Probing local quantities in a strongly interacting Fermi gas
and the construction of an ultracold Fermi gas apparatus
byRabin Paudel
B.A. Physics and Mathematics, Wabash College, 2010
M.S. Physics, University of Colorado, 2013

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2017This thesis entitled:
Probing local quantities in a strongly interacting Fermi gas and the construction of an ultracold
Fermi gas apparatus
written by Rabin Paudel
has been approved for the Department of Physics

Prof. Deborah Jin (deceased)

Prof. Eric Cornell

Prof. Cindy Regal

Date

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

iii
Paudel, Rabin (Ph.D., Physics)
Probing local quantities in a strongly interacting Fermi gas and the construction of an ultracold
Fermi gas apparatus
Thesis directed by Prof. Deborah Jin (deceased)

The nature of the normal phase of an ultracold Fermi gas in the BCS-BEC crossover regime
is an interesting and unresolved question. As interactions increase, while the many-body groundstate remains a condensate of paired fermions, the normal state must evolve from a Fermi liquid
to a Bose gas of molecules. In this thesis, I present a technique to spatially select a homogeneous
sample from the center of a trapped gas to explore this crossover. Combining this technique with
RF spectroscopy, we locally measure Tan’s contact as a function of temperature and compare tovarious many-body theories. In another experiment, we measure the distribution of single-particle
energies and momenta for a normal gas across the BCS-BEC crossover. We find that the data fit
well to a two-part function that includes a peak corresponding to fermionic quasiparticles and an“incoherent background” that is modeled using the dispersion of thermal molecules. I also describe
the construction of a new-generation Fermi gas apparatus.

Dedication

To my parents.

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Acknowledgements

My PhD would not have been possible without the support of so many people. I am gratefulto Debbie for giving me an opportunity to work in her lab. I feel very fortunate to have beensupervised by one of the most brilliant scientists of our generation. Over the years, I learned tothink like a physicist from Debbie. I have greatly missed her ability to clarify seemingly complex
ideas, her intuition and insight, and her troubleshooting skills. Losing her at such a young age is
a profound loss to our community.
When I started grad school, Tara Drake was patient enough to teach me how to work in an
ultracold atomic physics lab. Yoav Sagi joined the following year as a postdoc and made the labvery efficient and productive. I have learned so much physics and programming skills from Yoav.
Moreover, I have been able to count on both Tara and Yoav for scientific, career and personal
guidance during Debbie’s absence.
Building the new apparatus would have been difficult without the help of Roman. His focus
on details has made sure that our new apparatus is robust and very reliable. Vandy and Xin joined
our lab recently. I am sure that they will do great scientific work with the apparatus. I would also
like to thank the undergraduate students Sean Braxton and Josh Giles for assembling countlesselectronic circuit boards.
I would also like to thank the Jin/Cornell group – especially Steven Moses, Phil Makotyn,
Dan Gresh, Matt Grau, Cathy Klauss, Jake Covey and Rob Wild. Steven Moses has been a greatfriend to hike, ski, drink beer and talk physics with. Thanks to Tianlu Yuan and Jamey Szaley fortrekking to Everest Base Camp with me. I would also like to thank other JILAns Bryce Bjork, Yuval

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Shagam, Gil Porat, Rohan Singh, Bo Yan, Tim Langen, Martin and Vanessa Gaertner, Amodsen
Chotia, and Adam Reed for their friendship.
I would like to thank JILA machine shop sta↵ for designing so many parts for the new
apparatus. I could even count on Hans Green’s help at midnight on the vacuum chamber assemblyday. Thanks to James Fung-a-Fat for teaching me how to design and troubleshoot analog electronics
and feedback servos. I appreciate Krista Beck for all of her administrative work and helping mewith my paperwork.
Last, but not the least, I would also like to thank Eric Cornell, Jun Ye and Cindy Regal forsupervising me during the difficult times of Debbie’s absence and helping me finish my PhD.

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Contents

Chapter
1 Introduction

1

1.1

Basic scattering theory and Feshbach resonance . . . . . . . . . . . . . . . . . . . . .

2

1.2

The phase diagram of an ultracold Fermi gas . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Homogeneous Fermi gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4

Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2 Donut Beam technique to probe a homogeneous Fermi gas

8

2.1

Creating a donut beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Weakly interacting Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3

Experimental Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1

9

Aligning the beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4

Fermi Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5

How homogeneous is the gas? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6

The e↵ect of optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.7

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Probing a homogeneous strongly interacting Fermi gas
3.1

22

Tan’s Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.1

RF Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.2

Experimental Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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3.2

3.1.3

Contact versus Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.4

Obtaining the in-situ density distribution . . . . . . . . . . . . . . . . . . . . 30

3.1.5

How homogeneous is homogeneous enough? . . . . . . . . . . . . . . . . . . . 33

3.1.6

The e↵ect of the remaining density inhomogeneity . . . . . . . . . . . . . . . 35

3.1.7

Thermometry of a unitary Fermi gas . . . . . . . . . . . . . . . . . . . . . . . 35

The spectral function of a strongly interacting Fermi gas . . . . . . . . . . . . . . . . 40
3.2.1

Momentum-resolved RF spectroscopy . . . . . . . . . . . . . . . . . . . . . . 41

3.2.2

Experimental Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.3

Two-mode fitting function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.4

Energy Distribution Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.5

Contact vs interaction strength . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 New Generation Fermi Gas Apparatus Design
4.1

4.2

51

Vacuum chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1.1

Dispenser Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1.2

Second MOT-chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1.3

Science Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Pumping Speed Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2.1

Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.2

Conductance in our vacuum system setup and pressure estimates . . . . . . . 58

4.3

Pumping down to ultrahigh vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4

Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.5

Cart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.6

Magnetic Trap Description

4.7

Electronics for the magnetic trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.8

Dipole Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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4.9

Computer Control and Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . 73

5 Cooling to degeneracy

76

5.1

MOT Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2

First MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2.1

Calculating MOT Number from fluorescence . . . . . . . . . . . . . . . . . . 79

5.2.2

Six-level MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3

Push to the second MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.4

Second MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4.1

Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.5

Gray Molasses cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.6

Loading to the quadrupole trap and cart transfer . . . . . . . . . . . . . . . . . . . . 95
5.6.1

Trapping in a magnetic trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.6.2

Optical pumping and loading . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.6.3

Pros and cons of using a cart . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.6.4

Transfer efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.7

Evaporation in the QUIC trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.8

Optical Dipole Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.9

Thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.10 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6 Conclusion and Future Work

119

Bibliography

122

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TablesTable
1.1

Density and superfluid transition temperature of various Fermi superfluids . . . . . .

