. Note that the above formula ignores higher order effects suchas reabsorption, which will be discussed further later in this section.
The absorption imaging signal can be extracted using three images that are taken in succession. First, an imaging pulse is applied to the atomic cloud and the intensity transmission is imaged
onto the camera, which gives counts S̃a . A second ‘bright’ image is taken a few ms after the first(depending on the readout time of the camera) without atoms, so that the incident intensity can bemeasured with counts S̃b . A final ‘dark’ image is then taken without the imaging probe light andwithout atoms, which measures the background counts due to dark counts and stray light, Sof f set .
This image is then subtracted from the bright (Sb = S̃b − Sof f set ) and atom (Sa = S̃a − Sof f set )frames before calculating the number of atoms per pixel according to Equation 3.3. Fig. 3.19
shows an absorption image of our cloud after evaporation with roughly 5 × 105 atoms. Fitting thisimage to a Fermi-Dirac distribution allows us to extract the temperature of the gas according to

70

Resid.

Column density (a.u.)

1.0
10
0
10

0.8
0.6
0.4
0.2
0.0

Fermi-Dirac
Thermal

0.1
0.0
0.1
0.0

2.5

5.0

10 0 10v (mm/s)

7.5 10.0 12.5 15.0
Velocity (mm/s)

Figure 3.19: Fit of the atomic cloud to a Fermi-Dirac and a thermal distribution after the Fermi
gas is released from the crossed optical dipole trap. The Fermi-Dirac distribution minimizes theresiduals of the fit. The Fermi-Dirac fit extracts a T /TF = 0.08. Inset: Absorption image of an
expanded Fermi gas with a 10 ms time-of-flight.

Equation 2.19; here T /TF = 0.08.
A calculation of the effective atomic cross-section must include the relevant hyperfine manifoldof the ground and excited states, as well as the optical pumping dynamics throughout the imaging
pulse. We perform imaging on the broad (Γ = 2π × 30.4 MHz) 1 P1 state, where the large scattering
rate allows high SNR images. We also have access to spin-resolved absorption imaging using the‘red’ narrow 3 P1 transition. For 1 P1 , the excited hyperfine manifold is unresolved so that all excitedstates are necessary to include in the calculation. Similar to Ref. [120], we calculate the effective
cross section for 1 P1 using a rate equation model, and confirm its validity with a master equation
2

The effective pixel area A is equal to the camera’s pixel size multiplied by the imaging resolution.

71
calculation. For linearly polarized light on-resonance with the 1 P1 F = 11/2 state, a saturation
parameter of s = 0.17, and a 3 µs imaging pulse, the cross-section is 0.49 times the bosonic crosssection of 3λ2 /2π, where λ is the imaging probe wavelength. Imaging pulses of only a few µs ensurethat blurring due to the directional absorption and diffusive emission of the imaging light is kept
small. For a 1 µs pulse time, the atoms undergo a 500 nm displacement assuming a fully saturated
transition.
Experimental imperfections such as polarization impurity can modify the effective cross section from the theoretical result. We perform two separate measurements to double check the
calculation. First, we measure the local density fluctuations of a thermal gas. As detailed in Chapter 2, a thermal gas exhibits Poisson statistics, with ∆N 2 /N = 1. By analyzing the noise versus
mean atom number in different regions of the gas, we can extract the atom number which provides
a cross-check for the effective cross-section. A thermal gas can easily be created using for example
parametric heating on evaporated samples. Additionally, quantum projection noise can be used
to extract the atom number. Measuring the noise between two regions of the atomic cloud in a
3D optical lattice for a fixed pulse time and dead time provides a calibration of N . All of thesemethods agree on the effective cross-section within 20%. These measurements can be performed at
low probe intensity, where only the first term in Equation 3.3 contributes to the result, so that A/σ0
can be extracted. From a separate measurement of the effective pixel size A (see subsection 3.2.5),
the cross-section can be calculated. The probe intensity can then be varied and the coefficient infront of the second term adjusted so that the atom number remains fixed for different incident
intensities.

3.2.2

In-situ imaging

In-situ, atomic densities are much higher and collective effects such as multiple scattering anddipolar shifts can distort the absorption imaging signal. Typically we think of emitters as being
independent, so that the light scattered by individual atoms does not interact with the rest of thesample. This approximation breaks down at the high in-trap densities of over 1014 atoms/cm3

72

we demonstrate after evaporation. Light that is scattered off single atoms can be reabsorbed byother atoms, which destroys the spatial information of the scattered photons and thus scrambles
the imaging signal. Absorption imaging thus does not work well at high atomic densities, and otherimaging techniques must be considered.
Phase contrast imaging as well as dark ground imaging [121] extract the atomic density
distribution by looking at the phase change of highly detuned light across the atomic sample.
These techniques prevent optical absorption of the imaging light and thus avoid the common pitfalls
associated with absorption imaging. An alternative technique is high intensity fluorescence imaging.
Here, a high intensity probe beam is applied to the cloud on-resonance, and the spontaneously
emitted light is collected off-axis. If the number of probe photons is much higher than the numberof atoms, atoms are much more likely to absorb a photon from the probe beam as opposed toa re-scattered photon from another atom. High-intensity fluorescence imaging has been used toaccurately extract the density distribution of a MOT with densities as high as 1012 atoms/cm3[122]. We use it to accurately image in-situ Fermi-dirac density distributions with densities higher
than 1014 atoms/cm3 .
An example of in-trap and time-of-flight density images extracted using absorption imaging(left column) and high-intensity fluorescence imaging (right column) are shown in Fig. 3.20. At
central densities above 1014 atoms/cm3 , the optical depth is greater than 100 despite the apparent
reduced optical depth shown in the absorption images. Collective effects such as superradiancealong the probe beam cause the apparent optical depth to decrease. The apparent optical depth
increases quickly as the atoms are released from the trap and the density decreases, but at 500 µs
time-of-flight, the apparent optical depth only reaches 5. In contrast, high-intensity fluorescenceimaging correctly models the in-situ optical depth of 150, and the expected decrease in the opticaldepth after time-of-flight. The intensity of the fluorescence beam is set by looking at the number ofdetected photons on a camera when the probe intensity is varied. Around 1000 I/Isat , the number

73

Figure 3.20: Measured optical depth of an atomic cloud with densities greater than 1014 atoms/cm3 ,captured both in situ (top row) and after 500 µs time-of-flight (second row). Absorption imaging(left column) incorrectly extracts the optical depth due to collective effects such as superradiance.
The optical depth is conversely accurately extracted using high-intensity fluorescence imaging (rightcolumn). The images are extracted using a high resolution side imaging setup with an NA = 0.1

of photons saturates. As expected, we additionally measure that the total fluorescence counts,
integrated over the entire cloud, does not change after the cloud is expanded in time-of-flight.
The expected number of camera counts is given byS=Γ
× tp × q × g × N × SA ,
2

(3.4)

where tp is the imaging pulse time, q is the quantum efficiency, g is the camera gain, N is theatom number, and SA is the collection angle of the imaging system. Practically, it is challenging
to precisely extract the solid angle of the camera. We alternatively calibrate the fluorescence74
signal by using absorption imaging on expanded gas-clouds, where atom numbers can be precisely
determined. For a 5 µs pulse time, we expect 8 counts per atom on the Princeton camera (seesubsection 3.2.3). The atom number we extract with this method agrees with Equation 3.4 using
our calculated solid angle within 30%.
Fluorescence images are acquired in a single shot. However, background counts due to thecamera offset and stray light need to be removed to accurately extract the number of counts. For astable experiment where stray light does not drift over time, many ‘dark’ frames can be taken in a
separate experimental run and averaged before subtracting it from individual fluorescence images.
This allows greater SNR since it reduces the read noise of the processed image (see subsection 3.2.4).
We have found that when trying to extract spatial density distributions with on average less than
one photon per pixel, as will be shown in Chapter 5, it is necessary remove the average dark counton a pixel-by-pixel basis, since the mean dark count varies slightly across the CCD.
After time-of-flight, the imaged atoms are spread over many pixels and the fluorescenceimaging signal is more affected by the camera’s read noise. As such, it is easier to obtain high SNR
images using absorption imaging after time-of-flight.

3.2.3

Imaging paths

Our experiment contains four different imaging paths to access a range of resolutions andfield of views along different angles, as shown in Figs. 3.1, 3.2, and 3.3. Our highest-NA imaging
axis is along the vertical direction, where the 90 mm distance between the atoms and the 2 inch
wide objective lens provides us with an NA = 0.23 and a magnification of 13 (labeled ‘Princeton
Kuro’). We also have a low resolution camera with a wider field of view from the top (‘Mako 1’)that allows us to image for example the entire ODT prior to evaporation. The Princeton camera isa back-illuminated CMOS sensor with 1200 x 1200 pixels that are 11 µm × 11 µm with a quantum
efficiency of 85% at 461 nm. It has a specified read noise of 2.1 e− at a conversion gain of 1.59
ADU/e− with a 200 MHz readout rate/12 bit mode (ADU means analog-digital-units, that is,the conversion from electrons to digital counts). This camera is quite fast because each row is75
simultaneously read out in parallel; the 200 MHz readout rate implies the one million pixels canbe read out in around 10 ms. This makes it ideal for taking multiple images in fast succession, andreduces the amount of stray light that hits the detector during the readout.
From the side we have a high-resolution axis with an NA = 0.1 and a magnification of 11(‘Andor iKon’), while the low-resolution camera (‘Mako 2’) provides a wider field of view and moreeasily allows us to look at expanded gas clouds over a range of time-of-flights. The Andor iKon-M
934 is a CCD camera with 1024 x 1024 pixels that are 13 µm × 13 µm each. The quantum efficiency
is 80% at 461 nm. At a readout speed of 1 MHz (1 full frame per second), it has a read noise of 8e− , and a conversion gain of 0.5 ADU/e− with a 16 bit ADU resolution.
All of these cameras are set up to perform absorption imaging or fluorescence imaging on the
1P

1 transition, where the fast decay allows many scattering events in µs timescales so that high

SNR images can be obtained. We are also equipped with a red absorption path from the side sothat we can perform spin-resolved measurements on the 3 P1 transition.
The high intensities we use for fluorescence imaging require a tightly focused imaging beamwith a large power. We use a commercial Toptica system that delivers up to 30 mW of power to theexperiment for this purpose. The probe beam we use for fluorescence imaging is shown in Fig. 3.1(labeled ‘flour. imaging’). The beam is focused onto the atoms with a waist of 100 µm and is on
axis with the ‘Mako 1’ camera shown in Fig. 3.3. High-resolution fluorescence images can be takenusing the horizontal imaging system labeled ‘Camera Andor iXon’ in Fig. 3.2 (NA = 0.1) and thevertical imaging system labeled ‘Camera Kuro Princeton’ in Fig. 3.3 (NA = 0.23).
To switch between absorption imaging and fluorescence imaging, we use a flipper mirror on
the bottom mezzanine of our experiment. The small beam waist we use for fluorescence imaging
is not suitable for absorption imaging over a wide field of view. We thus switch between this fiber
port and a low resolution absorption imaging port with a flipper mirror. The low resolution fiber
port has a mm beam waist so that it can image large atom clouds, for example in the XODT.
Absorption imaging axes are also implemented for the high-NA side (labeled ‘absorption probe’)
and vertical (labeled ‘High. res. abs. imag.’) axes. For the vertical path, care is taken to reduce

76
the number of imaging fringes, as discussed in the following subsection.

3.2.3.1

Imaging fringe mitigation

Fringes that show up in the imaging system can distort the atomic density distribution andadd noise to the extracted atom number, and thus are essential to remove for accurate imaging.
Because absorption imaging requires multiple shots to perform the analysis, the technique is sensitive to varying fringes in the imaging system. Vibrations that are on the order of 10s of Hz cancause fringes due to either dust particles, residual etalons, or pointing fluctuations of the probe
beam to move between the atom and bright frames, such that the artifact remains in the analyzed
image.
To ensure we have high quality images with few fringes, the optical beam path needs carefulconsideration. First, the high-resolution vertical beam path (labeled ‘high res. abs. imag.’) is
made as short as possible, with no optics close to the edge of the optics table where there are moreenvironmental perturbations. This helps minimize the influence of pointing fluctuations at thefiber tip. In addition, few optical elements are used to reduce the number of possible etalons andpotential dust particles in the optics path. Etalons are particularly insidious in imaging systems,
since even a 100:1 intensity ratio can produce a fringe contrast of 20%. AR coatings are thus not
sufficient to completely remove this noise source. We tilt our imaging beam by 5 degrees withrespect to the viewports to avoid reflection-induced interference. The CCD is also tilted and has an
appropriate filter so that no stray light hits the camera. Care is taken in choosing the collimation
package of the imaging fiber so that a Gaussian beam exits the fiber. Typically monochromaticlenses are better for this purpose than molded aspheres, which can cause residual structure in theemitted beam.
At the output of the imaging laser (not shown in the optical beam path diagrams), we use
AOMs to switch on and off the imaging light and a shutter right before the fiber to block any
unwanted light. The shutter is opened a few ms before the AOM is switched on to produce a clean
pulse of imaging light at the atoms. During this time, zeroth order leakage of the AOM that is

77
coupled into the fiber can heat the atoms. We have found that it is desirable to have two switching
AOMs in series to reduce zeroth order light leakage to much less than a photon per atom over thesefew ms. RF electronics such as switches should also be properly terminated. Finally, if one is using
an injection locked laser for the imaging light, it is desirable to have on hand a diagnostic tool suchas a Fabry Perot cavity to ensure the seeding is adequate. We have found it is best to avoid thisissue entirely by not using injection locked lasers for our imaging light.

3.2.4

Measuring camera parameters

To get an accurate calibration of the atom number and fluctuations, it is necessary to measurecamera parameters such as the read noise and the gain. In a typical CCD camera, a two dimensional
array of pixels measures the intensity distribution across the array. Photons are converted toelectrons based on the quantum efficiency, q, and are then amplified by a gain factor g and readout on a pixel by pixel basis. The camera is thermoelectrically cooled to reduce the dark current
below other relevant noise sources. However, every camera has noise associated with the readout
process of the electrons. For a good camera, the read noise r is typically a few electrons. It canbe measured by looking at the pixel counts in a dark image, that is an image without atoms andwithout light, where the standard deviation of pixel counts S is given by ∆Sread = g r.
The CCD is also subject to photon shot noise, given by ∆n2p /n̄p = 1 with n̄p average photons.
The conversion of photons to electrons follows a binomial distribution with a success probability
given by the quantum efficiency, so that ∆n2e = n̄e where n̄e = q n̄p is the average number ofelectrons. The electrons are then amplified deterministically by the gain, giving a variance in the2number of counts due to shot noise ∆Sshot= g S̄ where S̄ = g q n̄p . The camera gain can thus bemeasured by looking at the noise of an illuminated CCD. CCDs are not necessarily illuminateduniformly, so that the analysis requires two frames to be taken and subtracted. A difference image
is calculated as Sdif f = Sa − Sb , where both frames are without atoms but with equivalent lightillumination3 . The read noise and shot noise are independent, so that total noise in a given frame
3

It is usually necessary to provide a normalization in the total counts between the two images to correct for slight78

can be calculated as
2
2
2
2
∆Sdif
f = 2∆Sread + 2∆Sshot = 2g S̄ + 2(g r) ,(3.5)

where the factor of two is due to the independent noise from two separate images and S̄ = S̄a = S̄b .
Note that if separate dark frames are subtracted from each image, the added read noise must
be included in this equation. If the same dark frame is used, however, the frame cancels in the2difference image. The camera gain thus appears as the slope of a line that plots ∆Sdif
f versus S̄.

Fitting the offset provides a consistency check of the read noise determined from the dark image
measurement. An example of this is given in Fig. 3.21, where 700 images were used to extract acamera gain of c = 1.63 ± .002 ADU/e− and a read noise of r = 2.4 ± .01 e− , consistent with the
Princeton Kuro data sheet and an independent calibration of the read noise using a dark frame.
Imaging artifacts such as residual fringes can increase the measured noise in the difference image.
The data in Fig. 3.21 shows measurements for both coherent and incoherent light, demonstrating
there is no added noise in the system due to interference effects. Data was taken for each gain
setting, but for Princeton’s 16 bit ‘gain combining’ mode, it was found that the gain was nonlinear.

3.2.5

Measuring imaging parameters

Two other parameters that must be measured are the imaging magnification and resolution.
The magnification can be measured using Kapitza-Dirac scattering [123, 99]. Here, lattice lightis quickly pulsed on which populates momentum states 2n~k, where n is an integer and k is thelattice wavevector. After time of flight, these momentum wavepackets separate, and the distance
between them can be used to accurately calibrate the magnification. For our side imaging system,
the magnification can also be calibrated by measuring the distance that the atomic cloud falls undergravity for a variable time-of-flight.
The imaging resolution can be extracted by measuring spatial correlations in a system withzero correlation length. This can be achieved by looking at the spatial correlation of independent
variations in the total intensity between the shots.

79

10000

Fit to incoherent data
Coherent light
Incoherent light

S2diff

8000
6000
4000
2000
0

0

500

1000

1500
S

2000

2500

3000

2
Figure 3.21: Variance of the difference image ∆Sdif
f versus the mean of the bright image S̄. Fitting
the data to a line allows one to extract the camera gain through the slope, and the read noise throughthe offset. Measurements are taken with both incoherent and coherent light to confirm the absenceof excess noise when using coherent light.

atoms in a 3D optical lattice [32, 124]. In a lattice, correlations in the excitation fraction canbe measured by performing Ramsey spectroscopy on the atoms. After measurement, the atoms
are projected into either the ground or excited state which produces binomial noise given by the
QPN. A finite imaging resolution creates spatial correlations in the image (not, of course, in theatoms), which can be calculated by looking at the average correlation between pixels that are aDE
certain distance j apart after standardizing the data: (pie − p̄e )(pei+j − p̄e ) /varpe , where pie is theexcitation fraction at pixel i, p̄e is the average excitation fraction over pixels i, and hi refers to an
average over all pixels i. The resolution can then be extracted by plotting the correlation versus
distance j and fitting the 1/e2 distance. A similar measurement can be performed by looking at
the spatial dependence of the number of atoms in a thermal gas, where Poisson statistics dictates∆N 2 /N = 1. In either case, it is crucial to double check that the resolution degrades as the image
is defocused. If the imaging system is outside the depth of field, the correlation will produce the80resolution defined by the NA of the imaging system, which is misleadingly given by the diffraction
limit, and will be rather insensitive to changes in the focus.

3.3

Stability transfer
Our ultrastable laser locked to a Si resonator is centered at 1542 nm, where high quality

telecom fiber components and fiber lasers exist. The transparency window of Si starts around 1µm, preventing construction of the laser at the 698 nm 1 S0 → 3 P0 clock transition. To bridge
the frequency gap between 1542 nm and 698 nm while maintaining the high Q optical stability of
the ultrastable laser, a frequency comb is used. In this section, we summarize our frequency comb
architecture and stability transfer measurements from the ultrastable laser to the 3D optical latticeexperiment. In subsection 3.3.1 we briefly introduce frequency combs. Subsection 3.3.2 details thestability transfer setup, and subsection 3.3.3 summarizes measurements of the frequency comb’s
instability. Finally, in subsection 3.3.4 we discuss measurements of the fiber noise that is introduced
when the clock light is delivered to the 3D optical lattice experiment.