2

5.1

Evaporation stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

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FiguresFigure1.1

Phase diagram of the ultracold Fermi gas. See text for explanation. This figure is
reproduced from Ref. [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

4

BCS-BEC Crossover. On one side of the resonance, the atoms are correlated in
momentum near the Fermi surface to form a superfluid as in Cooper pairs in BCS
superconductors. On the other side of the resonance, dimers of two atoms are condensed as in BEC of bosonic atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3

4

Momentum distribution of a harmonically trapped Fermi gas. Even as we lower
the temperature of the gas, we do not see a sharp Fermi surface as expected for an
ultracold Fermi gas due to density inhomogeneity. The figure is reproduced from [2].

2.1Di↵raction grating patterns we use to produce Laguerre-Gaussian beam (top) and thecorresponding measured cross-section of the intensity profile at the far field (bottom).

2.2

6

9

Setup to create a Laguerre-Gaussian beam. The beam is focused onto the atom
cloud after the last lens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3

Optical pumping transitions for the donut beam. Depending on the experiment, weeither use |F = 9/2, mF =5/2i ! |F 0 = 5/2, mF =7/2i ! |F 0 = 5/2, mF =5/2i or |F = 9/2, mF =3/2i transition. Both of these transitions shelve atoms

to the upper hyperfine level in the ground state invisible to the probe beam. . . . . . 11

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2.4

Alignment of the donut beam by scanning to position to see a “W” shape in theatom number corresponding to the cross-section of the beam. We scan the motorized
mirror shown in Figure 2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5

Alignment of the donut beam by looking at the slosh and breathe of the cloud atdi↵erent hold time after a TOF. When the donut beam is aligned to the center of
the atom cloud these e↵ects are minimized. . . . . . . . . . . . . . . . . . . . . . . . 14

2.6

Momentum distribution for a weakly interacting Fermi gas after optical pumpingwith a donut beam. The distribution is an average of 12 images after selecting thecentral 16% of a harmonically trapped gas. The data is normalized to have the area
under the curve to be equal to 1. The solid line is the fit to the Fermi distribution
with a fixed T, kF is the only fit parameter. The inset shows the distribution
without optical pumping. The dashed line in the inset shows the expected momentum
distribution for a trapped gas with the temperature T. . . . . . . . . . . . . . . . . . 17

2.7

We look at reduced thesquared (a) of the fit and the fit parameter kF (b) as

a function of fraction probed. We find that the reduced

squared is lowered aslower fraction of atoms are probed, which means the system is better described as ahomogenous gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.8

Measured T/TF vs the fraction of atoms probed. Here, we fit the measured momentum distribution to a homogeneous gas distribution with two free parameters,
T /TF and kF . The density inhomogeneity of the probed gas results in T /TF thatis much larger than expected from the calculated average density of the probed gas(solid line). A sharp Fermi surface, characterized by a small fit T /TF , emerges as
the fraction of atoms probed decreases. The dashed line shows the result of fitting
to model calculations of the probed momentum distribution, which agrees well withthe data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

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2.9

Modeling the spatially selective optical pumping. We compare the normalized momentum distribution of the central 38% of the atoms to three di↵erent models (dotted, solid, and dashed lines; see text). The data (circles) are obtained from an averageof four images. We find that the attenuation of the hollow light beams (inset) does
not strongly a↵ect the predicted final momentum distribution when probing a small
fraction of the gas. (Inset) We take images of the cloud after a short (1.3 ms) expansion and compare data with the horizontal hollow light beam and without any
optical pumping in order to measure the fraction of atoms probed (circles) vs z/zFq
2EF,trap, where zF = m(2⇡⌫2 . For this data, the fraction probed is 71%. The predictionz)of our model (solid line), which includes attenuation of the hollow light beam as itpropagates through the cloud, agrees well with the data. . . . . . . . . . . . . . . . . 20
3.1Time sequence of the experiment. The magnetic field is ramped from 203.4 G,
where the atoms are initially prepared, to the Feshbach resonance. The hollow lightbeams are turned on 280 µs before trap release; initially, the beam that propagates
perpendicular to the long axis of the cloud is pulsed on for 10 µs followed by 40 µsof the second beam. The line shape is measured using an rf pulse with a total
duration of 100 µs and a gaussian field envelope with= 17 µs, centered 180 µs

before trap release. The cloud expands for 3 ms before being detected by absorption
imaging. To improve the signal-to-noise ratio, we remove the remaining atoms fromthe |9/2, 9/2i and |9/2, 7/2i states and then transfer the outcoupled atoms in the|9/2, 5/2i state to the |9/2, 9/2i state, where we image on the cycling transition[3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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3.2

An rf line shape for the unitary Fermi gas at T /TF = 0.25 with 30% of the atoms
R1
probed. The solid (red) line is a fit to Eq.(3.3) with the normalization 1 (⌫)d⌫ =0.5, due to the 50% 50% spin mixture. The inset shows the same data multiplied by

23/2 ⇡ 2 ⌫ 3/2 . We make sure the rf pulse induces only a small perturbation, by setting
its power to well below the value where we see the onset of saturation of the numberof outcoupled atoms. The measurement at di↵erent frequencies is done with di↵erent
rf powers, and when analyzing the data, we linearly scale the measured number ofatoms outcoupled at each frequency to correspond to a common rf power. . . . . . . 28
3.3

The contact of a nearly homogeneous sample (about 30% of the trapped atoms
probed), versus T /TF at unitarity (black circles). The shaded area marks the superfluid phase transition, with some uncertainty in its exact position (Tc /TF =0.160.23) [4]. As a comparison, we plot the gaussian pair-fluctuation NSR model(GPF) [5], the self-consistent t-matrix model (GG) [4], the non-self-consistent tmatrix model (G0 G0 ) [6], the 2nd and 3rd order virial expansion [5], a quantum
Monte-Carlo calculation (QMC) [7], and the contact extracted from a thermodynamic measurement done at ENS [8]. The error bars represent one standard deviation. The inset shows the high temperature behavior of the contact, where we findgood agreement with the virial expansion. . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4

Hydrodynamic expansion at unitarity. We start with a weakly interacting gas with⇠ 90, 000 atoms per spin state at T /TF = 0.12 and ramp adiabatically to the Feshbach resonance field. We fit the cloud with a Thomas-Fermi distribution after avariable expansion time and extract the rms widths,(t), in the radial and axial

directions. For the data, we de-convolve the measured width with a gaussian point
spread function with an rms width of 2.9 µm, to account for the finite resolution of
the optical system. The data is normalized by the initial cloud size, which is 33.4
µm and 2.8 µm in the axial and radial directions, respectively. The solid lines are
the numerical solution of the hydrodynamic equation. . . . . . . . . . . . . . . . . . 32

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3.5

Contact versus the fraction of atoms probed for a gas with T /TF = 0.46 at thecenter of the cloud. In the main plot, the measured contact (squares) is normalized
in respect to the trap kF , and is compared to the predictions of several theoreticalmodels (lines) using the local density approximation. The measured contact increases
as we probe fewer atoms at the cloud center, where the local density is largest. The
inset shows the contact normalized by the average kF of the probed atoms (squares),compared to theoretical predictions of the homogeneous contact at the average T /TF(lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.6

Comparison of the homogeneous contact calculated by several theoretical models(solid lines) and the contact averaged over the remaining density inhomogeneity
when probing the central 30% of the cloud for the same models (open symbols).
The excellent agreement of the points and the lines shows that the e↵ect of theremaining density inhomogeneity on the contact data can be neglected. See text for
the explanation of the theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.7

Comparison of di↵erent thermometry methods. The x-axis is the temperature, T /TF ,where TF is the trap Fermi temperature, as extracted from the release energy. The
y-axis is the temperature we get from the two other thermometry methods (see text
for more details). The dashed line is y = x. . . . . . . . . . . . . . . . . . . . . . . . 39

3.8

Momentum-resolved RF spectroscopy. The atoms in one of the strongly interacting
spin states are outcoupled to a third spin state. Since the outcoupled atoms haveweak interaction with the atoms in remaining spin states, the momentum distribution
and energy of these atoms can be measured after a balistic TOF expansion. The
donut beam technique is employed on the outcoupled atoms right before TOF toselect the homogeneous sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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3.9

Momentum-resolved RF spectroscopy data taken at T > Tc (see text) as a functionof interaction strength. The color represents the probability distribution of atoms
at a given E and k in the strongly interacting gas. The white line is the quadratic
dispersion of free particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.10 The quasiparticle residue Z and the e↵ective mass m⇤ as a function of the interaction
strength. Both of them are the fit parameters in Equation 3.15. . . . . . . . . . . . . 47
3.11 Energy Distribution Curves at 1/kF a =0.08. Each panel is the cut of the spectral

function at the given k/kF . The circles are data and the lines are the fits. . . . . . . 49
3.12 The contact density per particle (in units of kF ) for a homogeneous Fermi gas above
Tc is shown versus (kF a) 1 . The contact measured from the tail of the rf lineshape(blue circles) agrees well with the contact extrapolated from the fits of the PES data
(red squares). For comparison, we also plot the BCS (dashed black line) and BEC(dashed magenta line) limits, given by 4(kF a)2 /3 and 4⇡(kF a) 1 , respectively [9],the non-self-consistent t-matrix at T = 0 (dotted blue line) and its Popov version
at Tc (dash-dotted red line) [6], and the self-consistent t-matrix model at T = 0(double-dotted green line) [10]. We find that the contact measured above Tc agrees
well with the T = 0 theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1

Six-way cross glass chamber which contains the atomic source for the system. The
potassium atoms are collected in a 3D MOT in the first stage of the experiment. . . 52

4.2

CAD Drawings of the apparatus. Left: Top view, Right: Side View

. . . . . . . . . 54

4.3

The Science Cell: the cell is manufactured by Precision Glassblowing in Denver andhas seven 0.625” windows and two 1.7” that are double AR-coated at around 767
nm and 1064 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4

Transmission data for the science cell windows. The windows are AR coated for 767
nm and 1064 nm light on both sides. The baseline measurement was done without
any windows in place. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

xvii
4.5

Oven construction for vacuum bakeout. Heating plates are placed underneath theapparatus. We make an oven around the chamber using bricks and cover withthermally insulating blankets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.6

Pressure log during the first vacuum bakeout. . . . . . . . . . . . . . . . . . . . . . . 61

4.7

D2 Laser setup. We set up two DBR lasers for repump and trap light. The repump
laser is locked to the crossover transition of 39 K atoms using a spectroscopy in avapor cell. The trap laser is o↵set locked to the repump laser using a photodetector.

4.8

Tapered Amplifier setup.

4.9

40 K D2 transition lines. The frequencies on the right side are in MHz.

62

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
. . . . . . . . 65

4.10 The top view of the QUIC trap. The quadrupole pair (orange), the fast-B coil(yellow), Io↵e (light yellow) are visible. . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.11 B-field profile along the direction of Io↵e-coil while increasing the Io↵e coil current.
Quadrupole current = 380 A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.12 Current Servo electronics. This servo is used to control the gate voltage of MOSFETs/IGBTs for MOT coils, shim coils, quadrupole coils and the Io↵e coil. . . . . . 70
4.13 Circuit diagram for cart quadrupole trap and QUIC trap. Left: H-bridge setup to
switch between science quadrupole trap and Feshbach coils. Right: Switching circuit
between the Io↵e coil and the cart quadrupole coil. The element H are the Danfysik
IT 700-S, used for in-loop and out-of-loop current sensors for servoing and monitoring
purpose respectively. Relays are the Kissling (29.511.11) switches. IGBTs used inthese circuits are SKM400GA12V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.14 Dipole trap setup. Left: Setup for horizontal dipole trap to produce 25 µm beam
waist. Right: Setup for vertical dipole trap to produce 150 µm beam waist. . . . . . 72

xviii4.15 Layout of the computer control. We use one main computer for FPGA and GPIB
control. Three other workstations are used to acquire data from PixelFly/Princeton
Instrument cameras, and to send commands to Versatile Frequency Generator (VFG),
Parker Motor and dipole trap waveform. The computers are named after potassium
isotopes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.1

a. A sketch of a magneto-optical trap (MOT) setup. Three orthogonal beams are
passed through the chamber and are retro-reflected in combination with quarter wave
plates to form a+

setup. We run current in opposite direction in the two coils

to form a quadrupole magnetic field. b. The basic energy level diagram for a simple
MOT transition, which shows shift in energy level as a function of position. . . . . . 77
5.2

Fluorescence measurement of the first MOT. A 2f ! 2f imaging system is setup
to collect the fluorescence of the first MOT. By knowing the solid angle of the lens
subtained on the atoms, and intensity and detuning of the MOT beams, we cancalculate the number of atoms collected in the MOT. There’s also a security camera,
which is sensitive to near IR light, to view the MOT for troubleshooting purpose. . . 78

5.3

Potassium yield vs dispenser current. We raised the dispenser current and waited
for a few hours to measure the fill level and the half fill time. The measurement istaken with a photodiode.