3.3.1

Frequency comb technology

A frequency comb is a phase-stabilized mode-locked laser. The series of optical pulses at theoutput of a mode-locked laser is the result of the coherent addition of many optical modes of aresonator over 10s of nm in bandwidth, which are locked together as a result of nonlinearity in thelaser [125]. The series of pulses are not only regularly spaced, but are also phase coherent with one
another. In the frequency domain, this corresponds to many ‘comb’ teeth, as shown in Fig. 3.22.
Each comb tooth is defined by the equation [126, 127, 128]νn = n × frep ± f0 ,(3.6)

where n is an integer and frep = 1/T = c/L is the repetition rate, T is the time between pulses,
and L is the mode-locked laser round-trip cavity length. The carrier-envelope offset is given by f0
and is related to the phase slip per pulse ∆φ, which arises due to a difference between the group

81

Figure 3.22: Time and frequency representation of a frequency comb. In the time domain, a seriesof pulses regularly spaced by time T are generated from the output of a mode-locked laser. These
pulses are the result of the coherent addition of many optical modes of a mode-locked laser. The
carrier offset-frequency, defined as zero here, is due to a phase slip between the carrier and envelope
each pulse. frep can be locked to a microwave reference or a laser. The carrier-envelope offset f0
must also be stabilized. Figure adapted from Ref. [1].

and phase velocity in the mode-locked resonator. The carrier-envelope offset is related to the phaseslip per pulse throughf0 =1
frep ∆φ ,
2π

(3.7)

where ∆φ = (1/vg − 1/vp )Lωc where vg (vp ) is the group (phase) velocity, and ωc is the carrier
frequency. The difference between group and phase velocities arise from dispersion inside the82
mode-locked resonator.
The only two free parameters that describe every comb tooth are frep and f0 . These twoparameters can be robustly stabilized. Many applications that do not require state-of-the-art
precision lock frep directly to a microwave source. The repetition rate is generally an accessible
microwave frequency between 100 MHz and 1 GHz. However, the noise on frep gets amplified by n,which is on the order of 106 . As a result, it is more beneficial to lock a single comb tooth directly
to a stable laser.
The carrier-offset phase is separately stabilized and can be detected using a self-referencing
technique called an f-2f interferometer [129]. Here, a low frequency part of the optical spectrum isfrequency doubled using second harmonic generation. A heterodyne beat is then formed between
the doubled spectrum and a high-frequency portion of the comb spectrum to produce f0 , sincef0 = 2vN − v2N = 2(N frep + f0 ) − (2N frep + f0 ) .

(3.8)

Self-referencing requires the comb to span an optical octave. Mode-lock laser bandwidths canonly achieve 10s of nm bandwidth, meaning external frequency broadening must be performed.
Frequency broadening is generally achieved in optical fiber through the process of self-phase modulation, which uses the intensity dependence of the nonlinear index of refraction (the Kerr effect).
Optical fibers allow large intensities to be generated since they confine light in small mode-waists.
However, chromatic dispersion gradually widens the pulse duration, reducing the peak intensity ofthe pulses and thus limiting the maximum achievable frequency broadening. Engineered photonic
crystal fiber [130] circumvents this problem by providing zero dispersion near the carrier frequency
while supporting a fundamental spatial mode. It was the introduction of these optical fibers thatallowed the first demonstration of an optical octave and subsequent direct detection and locking of
f0 [131, 132].
With the repetition rate locked to an ultrastable laser and the carrier offset phase stabilized,
the stability of the ultrastable laser is then transferred to the entire comb spectrum. One canthink of this as converting an ultrastable laser at a single frequency into hundreds of thousands of83
ultrastable lasers that span frequencies of over 100 THz. The stability of the Si cavity can thus
be transferred to the comb tooth at 698 nm, on resonance with the clock transition. A laser at
698 nm can then be phase-locked to the comb and delivered to the optical lattice clock throughphase-stabilized optical fibers, as discussed in the following subsections.

3.3.2

Stability transfer frequency chain

We use a commercial Menlo Er:fiber frequency comb to transfer the stability of the ultrastablelaser at 1542 nm to 698 nm. The comb is based on an Er-doped femtosecond all-PM fiber modelocked laser [133] with a repetition rate of 250 MHz and a center wavelength around 1560 nm. The
comb has over a 1 MHz bandwidth on both f0 and frep due to high-bandwidth EOM actuatorsthat have minimal crosstalk [134]. The almost-independent control of the two actuators is achieved
using an interferometer implemented with polarization optics, where the amplitude between the twoarms is adjusted to achieve a change in the carrier envelope without altering the center frequency.
The comb is equipped with additional low-bandwidth piezo and thermoelectric actuators that allow
the comb to stay locked for multiple months.
The frequency comb is a multi-branch comb, where the output of the mode-locked laser is split
between different ‘branches’ that serve different optical purposes as illustrated in Fig. 3.23. Eachbranch contains a separate Er:doped fiber amplifier (EDFA) and highly nonlinear fiber to maximize
frequency broadening at a particular optical frequency. Our frequency comb contains three separate
branches. The first arm is dedicated to detecting f0 through an f-2f interferometer. A second arm
is used to provide a frequency reference at 813 nm, which is our lattice wavelength. Approximatelya 1 Hz frequency shift of the clock transition occurs if the lattice laser is off the magic wavelengthby 1 GHz per 100Erec lattice depth for a combined scalar and tensor Stark shift of an atom with|mF | = 9/2. Stabilizing the lattice lasers to below a kHz is easily achievable with this system. Since
the broadened comb spectrum spans roughly 1 µm to 2 µm, it is necessary to double the comb
light using a periodically-poled lithium niobate (PPLN) crystal, and 25 mW of power is provided
in a 3 nm bandwidth at the free-space output of the crystal. The light then goes into a home-built

84
Sr1

f-2f
interferometer

FSB
FSB

BPF
Mode-locked
oscillator
250 MHz, 1560 nm

EDFA & HNLF

1542/698 branch

FSB

90/10
FS
FM

FSB

813 nm
FBG

1542 nm

FSB

698 nm

FSB

Si4

90/10
FS
FSB
FM
90/10
FSB

Si3

AOM

PPLN

Si3/Si4 BDU
FSB

813 branch
EDFA & HNLF

FSB

FBG

1542 nm
PPLN

Sr2

Lattice BDU+1

30 MHzBeamdump

Si3/MJM BDU
MJM
From distribution center

Figure 3.23: Outline of the Menlo frequency comb we use for stability transfer of the Si3 cavity toother wavelengths relevant for Sr. Separate branches are used for each desired frequency output.
We have branches that stabilize f0 , our 813 nm lattice lasers, and a custom branch with minimal
differential noise to transfer the stability of Si3 (the 124 K cavity) to 698 nm, the 1 S0 → 3 P0 clocktransition. Our frequency comb is locked to Si3, and the beat between the frequency comb and
MJM (a pre-stabilized cavity at 698 nm) provides the error signal to phase-lock MJM to the comb.
Heterodyne beats between Si3 and Si4 (the 4 K Si cavity), and between the frequency comb andSi4, also occur inside the Menlo comb box. Acronyms are: erbium-doped fiber amplifier (EDFA),
highly nonlinear fiber (HNLF), periodically-poled lithium niobate (PPLN), bandpass filter (BPF),
beat-detection unit (BDU), pm fiber splitter (FS), pm fiber splitter with fast axis blocked (FSB),
fiber bragg grating (FBG), faraday mirror (FM), acousto-optic modulator (AOM). All fibers andfiber components are polarization-maintaining (PM).

beat detection unit (BDU), which contains PM fiber splitters to combine the comb light with thelattice lasers, and a high-bandwidth balanced detector to detect the heterodyne beat. Note that
because of the doubler, the comb teeth follow the equation νn = 2nfrep ± 2f0 . Heterodyne beats
are generated separately with two different lattice lasers to provide a stable frequency reference foreach Sr experiment. We achieve 30 dB - 40 dB SNR with 100 kHz resolution bandwidth for thesebeats.

85

The last branch is used generate beatnotes at both 1542 nm and 698 nm. Outputs centeredat these two different wavelengths were implemented in a single branch to reduce the differential
noise between the two spectral regions. The comb light around 1542 nm is split off right before
the HNLF, while the output of the HNLF goes into a PPLN to generate 5 mW of power in a 3
nm bandwidth centered at 698 nm. In both cases, the comb light is then filtered with a 0.5 nm
bandwidth fiber bragg grating (FBG) and combined on a beam splitter with a cw laser to generatea heterodyne beat that is detected using a balanced detector. The 1542 nm output of the branchforms a beat with the Si3 light (the 124 K Si cavity), which is transferred over to the Menlo optics
table using a phase-stabilized fiber link. The fiber noise reference is provided by a PM Faraday
mirror that is inside a metal, temperature-controlled box. The 698 nm comb light forms a beatwith MJM, which is a pre-stabilized laser at 698 nm [35]. The fiber noise cancellation between
MJM and the frequency comb uses two AOMs to form the beat. The first AOM before the fiber
provides the actuator for the fiber stabilization. A unique frequency from the fiber noise referenceis guaranteed using a second static AOM right before the fiber noise reference mirror. This ensuresthat the servo only sees the reflection from the mirror and is insensitive to spurious reflections
from other micro-optics, fiber tips, or fiber splices. We are able to achieve 50 dB SNR at 100 kHz
resolution bandwidth for both the Si3-comb beat and the MJM-comb beat.
We additionally take spare light from the comb at 1542 nm to generate a beat with Si4 (the4 K Si cavity) in a separate Si3/Si4 BDU. The comb light is similarly filtered using a FBG before
being combined with the fiber-stabilized Si4 light to generate a beat. We also create a second beat
between Si4 and Si3 inside this BDU.
The full stability transfer schematic is shown in Fig. 3.24. Light from Si3 is delivered to the
Er comb using a phase-stabilized link, and is used to lock the comb at 35 MHz by actuating on frep .
We additionally lock f0 to a referenced microwave signal at 35 MHz. This transfers the stability of
Si3 to every comb tooth, providing a stable reference for our lattice lasers and our pre-stabilized

86
High-NA
imagingsystem

AOM2

AOM3

A
O
M -1

A
O
M +1
Fluorescence-1
AOM1 AOM

FNC
10 μm

A
O
M

+1

-1

30
MHz

Fiber noise
canc. (FNC)

AOM

fSi3
21 cm
Silicon1542 nm“Si3”

Er comb

0

+1

87

A -1
O
M

A
O
M

FNC

FNC

fFNC

fFNC

fsteerAOM-1

240 m
-1

frep

F
N
C

fULE
fb,1542
35 MHzf0 = +35 MHz

Sr atoms

Fluorescence

AOM

40 cm
ULE

PMT
mF = -9/2

fb,698+9/2

Error Signal

fdemod698 nm

“MJM”

fSi3

Atom servo

Figure 3.24: Stability transfer of the Si3 cavity to the 1D and 3D optical lattice clocks. A frequency
comb is utilized to do the transfer. The comb is locked to Si3, and a pre-stabilized ULE cavity at
698 nm is locked to the comb. The stabilized light of the ULE cavity seeds two injection locked
lasers (not shown) before being distributed to the Sr experiments and the frequency comb. The
fiber links between the comb and the Si3/ULE laser, and the Sr experiments and the distribution
center, are phase-stabilized.

laser at 698 nm. To complete the stability transfer process, the MJM laser is then phase-locked
to the comb. The MJM ultrastable laser’s output first goes to a distribution center with a phasestabilized fiber link. The light is used as a seed to injection lock two different 30 mW diodes, which
are then distributed to the various Sr experiments and to the frequency comb. The MJM laser has
1 × 10−16 fractional frequency instability from 1 s to 10s of s [42], and good high frequency noise.
Si3, on the other hand, has improved Brownian thermal noise which is the dominant noise source
up to around 2 Hz. To maximize the noise performance between the two cavities, we lock MJM tothe comb with a bandwidth around 1 Hz. This optimizes the noise performance of the combined
Si3-MJM cavity as determined through simulations of the Dick Effect, described in Chapter 1. To
actuate on the MJM laser, we sum the error signal from the MJM-comb beat into the fiber noiseFN
reference

FNC
MJM-comb
beat

BPF

fdemod1/8

phase-freq 22.082 MHz
det
R
140 MHz

Loop filter

Low-noise
VCO

Loop filter

fULE

L

R

VCO

L

1/2

R

87

fAOM

FN beat

155 MHz

15 MHz“MJM”
Figure 3.25: Electronics demonstrating how we lock MJM to the frequency comb. The MJM-comb
beat provides an error signal which we use to steer the demodulation frequency of the MJM fiber
noise servo. Both the fiber noise and the MJM-comb servo are closed feedback loops.

frequency reference, as shown in Fig. 3.25. The beat is bandpass filtered, amplified, and divided
by a factor of 8 before being demodulated and sent into a loop filter. The output of the loop filter
is then sent into a low-noise VCO, which is mixed with a local oscillator to provide the fiber noise
frequency reference at the correct frequency. The fiber noise reference is mixed with the fiber noise
beat and sent to a loop filter that drives an internal VCO, which powers the MJM fiber noise AOM.
All of the fiber noise servos and the phase locks with the frequency comb are closed feedback loops
so that the Si3 phase is properly transferred to MJM.

3.3.3

Differential noise

To not introduce additional noise in the stability transfer, it is essential to reduce differential
noise between the 1542 nm path and the 698 nm path. Any noise added between the 1542 nm and698 nm branches of the comb, in the distribution center, or in the transfer of the light to the Sr
experiments, will degrade the pristine stability provided by the Si cavity. A critical potential noise
source in the frequency comb is from unstabilized fiber, which can add noise caused by vibrations
or temperature fluctuations. To reduce the amount of differential noise between the two paths,

88
(a)

(b)

Figure 3.26: (a) Picture of the frequency comb, including the outer Al box, the 2 inch breadboard,and the inner box that goes around a smaller Al breadboard that is water-cooled and clamped
to the middle-sized breadboard. (b) Optical components of the frequency comb, inside the innerbox. Out-of-loop fiber components are housed in Al boxes that are thermo-electrically stabilized
to within 10 mK.

89
the fiber lengths are made as short as possible. The 1542 nm path is picked off in the comb as
close to the 698 branch as possible, so that the differential fiber path in the 1542/698 branch isonly 10-15 cm. Fiber components in the Si3/MJM BDU are spliced with roughly 10 cm leads.
In addition, all fiber components in the BDU are housed within a metal box that is temperature
controlled with a 10 mK temperature stability. The fiber leads between the 1542/698 branch andthe Si3/MJM BDU box are only a few cm long. Furthermore, all fibers in the frequency comb
are polarization-maintaining (PM) to reduce fluctuations due to polarization rotation, and all fiber
splitters have the fast axis blocked. The box has foam on top and the sides to reduce vibrational
coupling.
The frequency comb sits on top of a water-cooled Al breadboard that is tightly clamped to a 2
inch breadboard that sits on top of the larger optical table with small viton feet. We use a low-noise
Lauda ECO RE 415 S chiller at the minimum flow rate to provide the water cooling. We confirmedthat the chiller did not add vibration noise by looking at the coherence between the Si3-MJM beatand an accelerometer placed on the comb table. The water-cooled breadboard is surrounded bya plastic box with foam on the lids. Furthermore, the 2 inch thick breadboard is surrounded bya large Al box with conical foam inside to damp vibrations. Temperature fluctuations inside thelarge Al box are sub-100 mK over the course of a day. The various boxes are shown in Fig. 3.26.
The large Al box around the Sr2 (3D optical lattice) experiment was modeled after the box aroundthe frequency comb. The Menlo comb is equipped with a number of heavy electrical cables topower the pump diodes, provide feedback to the repetition rate and the offset beat, etc. We clamp
the cables with viton on the main large optical table to dampen their vibrations. We furthermore
connect ribbon cables for the pump diodes and light flexible cables for the rest of the cables tolessen the impact of cable vibrations on the comb.
The added instability of the stability transfer process due to the frequency comb is assessedby performing a comb comparison with an identical reference system [135]. Each comb is locked to
a common local oscillator at 1542 nm, and the repetition rates are set to the same value. The offset
beat f0 is also phase-locked but differs between the two combs by 5 MHz so that the heterodyne

90

Figure 3.27: Differential noise between two identical frequency combs. Each comb is locked to acommon reference at 1542 nm, and a heterodyne beat is formed at 698 nm that is then counted ona frequency counter.

beat at 698 nm is 10 MHz after the PPLN crystals. The heterodyne beat is then counted using
a dead-time free lambda-type frequency counter with a 100 ms gate time. Because the repetition
rates are equal for the two systems, all comb lines within the 2.5 nm output of the PPLN crystal
contribute to the measured signal. The results of the measurement are shown in Fig. 3.27. The
instability due to the frequency comb is 1.6 × 10−18 at 1 s, well below the local oscillator noise,
while the measured frequency offset of the beat is 1.5 × 10−19 .
This measurement was performed with Menlo’s commercial BDU units. The instability of our

91

home-built BDU units were measured separately. To measure the noise, the comb light is replacedso that a cw-cw beat is formed with the same local oscillator. To measure the noise in the 1542
nm out-of-loop fiber, Si3 light is split off before the fiber link and a separate phase-stabilized fiber
link is sent to the comb input port of the 1542 nm BDU (prior to the circulator in Fig. 3.23). The
fiber noise AOMs are offset in frequency so that a heterodyne beat is formed at 10 kHz, which issubsequently counted. The results, as seen in Fig. 3.28 (a), show an instability of 4 × 10−18 at 1s averaging time. We further measured the frequency offset over the course of a week, where wesaw variations in the center frequency of 2 × 10−19 . A similar experiment is performed on the 698
nm out-of-loop fiber path in the BDU. A separate path on the distribution center is used, witha separate phase-stabilized link, to generate the cw-cw beat. Results are shown in Fig. 3.28 (b),where the instability is 5 × 10−18 at 1 s averaging time. The frequency offset over the course of aweek varied at 5 × 10−19 . The noise is well below the Si cavity, ensuring that any degradation of thestability transfer due to the frequency comb is negligible. Note that these measurements provide an
upper bound of the instability of the home-built BDU, since the measurement includes noise fromthe phase-stabilized fiber links and any out-of-loop noise on the distribution center between the two
698 nm paths. The noise of the 698 and 1542 fiber paths in the BDU are measured independently,
and thus does not take into account that some noise could be common-mode between the two fiber
paths.

3.3.4

Optical path length stabilization

The 698 nm light, locked to the Si3 ultrastable laser, is transferred to the Sr experiments using
phase-stabilized fibers [136]. The link operates in a pulsed fashion so that light is only delivered
to the atoms throughout the coherent interrogation, and is extinguished during the dead time andreadout time. This pulsed operation is a known source of phase transients [137]. To overcome thisissue in previous work, the 0th order transmission of the switching AOM was used as the fiber
noise reference, while the 1st order diffracted beam was directed to the atoms [36]. This ensured

92
(a)

1542 out-of-loop fiber(b)

698 out-of-loop fiber

Figure 3.28: Instability of the out-of-loop fiber in the home-built BDU boxes. The measurements
include fiber noise from the phase-stabilized links.

that RF-induced heating of the AOM was mostly common-mode, and that the fiber-noise servo
was always engaged. The 0th order beam was reflected right before the lattice retro mirror so thatboth the 0th and 1st order beams sampled roughly the same optical path. This method reduced
the frequency shift uncertainty to below 10−18 [138], and the out-of-loop path length between the
0th and 1st order was small enough to successfully transfer a clock with an instability of 1 × 10−16
to the atoms.
However, we found that the differential noise between the 0th and 1st order paths was above
the level of the Si3 clock laser instability despite the only 10 cm path length difference. The outof-loop noise was made worse by the lack of a tight enclosure around the experiment during thattime. Differential path length changes between the 0th and 1st order beams can be driven by
thermal changes in the AOM and air currents. Furthermore, reflections from the fiber tip or thefront surface of the AOM can add noise if the 0th order light is used as the phase reference. To test
the out of loop noise, a heterodyne beat was formed between the 1st order paths of two separate
phase-stabilized links with an out-of-loop path length difference of 10 cm. We saw a significantly
reduced out-of-loop noise when the 1st order light of the AOM was used as the phase-reference as93

Figure 3.29: PSD of the out-of-loop fiber noise measurements. When using the 0th order as thefiber noise reflection but the 1st order to perform spectroscopy, the differential noise between thepaths is above the Si3 noise floor. When using the 1st order to provide the phase reference andperform spectroscopy, the noise is much smaller.

opposed to the 0th order, as shown in Fig. 3.29.
To circumvent these issues, in Ref. [3] we used the first diffracted order of the AOM for boththe fiber phase noise reference and for spectroscopy. A wedged beam sampler is placed before thelattice retro mirror to provide a 10% back-reflection for optical path length stabilization. A shutter
is placed after the beam sampler to keep clock light from hitting the atoms while keeping the servo
engaged. To prevent phase transients from impacting the clock interrogation, the clock light isdetuned by 1 MHz 7 ms before the Rabi pulse and the shutter is opened. Any phase transientsfrom re-engaging the servo occur while the clock light is detuned, that is when the atomic sensitivity

94

Distribution center

Sr2
Dichroic
R 813
T 698

0th order
FN servo

shutter

FNC

RF splitter

Cu

sto

m

mo
un

t

Atoms

Injection lockseeding

BS77.76 MHzBS

1st order
FN servo
reference

0th order
FN servo
reference

Iris

Intensity
servo

-1

Serv
o
PD

λ/4

λ/2

bo
x

Len
s

FN

Gla
n

insertable
ND filters

AOM1

MJM

Clock
Laser
PBS

1st order
FN servo+1
41.76 MHz

FNC

100 MHz
AOM2

AOM3

-1
Iris

A

JIL
m

r
irro

irro
r

LA

1"

1"

m

JI

Figure 3.30: Schematic of the updated fiber noise stabilization setup. Both a 0th order and a 1storder servo are utilized. The 0th order servo, which provides actuation on the RF of AOM1, is
continuously on, and provides phase-stabilization for the majority of the fiber link. The 1st order
servo removes any differential noise between the 0th and 1st order paths. This servo is turned off(by shutting off AOM3) right before spectroscopy, while the shutter is opened.

function is negligible. After the servo settles, the frequency is then linearly swept onto resonanceover 3.4 ms. The fractional frequency shift due to the sweeping of the light onto resonance iscalculated to be below 2 × 10−19 for a 600 ms pi pulse.
For Ramsey instead of Rabi spectroscopy, the clock light must be extinguished during thedark time to avoid a significant clock shift. We thus devise an alternative fiber noise locking
scheme that A.) adds minimal phase noise to the clock light delivered to the atoms, B.) reducesphase transients when the clock light is pulsed on/off, and C.) provides maximal frequency agilityon the Bloch sphere. The optics and general locking schematic are shown in Fig. 3.30. A fiber noise

95

servo is implemented using the 0th order light of AOM3 as the fiber noise reference, and a separate
servo takes care of the differential noise between the 0th order and 1st order paths. For this second
servo, the diffracted 1st order light from AOM3 is used as the fiber noise reference. The 0th order
servo steers the RF drive of AOM1, while the 1st order servo steers AOM3. We additionally add
another AOM, AOM2, after the fiber so that the 0th order servo fiber noise reference is not at thesame frequency as the reflection off the fiber tip. The 0th order servo is continuously on, providing
phase-stable clock light at the 10−16 level.
The 1st order servo remains activated for the majority of the cycle by having the shutter
closed so that the light does not hit the atoms. Before performing spectroscopy, AOM3 is shutoff, the shutter is opened, and then AOM3 is switched on again to create a clean pulse of light forspectroscopy. Opening the shutter takes around 3 ms. During this time, AOM3 is off, and the 1storder fiber noise servo is disengaged. The key challenge is then to make sure that re-engaging this
servo does not introduce significant phase transients. One may think that you could just use an
integrator hold feature on the servo during this time, but we measured significant phase deviations
with this method.
Instead, we feed in a synthetic fiber noise beat into the servo while AOM3 is turned off. The
electronics, shown in Fig. 3.31, show how we provide a reference with the right phase and frequency
to the servo box. The solution presented in Fig. 3.31 solves issues A.) and B.) discussed above.
When AOM3 is on, the fiber noise beat is connected to the input of the servo, and is phase-locked
to a reference (fref ). The error signal steers an internal VCO that is used to drive AOM3 (‘1Fout’). The VCO output is centered at 77.76 MHz, so that we need to mix the signal with a function
generator at 36 MHz to get to the right frequency for AOM3.
Right before the shutter is opened, a TTL shuts off the RF drive of AOM3 and simultaneously
switches the input of the fiber noise servo to the synthetic beat signal. The correct frequency isprovided by fref , but the phase needs to be locked so that it is not sensitive to environmental

96

Func. Gen
@116.48 MHz FM

Fiber noise
beat

REF. SOURCESIGNAL IN

Func. Gen
@36 MHzTTL LOW

FNC

fREF = 272 MHz

TTL switch

REF. IN~77.76 MHz

1F OUT
TTL
switch
41.76 MHz TTL
LOW
AOM 3

TTL
HIGH

2F OUT
TTL
switch

f

TTL
Det.
LOW

Loop Filter

SYNTHETICBEAT SIGNAL
TTL

Figure 3.31: 1st order fiber noise servo electronics, which reduces phase excursions when the servo
is re-engaged. While AOM3 is on, the 1st order diffracted beam provides a reference for the fiber
noise beat, which actuates on the RF to AOM3. AOM3 is shut off before spectroscopy using a TTL.
During this time, a synthetic beat signal is fed into the fiber noise input. To get the right phaseand frequency input, a servo is used on the synthetic beat signal before AOM3 is extinguished toproperly reference it to fref .

perturbations such as temperature changes. This is achieved using a phase lock loop. The referenceis mixed with a function generator using a phase detector, and the output passes through a loop
filter into the FM modulation port of the function generator. Temperature changes can also alter
the mixer output inside the servo box. As a result, it is necessary to keep the fiber noise servo aspart of the phase loop to prevent accrued error. We use the 2F output of the box for this purpose,
which is derived from the internal VCO. The function generator is first mixed with the 2F output
before being combined with the reference on the phase detector. The output of the mixer is then97With synthetic
beat phasestabilization

No synthetic
beat phasestabilization

Figure 3.32: Phase transients of the 1st order fiber noise servo once AOM3 is powered and the fiber
noise beat is connected to the servo. 3V corresponds to 90 degrees. Here, the purple traces show
residualphase
hiccupsthe TTL
that turns
AOM3 on, and simultaneously switches the servo input to the fiber noise beat.
CH2
phase
monitor:90 degrees
The blue traces show3V
the->in-loop
error of the fiber noise servo. The top and bottom traces show thephase deviation without the feedback loop on the synthetic beat’s phase. The center trace shows
the reduced phase deviation when the loop is engaged. The magnitude of the phase transient withthe synthetic beat phase stabilization enabled corresponds to an AOM phase chirp frequency shiftof below 10−19 for a 450 ms Ramsey dark time and 15 ms π/2 pulses.

used as the synthetic beat signal while AOM3 is off.
To ensure the right phase is fed into the servo once AOM3 is extinguished, the phase lock
loop is activated while AOM3 is on. To set the phase to the right value in the servo, the BNC cable
length between fref and the phase detector is varied to minimize the phase transients once theservo is reengaged with the real fiber noise beat. Finally, when AOM3 is off, we put an integrator
hold on the clock light intensity servo to avoid integrator wind-up.
The phase transients that occur once AOM3 is re-activated with and without the phase sta-

98bilization of the synthetic beat signal are shown in Fig. 3.32. With the phase stabilization engaged,
we numerically calculate that the deviations correspond to a sub-10−19 clock shift assuming a 450
ms Ramsey dark time and 15 ms π/2 pulses.
The solution presented thus far in Fig. 3.31 solves issues A.) and B.) discussed above. However, the VCOs in the fiber noise servos have a limited bandwidth of +/- 100 kHz. To increase
our scan range so that we have a solution that solves A.), B.) and C.), we simultaneously jump
the AOM3 frequency and the reference frequency. This is accomplished, with a few additional
electronics, by changing fref , as shown in Fig. 3.33. Instead of a single frequency synthesizer asshown in Fig. 3.31, the AOM3 frequency is mixed with fref and another function generator. The
synthetic beat signal phase lock also needs to be jumped, so fgen is split into two synthesizers - fref
and another signal generator. fref is then common to the derived AOM3 frequency, the reference
frequency for the fiber noise servo, and both inputs of the phase detector, so that a different spectroscopy frequency can be achieved by changing fref without significantly disturbing the fiber noise
servo. This also allows the clock light frequency to be quickly adjusted, since it mostly removes
the change in fref from the error signal.
Any vibrations from the dichroic onwards are common-mode between the lattice and theclock laser. Path length fluctuations between the 1st order FN reference and the dichroic however
are uncompensated and must be minimized. To accomplish this, we implement a rigid mounting
structure between the two mirrors, as shown in Fig. 3.34. The mirrors are mounted into thestructure of the box and the box is hermetically sealed. Furthermore, ND filters can be inserted
using a motorized actuator so that the intensity of the clock light at the atoms can be changed
by many orders of magnitude without changing the fiber noise beat amplitude. An SRS shutter isimplemented inside the box as well and is mounted on a translation stage for alignment purposes.

99

Func. Gen
@2F

REF. SOURCEFM
Fiber noise
beat

fREF = 272 MHz

TTL LOW

FNC
SIGNAL IN~77.76 MHz

Func. Gen
@100 MHzTTL switch

REF. IN
1F OUT
divide
by 2

TTL
HIGH

2F OUT
TTL
switchTTL
switch

f

TTL
Det.
LOW

Loop Filter

41.76 MHz TTL
LOW
AOM 3

SYNTHETICBEAT SIGNAL
TTL

Figure 3.33: Full electronics for the 1st order fiber noise servo, that reduces phase transients whenreengaging the servo and provides maximal frequency agility. The electronics feed in a synthetic
beat signal with the right phase and frequency when AOM3 is shut off, right before spectroscopy.
The electronics also include the ability to scan the beat frequency over a wide range by feeding
forward on fref and the beat signal.

100

(a)

(b)

0th order
FN ref.
Shutter
ND filter
actuator

Figure 3.34: Fiber noise box. The first mirror of the box contains the 1st order FN reference, andthe last mount contains the lattice retro mirror (a dielectric that reflects 813 nm, transmits 698nm) that provides a common phase reference for the lattice light and the clock light. ND filters
can be inserted inside the box with a motorized actuator, and an SRS shutter is also mounted tothe top of the box.

Chapter 4

Thermodynamics of a deeply degenerate SU(N )-symmetric Fermi gas

The research described in this chapter has been reported in the following publication:
Lindsay Sonderhouse*, Christian Sanner*, Ross B. Hutson, Akihisa Goban, Thomas Bilitewski,
Lingfeng Yan, William R. Milner, Ana M. Rey and Jun Ye, “Thermodynamics of a deeply degenerate SU(N)-symmetric Fermi gas,” Nat. Phys. 16, 1216–1221 (2020). * (these authors contributed
equally) [51].

A Fermi-degenerate 3D optical lattice clock has shown a record instrinsic QPN stability of√
1.5 × 10−17 / τ . However, due to the extended time spent preparing a Fermi gas of more than
10 s, the corresponding laser noise is high - with the Si cavity’s 4 × 10−17 thermal noise floor and√a few second pulse time, the Dick Effect stability is above 10−16 / τ . To make the 3D latticeclock competitive as an independent clock, it is necessary to reduce the preparation time, which isdominated by the evaporative cooling step to reach degeneracy.
Pioneering work by Deborah Jin showed that Pauli blocking reduces the efficiency of evaporative cooling at low temperatures, making it challenging to reach deep degeneracy. As a result, realizing degeneracy in fermions has spurred numerous independent studies over the past two decades
by exploiting two-component spin mixtures near a Feshbach resonance. In this chapter, we detailexperiments to study SU(N ) interactions [139, 140] in a 87 Sr Fermi gas with N ≤ 10, and utilize
the enhanced interactions for rapid evaporation. While recent studies have started to investigate
the intriguing properties of SU(N ) quantum matter, most of the effort so far has been concentrated

102
on the investigation of lattice-confined gases [141, 139, 142, 143, 144, 54, 145]. On the other hand,experiments probing the role of SU(N ) interactions in a regime where a Fermi liquid description isaccurate have been limited to non-degenerate [62] or only slightly degenerate [146, 147, 148, 149]
gases. Here, we explore a deeply degenerate regime where N Fermi seas coexist and fundamentally
modify the system’s thermodynamics (Fig. 4.1).
We first outline the challenges of evaporating fermions and motivate using many components
with SU(N ) symmetry during evaporation. We then detail our preparation scheme, and demonstrate an unprecedented short time to reach Fermi degeneracy with a temperature T /TF = 0.22,
where TF is the Fermi temperature, achieved in just 0.6 s of evaporation with a laser-cooled sample. Our preparation scheme concludes with a new technique to spin-polarize degenerate Fermi
gases without causing light-induced heating, allowing efficient preparation of a low entropy spinpolarized Fermi gas. We then characterize the enhanced interactions caused by a many-component
SU(N )-symmetric gas. In the deeply degenerate regime, we show through precise measurements ofdensity fluctuations and expansion dynamics that the large number N of spin states under SU(N )symmetry leads to pronounced interaction effects in a system with a nominally negligible interaction parameter. We conclude with a technique to extract T /TF in an interacting gas by fitting
the full density profile of the gas, and with it are able to demonstrate thermometry accurate to
one-hundredth of the Fermi energy using a single experimental image.

4.1

Cooling fermions to degeneracy
Cooling fermions to degeneracy has historically been more challenging than cooling bosons

due to the reduced interaction between identical fermions at low temperatures. Evaporation, aworkhorse technique that is used to achieve quantum degeneracy in atomic gases [150, 151], requires
atoms to rethermalize through collisions at each step of the evaporation process. As outlined in
Chapter 2, the dominant interactions at ultralow temperatures in quantum gases occur via s-wave
contact scattering, which, for identical fermions, vanishes due to the anti-symmetry of fermionic
wavefunctions. As a result, spin-polarized fermions predominantly interact through higher order

103
p-wave interactions. At low temperatures these interactions are suppressed by the height of thecentrifugal barrier (10s of µK in the 1 S0 state of 87 Sr). Fermi-degeneracy requires temperatures of100s of nK. Identical fermions can thus not be evaporatively cooled to degeneracy.

4.1.1

Common techniques

Thermalization throughout evaporation requires collisions between distinguishable atoms.
There are two common techniques to achieve this. First, multi-component fermionic gases can beused. S-wave collisions are then allowed between different components. This was first achieved in
1999 in the group of Debbie Jin using a Fermi gas consisting of atoms in two magnetic sublevelsof 40 K [70]. However, they observed that the cooling efficiency decreased when the gas became
sufficiently degenerate, since rethermalizing collisions were blocked if the final states were occupied
by another atom. This limited the efficiency of evaporative cooling at degenerate temperatures.
Many cold fermion clouds are also created through sympathetic cooling with a bosonic atom.
If bosons are co-loaded into the optical or magnetic trap, they can efficiently evaporate and thefermions will sympathetically cool through collisions with the bosons. Popular combinations include
7 Li and 6 Li [68], 6 Li and 23 Na [152, 153] and 40 K with 87 Rb [154].

There is no Pauli blocking ofcollisions with a Bose-Fermi mixture, which may improve the evaporation efficiency. However,additional loss can arise due to three body recombination. Extra experimental complexity is also
required to create both a Fermi and Bose gas in the same apparatus. Furthermore, the spatial
overlap between the BEC and Fermi gas can be poor due to Fermi pressure, which increases thesize of the Fermi gas compared to BEC [68]. At this time, Fermi degeneracy has been achieved
through evaporation in a range of other atoms and molecules, including but not limited to 6 Li [155],
173 Yb [156], 171 Yb [112], 87 Sr [157, 158, 49], 167 Er [159], 3 He [160] and KRb [161].

Although these techniques have allowed systems to reach Fermi degeneracy, evaporation istypically slow, 10s of seconds. This is because rethermalization, which requires a few collisions per
particle, is still limited despite the described increase in the s-wave collisional parameter. For a
two-component gas of equal densities, the elastic collision rate is given by Γ = nσvrel , where n is

104
the single component density, σ = 4πa2 is the collision cross section, a is the s-wave scattering
length, and vrel is the relative velocity between the two colliding particles. The latter, of course,pdecreases as the temperature is reduced. For a 3D Gaussian distribution, vrel = 16kB T /πm.
To increase the collision rate (and thus the rethermalization rate), one can increase the scattering
length a, through for example a Feshbach resonance. However, in alkaline-earth atoms such as Sr,
only optical Feshbach resonances exist which are known to be lossy [162]. As a result, one relies on
increasing the density of atoms. This however has its own limits - light-assisted collisions limit theattainable initial density [106]. In addition, once degeneracy is reached, Fermi pressure reduces theattainable density per component.

4.1.2

Exploiting SU(N ) symmetry for cooling

Binary mixtures are commonly used to reach degeneracy. They are well studied and versatile
platforms for quantum simulation capable of exploring a wide parameter space [163, 164]. However, the additional freedom in choosing the number of spin components, N , offers unique, largely
unexplored opportunities. Using more components enhances the collision rate, and in the limit oflarge N , the gas should behave like a boson [147]. The collision rate, given by Γ = (N − 1)nσvrel ,increases linearly with the number of components, N . In the 1 S0 state, 87 Sr has 10 nuclear spincomponents. Compared to a 2 component mixture, where each atom only interacts with 50% of
the gas, an atom interacts with 90% of the particles in a 10 component gas. This increase in thenumber of collisional partners enhances the collision rate.
Alkaline-earth fermionic atoms have SU(N ) symmetry in the nuclear spin states of the 1 S0ground and 3 P0 clock state, where N can be as large as 10. This symmetry arises from thedecoupling between the nuclear and electronic degrees of freedom (I · J = 0) intrinsic to theatom’s internal ground state manifold and gives rise to SU(N )-symmetric two-body interactions,characterized by a single nuclear-spin-independent scattering length, a [139, 140]. SU(N ) symmetry
is an untapped resource for cooling [142, 165, 166, 167]. Absent of SU(N ) symmetry, a 10 component
gas would experience spin changing collisions, which cause heating and loss of atoms. Inelastic spin

105

Bosons

SU(N) FermionsEF

FermionsEF

Figure 4.1: Unlike bosons which can occupy the same state, indistinguishable fermions must separate into different energy levels. SU(N ) fermions, on the other hand, can have N particles perstate. In a given level, each particle has (N − 1) distinct partners, as shown in the top right, andinteractions are correspondingly enhanced.

changing collisions are forbidden under SU(N ) symmetry, and thus the enhanced collision andthermalization rate provided by the increased number of components can be fully taken advantage
of. SU(N ) symmetry thus provides a tool to efficiently remove entropy from the system.