5.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

MOT fill level vs beat frequency. The y-axis is normalized such that it is proportional
to the atom number. The resonance is at 682 MHz. . . . . . . . . . . . . . . . . . . . 80

5.5

MOT fill level vs push frequency detuning.

. . . . . . . . . . . . . . . . . . . . . . . 83

5.6

MOT fill level vs push beam power. . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

xix
5.7

Schematic of the second MOT beams. The dark-repump is formed by imaging a
1.6 mm aluminum dot on to the atoms by using F = 300 mm and F = 1000 mm
lens setup as seen in the figure above. The bright repump is turned on only during
optical pumping stage. D1 beam consists of both trap and repump light out of thefiber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.8

MOT fill level vs push time. a. with dark SPOT repump b. without dark SPOT
repump. We notice a significant increase in atom number with the dark SPOT
repump beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.9

Lifetime of atoms in the second MOT. We load the atoms to the MOT and turn o↵
the push beam and wait. We see a two clear timescales, a fast timescale of about 8s
corresponding to density dependent losses and a slower timescale corresponding tobackground losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.10 Lifetime of atoms in the quadrupole trap in the second MOT chamber. We load
the atoms into a quadrupole trap and wait there for a certain time(dark time), thenreload the atom to a MOT and measure the recapture fraction. This lifetime is lessthan the MOT lifetime because atoms are more sensitive to losses due to scattered
light in a quadrupole trap. Also, the magnetic trap is more shallow, thus vulnerable
to grazing collisions with background gas. . . . . . . . . . . . . . . . . . . . . . . . . 88
5.11 Gray molasses cooling. The cooling is a combination of velocity-selective coherent
population trapping and Sisyphus cooling. . . . . . . . . . . . . . . . . . . . . . . . . 90
5.12 D1 transition lines for gray molasses cooling in 40 K. The numbers on the rightcorrespond to the frequency shift of the hyperfine levels in MHz. . . . . . . . . . . . 92
5.13 Number (black square) and temperature (green circle) of the atoms transferred tothe science cell as a function of D1 trap frequency

while keeping= 0. Notethat zero is suppressed for temperature. The reason we don’t see a drastic change
in temperature is explained in the main text. . . . . . . . . . . . . . . . . . . . . . . 93

xx
5.14 Number (black square) and temperature (green circle) of the atoms transferred tothe science cell as a function of frequency di↵erence between trap and repump beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.15 Number(black square) and temperature (green circle) of the atoms transferred tothe science cell as a function of D1 molasses pulse duration. . . . . . . . . . . . . . . 94
5.16 Number (black square) and temperature (green circle) of the atoms transferred tothe science cell as a function of D1 molasses power (trap plus repump). Repump
beam power is 1/10th trap beam power. . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.17 Finding the right transition for optical pumping. We vary the optical pumping
frequency, load the atoms to the quadrupole trap, hold for a short time and measure
the MOT recapture fraction. The four lines corresponds to the energy of the Zeeman
levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.18 We vary the optical pumping pulse duration and measure the recapture fraction. We
find out that the recapture fraction saturates beyond 80 µs . . . . . . . . . . . . . . 96
5.19 Optical pumping transitions needs a+ light. Here, we vary the polarization of lightusing a quarter-wave plate to find the right polarization. . . . . . . . . . . . . . . . . 97
5.20 We vary the optical pumping power to get the highest number of atoms into thequadrupole trap. We vary the voltage driving the AOM frequency driver. 0.15V
corresponds to 13.5 µW of optical power and the power varies linearly with voltage
in this regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.21 We measure the transfer efficiency of the atoms as a function of transfer distancethrough the transfer tube to the science cell. The red points are with D1-graymolasses cooling and the green points are without molasses cooling. The blue line isthe diameter of the transfer tube. The center of the science cell is 54 cm away. We
find that the transfer efficiency is only limited by the vacuum lifetime in the caseof D1-cooled atoms. Without D1-cooling, we see that most of the atoms are lostcoming right out of the chamber going through the narrow part of the tube. . . . . . 100

xxi
5.22 We measure the trapping frequency in the QUIC trap by sloshing the atoms with afield gradient pulse and measuring the center position in TOF as a function of holdtime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.23 We vary the current in the Io↵e coil and measure the trapping frequency. . . . . . . 102
5.24 We measure the bias field of the QUIC trap by measuring the temperature of thecloud at the end of the evaporation and linearly extrapolating to find the frequency
where the temperature goes to zero. Depending on this frequency, we can calculate
the magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.25 We vary the current in the Io↵e coil and measure the bias field. For a good evaporation, we would like to have a bias field of a few Gauss. . . . . . . . . . . . . . . . . 103
5.26 We measure the axial trapping frequency of the QUIC trap by sloshing the atoms in
the axial direction with a field gradient pulse and measuring the center position in
TOF as a function of hold time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.27 Elastic cross-section vs temperature. The plot is reproduced from [11]. The filled
circles are the p-wave scattering cross-section for spin polarized cloud and the open
circles are the s-wave cross section for mixed spin states. We see that the p-wave
cross-section is large down to 100 µK and plummets as the temperature is reduced.
The s-wave cross-section is relatively flat as a function of temperature. . . . . . . . . 105
5.28 Zeeman sublevels of the ground levels in potassium. The evaporation microwave
transfers the atoms in |F = 9/2, mF = 9/2i and |F = 9/2, mF = 7/2i to F = 7/2
hyperfine level, where they become high-field seeking and leave the trap. . . . . . . . 105
5.29 The evaporation trajectory of the atoms plotted as Log T vs Log N. For a good
evaporation which increases the collision rate, we need the slope of this to be greaterthan 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.30 The evaporation trajectory of the atoms plotted as temperature vs final microwave
frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.31 The evaporation trajectory of the atoms plotted as Log PSD vs Log N. . . . . . . . . 107

xxii
5.32 a. Measurement of the radial trapping frequency by modulating the horizontal dipole
trap beam. We see a loss in number at twice the trapping frequency and a smallfeature at the trap frequency. We have done a similar measurement to measure
the trapping frequency in the axial direction by modulating the vertical dipole trap
beam. b. Radial trap frequency vs power of the dipole trap beam. The red line is
the fit to P 1/2 and the black line is from calculation. . . . . . . . . . . . . . . . . . . 109
5.33 Evaporation trajectory for the optical trap evaporation as we lower the horizontal
dipole trap intensity. We see that the slope of Log T/Log N is 1.8 which means theevaporation in the dipole trap is much better than that in the magnetic trap. At
certain point, we see the evaporation saturates: we see a loss in number without
seeing a loss in temperature. This is because for this evaporation we started with animbalanced gas in mF = 9/2 and mF = 7/2. We lost all of the minority mF = 7/2
atoms and only mF = 9/2 remain. To get to colder temperature, we had to startwith a 50/50 mixture of spin states. . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.34 An example of a 2D surface fit of a degenerate cloud to an FD fit in 5.37. . . . . . . 113
5.35 A comparison of the temperature of the gas measured in two di↵erent ways. The xaxis is the T /TF extracted from the fugacity of the cloud using an FD fit. The y-axis
is the T /TF extracted by measuring the temperature fromof the cloud (Equation