4.2

Rapid cooling of fermions to degeneracy
Our preparation scheme (Fig. 4.2a) begins with standard laser cooling techniques developed

for alkaline-earth atoms, as described in [104] and in Chapter 3. After two stages of laser cooling,roughly 107 atoms are cooled to 2 µK in a far-off-resonant crossed optical dipole trap (XODT) with

106
a

Time
Fermi-degenerate
gas preparation

Load
opticallattice

Coherent interrogation

2.3 s

0.4 s

up to 12 s

Clock spectroscopy

b

Read
state
0.025 sLasercooling

Transparency
beam

Spin
preparation

Evaporation

0.6 s

0.4 s

0.1 s

1.2 s

TenS4
0.01 syReservoir
Dimpleg, -zx

Figure 4.2: a, Timing diagram showing stages of the experimental sequence. The times are optimized for clock operation in the band insulating regime. Single spin samples with 20,000 atoms at
T /TF = 0.2 are prepared in a 3D optical lattice in under 3 s. b, Atoms are confined in a crossed
optical dipole trap consisting of a reservoir and dimple trap, and accumulate in a dimple after a“transparency” laser is applied. The transparency and TenS4 lasers overlap with the dimple, andform a 10◦ angle with respect to gravity.

a sheet-like geometry [49, 157]. A vertically oriented round optical dipole trap (VODT) forms a
dimple in a horizontal optical sheet potential that is provided by an elliptically shaped horizontal
optical dipole trap (HODT) (Fig. 4.2b). The HODT provides support against gravity and therefore
determines the effective trap depth. The stronger confinement in the horizontal plane afforded by
the dimple and the large vertical trap frequencies provided by the HODT provide a high initialatomic density, which is ideal for efficient evaporation.

4.2.1

Transparency beam

Here we provide a brief summary of the transparency beam, which is discussed in detailin Chapter 3. The density obtainable in a MOT is generally limited by inelastic light-assisted
collisions [106] and reabsorption of the cooling light, which leads to an effective repulsion between
atoms [107]. To further increase the density inside the dimple beyond these limits, we locally apply
an additional laser that renders atoms inside the dimple region transparent to MOT light, a methodthat was adapted from bosonic 84 Sr [109]. The beam spatially overlaps with the VODT but has
a slightly larger waist. This “transparency” laser shifts the cooling light out of resonance, and ahigh density can then be reached inside the dimple, where atoms collect. Atoms in the dimple

107
a

b
25 GHz3

S1

Transparency
beam
F=7/2
F=9/2

3

P1

266 MHz

F=11/2
TenS4
1

S0
d

+9/2
+/- 5/2+/- 3/2

Population

c

+7/2

+/- 1/2

Figure 4.3: a, Level diagram depicting the energy transitions for the transparency beam andTensor Stark Shift Spin Selector (TenS4). b, Time to reach T /TF = 0.12 for different numbers of
nuclear spin states N participating in evaporation. Each trajectory is prepared with 9.6(8) × 105
atoms per spin state and T /TF = 2.0(1). The gray band denotes a 1/(N − 1) scaling based on
the number of initial collisional partners, given a ±20% change in the atom number. The error
bars are upper bounds, reflecting the resolution of the sampling grid. c, Combined optical andgravitational potential after application of the TenS4 laser. The beam creates a spin-dependent
modification of the potential, and only atoms in spin state mF = +9/2 (purple line) are supported
against gravity. Atoms in mF = −9/2 and −7/2 are removed through optical pumping. d, Spinpurity after application of the TenS4 laser for 10 ms. Atoms are loaded into a 3D optical lattice
and the spin state population is determined using selective excitation on the ultranarrow 1 S0 – 3 P0transition. Gray triangles correspond to spin states that are removed through optical pumping.
The error bars denote the standard error of the mean (s.e.m.) and are smaller than the marker
size.

thermalize with atoms in a large reservoir part of the trap, which are continually being cooled by

108
the MOT light.
The transparency beam has allowed the production of Bose-Einstein condensation without
additional evaporation [109]. However, for fermions additional complications can arise. Compared
to bosonic strontium, 87 Sr has additional hyperfine structure, creating a different AC Stark shift
for each nuclear sublevel. Transparency from both the trapping and stirring second stage MOT
lasers is also required [110]. Accordingly, we use transparency light that is blue-detuned by 25 GHz
from the 3 P1 − 3 S1 transition (Fig. 4.3a). This provides an ample shift for all nuclear spin states
and keeps spontaneous scattering events beyond relevant experimental time scales. After 400 ms
of cooling with the transparency beam, the number of atoms in the dimple saturates with 5 × 106
atoms, i.e. about 50% of the total atom number, a temperature of 2 µK, and T/TF = 1.9. Lower
temperatures can be achieved by adjusting the MOT light, but without further improving the phasespace density.

4.2.2

The SU(N ) edge

The very dense and almost degenerate sample can then be further cooled via forced evaporation. Spin relaxation is absent due to the SU(N )-symmetric nature of 87 Sr in the 1 S0 ground state;
however, over the timescale of seconds we observe a decay of the sample which is well described bya 3-body loss process. The lack of spin relaxation and the small 3-body loss coefficient are further
advantageous features of cooling using 87 Sr atoms in comparison to alkalis.
To determine the three-body loss coefficient k3 , we load a thermal gas with temperature
T = 1.45 µK into the dimple part of the recompressed dipole trap. Starting from an initial central
density of n = 3.9 × 1014 cm−3 we measure over the next ten seconds a decay of the total atom
number Ñ (t) as a function of the holding time t. The observed atom loss is modeled as
Z t
Ñ (t) = Ñ (0) − k3Zdτn3 (r, τ )dV ,

(4.1)

0

where n(r, t) is the total atomic density at position r and time t. The data and corresponding
fit are given in Fig.4.4. We find a three-body loss coefficient of k3 = 4.7(1.2) × 10−30 cm6 /s.

109

data
three body loss fit

Atom number N (×106)

2.2
2.0
1.8
1.6
1.4
1.2
0

2

4

6
8
Hold time (s)1012Figure 4.4: Three-body loss measurement, which is used to extract k3 .

Interestingly, this is a factor of 2 larger than the recent lattice-based 3-body measurement thatshowed agreement with a universal Van der Waals model [54]. Discrepancies in 3-body loss between
bulk gas measurements and predictions have been seen before [168, 169]. These discrepancies
indicate the challenge of accounting for inhomogeneous density profiles in a bulk gas measurement.
Under our experimental conditions, single and two-body contributions are expected to be negligible
over a time interval of ten seconds. Using a corresponding multivariate model, we find both to bestatistically insignificant.
Before quantum degeneracy, the elastic collision rate [156] for a balanced spin mixture witha spatially averaged single-spin density n̄σ is proportional to (1 − 1/N ) N n̄σ . Assuming a constantatom number per spin state, it is thus advantageous to have all 10 spin states populated. We reach

110

T/TF

100
10 1
10 2

0.0

0.5

1.0
1.5
Time (s)

2.0

2.5

T/TF

100
10 1
10 2

1.2 s
0.6 s
2.4 s
104

105
Atom Number/spin state

106

Figure 4.5: Evaporation trajectories for three different total times, 0.6 s (orange), 1.2 s (blue), and2.4 s (green). The time constant of the ramps are adjusted to increase the total evaporation length
while maintaining the same final trap depth. The top plot shows T /TF as a function of time, andthe bottom T /TF versus atom number per spin state.

an initial collision rate of 1000 s−1 with N = 10. Evaporation begins at a trap depth of 20 µK withtrap frequencies in the dimple of (νr , νz ) = (100, 800) Hz. The HODT intensity is then reduced in a
two-stage ramp down to a final trap depth of a few 100 nK with trap frequencies of (100, 200) Hz.
After 600 ms of evaporation, we reach T /TF = 0.22 with 3 × 104 atoms per spin state. Slower
evaporation leads to lower temperatures, and we achieve T /TF = 0.07 with 5 × 104 atoms per spin

111
(a)

(b)

(c)

Figure 4.6: a, Density profile of a Fermi-degenerate gas after 10 ms time-of-flight with 10 spin states
and T /TF = 0.07. b, Residuals after fitting the cloud to a Maxwell-Boltzmann distribution. Colorbars in each subplot correspond to the number of atoms per 1.37 µm2 . c, Radially averaged fit ofthe density profile given in a to a Maxwell-Boltzmann distribution and a Fermi-Dirac distribution.
The thermal distribution has larger residuals than the Fermi-Dirac fit.

state after evaporating for 2.4 s. Trajectories of three different lengths are given in Fig. 4.5 1 . At
such low temperatures, the normally modest deviation from a thermal distribution is striking, seeFig. 4.6. Such fast trajectories mark a considerable improvement over previous evaporation results,
where evaporation stages took around 10 s [49].
We observe an approximate 1/(N −1) scaling of the total evaporation time with the number ofspin states participating in evaporation as shown in Fig. 4.3b, reflecting the reduction in collisional
partners for smaller N . Here, each sample is prepared with the same atom number per spin state
and T /TF , and is measured after reaching T /TF = 0.12. The final atom number per spin state is
reduced by roughly a factor of two as N is decreased.

4.3

Spin Manipulation
Here we summarize the TenS4 technique, which is discussed in more detail in Chapter 3. Inorder to manipulate the spin composition of the atom sample and prepare a spin-polarized gas,
we apply a spin-selective optical potential to the atoms after evaporation. While past procedures
1

The data shown in Fig. 4.3 (b) and Fig. 4.5 are extracted using a full density profile fit that includes a mean-fieldinteraction term, as described later in this chapter. Corrections to T /TF are around 20% at the lowest temperatures.
Using a non-interacting fit, our minimum extracted T /TF is 0.05.

112
have used optical Stern-Gerlach techniques to separate out spin states during time-of-flight [111,112, 113], our method, the Tensor Stark Shift Spin Selector (TenS4), creates a spin-selective force
on the atoms from the tensor Stark shift of a laser while the atoms remain trapped in the XODT.
Atoms with the same |mF | experience a small differential force due to an applied magnetic field of
5 G. This magnetic field is too small to fully remove mF = −9/2 and −7/2. We thus conventionally
remove these spins via optical pumping prior to evaporative cooling. For the SU(N ) measurementsin the following section, however, we always use all 10 nuclear spins. The TenS4 beam is offset fromthe atoms such that the AC Stark shift varies across the atomic sample by hundreds of nK, whichcauses a spin-dependent modification in the combined optical and gravitational potential of theatoms (Fig. 4.3c). The TenS4 laser is blue-detuned from the 3 P1 , F = 11/2 transition by 266 MHz(Fig. 4.3a), where the polarizability from 3 P1 , F = 11/2 cancels the polarizability from 3 P1 , F =9/2 for nuclear spin state mF = +9/2. As a result, atoms with mF = +9/2 are unaffected by the
TenS4 laser while all other spin states feel a repelling force.
The spin purity after applying the TenS4 laser for 10 ms is measured by loading the atoms
into a deep 3D optical lattice. The spin population for each nuclear spin state is then read out
through selective π-pulse excitations on the clock transition. We measure 92% of the atoms in thetarget mF = +9/2 state, as shown in Fig. 4.3d, and an atom number of 3.3 × 104 after applicationof the TenS4 beam, in rough agreement with 1/8th of the initial atom population of 2.5 × 105 .
The temperature of the sample heats by only ∼10%. Our technique provides spin-state selectivity
without optical excitation and as a result does not cause light-induced heating, overcoming issues
typically associated with optical pumping schemes. The spin distillation technique enables us toload a single spin with T /TF = 0.2 into a 3D optical lattice with a total preparation time under 3 s(Fig. 4.2 a).

4.4

Characterization of SU(N )-enhanced interactions
Having prepared a high-density deeply degenerate SU(10) gas, we demonstrate in this sectionthat a nominally very weakly interacting quantum system with interaction parameter kF a  1,113where kF is the Fermi wave vector, can develop striking interaction effects due to SU(N ) enhancement. The nuclear spin degree of freedom substantially modifies the character of the gas towards
an interacting multi-component Fermi liquid with subtle consequences for correlation analysis andthermometry. We first describe our experimental measurements of the compressibility, the expansion dynamics, and anisotropic features of expanded gas clouds, and conclude with a summary of
the interacting theoretical model we use to describe the measurements.

4.4.1

Isothermal compressibility

In order to investigate this intriguing quantum system and illuminate the role of SU(N ) symmetry, we perform measurements that characterize the system’s thermodynamics. A key quantity
in this context is the isothermal compressibility κ = n12 ∂n∂µ , where n = N nσ denotes the particle
density and µ the chemical potential. For 87 Sr with a = 97aBohr , the contact interactions are
repulsive and one expects a decreased compressibility compared to an ideal Fermi gas with compressibility κ0 . For a homogeneous gas in the zero temperature limit, κ0 = 3/(2N nσ EF ) with Fermi
energy EF and Fermi wave vector kF = (6π 2 nσ )1/3 . One finds to first order in kF a that [170]κ0 /κ = 1 + (N − 1) 2kF a/π .

(4.2)

Therefore, the symmetric N -component system is effectively (N − 1)-fold more repulsive than atypical two spin component Fermi liquid [171]. The favorable scaling with number of internal levels
has to be contrasted with the weak dependence of the compressibility on the atom number per spin
1/6

state in a harmonically trapped gas, where kF ∝ Nσ . All experiments reported in the following
were carried out with spin-balanced 10-component samples. By focusing on this maximum N-limit,
we achieve the most pronounced interaction effects and can perform the most sensitive experimenttheory comparisons.
Experimentally, we access the compressibility of the gas by measuring its local density fluctuations. Fluctuations, either thermal or quantum, are the drivers of phase transitions, and are

114

40000

Thermal gas
Degenerate gas
N2 = N
T/Tf = 0.13, non-interacting
T/Tf = 0.16, interacting

35000

Variance N2

30000
25000
20000
15000
10000
5000
0

0

5000 10000150002000025000300003500040000
Atom number N

Figure 4.7: Density fluctuations after 11.5 ms time-of-flight for a degenerate cloud with N = 10
nuclear spin states (red circles). The data is fit using an SU(N ) interacting model to extract
T /TF = 0.16 (red solid line), with shading representing a 2σ uncertainty of ±0.02 T /TF . Fitting the data instead to a non-interacting ideal Fermi gas gives T /TF = 0.13 (blue dashed line),showing an interaction-induced suppression of ∼20%. The difference between the interacting andnon-interacting fits is much less than the scatter in the data, highlighting the indistinguishability
between interacting and non-interacting systems by measuring density fluctuations alone at thegiven signal-to-noise ratio. The total density fluctuations are 25% of that of the thermal gas. A
thermal cloud (purple triangles) reproduces Poisson statistics with ∆Ñ 2 /Ñ = 1 (purple dot-dashedline). Each data point is obtained by looking at the atom number variation and mean in a subregionof the cloud for a series of images.

sensitive to the underlying phase of matter, its quasi-particles and interactions. The fluctuationdissipation theorem states that the thermally driven fluctuations of a thermodynamic variable arefundamentally related to the conjugate external force through the susceptibility [84]. Consideringa small subvolume of the gas cloud containing on average Ñ atoms, the corresponding generalized
force is the local chemical potential µ. The fluctuation response theorem then states that∂ Ñ /∂µ =1(∆Ñ )2 .
kB T

(4.3)

115
Through Maxwell’s relations we can arrive at ∂ Ñ /∂µ = κÑ 2 /V [85]. The relative number fluctuations η = ∆Ñ 2 /Ñ are therefore related to the susceptibility ∂ Ñ /∂µ via η = nkB T κ, where kB is
the Boltzmann constant. The compressibility elucidates the underlying statistics of the gas. The
equation of state of a classical ideal gas dictates η = 1, reflecting the Poisson statistics of a thermal
sample. A Fermi gas on the other hand is incompressible due to Fermi pressure. As a result, one
expects a reduced compressibility compared to a thermal sample; for a deeply degenerate ideal
Fermi gas, η = 3/2 T /TF . These sub-Poissonian fluctuations reflect the degeneracy pressure in thegas.

4.4.1.1

Virial expansion

To theoretically derive the compressibility under SU(N ) interactions at finite temperature, avirial expansion technique can be used in addition to directly calculating the density distributionof the gas (see Section 4.4.4) [172, 173, 174]. Here, the grand partition function of a many-bodyquantum system is expressed using a series of perturbative functions that depend on the statisticsof the gas. We start with the expression for the chemical potential of a homogeneous Fermi gas at
low temperature and with weak interactions [175]:
π2
4µ(n, T, a) = EF 1 − (T /TF )2 +(N − 1)kF a12
3π
4(11 − 2 ln(2))
2+(kF a) (N − 1) + CT 2 a2 ,
15π 2

(4.4)

where C is a constant independent of n. The compressibility, κ = n12 ∂n∂µ , can then be evaluatedfrom the dependence of n on the Fermi parameters.
For simplicity we only keep terms up to first order, to obtain for the number fluctuations∆Ñ 2 /Ñ = kB T nκ =

3
T /TF
.
2
2 1 + π (kF a)(N − 1)

(4.5)

This suggests that even in the kF a  1 limit the interaction effects become non-negligible due tothe (N − 1)-fold SU(N ) enhancement.

116
4.4.1.2

Density fluctuation measurements

Our density fluctuation measurements are performed on expanded gas clouds. After abruptly
turning off the harmonic confinement (νr = 130 Hz, νz = 240 Hz) the quantum degenerate sample
that contains in total 10 × 59000 atoms such that kF a = 0.07 freely expands over 11.5 ms. We
then obtain line-of-sight integrated density profiles via absorption imaging. Following the protocol
described in [86, 87] we run this experiment in a repeated fashion. Each image is broken up into
small subregions, and the images are normalized such that the total atom number is equal for each
image. For each projected subregion of the cloud containing on average Ñ atoms we can measure
the statistical variance ∆Ñ 2 . Fig. 4.7 shows the results obtained from 400 individual images
together with a calibration line derived from noise measurements on a thermal gas. Pronounced
noise suppression down to about 25% of thermal noise in the center of the sample indicates thatthe gas is deeply in the quantum regime.
To quantitatively interpret the noise data beyond first order including both Fermi pressureand SU(N )-enhanced interactions, we calculate the expected line-of-sight integrated number fluctuations based on a kinetic approach [176, 177], using the collisional Boltzmann-Vlasov equation witha mean-field interaction term (see Section 4.4.4 for more details). The Boltzmann-Vlasov equation
describes the evolution of the semi-classical phase-space distribution f (r, p) with position r andmomentum p:


∂t +


p
· ∇r − ∇r [U (r) + VMF (r)] · ∇p f = Ic (f ) .
m

(4.6)

The phase-space distribution evolves due to ballistic motion of particles (second term), the forcesP
due to the harmonic trapping potential, U = m/2 i (2πνi )2 ri2 , the mean field interactions, VMF =g(N − 1)n with g = 4π~2 a/m, and the collisional integral, Ic (f ) [177, 96]. Solving the BoltzmannVlasov equation in equilibrium and for finite temperature allows us to obtain the real space density,n(r), from which we can compute the compressibility, and thus the number fluctuations in trap andafter time of flight. By fitting this model to the observed fluctuations, we extract a temperature
T /TF = 0.16 ± 0.01. At these low temperatures, Ic (f ) plays no role.

117

To illustrate the interaction-induced compressibility change we additionally fit a noninteracting model to the noise data, which gives an apparent T /TF = 0.13 ± 0.01, indicating a ∼20%
compressibility reduction due to interactions. More precisely, we find that the compressibility inthe center of the trap is reduced by 18% compared to a non-interacting gas at the same density
and temperature. This percentage is comparable to the ratio of 21% between the interaction energy in a small volume V at the centre of the cloud, g/2 (N − 1)n2σ V , and the total energy of anoninteracting Fermi gas at the same density, 3/5 EF nσ V . Clearly, without prior knowledge ofthe interaction parameter, one cannot distinguish between a colder or more repulsively interacting
system. Repulsive interactions mimic Fermi statistics, each acting to reduce the density where it ishighest, in the center of the gas. Having a full thermodynamic description at hand we also perform
global profile fits of the acquired images to numerically calculated density distributions and find
T /TF = 0.17 ± 0.01. That is, using the self-consistent solution of Equation 4.13 after time of flightprovides us with a density distribution, which we can use to fit the full density profile of the gas with
T /TF as a free parameter. This approach, which we discuss in more detail later in Section 4.4.3(and as shown in Fig. 4.9), allows measurement of the temperature in the presence of interactions
in a single measurement. This is in contrast to measuring local density fluctuations, which requires
taking hundreds of images to get good statistics. The temperature derived by fitting the full density profile is in good agreement with the temperature derived from density noise measurements.
We want to emphasize again that even though the two underlying physical mechanisms that leadto the observed suppression of density fluctuations are fundamentally different, it is impossible todistinguish these two contributions by performing density noise measurements alone at the given
signal-to-noise ratio.

4.4.2

Expansion dynamics

To unambiguously distinguish between temperature and interaction effects, we study the
expansion dynamics of the cloud after being released from the trap. Interactions provide an ad-

118
b

a

ttof = 0.5 msdExpansion timecttof = 8.5 ms

ttof = 3.5 msef

Figure 4.8: a, b, c, Line-of-sight integrated atomic density for 0.5 ms, 3.5 ms, and 8.5 ms timeof-flight (ttof ) expansion times. The colorbar’s unit corresponds to the number of atoms per 1.37
µm2 . d, Aspect ratio of a cloud of cold atoms with N = 10 nuclear spin states released from an
optical dipole trap for variable expansion times (red circles). After ∼3 ms, the aspect ratio passes
through unity (black dashed line), a clear signature of interactions in the gas. At long times, thesample approaches an aspect ratio of 1.12 (grey line). The sample has an initial trap asymmetryof νz /νr = 6.4. In panels d - f, the expected aspect ratio is shown for N = 10 (red dashed line), N= 7 (purple dotted line), N = 4 (green dash-dotted line), and N = 1 (blue solid line) nuclear spinstates. e, Aspect ratio versus initial trap asymmetry of a degenerate gas for N = 10 (red circles)
and N = 1 (blue squares) spin states after time-of-flight expansion for 15.5 ms. f, Aspect ratio
versus temperature. Data is shown with roughly the same atom number per shot. The data is fit
using an interacting model that includes both a mean-field interaction and an additional collisional
term. Neglecting the collisional term fails to explain the results (gray dashed line). All error bars
represent the s.e.m. and are smaller than the marker size.

ditional release energy during expansion that is mapped to momentum after long time-of-flight.
Interactions preferentially push atoms along the direction of the largest density gradient. As aresult, the initial trap asymmetry is inverted and this conversion produces an anisotropic distribution after long time-of-flight, see Fig. 4.8a-c. The expansion can be described via scaling solutionsof the time-dependent Boltzmann–Vlasov equation (see Section 4.4.4) [178, 179, 76, 180]. Unlike
in trap, the effect of interactions on the density after expansion cannot be partly captured by alower temperature, and result in a non-unity aspect ratio at long times. This is in contrast to a

119

non-interacting gas, where expansion occurs ballistically so that at after long time-of-flight, thecloud reflects the isotropic momentum distribution even if the confining potential is anisotropic.
Fig. 4.8d displays the aspect ratio of a SU(10) atom cloud measured after variable expansion
times ttof out of a harmonic trap with νr = 125 Hz and νz = 800 Hz. The sample contains
50,000 atoms per spin component at T /TF = 0.16. Initially, the atom cloud reproduces the trap’s
asymmetry. As νr,z ttof becomes larger than one, the spatial density distribution is more and moredetermined by the momentum distribution in the gas. Observing an inversion of the aspect ratio
beyond 1 is an unambiguous signature that interactions have modified the isotropic momentum
distribution during time-of-flight [97].
To further explore this behavior, in Fig. 4.8e we present measurements of the expanded cloud
aspect ratio (ttof = 15.5 ms) as a function of the confinement asymmetry for a 10-component gas
and a spin-polarized gas, both at T /TF = 0.16. In the non-interacting case N = 1 (blue data
points), the aspect ratio is always 1 as expected in the long time-of-flight limit. Finally, in Fig. 4.8f
we show the dependence of the observed cloud aspect ratio on T /TF for a fixed initial confinementwith νr = 130 Hz and νz = 725 Hz at a fixed ttof = 15.5 ms. Atomic interactions add a mean fieldterm ∝ a and a collision term Ic (f ) ∝ a2 , see Equations 4.10 and 4.12. The latter, however, is only
relevant in the presence of high collision rates, and, in fact, its pronounced effect is observed when
the gas is relatively hot (Fig. 4.8f). In comparison, at low temperatures the interaction energy andkinetic energy become comparable, and the collisional rate is suppressed by Fermi statistics.
All measurements are well reproduced by our quantitative model, see Section 4.4.4. Since
our model allows us to extract the density distribution of the gas, we can also easily extract the
aspect ratio. To emphasize the role of N in modifying the dynamics, we also plot in Fig. 4.8d-f the
behavior expected for N = 1, 4, 7, 10 spin states using the validated theoretical model. Because
the aspect ratio measurements, which are extracted through Gaussian fits, are fairly immune tomost imaging artifacts, they can be exploited to perform precise thermometry of the interacting

120

Figure 4.9: a,b,c, Line-of-sight integrated atomic density n(x, z) after time-of-flight for a single
shot of the experimental data, non-interacting theory, and interacting theory, respectively. The
colorbar’s units represent the number of atoms per 1.37 µm2 . Images are shown with an initialtrap asymmetry of νz /νr = 1.8 and T /TF = 0.17. d,e,f, The anisotropy of the cloud, definedas n(x, z) − n(z, x), is shown in the second row, where lobes are clearly visible for clouds withinteractions. To improve the signal-to-noise ratio the experimental image is first symmetrized byreflection along the x and z-axes. g,h,i, If the anisotropy is integrated along one direction, peaks
symmetric to the center of the gas appear for the interacting distribution that are sensitive totemperature, while the non-interacting signal shows a different signature that displays only weaktemperature dependence. Here, the red lines show the integrated anisotropy of the images in (d-f),
while the blue dashed lines in (h) and (i) show the integrated profile for a 50% higher temperature.

Fermi gas.

121
4.4.3

Anisotropic density features

Beyond their directly visible manifestation through cloud ellipticity one can also identify
interaction-modified expansion dynamics in the full density profile of the gas by carefully inspecting
high-signal-to-noise absorption images. Fig. 4.9 illustrates that a presumably round line-of-sightintegrated density profile n(x, z) still contains a systematic interaction signature. After a time-offlight time of 11.5 ms, we observe density profiles that at first sight appear circularly symmetric(first row) for the experimental data (first column), and the noninteracting (second column) andthe interacting (third column) model. Interactions are revealed in the transpose-anisotropy of thedensity distribution, defined as n(x, z)−n(z, x), shown in the second row. The transpose anisotropy
removes any common mode features between the two axes. Both the experimental data and theinteracting model exhibit pronounced lobes that are not visible in the non-interacting case. Finally,a one-dimensional measure of this anisotropy can be defined by integrating over one of the axesR
( dz(n(x, z) − n(z, x))) as shown in the third row. In the integrated transpose-anisotropy weobserve peaks symmetric to the centre of the cloud whose height are sensitive to T /TF for theexperimental data and the interacting model, whereas in the noninteracting case this anisotropy
is significantly reduced, inverted and insensitive to temperature. The small residual anisotropy in
the non-interacting theory reflects the initial asymmetric trap, whose effect on the cloud after longtime-of-flight is minimal. These results reveals the interacting nature of the Fermi gas in a single
absorption image and can serve as a precise temperature probe. In practice, to extract T /TF in
an interacting gas, we minimize the least squares residuals between the integrated anisotropy of anexperimental image and the simulated image with T /TF used as a single free parameter.

4.4.4

SU(N ) interacting model

In this section, we detail the theoretical model used to describe the density fluctuation andanisotropic expansion measurements. Study of a multi-component interacting system with SU(N )symmetry is a logical extension of studying interacting two-component systems, as detailed in122
Chapter 2. To study the collective behaviour of the quantum gas, we use a kinetic approach[176, 177]. Interactions can be approximated as two-body s-wave contact interactions. The totalinteraction energy is given byHint =

gX
2 0

Z

d3 rψ̂σ† (r)ψ̂σ† 0 (r)ψ̂σ0 (r)ψ̂σ (r)σ,σgN (N − 1)=2Z

d3 rn2σ (r) ,(4.7)

where g = 4π~2 a/m, the sum includes all N spin components, and in the final step we assume
the single component density nσ (r) = ψ̂σ† (r)ψ̂σ (r) is the same for each component. Note that thisequation is drastically simplified compared to a general multi-component interaction Hamiltonian
due to SU(N ) symmetry2 .
The total Hamiltonian for a single component is thenZHσ =d r ψ̂σ† (r)
3



Z


∇2
−
+ U (r) ψ̂σ (r) + d3 rψ̂σ† (r) g(N − 1)nσ (r) ψ̂σ (r) ,
2m

(4.8)

where the first term describes the kinetic and potential energy of the fermions, which are confinedPby a harmonic trapping potential U = m/2 i (2πνi )2 x2i . The above equation looks like the Hamiltonian for a single species in an external potential with an additional effective potential given by
the mean field energy:
VM F = g(N − 1)nσ (r) ,(4.9)

which is (N − 1) more repulsive than a two-component gas (see Chapter 2).
To look at the dynamics, we use the semiclassical transport Boltzmann-Vlasov equation:


∂t +


p
· ∇r − ∇r [g(N − 1)nσ (r) + U (r)] · ∇p f = Ic [f ] .
m(4.10)The single component phase-space distribution f evolves due to ballistic motion (second term),and from forces due to the harmonic trapping potential U (r) and the mean-field interactions withstrength VM F = g(N − 1)nσ (r). Finally, collisions during expansion are described by the integral
Ic [f ] [177].
2

For a theoretical description of a multi-component Fermi gas without SU(N ) symmetry, see Ref. [181].

123
Collisions can be approximated using the relaxation time approximation as [96]Ic [f ] = −f − fle,τ(4.11)

where f approaches the local equilibrium state fle over a characteristic relaxation time τ . We canfollow [94, 182] and approximate the relaxation time as(2πντ )

−1

= (N − 1)4
5 31/3 π


Ñ

1/3

a
aho

2
FQ (T /TF ) ,(4.12)

where FQ (T /TF ) is a universal function given by an integral [94]. At low temperatures, collisions
are Pauli blocked and thus FQ vanishes as (T /TF )2 , a signature of Fermi statistics. FQ is of order
1 in the intermediate temperature regime, and vanishes as (T /TF )−1 at higher temperatures. It isthus only in the intermediate temperature region where collisions play a significant role.
To obtain the phase space density in-trap, we solve the Boltzmann-Vlasov equation in equilibrium (i.e. with Ic [f ] = 0 and ∂f /∂t = 0). This gives us
1f (r, p) =
e

β



p2
+U (r)+g(N −1)nσ (r)−µ
2m

.



(4.13)+1

However, since this equation depends on the single component density nσ (r) =Rd3 pf (r, p), it(2π~)3has to be solved self-consistently to get a closed-form solution. This is achieved iteratively. First,
we ignore the interaction term which gives us f0 . Since for a given atom cloud, µ is unknown, µ0 is
R 3 rd3 pthen determined from d(2π~)
3 f0 (r, p) = Nσ where Nσ is the number of atoms per spin species. The
atom number can be measured through standard absorption imaging. For iteration number i, ni
can then be computed from fi−1 to update VM F,i . This defines f (µi , VM F,i ), where at each step µi is
determined through normalization to the atom number. We then set fi = αf (µi , VM F,i )+(1−α)fi−1
with α = 0.9. Iterating this procedure leads to convergence in 5-10 iterations for the parameters
we consider. From Equation 4.13 we can then calculate, given the mean-field interaction, the fulldensity profile n(r) = N nσ (r) and the corresponding T /TF of the gas. However, due to higher
order dipolar interactions caused by the absorption imaging probe beam, the density profile cannot
be accurately fitted in-trap. As a result, we expand the gas before measurement.

124
4.4.4.1

Time of flight expansion

To study the dynamics, we make a scaling ansatz for the distribution function [183, 96, 93,
184, 180]. Our ansatz isf (r, p, t) = Q

1
1/2

j (λj θj )

f0

ri 1
,
(pi − mλ̇/λi ri )λi θ1/2!
.

(4.14)i

The parameters λi and θi are scaling parameters along the ith direction that vary in time. Following[96], we take the moments of ri pi and p2i to obtain a closed set of differential equations for the scaling
parameters λi , θi :θi
λ̈i + (2πνi )2 λi − (2πνi )2 + (2πνi )2 ξi
λi

θi
1
− Q
λi λi j λj!
= 0,(4.15)andθ̇i + 2λ̇iθi = − θi − θ̄ /τ ,λi

(4.16)−1)hnσ i
accounts for the mean-field interaction with h· · · i being phase-space
where ξi = g/2(Ng/2(N−1)hnσ i+hp2i i/m
Paverages with respect to the equilibrium distribution and θ̄ = 1/3 i θi .

To study the expansion after switching off the trap, the second term in Equation 4.15 is set to
0. The differential equations are then solved for the scaling parameters, which when plugged into thescaling ansatz yield the phase-space distribution after time-of-flight. With the phase distribution
distribution, we can then calculate the full density profile of the gas. The compressibility can becomputed directly from the density profiles. In a harmonic trap the chemical potential varies asµ(r) = µ − U (r). Thus, to calculate the compressibility, the derivative with respect to µ can be−1∂nreplaced by a derivative with respect to one of the spatial directions, giving ∂n∂µ = m(2πνi )2 ri ∂ri .

4.5

Concluding remarks
We have demonstrated that SU(N ) symmetry substantially enhances interaction dynamics

in a quantum degenerate Fermi gas. The many-body problem for the dilute repulsively interacting
Fermi gas can be solved exactly, and we have shown with high precision how the additional spin

125
degree of freedom systematically modifies thermodynamic properties in the bulk gas. This opens
the path for future quantum simulators to systematically explore SU(N )-symmetric Fermi systems
in periodic potentials.
The enhanced SU(N ) interactions allow us to reach ultralow temperatures in record speed.
Rapidly creating a spin-polarized degenerate sample is an important milestone for the realization ofa band-insulator atomic clock [55]. With a total preparation time under 3 s, Dick effect contributions√
are in the mid-10−17 / τ , and are thus on-par with the most stable intercomparisons to date [3].
This work sets the stage to prepare a low-entropy 3D band insulator with a spin-polarized Fermi
gas. Ongoing work is expected to push the lattice clock coherence time to 10s of seconds.

Chapter 5

Pauli blocking of atom-light scattering

The research described in this chapter has been reported in the following publication:
Christian Sanner*, Lindsay Sonderhouse*, Ross B. Hutson, Lingfeng Yan, William R. Milner, and
Jun Ye, “Pauli blocking of atom-light scattering,” Preprint at https://arxiv.org/abs/2103.02216(2021). * (these authors contributed equally) [185].

Radiative relaxation of an excited quantum system is a ubiquitous phenomenon: it makes
fireflies glow, underlies the radiative recombination of electrons and holes in light-emitting diodes,
and can be observed as gamma decay of nuclear isomers. The intimately related phenomenon oflight scattering involves minimally populated excited states that often assume a virtual character[60, 186]. The blue sky is a direct manifestation of such a second-order excitation-emission processwith air molecules scattering sunlight. What all these light-matter interactions have in common isthat the radiative decay depends on the final density of states for the joint emitter-photon system.
In particular, for the photon channel it has been demonstrated [187, 188, 189] that manipulation of
the density of vacuum modes through the use of an electromagnetic resonator modifies the emission
and scattering of light. This Purcell effect is now widely used in nanostructured devices [190].
More than 30 years ago it was suggested [191] that constraints imposed by quantum statistics
on an atomic medium could also modify spontaneous emission and light scattering. Fermi statistics
requires the total wavefunction of a fermionic system to be antisymmetric, giving rise to the Pauli
exclusion principle that forbids indistinguishable fermions from occupying the same internal and127
external quantum states. Accordingly, if a sufficient number of ground state fermionic atoms occupy
all available external motional states into which an internally excited fermionic atom has to decay,
this decay process will be blocked. The prospect to manipulate light scattering properties and toquantum engineer the natural lifetime of an excited atomic state by embedding it inside a Fermi
sea has triggered many theoretical studies and proposals [192, 193, 194, 195, 196, 197, 72, 198, 199],
but up to now this fundamental modification of atom-light scattering by Fermi statistics has notbeen observed, complicated by atomic properties and competing collective radiative behavior.
Spontaneous emission and light scattering are not synonymous. Even though the usual descriptions [186] of both processes involve the relaxation of an excited state (a populated atomic
energy eigenstate and a virtual intermediate state, respectively), their properties and decay dynamics are different. In a noninteracting Fermi gas, however, the unavailability of final momentum
states is the sole cause for single-particle modifications of radiative (re)emission properties andtherefore equally affects all atom-light processes that transfer randomly directed recoil momentum
to the atom [197].
In the following, we report the first direct observation of Pauli suppression of light scattering
using a quantum degenerate Fermi gas of strontium atoms. We confirm that the suppression
becomes stronger as degeneracy is increased and when the Fermi energy approaches the photon
recoil energy. By angularly resolving the photon scattering rate, we measure up to a factor-of-two
reduction in comparison to the natural value determined from a thermal ensemble. This striking
manifestation of Fermi statistics connects for the first time the fundamental radiative propertyof atoms to their motional degrees of freedom subject to quantum statistics. The consequencesof Pauli blocking of atomic motion have been demonstrated earlier, including the suppression ofcollisions [71], the direct observation of Fermi pressure [68], the onset of Hanbury Brown-Twiss
anticorrelations [79, 80], local antibunching [200, 19, 88], the suppression of chemical reactions
between molecules [201], and the formation of Pauli crystals [81].
In this chapter, we reintroduce the concept of Pauli blocking in section 5.1. In section 5.2
we explain our experimental strategy to observe Pauli blocking using a light scattering experiment128with a weakly excited Fermi gas. We then detail our experimental results in section 5.3, and finallyconclude in section 5.4.