5.37) and the TF from number and trapping frequency. The line has a slope of 1. . . 114
5.36 Imaging setup for MOT # 2. We can do either absorption imaging or fluorescenceimaging with this setup. The setup has a magnification of 1. There’s also a 50/50
beam splitter which allow us to look a the MOT fluorescence with a security camera. 116
5.37 Imaging setup for imaging atoms in the science cell in the radial direction. The
probe light is sent in the same direction as the horizontal dipole trap. . . . . . . . . 116
5.38 Stern Gerlach imaging of the atoms in di↵erent Zeeman state. Note that this image
consists of two OD images stitched together, one for positive spin states and anotherfor negative spin states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Chapter 1
Introduction

Strongly correlated systems are found in di↵erent forms and sizes. High Tc superconductors,
superfluid 3 He, quark-gluon plasma, neutron stars are all examples of strongly correlated quantum
systems [12]. Compared to other strongly interacting systems, ultracold Fermi gases are a littledi↵erent in the sense that the interaction strength in these systems can be tuned easily from weak
to strong in a table-top experiment setting.
Since the advent of laser cooling and the production of the first BEC, interest in quantum
gases has increased significantly [13]. Clean systems, controllable dimensality, and tunable interactions make quantum gases quite attractive to study many-body physics [14]. BECs or degenerate
Fermi gases of 13 atomic species have been created and hundreds of groups are studying themaround the world. In 1999, the fermionic isotope of potassium was cooled to degeneracy by ourgroup [15]. Then, in early 2000s, using Feshbach resonance, these atoms were paired up either as
Cooper pairs such as in superconductors or as dimers, to form a fermionic superfluid. That opened
an even more new, interesting, and challenging opportunity that is to study strongly-interacting
fermions [16]. These superfluids are unique because the critical temperature for the superfluid
transition is quite high (compared to TF ). Table 1.1 below lists a comparison between the density
and superfluid transition temperature Tc of ultracold Fermi gases and other Fermi systems [17].

2Table 1.1: Density and superfluid transition temperature of various Fermi superfluids
SystemConventional superconductors
High Tc superconductors
3 He
Ultracold Fermi gas of 40 K atoms

1.1

Density(cm 3 )
1023
1023
1022
1013

Tc (K)
1 10⇠ 70
3 ⇥ 10 3
10 7

Tc /TF
10 5
10 2
10 3
0.17

Basic scattering theory and Feshbach resonanceIn a simple two-body picture, the collision process between two atoms can be reduced to thesolution of the Schrodinger equation of the relative motion [18, 19, 20]. For low energy scattering,when an incoming wave of the form eikz scatters o↵ of a potential of size r0 , the outgoing wave has
the form,(r) = eikz + f (k)

eikr
,
r

(1.1)

where f (k) is the scattering amplitude. For the case of low k, f (k) can be expanded as,
1
k cot 0 (k) + ik
1
=
,
1
a
r0 k 2 /2 + ik

f (k) =

where a is the s-wave scattering length which is related to the phase shift

(1.2)
(1.3)
0 (k).

For positive

scattering lengths, there exists a bound state with binding energy,
✏b =

~2
2ma2

(1.4)

When the energy associated with the two atoms in one channel (open channel) equals with theenergy of a bound molecular state in a di↵erent channel (closed channel), a scattering resonanceknown as the Feshbach resonance occurs. If the magnetic moments of atoms in the two channels
are di↵erent, we can change the energy di↵erence between the two states by tuning the magnetic
field. The scattering length tuned by that process around the resonance is given by [18],✓a = abg 1

B

B0

◆

.

(1.5)

3
Here, B0 is the position of the resonance, abg is the background scattering length, andis thewidth from the resonance position to the field where the scattering length is zero.

1.2

The phase diagram of an ultracold Fermi gas
The physics described by the phase diagram in Figure 1.1 is known as the BCS-BEC crossover[1]. The crossover is a powerful idea connecting two physicals theories into one first experimentallyshow in a degenerate Fermi gas [16, 13]. The x-axis is the interaction strength described by thedimensionless quantity 1/kF a, where kF = (3⇡ 2 n)1/3 is the Fermi momentum of the gas, and n
is the total density of the gas. At resonance, when 1/kF a = 0, the gas is said to be at unitarity.
Since the scattering length diverges, the only relevant length parameter that describes the gas is
the density. The y-axis in the diagram is the temperature parameterized by the Fermi temperature
TF . The gas below the critical temperature (Tc ) is in the superfluid phase. Depending on the sideof the resonance, the superfluid can either be like correlated electrons as described by BCS theoryof superconductivity or a Bose Einstein Condensate of K2 dimers (Figure 1.2). The normal phase
is a bit more interesting to explore as various theories do not agree with each other. The normal
phase atoms on the BCS side of the resonance behave like quasiparticles described by the Fermi
liquid theory. As the interaction is increased, they pair up and form uncondensed dimers on the
BEC side of the resonance. However, the behavior of the gas near the resonance is controversial
among the community [21, 8]. Some theorists propose a phase similar to pseudogap phase in highTc superconductors involving many-body pairing of atoms above the critical temperature. The
pairing mechanism is many-body as it requires Fermi surface as in Cooper pairing as opposed to
two-body dimer pairs. However, other theorists reject this notion of pseudogap physics for a unitary
Fermi gas and claim this phase can be described Fermi liquid theory. During my PhD research,
we have explored this regime of the strongly interacting Fermi gas using RF spectroscopy to better
understand the normal phase behavior of the gas.

4

Figure 1.1: Phase diagram of the ultracold Fermi gas. See text for explanation. This figure is
reproduced from Ref. [1].