5.1

Pauli blocking in a large Fermi sea
Fig. 5.1 illustrates the key concept of the experiment and introduces the relevant energy scales[178]. An ensemble of N harmonically confined identical fermions of mass m forms a quantum
degenerate Fermi sea with close to unity occupation of the oscillator states if the thermal energy
kB T is small compared to the Fermi energy EF = (6N )1/3 ~ω. The three-dimensional confinement ischaracterized by the mean trap frequency ω = (ωx ωy ωz )1/3 , and kB and ~ denote the Boltzmann andreduced Planck constants, respectively. The Fermi temperature TF and Fermi wavevector kF aredefined via EF = (~kF )2 /(2m) = kB TF . If an atom inside the Fermi sea absorbs a photon carrying
momentum ~kabs , it gains a corresponding recoil energy ER = (~kabs )2 /2m. Here, we consider thecase of weak confinement with ~ω << ER and treat Rayleigh scattering as a two-step momentum
transfer process [72, 202]. Upon photon re-emission the atom experiences a randomly directed
second momentum kick ~kemi where |kemi | = |kabs | = kR , resulting in a total momentum transfer~k = ~kabs + ~kemi . If, however, the corresponding motional state is already occupied by anotheratom within the Fermi sea, this decay channel is blocked, and light scattering will be suppressed.
The relative temperature T /TF and the wavevector ratio k/kF , where ~kF = (2mEF )1/2 is themomentum space radius of the Fermi sea, determine the density of available final momentum statesand hence the degree of blockade. Following Fermi’s golden rule one finds using a local density
approach [195, 72] a relative scattering rate of1
R
S(k) =

d3 p d3 q ni (p, q) [1 − nf (p, q)]R.
d3 p d3 q ni (p, q)(5.1)

Here the integrals cover the six-dimensional phase space spanned by three momentum dimensions p and three real space dimensions q. The initial and final state phase space cell occupations
1

This equation simplifies to Equation 2.27 for a harmonically trapped gas.

129

A

B
EF
kemi
α
kabs

k
ER
kBT
} ħω

|e >

Incoming
photonkF|g >EmittedphotonholedαSuppressionC

EF = 1 ER

EF = 10 ER

Figure 5.1: Light scattering on atoms embedded inside a Fermi sea.
(A) Indistinguishable fermions obey the Pauli exclusion principle. If the thermal energy kB T issufficiently low they fill almost all available harmonic oscillator states up to the Fermi energy EF
with near-unity occupation. An excited atom (orange) acquires a recoil energy ER when returning
to its internal ground state (blue). (B) In momentum space, the atoms form a Fermi sea occupying
most states up to the Fermi momentum ~kF . Light scattering with re-emission along α and total
momentum transfer ~k can happen only if the final momentum state is not occupied by anotherground state fermion. A detector covering a solid angle dα registers the emitted photon. (C) Pauliblocking leads to a characteristic angular distribution of scattered photons in the deeply degenerate
regime (here T /TF = 0.1). For EF ∼ ER (left sphere) scattering is preferentially suppressed in asmall cone around the forward direction, while EF >> ER (right sphere) causes strong suppression
for all scattering angles α. The suppression factor specifies the scattering rate relative to a nonPauli-blocked sample.

130
are given by ni = nF D (p, q) and nf = nF D (p + ~k, q), where nF D is the Fermi-Dirac distribution
for a harmonically trapped gas, that isnF D (p, q) =1
1 + ξ −1 exp[ Σi p2i /(2m) + Σi mωi2 qi2 /2/kB T ].

(5.2)

The index i runs over all three dimensions and ξ is the fugacity related to T /TF through
1/Li3 (−ξ) = −6(T /TF )3 , where Li3 is the trilogarithm function. The expression for S(k) counts
all available final momentum states for a given momentum transfer ~k and averages over all initialstates within the Fermi sea.

5.2

Light scattering experimental strategy
The above analysis suggests two pathways to observe pronounced Pauli suppression of lightscattering. One can either prepare a Fermi gas with EF >> ER so that a significant decay blockade
is obtained for all emission directions, that is, for any momentum transfer up to the maximum
2~kR (homogeneously colored right sphere in Fig. 5.1 C). Or one can relax this requirement andselectively observe only scattering events with a small momentum transfer so that EF ∼ ER is
sufficient. The second approach, which we take here (gradient-colored left sphere in Fig. 5.1 C), isstraightforwardly realized in a small-angle light scattering configuration where a small number ofatoms within the Fermi sea are optically excited and a subset of the scattered photons is collectedunder a shallow angle α with respect to the excitation beam corresponding to a momentum transferof ~k = 2~kR sin(α/2).
A multitude of effects besides quantum statistics can influence radiation dynamics in a dense
ensemble of emitters. Coherence, either externally imprinted or spontaneously established, canlead to super- and subradiant collective states that correspondingly exhibit super- and subnaturalradiative lifetimes. Dicke superradiance [203, 204], radiation trapping [205], multiple scattering[206], and other forms of coherent or incoherent collective scattering [207] all critically dependR
on the integrated optical density OD = σndl of the atomic gas. This attenuation parameter,
derived from the single-atom scattering cross section σ and density n, where l is measured along

131

the direction of the incoming light, defines a parameter region where the gas is optically thin
(OD << 1) and single-particle scattering dominates over collective effects. Given that on resonance,
σ is on the order of the squared optical wavelength λ2 = (2π/kR )2 , and kF is tied to the 3D
peak density via n = kF3 /6π 2 , it is not possible to satisfy the Pauli blocking criterion kF ∼ kR
without violating the small OD requirement. Indeed, typical optical densities encountered in atomic
Fermi gas experiments easily exceed 100. In this optically thick regime multiple photon scatteringstrongly affects the light propagation inside the sample, as evidenced with the resonant fluorescence
image displayed in Fig. 5.2 B. Using off-resonant light reduces the effective scattering cross section
and renders the atom cloud weakly absorbing for the incoming probe light at sufficient detuning.
This, combined with differential observation strategies and avoiding light detection in the forward
direction, minimizes the influence of collective scattering dynamics. Furthermore, unlike Pauliblocking, these competing effects are not dependent on quantum statistics and can therefore beobserved using a thermal gas [26], instead of a quantum degenerate gas, to provide a baseline.
Our experiment starts with the preparation of a 87 Sr Fermi gas as described in reference [54].
The 1 S0 , F = 9/2 ground state is split into ten magnetic spin states mF = −9/2, ..., 9/2. Here F isthe total angular momentum of the nuclear spin. This fully thermalized 10 component sample [51]contains 18 000 atoms per spin state confined in a crossed optical dipole trap with maximum radial
trap frequencies of ωx = ωy = 2π × 120 Hz and an axial confinement with ωz = 2π × 506 Hz. This
leads to a Fermi energy of EF = 440 nK for each of the ten Fermi seas and we reach temperatures
down to 0.1TF . The small inter-component s-wave scattering length a = 97a0 , where a0 is the Bohrradius, makes the Fermi gas weakly interacting with kF a = 0.06, which is scaled by the number ofspin states, increasing the total energy of the gas by less than 7%. These weak repulsive interactionsin principle affect S(k), but their differential effect is negligible [51] at the measurement precisionachieved in this study. Under suitable conditions, any optical excitation returning to the groundstate should experience a decay blockade. We perform specific measurements on the 1 S0 − 1 P1

132

A

On resonanceB
Camera 1

1

P1

Δ = 40 Γ

Detunedλ = 461 nm
Γ/2π = 30.4 MHz
NA = 0.23

10 μm

1

S0
10 μm

24º

72º
z
y

NA = 0.1
x
Camera 2

B
Blue
probe

Figure 5.2: Experimental Setup.
(A) Off-resonant probe light excites 87 Sr atoms inside a Fermi sea. Spontaneously re-emitted
photons are collected simultaneously along two imaging axes under angles of 24◦ and 72◦ , with
their numerical apertures (N A) shown respectively. Small scattering angles correspond to a small
momentum transfer with k/kF < 1, whereas the transversal observation detects photons fromscattering events with k/kF > 1. The circularly polarized probe beam has an intensity of 5Isat
where the resonant saturation intensity is Isat = 41 mW/cm2 . (B) On resonance the atomic cloudis optically thick for the probe beam and the image formed on Camera 1 displays a hole in thecloud center due to multiple scattering. At a detuning of ∆ = 40Γ the atom cloud is optically thinand the corresponding image resembles the atomic density distribution. The detuned frequency isused in the Pauli blocking experiment. A magnetic bias field of 3 Gauss applied in the horizontal
plane along the y-direction defines the atomic quantization axis.

133transition at 461 nm with a natural linewidth of Γ = 2π × 30.4 MHz [208] and a recoil energy of
ER = 520 nK. Employing a continuous weak-drive Rayleigh scattering scheme, the Fermi gas is
exposed for 1 µs to a 1.2 GHz detuned drive beam that causes on average < 10% of the atoms toundergo an excitation cycle. No evidence for inelastic scattering is observed.
For the given drive beam and atom cloud parameters the sample is optically thin with aneffective OD of 0.02, corresponding to a forward transmission of 98%. As illustrated in Fig. 5.2, twodetectors collect scattered photons simultaneously under off-axis angles of α1 = 24◦ and α2 = 72◦ .
These operational parameters are chosen to simplify the interpretation of our measurements. First,
as discussed above, the detuning from resonance by 40Γ eliminates multiple scattering dynamics andfurthermore avoids refractive lensing contributions [121] far off-axis. Second, the low excitation rate
keeps the Fermi sea intact throughout the probe pulse exposure. Third, operating in the weak-drive
limit ensures that there is no inelastic scattering contribution beyond the recoil-induced energy shift,i.e., Mollow triple-peaked fluorescence spectra and other strong-drive effects are negligible [186].

5.3

Experimental results
To systematically explore how quantum degeneracy affects light scattering, we start with adeeply degenerate gas of T /TF = 0.1 and gradually heat it up to T /TF = 0.7 through parametricconfinement modulation while keeping the atom number and Fermi energy constant. Under these
conditions photons scattered off of the weak probe pulse are counted. To satisfy the requirement
to minimally disturb the Fermi gas by the probe, the number of collected photons along the two
detection axes is correspondingly low. Even for the axis with a high N A = 0.23 objective lens fewer
than 200 photons are collected at full quantum efficiency. By operating the gated CCD detectors ina hardware binning mode that maps all detected photons into a 3x3 superpixel array, we spatiallyand temporally isolate the signal from background contributions and maintain low readout noise.
The measured photon counts are compatible with ab-initio estimates within 30% based on thereported drive parameters and detection efficiencies. All relevant thermodynamic parameters are
independently assessed through measurements on expanded gas clouds after time of flight.

134

5.3.1

Temperature and Fermi energy dependence

The results of Pauli-suppressed scattering are shown in Fig. 5.3A. Under the shallow offaxis angle of 24◦ , where k/kF = 0.45 for our Fermi sea, we find a strong dependence of thephoton counts on T /TF . To properly normalize the detector signal, i.e., to convert the registeredphoton counts to a suppression ratio without introducing an arbitrary scaling factor, it is necessary
to prepare an equivalent reference sample that is not Pauli-blocked. Because the weak opticalconfinement does not maintain a constant atom number for T /TF > 1, we devise an alternative
method to eliminate Pauli blocking: At 10 µs before applying the actual probe pulse we expose
the Fermi gas to a 5 µs long pre-pulse that destroys the Fermi sea by randomly exciting atoms to
momentum states beyond kF . Consequently, light scattering is no longer Pauli suppressed during
the subsequent probe pulse. All blue round data points in Fig. 5.3A/B are normalized to a common
reference photon count obtained from a single pre-pulse exposed sample (see below). Numericallyintegrating the expression for S(k) at all probed temperatures reveals good agreement between
experiment and theory. The simultaneously acquired measurements at an observation angle of72◦ do not show a pronounced suppression and only exhibit a weak temperature dependence, as
expected for k/kF = 1.27 since most final momentum states lie outside of the Fermi sea.
To further verify that Pauli blocking is the mechanism responsible for the observed scattering
behavior, we study the dependence on kF /kR by varying the confinement while keeping atom
number and T /TF = 0.13 constant, as displayed in Fig. 5.3B. Even at the shallowest confinementwith kF /kR = 0.57 (EF /ER = 0.32), the momentum transfer along the 24◦ axis amounts to only
k/kF = 0.74 so that we still observe substantial suppression of light scattering. This is in contrast
to the 72◦ case, where k/kF = 2.07 at kF /kR = 0.57. If k/kF > 2 Pauli blocking is negligible atany temperature so that we normalize all photon counts acquired under 72◦ (red data points in
Fig. 5.3A/B) with respect to this reference point (solid red circle in Fig. 5.3B).

135
A

B
kF / kR = 0.93 ± 0.02

Suppression

1.0
0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5
0.1

0.2

0.3

0.4
T / TF

0.5

T / TF = 0.13 ± 0.01

1.0

0.6

0.7

0.6

0.7

0.8
kF / kR

0.9

1.0

C
1.0

0.8

k / kF

1.5

1.0

0.6

0.5

Suppression

2.0

0.4

0.1 0.3 0.5 0.7

T / TF

Figure 5.3: Suppression of light scattering in a 87 Sr Fermi gas over a range of temperatures and Fermi momentums.
All measurements are performed with a 10-component Fermi gas containing 18 000 atoms per spinstate. The scattering behavior is distinctly different for the two observation angles of 24◦ (bluecircles and squares) and 72◦ (red circles). Raw photon counts are normalized with respect to measurements on non-Pauli-blocked reference samples (see main text). Each circle data point is derived
from 150 iterations of the experiment, while each square point results from 50 experimental runs.
Solid theory curves are calculated with no free parameters. The widths of the theory lines reflect
the experimental uncertainties of Fermi energy and temperature. The error bars are purely statistical and indicate one-standard-deviation confidence intervals. (A) At a constant Fermi wavevectorof kF /kR = 0.93 (EF /ER = 0.86), the atom ensemble’s scattering cross section decreases as thegas approaches deep quantum degeneracy. The suppression observed under 24◦ is pronounced andreaches 50% at T /TF = 0.13. In contrast, under 72◦ , the suppression is negligible. (B) At constant
T /TF = 0.13, kF is varied by adiabatically changing the confinement. A larger kF results in astronger suppression. (C) The data reported in A and B are measured along 4 trajectories (dottedlines) through the parameter space spanned by k/kF and T /TF . Depending on the scattering angle
k varies between 0 and 2kR . Light collected under an off-axis angle of 24◦ corresponds to a momentum transfer ~k < ~kF for the given Fermi gas, leading to substantial reduction of the density ofavailable final states. On the contrary, for the 72◦ collection angle, the corresponding momentum
transfer ~k > ~kF . Thus most final states are not blocked and scattering is not suppressed.

136

5.3.2

Pre-pulse application

The additional blue square data points displayed in Fig. 5.3 were normalized by applying
a pre-pulse separately for each data point and are in good agreement with the common-mode
normalized measurements. In particular, with a 10 µs wait time before the probe pulse, the atomic
density remains essentially the same with or without the pre-pulse. This ensures common-mode
cancelation of collective scattering dynamics. Below we present light scattering measurements withvariable pre-pulse durations for a deeply degenerate (T /TF = 0.11) and heated (T /TF = 0.58)
Fermi gas. This data confirms the scattering-induced destruction of the Fermi sea over the courseof the pre-pulse application.
The suppression factors reported in Fig. 5.3 as red and blue circles were obtained by normalizing the measured photon counts against a common reference count value. Residual total atom
number and probe pulse intensity fluctuations were below 10% over the course of the data acquisition. For the 72◦ detection axis (red circles) the measurement at kF /kR = 0.57 (red solid circle),which is outside the Pauli-blocking regime with S > 0.99, directly provided the reference count.
The photon counts detected along the 24◦ axis were normalized against a reference count measured
by exposing the Fermi gas to a 1.2 GHz detuned pre-pulse to create holes in the Fermi sea before
applying the actual probe pulse. Precise detection gating with an interline transfer CCD (Camera
1) and a short dark time of 10 µs between the two pulses avoided contamination of the probe signal
with pre-pulse fluorescence while keeping the atomic density distribution unaffected. The pre-pulse
intensity was calibrated such that each atom scattered on average more than one photon during a
5 µs long pulse. To directly verify that this exposure destroyed the Fermi sea and eliminated thePauli blockade, we systematically varied the pre-pulse duration and observed for two samples at
T /TF = 0.11 and T /TF = 0.58 how the probe signal saturated at the non-blocking reference level
for longer pre-pulses (Fig. S1). A pre-pulse duration of 5 µs was chosen to obtain the reference
count along the 24◦ detection axis.

137

0.58 TF

0.11 TF

Figure 5.4: Photon scattering from a Fermi gas after exposure to a pre-pulse of variable
duration.
For the chosen pre-pulse detuning and intensity, the Pauli blockade is destroyed after exposure
for a few µs. The scattering signal from the deeply degenerate sample (green squares) increases
by almost a factor of 2 while the barely degenerate sample (purple squares) shows only minimal
increase, as expected for a Fermi sea with kF /kR = 0.93 under an observation angle of 24◦ . Datain the plot is normalized relative to the mean counts detected for 4 and 5 µs pre-pulse durations.

5.3.3

Spatially resolved scattering rate

Spatially resolving the origin of the scattered photons within the Fermi gas, in addition tocounting the total number of scattered photons along a given direction, provides a picturesquerevelation of the Pauli blocking mechanism. For this purpose, we modify the 24◦ high-NA axis
to deliver magnified images of the atom cloud and employ a non-binning CMOS camera with a

138

low readout noise of 2 photoelectrons. The cylindrically symmetric atomic cloud at T /TF = 0.12
with a diameter of about 20 µm is projected along the z-axis into a 2D image with 0.9 µm wide
pixels. Because a single pixel collects on average less than 1 photon, it is necessary to average
hundreds of frames in order to derive a finely resolved scattering profile. Furthermore, we radiallyaverage the mean image to obtain the radial profile (blue data points) shown in Fig. 4. Residual
optomechanical drifts in the optical setup cause small displacements of the center of mass positionof the cloud during the frame averaging period. This, together with the finite resolution of theimaging system, leads to an effective 1/e2 pixel blurring on the order of 3 µm. To compare theobserved scattering profile to theory predictions without introducing a free scaling parameter, we
independently acquire in-situ density profiles of the Fermi gas through the same imaging setup via
high intensity fluorescent imaging [122]2 . Using the frame-wide summed total photon counts andthe measured global suppression ratio at T /TF = 0.12 (Fig. 5.3A), we then properly scale the radial
density profile to reflect the scattering profile expected without Pauli blockade for the weak probe
beam (purple curve). Momentum space integration plus one-dimensional line-of-sight integration ofS(k) yields a radially resolved suppression ratio for us to determine the expected scattering profile
with Pauli blockade (blue curve). Except at the center of the cloud, where radial averaging does
not significantly improve the signal-to-noise ratio, we find good agreement between calculated andmeasured profiles. Towards the outer rim of the cloud the local Fermi energy drops so that lightscattering is not suppressed anymore; the local suppression ratio will approach the thermal gas
limit of 1. This happens, as seen with the blue theory curve, only in the outermost region where
the density is so low that the signal-to-noise level is insufficient to reliably determine a suppression
ratio.
2
Also see Chapter 3. A separate manuscript is in preparation to further describe our high intensity fluorescenceimaging method.

139

no Pauli blocking

r
10 μm

Pauli blocking at 0.12 TF

Figure 5.5: Spatially resolved light scattering from a trapped Fermi gas at T/TF = 0.12.
Radially averaging the spatially resolved mean signal (inset) from 1100 individual images obtained
along the z-axis yields a radial light scattering profile (blue data points). In-situ column density
images, separately obtained using a high intensity fluorescent imaging technique, are used to predict
the scattering signal for a non-degenerate gas (purple curve). The spatial profile of light scattering
calculated for the T /TF = 0.12 ensemble (blue curve) agrees well with the measured data.

5.4

Conclusion
In conclusion, we have reported a clear demonstration that Fermi statistics leads to strongly

modified light scattering in a quantum degenerate system. The presence of a Fermi sea alters
the final atomic motional mode spectrum and enables a direct observation of Pauli blockade oflight scattering. Interpreted from a many-body system perspective, this experiment probes the
structure factor [73] of a quantum degenerate Fermi gas. Defined as the Fourier transform of thespatial density-density correlation, the static structure factor S(k) characterizes the linear responseof a system to a perturbation with wavevector k. Accordingly, the suppression of light scattering

140
and the suppression of density fluctuations [86] in a Fermi gas are two interrelated manifestationsof the same fundamental many-body physics.
It will be exciting to go beyond the regime of Rayleigh scattering and directly measure a
prolonged excited state lifetime using properly prepared quantum states. The capability to altera fundamental decoherence mechanism will open new avenues for quantum engineered atom-lightinterfaces. In particular, custom designed Fermi reservoirs can protect optical qubits at local nodes
while facilitating cavity-free directional photon emission for efficient network connectivity. In thecontext of optical atomic clocks this work could enable spectroscopic interrogation times exceeding
the natural lifetime of the excited clock state and investigation of novel radiative properties ofatoms.

Chapter 6

Attempts to measure Pauli blocking of the natural lifetime using coherent
control on a narrow-linewidth transition

Quantum information science strives to use well-controlled, artificial quantum systems toadvance scientific understanding and foster the growth of new technology. Pauli blocking is aquantum tool that lies at the heart of this idea. A large Fermi gas made out of ultracold atoms isbuilt using a tabletop experiment to quantum engineer an atomic transition with a lifetime longer
than what nature would provide alone. Spontaneous emission causes information to escape into theenvironment and can thus destroy optical qubit states and ultimately limit spectroscopic resolution.
As a result, this tool could have tremendous applications in quantum technologies such as precision
metrology and quantum information processing.
The first observations of Pauli blocking of light scattering, which has now been measured in
multiple experiments [185, 209], demonstrated the ability to realize this long-sought theoretical idea
and established a scientific understanding of the fundamental and competing radiative mechanisms.
This was a crucial first step towards implementing Pauli blockade as a practical tool, but the steadystate scattering regime these experiments were performed in is not suitable for modern atomic clocks
and quantum information platforms, which require arbitrary coherent control of the atomic state.
In this chapter, we detail experiments that attempt to observe Pauli blocking of the natural
lifetime of an atomic state after atoms in a Fermi gas are coherently driven to the excited state.
We directly detect the lifetime of the atomic state by time-resolving the number of excited atoms,
as opposed to angularly detecting the scattered photons as was done in Chapter 5 [185], and142

in Refs. [209, 210]. Measurements are performed on the 7.4 kHz linewidth 3 P1 transition in a
multi-spin component Fermi gas of 87 Sr atoms. In Section 6.1, we discuss our technique to create
pronounced suppression using imbalanced Fermi gases. We then outline our experimental setup
in Section 6.2. In Section 6.3 we detail a systematic study of the competing radiative effects
present in this coherent-drive scheme by measuring the lifetime of a thermal gas of atoms. We
then present lifetime measurements at degenerate temperatures. Although we measure an angleaveraged lifetime that is up to 20% longer than the natural lifetime at ultralow temperatures, as
shown in Section 6.4, we will show that we could not fully rule out other systematic effects thatcome with coherently driving a system with a high optical depth (OD). An intriguing alternative is
to use systems with different geometries. In Section 6.5 we detail initial measurements performedin two-dimensional systems formed in a 1D optical lattice, where collective radiative properties aredesigned to be different than in a 3D gas. In fact, here the experimental results are consistent withtheory predictions based on Pauli blocking. However, we cannot fully support Pauli blocking withexperimental results alone just yet. Finally, in Section 6.6 we present an outlook.

6.1

Pauli blocking with imbalanced Fermi gases
The key concept of Pauli suppression is illustrated in Fig. 6.1 (a), which shows the Fermi

sphere in momentum space. A photon with initial momentum ~kabs and final momentum ~kemitransfers momentum ~k = 2~kR sin(α/2) to an atom, where kR = |kabs | = |kemi | and α is theangle between kabs and kemi . For a zero-temperature gas with a Fermi energy much larger than
the recoil energy, atoms in the center of the Fermi sea cannot decay since all available final states
are occupied. Decay can only occur if an atom is within ~|k| of the Fermi surface. At finite
temperature, the number of holes in the Fermi sea increases, and thus also does the number ofunoccupied states that an atom can decay to. As the density of holes is greater near the surface,√atoms within roughly ~|k| + 2mkb T can decay.

143

(a)

k
Incoming
photon

kemi
α
kabs

kF
Emittedphoton

dα

(b)
|e>

|g2>

|g1>

Figure 6.1: (a) Pauli blocking is maximized for atoms in the center of the Fermi sea. Atoms get a
directional momentum kick ~kabs from the absorbed light, and a diffusive momentum kick ~kemifrom the emitted light, giving a net momentum kick ~k in the center of the Fermi sea. An atom inthe center has the maximal suppression since it is least likely to find an unoccupied final state. (b)Exciting atoms in the center of the Fermi sea can be approximated using a three-level scheme withimbalanced Fermi gases. Two ground states with an imbalanced atom number are created and theminor Fermi gas is excited so that it predominantly decays into the major Fermi gas. Under the
optimal conditions this decay channel is blocked.

144
For maximal suppression, it is thus desirable to only excite atoms in the center of the Fermi
sea. As suggested in Ref. [72], this could be achieved with an excitation laser that is focusedmuch more tightly than the atomic cloud. Alternatively, one can create scattering between two
imbalanced Fermi gases, where the atom number in one gas is much less than in the other, as
shown in Fig. 6.1 (b). The two Fermi gases can be created for example by having two different
spin states with unequal population that co-exist in the same optical trap. Their corresponding
optical transitions are connected by the relevant Clebsch-Gordan coefficients. The minor Fermisea, which only samples the smallest energies of the major sea, can then be excited and made to
predominantly decay to the major Fermi sea. Compared with the tight probe beam technique, thishas the further advantage that the ground state Fermi sea is not perturbed during the excitation.
In this three-level scheme where the minor Fermi sea with fugacity ξi is excited and only decays tothe major sea with fugacity ξf , the scattering rate relative to a non-Pauli blocked sample is (seeEquation 2.27)
SF M (k)8S(k(α)) =
=1−
3
SM B
πEF i β 3

Z

3/2

da a

Z
dpz

1

1

,
1 + ξi−1 ea+p2z 1 + ξf−1 ea+(pz +kscal )2

(6.1)

√
where kscal = 2 βEF i kR sin(α/2)/kF i .
If the atomic population is detected, the total decay rate as opposed to the angle-dependent
decay rate is measured. In contrast to the Purcell effect, where the solid angle of the cavity is
too small to measurably change the atom’s total decay rate, for large Fermi energies and low
temperatures, Pauli blocking has the capability of changing the total decoherence rate of the atom(see the right sphere in Fig 5.1 (c) ). The total relative decay rate M is given by integrating over
all emission angles
Z π

Z
M=

S(k(θ))P (θ)dΩ = 2π

S(k(θ))P (θ) sin θdθ

(6.2)

0

where θ is the angle between the quantization axis and kemi , and P (θ) is the dipole emission pattern.
For linear polarization P (θ) = 3/8π sin2 θ, and for circular polarization P (θ) = 3/16π (1 + cos2 θ).
Note that with these definitions, θ is not necessarily the same angle as α. The direction of the145

quantization axis can change the amount of suppression you get. If the maximum of the dipole
emission pattern happens to coincide with the smallest k-vectors, more suppression will be achieved.
Extending this to a three-level system where decay from the excited state can either occur to themajor Fermi sea with blockade M or back to the initial state (which is unblocked), the total
suppression has to be weighted by the Clebsch-Gordan coefficients of the two possible decay paths,
that is the suppression
Mtot = | hFgmin , mgmin ; 1, q|Fgmin , 1; Fe , me i |2 + Mmaj ∗ | Fgmaj , mgmaj ; 1, q|Fgmaj , 1; Fe , me |2 .
(6.3)

6.2

Experimental setup
The preparation of our Fermi gas is described in Chapter 3. 87 Sr atoms in all 10 nuclear

spin states are evaporated to temperatures as low as T /TF = 0.1 with radial confinement as highas ωr = 2π × 100 Hz and a maximum axial confinement of ωz = 2π × 700 Hz. Imbalanced Fermi
gases are achieved by using atoms in two different nuclear spin states, mF = 9/2 and mF = 7/2, of
the 1 S0 ground state, as shown in Fig. 6.2. The number of atoms with spin mF = 9/2 is a factorof 10 smaller than the number of atoms in mF = 7/2, which is achieved using optical pumping to
remove most of the atoms in the mF = 9/2 state. At T /TF = 0.1, the final atom numbers give amaximum Fermi energy of 450 nK for the major component.
In order to directly measure the spontaneous emission lifetime of the excited state, we thencoherently excite the mF = 9/2 atoms on the Γ = 2π × 7.4 kHz linewidth 3 P1 transition at 689 nm,
where the 21.8 µs natural lifetime can be easily time-resolved. This leads to a recoil energy of 230
nK. An applied magnetic bias field of 1 G splits the excited state’s sublevels so that the atoms canbe coherently excited to the F = 11/2, mF = 9/2 state using a 5 µs π pulse with π-polarized light.
Due to the Clebsch-Gordan coefficients, without Pauli blocking, the excited atoms would decay tothe mF = 7/2 ground state with an 82% probability. However, the large ground state Fermi sea

146

…

mF = 7/2

0.82

mF = 9/2

mF = 11/2

1 F' = 11/2

1S

F = 9/2π

0.18…
mF = 7/2

3P

0

mF = 9/2

Figure 6.2: 3 P1 level scheme. Imbalanced Fermi gases with a 10:1 atom number ratio are createdusing two nuclear spin states of the ground 1 S0 state. Atoms in mF = 9/2 are then excited to the
3 P , F = 11/2, m = 9/2 excited state using a π pulse with π polarized light. The excited atoms
1
F
are Pauli blocked from decaying to the mF = 7/2 ground state. The numbers in boxes refer to theClebsch-Gordan coefficients squared.

largely blocks that decay channel for the minor excited Fermi gas.
The F = 11/2 state is used since it is the most magnetically sensitive 3 P1 hyperfine state,
which reduces the probability of off-resonant excitation. The combined effects of off-resonant excitation and polarization impurity are measured by varying the Rabi detuning of the probe, as shownin Fig. 6.3. At -0.5 MHz, the probe is on-resonance with the mF = 9/2 to mF = 11/2 transition.
These measurements give us an upper bound of the two effects of 2%. Given the 10:1 atom number
ratio, this implies a worst-case scenario where we excite 5 times more 9/2 atoms than 7/2 atoms.
The prediction given in Equation 6.3 is shown in Fig. 6.4 for variable T /TF and EF /ER .
Here it is assumed that the atom number ratio is 10:1 and that the Clebsch-Gordan coefficientsare as shown in Fig. 6.2. Given our incoming light is π-polarized so that kabs is perpendicular tothe quantization axis, α = π/2 + θ. All of the mF = 9/2 population is excited so that no blockade
occurs when atoms decay back to the mF = 9/2 ground state. As a result the optimal suppression
is determined by the Clebsch-Gordan coefficients; in this case the optimum Mtot = 0.18. For the

147

35000
Atom number

30000
25000
20000
15000
10000
5000
0
0.5

0.4
0.3
0.2
0.1
Laser detuning (MHz)0.0

Figure 6.3: Rabi scan as the frequency of the excitation laser is changed. Zero corresponds to themF = 9/2 to 3 P1 , F = 11/2, mF = 9/2 transition. Given the magnetic field of 1 G, the mF = 11/2
transition is on resonance at -.5 MHz.

Fermi energies and temperatures in our sample, we can expect a optimum Mtot = 0.4. The benefitof going to an imbalanced Fermi sea is shown in Fig. 6.5. At T /TF = 0.1 and EF /ER = 2.0, twoequally populated Fermi gases have a total suppression of 0.6, compared to 0.4 for an atom number
imbalance of 10:1.
In this coherent excitation, 3-level scheme, multiple observables exist to measure Pauli suppression. The lifetime of the atomic state can be determined by time-resolving the population of
excited atoms. In addition, if decay of excited atoms to the 7/2 ground state is blocked, more
atoms will decay to the 9/2 ground state. Pauli blocking can thus be thought of as modifying theSuppression

148

T/TF, Maj
Figure 6.4: Prediction of Pauli blocking suppression for a three-level scheme where the atom number
ratio between the two ground state Fermi gases is 10:1 and the Clebsch-Gordan coefficients for the
decay are 0.82:0.18. Suppression is shown for various temperatures T/TF and Fermi energies
EF /ER , quoted for the majority component.

branching ratio of the excited state. The number of atoms decaying to 9/2 as opposed to 7/2 canprovide a separate confirmation of the blockade process. Finally, the decoherence rate of the atoms
due to spontaneous emission can be measured by looking at the decay of Rabi oscillations. Below
we discuss how we measure each of these observables.
To achieve fast, high SNR measurements, atomic state readout is performed using the 30.4
MHz broad 1 P1 transition, where roughly 100 photons can be scattered in 1 µs per atom. Around
1% of the fluoresced photons are subsequently detected using a high resolution imaging axis with
an N A = 0.23. The experimental sequence to measure the lifetime is shown in Fig. 6.6 (a). First,
the mF = 9/2 atoms are excited to the 3 P1 , F = 11/2, mF = 9/2 state using a π pulse. After avariable time twait , a 10 µs pulse of high intensity 1 P1 light causes significant recoil heating to the149

Figure 6.5: Total suppression for a three-level scheme where the Clebsch-Gordan coefficients forthe decay are 0.82:0.18, and T /TF = 0.1 with variable EF /ER for the major component. Here thesuppression is shown for different atom number imbalances for the major component and minor
component (EF is quoted for the majority component in the legend).

ground state atoms, which are as a result removed from the trap. After a further 1 ms, all excitedatoms have decayed and the ground state atoms are removed from the image. A fluorescence imageis then taken during a 2 µs long 1 P1 readout pulse, where light is collected using a vertical camera
axis with an N A = 0.23. Wait times are varied from 1 µs to 200 µs, and 200 measurements arerecorded over 20 different wait times. The wait time is randomized for each run. The lifetimeis then extracted from the time constant, which is determined by fitting the entire data set to asingle exponential. This allows us to time-resolve the decay of atoms out of the excited state. We
originally used an alternative differential measurement technique that allowed us to measure theexcitation fraction every shot so that we were immune to fluctuations in the total atom number.
However, we found that our experiment was stable enough that this differential technique was

150
Readout

ExcitemF = 9/2

3P

1 F = 11/2

1P

1

Lifetime measurements

Blow out
GS

ExciteReadout

π
1S

mF = 7/2

mF = 9/2

0

1S

T-wait0~ 1 msExcitation fractionA

T-wait (us)B

Camera

z
x
y

Lens
B

NA = 0.23

𝑒̂Branching ratio measurements

Excite Excite

Blow outGSmF = 9/2

Readout

3P
1
F = 11/2

1S

200 usReadout

1 us~ 1 msmF = 7/2

0

mF = 9/2

Excite5°

Figure 6.6: (a) Strategy to measure the excited state lifetime and the branching ratio from the
mF = 9/2 excited state. Atoms are excited on the narrow linewidth 3 P1 transition using a π pulse
and the excited atoms are read out using the broad 1 P1 transition. The upper and lower rightpanels describe how we measure the lifetime of excited atoms and the branching ratio of excitedatoms back to mF = 9/2, respectively, and are discussed in the main text. (b) Experimental
setup. Excitation light probes a Fermi gas on-resonance with the 1 S0 , mF = 9/2 to 3 P1 , F = 11/2,
mF = 9/2 transition. The ground state population is read out by taking a fluorescence image using
the 1 P1 state, where the fluorescence is measured using a high NA imaging system.

unnecessary. To measure the decoherence rate (not shown), the same experimental procedure isused with twait = 1µs and a variable excitation time.
To measure the number of 9/2 atoms, the lifetime measurement can be repeated with
T-wait = 1 µs, before any atoms have decayed. The branching ratio is then measured by firstexciting all the atoms to 9/2 with a first π pulse, waiting 200 µs for these atoms to decay, and thenmeasuring the number of atoms that have decayed to 9/2 with another π pulse (see Fig. 6.6 a).
Comparing the total number of atoms that decay to mF = 9/2 before and after the initial excitation
pulse provides a measurement of the branching ratio on the 9/2 → 9/2 transition.

151

6.3

Optical depth considerations
We first measured a 28 µs lifetime of the excited state at T /TF = 0.1, 30% longer than thenatural lifetime. We soon discovered however that the longer-than-natural lifetime was insensitive
to the temperature of the sample, indicating that other mechanisms were governing the radiative
behavior. We also observed a faster-than-expected decoherence rate when Rabi flopping between
the mF = 9/2 ground state and excited state. At the high central densities over 1014 atoms/cm3and on-axis optical depths (ODs) around 10 for each spin component, collective effects such as
super and sub-radiance as well as multiple scattering influence the radiative properties of theatoms [205, 26]. In Chapter 5, we suppressed rescattering by detuning the drive frequency fromthe atomic resonance. This was a steady-state scattering experiment where atoms act like driven
dipoles and light is scattered at the drive frequency. In contrast, here we are coherently driving
atoms to the excited state and then extinguishing the drive. Atoms consequently spontaneously
emit at the natural non-AC Stark shifted resonance frequency, and the probability of reabsorption
is no longer reduced for a detuned drive. We thus sought to systematically study the competingmechanisms as the OD was changed for the different components, with the hope that we could
find a parameter regime where we weren’t sensitive to collective effects but still had an appreciable
Pauli blocking signal.
Despite the long lifetime measured at thermal temperatures, we found evidence for superradiance in numerous measurements. By gating the camera on axis with the excitation laser within 1 µsof when the excitation pulse ends, we detected > 10 times more photons than what was expected
through spontaneous emission alone. In addition, we measured a lifetime as short as 8 µs after
exciting 9/2 atoms if the 7/2 component was first removed. With the 7/2 component in place, we
also measured that up to 70% of the atoms returned to the mF = 9/2 ground state for a thermal gas
with T /TF = 2.0 despite a branching ratio of only 18%. By systematically varying the confinement
and atom number of the mF = 9/2 component, we found that fractionally fewer atoms returnedto the 9/2 ground state as the OD decreased, as shown in Fig. 6.7. The OD shown in this plot is152

Branching ratio to 9/2

0.8
0.7
0.6
0.5
0.4
0.3
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25
mF = 9/2 optical depth

Figure 6.7: Fraction of atoms that return to the mF = 9/2 ground state after the mF = 9/2 atoms
are excited to the 3 P1 , F = 11/2, mF = 9/2 state. Measurements are performed with a thermal
sample where T /TF = 2.0.

measured by counting the number of atoms in a central region of the Fermi gas and multiplyingR
it by the cross-section, that is OD = 3λ2 /2π n(0, 0, z)dz. We measured that the branching ratio
approached a constant value for an OD < 1. The offset from 0.18, as expected from the ClebschGordan coefficients, can be explained by residual excitation of the 7/2 atoms. Residual excitationof 7/2 atoms can occur through off-resonant excitation of the π-polarized probe light and due topolarization impurity, which can cause on-resonant excitation of 7/2 atoms to 9/2.
Once we reduced the mF = 9/2 OD below 1, all measurable superradiant behavior disappeared. Coherent Rabi flopping on the 9/2 atoms in a thermal sample reproduced the expected
profile given by the Rabi frequency and the spontaneous decay of the excited state, shown in Fig. 6.8,