Figure 1.2: BCS-BEC Crossover. On one side of the resonance, the atoms are correlated in momentum near the Fermi surface to form a superfluid as in Cooper pairs in BCS superconductors.
On the other side of the resonance, dimers of two atoms are condensed as in BEC of bosonic atoms.

5

1.3

Homogeneous Fermi gases
For a trapped gas of atoms in a harmonic potential, even though the atoms are in thermal

equilibrium, the Fermi temperature is di↵erent at di↵erent part of the trap due to density variation.
Any physical quantity of interest which depends on the scaled temperature, Q(T /TF ), will also varyacross the gas. A global measurement of that quantity will yield [22]Q̄ = N

1Z

Q[T /TF (r)]n(r)d3 r(1.6)

where n(r) is the density distribution and N is the total number of atoms. In general, it can bedifficult to interpret the measured quantity Q̄. Furthermore, if Q has any sharp features as afunction of T /TF or 1/kF a, it can be washed out for Q̄.
One example of this is the Fermi surface for a weakly interacting ultracold Fermi gas. It
is well known that a Fermi gas at T = 0 has a step in momentum (or energy) states called the
Fermi surface. At a finite temperature, this step is smoothed out and the width of the sharpness
depends on T /TF , where TF is the Fermi temperature which depends on density. However, themomentum distribution for a harmonically trapped Fermi gas does not show sharp features(Figure1.3). Although the center of the cloud is at ultracold temperatures, the outskirts of the cloud aree↵ectively at higher temperature, which means the corresponding Fermi surface is broader. When
averaged together, the resulting momentum distribution does not show a sharp feature.
To overcome the issue of density inhomogeneity, some groups have implemented “box” traps
of repulsive optical potential to create a homogeneous Fermi gas [23]. In the work describedin my thesis, we developed a method based on optical pumping to spatially select the almost
homogeneous center of the harmonically trapped gas. You can read more about this technique andthe experiments in Tara Drake’s thesis [24].

1.4

Thesis outline
In this thesis, I start by explaining the technique we use to spatially select the harmonicallytrapped atoms and probe the homogeneous sample from the sample of the trap, also colloquially6

Figure 1.3: Momentum distribution of a harmonically trapped Fermi gas. Even as we lower thetemperature of the gas, we do not see a sharp Fermi surface as expected for an ultracold Fermi gas
due to density inhomogeneity. The figure is reproduced from [2].

7
known as the “donut beam technique”. In Chapter 2, I will explain the method we use to create
the Laguerre-Gaussian beam needed for this technique. I will also describe the proof-of-principle
experiment we performed to observe the sharp Fermi surface for the first time in ultracold Fermi
gases.
In Chapter 3, I will describe the two experiments we performed that combined RF spectroscopy with the donut beam technique. In one experiment, we measured Tan’s contact at unitarity for a homogeneous Fermi gas as a function of temperature. We compared our measurement withvarious many-body theories. This measurement provides a good benchmark for improving various
theoretical tools. We followed this work with the measurement of spectral function of the normal
phase of the strongly interacting Fermi gas across BCS-BEC crossover. We also analyzed the datawith a two-part function intuitively comparing BCS and BEC physics and extracted quantities likequasi-particle residue, e↵ective mass, Hartree energy shifts, and chemical potential.
In Chapter 4 and 5, I will describe the construction of the new Fermi gas apparatus. I willend the thesis with an outlook of experiments that can be done with interacting Fermi systems.
Since the fall of 2016, the Fermi gas apparatus that I built during my PhD has been converted toa strongly interacting BEC apparatus.

Chapter 2
Donut Beam technique to probe a homogeneous Fermi gas

Ultracold quantum gas experiments are usually performed in a harmonic trapping potential.
For magnetic traps such as the QUIC trap and Io↵e-Pritchard trap, the potential near the bottom of
the trap is harmonic. Similarly, an optical potential created by a Gaussian laser beam is harmonicnear the bottom of the trap. Such systems have allowed the ultracold community to perform many
important measurements and qualitative observations over the last two decades [13, 25]. However,
due to the inherent density inhomogeneity of the trapped gas, time-of-flight measurements are trap
averaged. Quantitative analysis on such data can sometimes be quite challenging or sometimes
even misleading. Some groups recently have developed “box” traps formed by repulsive opticalpotentials to create homogeneous gases [26, 23]. We used a di↵erent route to tackle this problem.
Our method relies on optical pumping of the trapped gas using a hollow beam. In this chapter, I
will describe the technique we recently developed to probe the atoms only from the center of theharmonic trap where the density is homogeneous. We apply this technique to a weakly interacting
Fermi gas to directly observe the Fermi surface in an ultracold atomic gas for the first time. Muchof the content of this chapter is publish in Ref [27].

9

2.1

Creating a donut beam
A donut beam has a Laguerre-Gaussian (LG) mode. An LG mode for a radial index p(positive integer) and an azimuthal index l (integer) is given by Ref [28] as
p ! ✓ 2◆
r 2
2r
2
2
2
2
2
1
l
p
Ep (r, , z) = ( 1)Llp
E ikr z/2(zr +z ) e r /w e i(2p+l+1)tan (z/zr ) e il )
w
w2

(2.1)

where zr is the Rayleigh range, w is the Gaussian beam waist, tan 1 (z/zr ) is the Guoy phase of
the mode and Llp is the generalized Laguerre polynomial. For us, only the p = 0 mode is relevant.
The cross-section of such beam is a single ring and the intensity is
2P
I(r) =⇡w2 l!

l=1

✓

2r2
w2

◆l

l=2

2

2

e 2r /w .

(2.2)

l=3

Figure 2.1: Di↵raction grating patterns we use to produce Laguerre-Gaussian beam (top) and thecorresponding measured cross-section of the intensity profile at the far field (bottom).

These modes can be created in an expensive way by using a Spatial Light Modulator or in acheaper way by using a holographic plate. In our case, we use an absorptive di↵raction grating torealize LG beams. The pattern shown in Figure 2.1 is printed on a glass slide with chromium. The

10
0th order
1st order

Figure 2.2: Setup to create a Laguerre-Gaussian beam. The beam is focused onto the atom cloudafter the last lens.

three patterns correspond to the angular index l = 1, 2 and 3. To create LG beams, a Gaussian
beam is passed through the set of optics shown in Figure 2.2. An LG mode of l = 2 is obtained in
the far field after going through the focusing lens that focuses the beam at the atom cloud position.
We have two such setups to produce beams of waist 18 µm and 150µm in the horizontal and verticaldirections respectively.
We primarily use one of the two optical transitions for the donut beam technique. The
first transition is |F = 9/2, mF =|F = 9/2, mF =7/2i ! |F 0 = 5/2, mF =5/2i ! |F 0 = 5/2, mF =5/2i and the second one is

3/2i. Both of these transitions optically pump theatoms into the F = 7/2 hyperfine levels, which are invisible to the probe beam. Thus, only theatoms that not pumped and are at the center of the cloud are imaged after a TOF. Atoms in the
excited state decay by spontaneous emission with a branching ratio of 0.955 to the |7/2, 7/2i
ground state and 0.044 to the original |9/2, 7/2i state.