153

Excitation fraction

1.0
0.8
0.6
0.4
0.2
0.0

0

5

10

15

20

Pulse time (us)Figure 6.8: Rabi flopping on 1 S0 , mF = 9/2 → 3 P1 , mF = 9/2 transition, measured on a thermal
Fermi gas. The red line represents a fit using no free parameters and assuming the 21.28 µs natural
lifetime for the excited state.

where the theory line is plotted for a two-level system with no free parameters. This experimental
evidence suggests that superradiance predominantly occurs on the 9/2 → 9/2 transition, given by
the phase imprint from the driving field.
Creating a sample with an OD < 1 for the mF = 9/2 atoms can easily be achieved by using
fewer than 5000 atoms in 9/2 and by making our Fermi gas into a pancake, with tight axial
frequencies of ωz = 2π × 700 Hz and loose radial trap frequencies of ωr = 2π × 30 Hz. The cloudis then probed along the z-direction for the minimal OD.
Despite the measured branching ratios at low 9/2 ODs that are close to the theory expectation, we still observed long lifetimes in non-degenerate Fermi samples. While superradiance seemed

154

Lifetime (us)

25
24
23
22
21
0.5

1.0

1.5
2.0
mF = 7/2 optical depth

2.5

Figure 6.9: Lifetime versus mF = 7/2 OD. Measurements are performed with a thermal sample
where T /TF = 0.75 at a 7/2 OD of 2.7. The 9/2 OD is fixed at 0.7 for all measurements.

to be removed by creating a sample with a 9/2 OD < 1, collective effects can still occur along the
7/2 channel, where the large Fermi sea creates an OD up to 5 at low temperatures. In particular,
we suspect that multiple scattering is a dominant collective mechanism along the 7/2 channel.
The lifetime was systematically measured in a thermal sample for various 7/2 ODs, as shown in
Fig. 6.9. Here, the OD was changed by reducing the number of 7/2 atoms using optically pumping.
We observed that the lifetime approached the natural lifetime as the 7/2 OD was reduced below 1.
Here is where we run into a challenge for observing Pauli blocking of the spontaneous emission
rate on the 3 P1 transition without any collective radiative effects that obscure the signal. Reducing
the 9/2 OD removes collective effects on the 9/2 channel and simultaneously improves the Pauliblocking signal since fewer atoms in 9/2 creates a larger Fermi sea imbalance. In contrast, reducing
the 7/2 OD reduces the major component’s Fermi energy, which in turn degrades the Pauli blocking

155

depth

mF = 7/2 OD = 2.0

mF = 7/2 OD = 1.5

Figure 6.10: Measurements of the excited state lifetime for variable temperatures (here quoted forthe majority component) while the OD is kept relatively constant. The 7/2 OD varies from 2.0
to 1.5 as T /TF is increased. Here the 9/2 OD is 0.7, and T /TF is quoted for the 7/2 component.
Measurements of the lifetime performed for a thermal (T /TF = 0.75) gas at equivalent ODs yieldshorter lifetimes.

suppression.

6.4

Measurements with a cold sample
Given the measured dependence of collective radiative effects on the optical depth, we tookdifferential measurements where we compared the lifetime at low and high temperatures with afixed OD. Fig. 6.10 shows the lifetime as T /TF is increased, where the 7/2 OD changes from 2.0
to 1.5 across the measurement, and the maximum 9/2 OD is 0.7. The sample temperature isincreased using parametric heating of the optical dipole trap. Measurements at the same 7/2 and9/2 ODs but at a higher temperature show a reduced dependence on the OD, suggesting thatthe large differential effect we see at low temperatures is influenced by Pauli blocking. For thethermal samples, the 7/2 OD is varied by adjusting the number of atoms in mF = 7/2 using opticalpumping. The measured decay curves from the excited state closely follow a single exponential at

Excited atoms

156

4000
3500
3000
2500
2000
1500
1000
500
0
0

50

100
T-wait (us)

150

200

Figure 6.11: Full lifetime curve for a Fermi gas with a 7/2 OD of 2, a 9/2 OD of 0.7, and T /TF = 0.2
for the major component. The blue line is an exponential fit to the data, which gives a 1/e lifetimeof 25.7 ± 0.3 µs. Data points reflect the mean, and error bars are 1 standard deviation.

degenerate temperatures, as shown in Fig. 6.11 where a 1/e lifetime of 25.7 ± 0.3 µs is extracted.
We additionally measured the branching ratio from the mF = 9/2 → mF = 9/2 transition forvariable T /TF , where the 7/2 OD and 9/2 OD was kept fixed by adjusting the number of atoms
in each spin state using optical pumping. As shown in Fig. 6.12, as T/TF increased, we observedthat fewer atoms return to the 9/2 ground state, which is consistent with Pauli blocking on the
7/2 channel causes more atoms to return to the 9/2 ground state at cold temperatures.
These measurements are suggestive that Pauli blocking is influencing the lifetime of excitedatoms, but they rely on the assumption that the OD is the only important parameter for determining
how much the lifetime is influenced by collective effects. To collect stronger evidence to claim theobservation of Pauli blocking on 3 P1 , we looked into alternative methods to observe Pauli blocking.

157

Figure 6.12: Fraction of atoms returning to mF = 9/2 while the 7/2 and 9/2 optical depth are
kept fixed. The temperature is quoted for the 7/2 component. As the sample is heated, the ratio
decreases towards the non-Pauli blocked ratio. Here T /TF is quoted for the major component.

6.5

Measurements in a 1D optical lattice
Going to 2D has intriguing benefits for observing Pauli blocking in a coherent drive schemeas described here. First off, if the drive beam is along the tightest direction where the latticeconfinement is large, the initial recoil associated with absorbing a photon is taken up by the trap
instead of the atom, increasing the degree of suppression. Second, due to a change in the Fermi
energy and the density of states in 2D, enhanced Pauli blocking is expected for a fixed atom
number and radial trap frequency compared to 3D, and Pauli blocking can be observed at higher
temperatures. Third, multiple scattering may be reduced. Even though the OD can be large
along the radial direction, the probability that a photon will be spontaneously emitted along thisdirection is small - that is, the angle-averaged OD is quite small. Initial theory results that include
the effects of both Pauli blocking and collective dynamics using a master equation suggest thatmultiple scattering is in fact reduced in a 2D geometry1 . Doing measurements in a 1D opticallattice offers an alternative to a true 2D geometry if an experiment does not have immediate accessto the required lattice setup. In a 1D optical lattice, it is desirable to load as few layers as possible.
1

See an upcoming publication with Thomas Bilitewski and Ana Maria Rey.

(a)

z
g

158
(b)

B

ê
5°

excitation
probe

Figure 6.13: (a) Experimental setup to measure the lifetime in a 1D optical lattice. The opticallattice is formed along gravity, g, and the excitation probe polarization (ê is along the magnetic
field, which points into the page. (b) Lifetime of excited atoms in the 1D optical lattice versus
EF /ER . The 9/2 OD is 0.8, and the temperature is T /TF = 0.6 in the XODT when the 7/2 OD
is 5.5.

For EF = 1Erec , a single layer has an OD of 1.5 along the tightly confined direction. However, howcollective effects depend on the optical depth in a 2D gas and in a 1D optical lattice have not beenstudied.
Here we show measurements performed in a 1D optical lattice formed along the z direction,
where the probe beam is incident with 5 degrees with respect to z, as shown in Fig. 6.13 (a). Thep paxial trap frequency is νz = 44 kHz, giving a Lamb-Dicke parameter along z of η = νrec/νz = 0.33,
pwhere νrecis the recoil frequency of the probe laser. We use the scheme described in Fig. 6.6 (a)to measure the lifetime of the excited atoms. For these measurements, the 9/2 OD is 0.8 andωr = 2π × 150 Hz. The vertical extent of the cloud is 4 µm so that roughly 10 pancakes are loadedwith 1000 atoms in 7/2 in the central pancake at EF /ER = 1.4. The temperature is T /TF = 0.6 for

159
(a)

(b)
7/2 OD = 5

7/2 OD = 3

Figure 6.14: (a) Lifetime as the excitation pulse area is varied for a fixed 7/2 OD of 5 and a fixed
9/2 OD of 0.8 (b) Lifetime as the excitation pulse area is varied for a 7/2 OD of 3 and a 9/2 OD
of 0.8the 7/2 component. To achieve a temperature of T /TF = 0.6, the Fermi gas is loaded into the 1D
lattice and parametrically heated. The temperature is measured by fitting the expanded the Fermi
gas in time-of-flight after the lattice is adiabatically decreased to 0 Erec and atoms are transferredto the XODT.
Fig. 6.13 (b) shows the measured lifetime as EF /ER is varied for the 7/2 component. The
7/2 Fermi energy is varied by changing the number of 7/2 atoms using optical pumping. Similar
to the 3D case, we observe that the lifetime increases as EF /ER , or equivalently the 7/2 OD, isincreased. Contrary to the single 2D pancake case, in a 1D optical lattice, multiple scattering
could still occur between pancakes, which makes interpreting the results more difficult. However,at T /TF = 0.6, a Pauli suppression factor of 0.8 is expected in 2D. We measured the lifetime whilevarying the excitation pulse area at a fixed 7/2 OD of 3, and separately for a fixed 7/2 OD of 5,
as shown in Fig. 6.14. In both cases the 9/2 OD was 0.8. Interestingly, we observe no differencein the lifetime as the pulse area is varied, suggesting that superradiance is absent in our system.
This observation agrees with our measurements in a 3D gas, where we observed that superradiance160
disappeared once the 9/2 OD was less than 1.
While superradiance and multiple scattering have been extensively studied in 3D [205, 26],
little research has been performed in 2D. Initial theory results suggest that multiple scattering is also
suppressed in a 1D optical lattice.2

More systematic measurements at a variety of temperaturesand Fermi energies would be a fruitful future study to clarify the role of Pauli blocking and collective
dynamics in this system.

6.6

Outlook
To observe the effects of Pauli blocking with an excitation scheme that coherently controls

the atomic state, as discussed here, it is desirable to go to a 2D gas with a single layer. Here, theon-axis OD can be quite small and the Fermi energies large so that significant Pauli suppression
is expected without the influence of competing collective radiative mechanisms. The three-levelsystem we outline here has the advantage of having multiple observables to confirm Pauli blocking.
In addition to measuring the lifetime of the excited state, the branching ratios between the two
ground states and the excite state provide a new angle to observe the suppression. The techniquepresented in this chapter to measure the branching ratio was associated with systematic effects thatcaused residual excitation of 7/2 atoms, which limited us from observing the natural branching ratioof 0.18 at low ODs and thermal temperatures. A larger magnetic field would help with off-resonant
excitation, but polarization impurity, which can excite 7/2 atoms to the 9/2 excited state, will
always be on-resonance with the probe. Reducing the impurity below a percent is difficult. It
would thus be optimal to devise an alternative strategy to measure the branching ratio, perhaps
using optical Stern-Gerlach separation to measure the ground state spin population [113]. However,the spin states need to be spatially well-separated since small changes in the minority component
need to be detected at high SNR, which can be technically difficult since this component has
significantly fewer atoms than other spin states. Additionally, it is intriguing to think of measuring
Pauli blocking by looking at the coherence time after continuously Rabi flopping on the desired2

See an upcoming publication by Thomas Bilitewski, Ana Maria Rey, and others.

161
state. While effects such as multiple scattering can change the lifetime as atoms spend more time on
average in the excited state, it should not increase the coherence time. Our initial measurements ofthe coherence time did not lead to obvious results but more study into this effect could be fruitful.
Finally, to use Pauli blocking in 3D optical lattice clocks, it is desirable to measure Pauli blocking
in a 3D optical lattice as opposed to a Fermi gas [198]. This could be demonstrated using a similar
three-level scheme with two atoms per lattice site.

Chapter 7

Conclusion

In this thesis, we present multiple results that highlight how quantum gas engineering canenhance the performance of Fermi-degenerate optical atomic clocks. Here, we provide a briefsummary of the results reported in this thesis and provide an outlook on future research directions.

7.1

Summary
To improve the dead time of Fermi-degenerate atomic clocks, we described an experimentthat used multiple nuclear spin states with SU(N) symmetry during evaporation to increase coolingrates. The compressibility and expansion dynamics of the SU(N)-interacting Fermi gas were studied
in detail. This new cooling scheme allows us to significantly reduce the preparation time of Fermidegenerate 3D optical lattice clocks, in turn reducing the Dick Effect for such systems to mid√
10−17 / τ if coherently interrogated for a few seconds using a state-of-the-art ultrastable laser.
We additionally showed that Fermi-degenerate gases can be used to reduce the spontaneousemission and light scattering rates using Pauli blocking. Pauli blocking exploits Fermi statistics toprohibit decay of an excited atom to states already occupied by an identical fermion. By embedding
excitations within a degenerate Fermi sea, we measured up to a factor of two reduction in thelight scattering rate along a direction where the net momentum transfer from the excitation light
was reduced. Our method employed multiple precautionary measurements, including simultaneous
observation of the fluoresced light under different angles (with different net momentum transfer) andmeasurements with hole-poked Fermi gases at the same density, to conclusively rule out potential

163
competing radiative mechanisms. Our results agreed with a simple semiclassical model with no freeparameters besides a single common scaling factor for the data.
We also presented results from an experiment that attempted to observe Pauli blocking
using coherent spectroscopy on the 3 P1 narrow linewidth transition. We presented a series ofmeasurements in a bulk 3D Fermi gas and initial measurements in a 1D optical lattice. In theformer we carefully explored the dependence of the lifetime on the optical depth.
The work presented in this thesis showcases a common theme in atomic physics: improvements in our fundamental understanding of quantum science goes hand in hand with advancing
quantum technology. This connection between the very fundamental and the very practical is acornerstone of our advances in AMO physics, and highlights the interlaced connections between
academia, industry, and society at large.

7.2

OutlookIn the future, engineered light-matter interfaces could provide a new means to create next-

generation optical lattice clocks. Contrary to what is practical using common light-matter interfaces
such as an optical cavity, Pauli blocking provides an avenue to change the total decoherence rateof an atom. For large Fermi energies and low temperatures, the spontaneous emission rate canbe reduced for all emission angles, resulting in a measurable change in the total (angle-averaged)
spontaneous emission rate. Suppression larger than what we demonstrated can be achieved using
a higher Fermi energy, or if instead the gas is loaded into a deep optical lattice. If Pauli blocking
is implemented in a 3D optical lattice, it could be used to increase the coherence time of current
optical atomic clocks beyond the limit given by the natural lifetime of an atomic transition.
One could even imagine using this technique to provide more flexibility in the chosen atomic
clock transition. Instead of optimizing for insensitivity to a range of environmental parameters andsimultaneously a long natural lifetime for narrow spectroscopy purposes, one could pick an atomic
transition that is optimally insensitive to the environment, perhaps at the cost of a slightly shorter
natural lifetime. Pauli blocking (or other cooperative techniques) could then be used to push the164system into spectroscopic competitivity. In order to use Pauli blocking for practical devices, it isnecessary to do a bit more homework on how the physical mechanism can be realized using coherent
spectroscopy as would be necessary in a real quantum device. Of course the challenge with thismechanism is to find a way to remove cooperative scattering while still keeping other important
quantities such as atom numbers high.
One could also use these competing radiative mechanisms as the dominant means for creatingnext-generation atomic clocks. A subradiant array of atoms is an alternative means to engineer
longer lifetimes, the promise of which has been demonstrated using a 2D array of 87 Rb atoms [211].
Avenues to observe subradiance in fermionic atoms with a multilevel internal structure such as 87 Sr
have also been discussed theoretically [212].
Superradiant lasers [213] reference the linewidth of the laser to an atomic transition, for thecase of the clock transition in 87 Sr, 1 mHz. A continuous superradiant laser has the potential toreduce the reliance of local oscillators on length standards, which could be advantageous as creating
ultrastable resonators is a significant technical challenge. This system may one day allow the local
oscillator and atomic reference to be combined into one device, removing the need for a distributed
clock system.
By combining atomic clocks with optical cavities, one could demonstrate non-destructive
detection of the atomic state, reducing dead time substantially. The combination of non-destructive
measurement and spin squeezing using an optical cavity could create an atomic clock whose stability
that surpasses the state-of-the-art and is better than the Standard Quantum Limit [214]. In the end,
creating a system where each degree of freedom can be controlled and manipulated for the optimal
conditions will only provide more opportunities and competitive avenues for quantum devices.
On the accuracy front, it will be interesting to study dipole-dipole interactions in the 3D
optical lattice clock. The same physics that gives rise to cooperative scattering also implies theexistence of dipole-dipole coupling between emitters. A spin-polarized 3D optical lattice clock
removes contact interactions by making doubly occupied sites energetically suppressed by the
bandgap. However, there are still higher order long-range interactions due to dipole-dipole in-165
teractions, which could be enhanced given the regularity of the atom array and the sub-wavelength
spacing between emitters. At first glance, you might assume that such an interaction effect is verysmall given the strength roughly scales like the linewidth, in this case only 1 mHz. However, evenfrequency shifts at a fraction of the linewidth are now measurable in modern clocks. Diagnosing the
spatial dependence of the dipole-dipole shift using our high-resolution imaging system will provide
a deterministic avenue to ensure the minute shifts are due to dipole-dipole interactions instead of anumber of other potential systematics such as AC Stark shifts, BBR gradients, and Zeeman shifts.
After measuring the shift we can explore different lattice geometries [215] or spin components [216]to reduce this shift below a measurable quantity.
A Fermi-degenerate 3D optical lattice clock could also offer a clean way to measure higherorder ac Stark shift coefficients, since the trapping potential occupation number is well-controlled
in all three directions. One challenge is that the hyperpolarizability coefficient changes for lattice
beams that have a different polarization. Measuring the spatial dependence of such shifts could
provide a sensitive detection method.
Engineering longer lifetimes also requires us to actually be limited by the natural lifetime ofthe excited clock state, which is 118 s for 3 P0 [58]. Reducing the lattice depth along all three axes
to below 4 Erec is necessary to not be limited by Raman scattering, but one then needs to find ameans to have low tunneling rates [55]. An optical lattice with a larger lattice spacing can reduce
the tunneling rate, which was used in an optical tweezer experiment with 88 Sr to demonstrate a
coherence time of 20 s for an 150 atom tweezer array [57]. Tweezer clocks are an intriguing avenue
for future metrology but will need to solve scaling to a competitive number of atoms, as well as
demonstrate a robust systematic evaluation. Alternatively lattices can be formed by interfering
two beams at an angle to create a large lattice spacing that still operates at the magic wavelength
necessary for state-independent trapping. A near-unity filled optical lattice could significantly relax
the lattice spacing requirement as it suppresses tunneling between occupied lattice sites, althoughdistinguishability created through the clock excitation will continue to be an issue unless the latticespacing is made to be commensurate with the clock light.

166
The future of quantum precision metrology is quite bright — the combination of better control
over atomic systems and further improvements in measurement stability and accuracy promises
many exciting discoveries. Clock performance will one day reach the limit where new physics islimiting our measurements. This new physics could be fluctuating fundamental constants [16], dark
matter [23], or a tantalizing idea, maybe even one day gravitational waves [24]. Moving atomic
clocks outside the laboratory, and potentially into outer space, will provide even more opportunities
for discovery. Regardless, the scientific attitude of understanding fundamentally the underlying
system and its current limits will allow us to propose new solutions to better improve our devices,which in turn will lead to new scientific discoveries and to practical technologies for society.

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