2.2

Weakly interacting Fermi gas
The homogeneous Fermi gas is a widely used model in quantum many-body physics and is thestarting point for theoretical treatment of interacting Fermi systems. The momentum distribution

11

F’ = 5/2

42P3/2

Donut beamtransi<on

F = 7/2

42S1/2mF = -9/2

mF = -7/2

mF = -5/2

mF = -3/2

F = 9/2

Figure 2.3: Optical pumping transitions for the donut beam. Depending on the experiment, weeither use |F = 9/2, mF = 7/2i ! |F 0 = 5/2, mF = 5/2i or |F = 9/2, mF = 5/2i ! |F 0 =5/2, mF = 3/2i transition. Both of these transitions shelve atoms to the upper hyperfine level inthe ground state invisible to the probe beam.

12
for an ideal Fermi gas is given by the Fermi-Dirac distribution:n(k) =
e

⇣ 2 2
~ k
2m

1

⌘
µ /kB T

,

(2.3)

+1

where the n(k) is the average occupation of a state with momentum ~k, m is the fermion mass,µ is the chemical potential, kB is Boltzmann’s constant, and T is the temperature. Surprisingly,
to our knowledge, the momentum distribution of an ideal Fermi gas, with its sharp step at the
Fermi momentum, ~kF , had not been directly observed in experiments before 2012. For the vast
majority of Fermi systems, such as electrons in materials, valence electrons in atoms, and protons
and/or neutrons in nuclear matter, one always has an interacting system. A dilute Fermi gas of
atoms opens new possibilities with its low density, access to the momentum distribution throughtime-of-flight imaging, and controllable interparticle interactions. However, these trapped gases
have nonuniform density, which has prevented the observation of a sharp step in their momentum
distribution and, more generally, can complicate comparisons with theory.
If the change in the trapped gas density is small on the length scale of the relevant physics,
one can apply a local density approximation. Measurements can then be compared to theory byintegrating the prediction for a homogeneous gas over the density distribution of the trapped gas.
While the agreement between experiment and theory can be quite good, characteristic features
such as a sharp Fermi surface in k-space can be lost in trap-averaged data. For rf spectroscopy
and for thermodynamic measurements, recent work has used in-situ imaging of trapped gases
combined with knowledge of the trapping potential to yield results that can be directly comparedto homogeneous Fermi gas theory [13, 21, 29, 30]. However, this technique cannot probe themomentum distribution, which requires a sudden release of the gas from the trap followed byballistic expansion and imaging. We developed a method to measure the momentum distribution
locally in a trapped Fermi gas and present a direct observation of the Fermi surface in k-space.

13

2.3

Experimental Sequence

2.3.1

Aligning the beam

We align the donut beam onto the cloud by either looking at the number loss and the peak
OD or by looking at the slosh of the cloud in the trap after optical pumping. In the first method,
as we scan the position of the donut beam across the cloud we see a “W” shape in number loss
corresponding to the shape of the donut beam cross-section (Figure 2.4). At the center, we see thatthe peak OD of the could same as that without optical pumping. Once the beam is aligned thisway, we do a finer scan by optically pumping beam and waiting for a some time before releasing
them from the trap Figure 2.5. The slosh of the atoms is minimized for an aligned beam. As
the atoms slosh at the trapping frequency this method can be alternatively used to measure thetrapping frequency in the most non-perturbative way.

Number

80,00040,000

0

150

0

150

knob position (arb.)
Figure 2.4: Alignment of the donut beam by scanning to position to see a “W” shape in the atom
number corresponding to the cross-section of the beam. We scan the motorized mirror shown in
Figure 2.2.

We begin with a quantum degenerate gas of N = 9 ⇥ 104 40 K atoms in an equal mixture of
the |f, mf i = |9/2, 9/2i and |9/2, 7/2i spin states. The atoms are confined in a cylindricallysymmetric, crossed-beam optical trap characterized by a radial trap frequency ⌫r of 214 Hz and14

x center position (pixels)

a
380
375
370
365
360

0

3
6
time in trap after donut (s)

9

0

3
6
time in trap after donut (s)9bx width (pixels)

23
21
19
17
15

Figure 2.5: Alignment of the donut beam by looking at the slosh and breathe of the cloud atdi↵erent hold time after a TOF. When the donut beam is aligned to the center of the atom cloudthese e↵ects are minimized.

15
an axial trap frequency ⌫z of 16 Hz. To facilitate future application of this technique to probing
strongly interacting Fermi gases, we work at a magnetic field that is near a Feshbach resonancebetween the initial two spin states. We take data at B = 208.2 G where the scattering length abetween atoms in the |9/2, 9/2i and |9/2, 7/2i states is approximately

30 a0 [31], where a0 is

the Bohr radius. Here, the gas is very weakly interacting, with a dimensionless interaction strengthof kF a = 0.011.
Our measurements probe only the |9/2, 7/2i spin component. We use two intersecting
hollow light beams to optically pump atoms into a spin state that is dark to our imaging (seeFigure 2.3). The hollow light beams are resonant with the transition from the |9/2, 7/2i state to
the electronically excited |5/2, 5/2i state (see Fig. 2.3).
The first beam propagates along the vertical (y) direction, is linearly polarized, and has awaist of 186 µm. Given the elongated shape of the trapped gas, this beam is primarily spatiallyselective along the long axis (z) of the cloud. The second beam propagates along the axial (z)direction of the cylindrically symmetric trap, parallel to B, and is circularly polarized. This beam
has a waist of 16.8 µm and selectively optically pumps atoms based on their location along x andy.
To probe the momentum distribution of the central part of the trapped gas, we first turno↵ the trap suddenly and illuminate the atoms with the vertical hollow light beam, followed immediately by pulsing on the horizontal beam. The power in the beams in on the order of tens tohundreds of nW and is varied to control the fraction of atoms that are optically pumped out of the|9/2, 7/2i state. Each beam is pulsed on for 10 to 40 µs, with the pulse durations chosen such thatthe fraction of atoms optically pumped by each of the two beams is roughly equal (within a factorof two). The relative durations of the two pulses are chosen to optimize the relative removal of thebeams with respect to the density homogeneity of the remaining cloud and are typically 10 to 40µs. We have found the optimal homogeneity occurs when the vertical beam alone removes roughly
half of the total signal removed; for example, when 66% of our signal is removed, 30% is removedby the vertical beam alone. We then image the remaining atoms in the |9/2, 7/2i state after 10

16
ms or 12 ms time of flight. The imaging light propagates along the z direction and we apply aninverse Abel transform to the 2D image (assuming spherical symmetry in k-space) to obtain the
3D momentum distribution, n(k).

2.4

Fermi Surface
In Fig. 2.6, we show normalized momentum distributions measured with and without using

the hollow light beams. As was first seen in Ref. [15], the trap-averaged momentum distribution
for the Fermi gas is only modestly distorted from the Gaussian distribution of a classical gas.
The dashed line in the inset to Fig. 2.6 shows a fit to the expected momentum distribution fora harmonically trapped ideal Fermi gas, from which we determine the temperature of the gas
to be T /TF,trap = 0.12 ± 0.02. Here, the Fermi temperature for the trapped gas is given by
TF,trap = EF,trap /kB , where EF,trap = h(⌫r2 ⌫z )1/3 (6N )1/3 is the Fermi energy for the trapped gas
and N is the total number of atoms. After optical pumping with the hollow light beams so that weprobe the central 16% of the atoms, the measured momentum distribution (main part of Fig. 2.6)
has a clear step, as expected for a homogeneous Fermi gas described by Equation 2.3.
For a sufficiently small density inhomogeneity, the momentum distribution should look likethat for a homogeneous gas at some average density. To characterize this, we fit the normalized
distributions to the prediction for an ideal homogeneous gas (solid lines). The homogeneous gas
distribution is described by its temperature and density. We fix T to that measured for the trappedgas, which leaves only a single fit parameter, kF , that characterizes the density. The momentum
distributions are then plotted as a function of the usual dimensionless momentum, k/kF . The
momentum distribution of the central 16% of the trapped gas fits well to the homogeneous gas
result, while the trap-averaged momentum distribution clearly does not.

2.5

How homogeneous is the gas?
In order to quantify how well the measured momentum distribution is described by that of ahomogeneous gas, we look at the reduced

2 statistic in Fig. 2.7a. The reduced

2 is much larger

17

n(k)

1

n(k)

1

0
0

1

2

k / kF00

1

2

k / kF
Figure 2.6: Momentum distribution for a weakly interacting Fermi gas after optical pumping witha donut beam. The distribution is an average of 12 images after selecting the central 16% of aharmonically trapped gas. The data is normalized to have the area under the curve to be equal to
1. The solid line is the fit to the Fermi distribution with a fixed T, kF is the only fit parameter.
The inset shows the distribution without optical pumping. The dashed line in the inset shows the
expected momentum distribution for a trapped gas with the temperature T.

reducedF

2

15
10

(a)

5

kF / kF, trap

0
1
0.8
0.6

(b)
0

0.5
fraction probed

1

Figure 2.7: We look at reduced thesquared (a) of the fit and the fit parameter kF (b) as afunction of fraction probed. We find that the reduced
squared is lowered as lower fraction of
atoms are probed, which means the system is better described as a homogenous gas.

18
than 1, indicating a poor fit, for the trap-averaged data due the fact that the density inhomogeneity
washes out the Fermi surface. As we probe a decreasing fraction of atoms near the center of thetrap,

2 decreases dramatically and approaches a value of 1.6 for fractions smaller than 40%.

The single fit parameter kF characterizes the density of the probed gas and should increase
as we probe fewer atoms near the center of the trap. Fig. 2.7b displays the fit value kF , in units
pof kF,trap = 2mEF,trap /~. As expected, kF increases as the fraction of atoms probed decreases.
We have developed a model of the spatially selective optical pumping by the hollow light beam,

which is discussed below. The model result for the average local kF , hkF i, of the probed gas isshown with the solid line in Fig. 2.7b, and we find that this agrees well with the fit kF , even whenthe measured momentum distributions clearly do not look like that of a homogeneous gas. Using
the model, we calculate the variance

2 of the local k

hkF i ± . In the region where the reduced

F , and the shaded region in Fig. 2.7b shows

2 indicates that the measured n(k) fits well that for ahomogeneous gas (fraction probed < 40%), / hkF i < 0.08.

T / TF

0.4
0.2
0

0

0.5
fraction probed

1

Figure 2.8: Measured T/TF vs the fraction of atoms probed. Here, we fit the measured momentum
distribution to a homogeneous gas distribution with two free parameters, T /TF and kF . The
density inhomogeneity of the probed gas results in T /TF that is much larger than expected fromthe calculated average density of the probed gas (solid line). A sharp Fermi surface, characterized
by a small fit T /TF , emerges as the fraction of atoms probed decreases. The dashed line shows
the result of fitting to model calculations of the probed momentum distribution, which agrees wellwith the data.

19
Instead of fixing T to the value obtained from fitting the trapped gas momentum distribution,
we can also look at measuring the temperature by fitting to a homogeneous gas distribution, where
both kF and T /TF are fit parameters. In this case, a large density inhomogeneity that washes
out the Fermi surface will result in an artificially high fit value for T /TF . This can be seen in
Figure 2.8. For comparison to the data, the solid line shows the average T / hTF i for the probed
gas calculated using our model. Here, T is fixed and the dependence on the fraction probed comesfrom the fact that the average density, and therefore the average local TF , increases as we probe asmaller fraction of atoms that were near the center of the trapped gas. The fit T /TF approaches
the average value from the model as we reduce the fraction of atoms probed, and for < 40% probed,
the two are consistent within our measurement uncertainty. For the smallest fraction probed (datashown in Fig. 2.6), the best fit value is T /TF = 0.14 ± 0.02. As a check of the model, we canalso calculate n(k) for the probed gas and fit this to the homogeneous gas distribution; the results(dashed line in Fig. 2.8) agree well with the data.

2.6

The e↵ect of optical pumping
In modeling the e↵ect of optical pumping with the hollow light beams, we assume that only

atoms that do not scatter a photon are probed. The probability to scatter zero photons from eachbeam is taken to be Pi = exp(i ⌧i), where ⌧i is the pulse duration and the subscripts i = 1, 2

denote the two hollow light beams. The photon flux is given byposition-dependent intensity, c is the speed of light, andoptical absorption cross section, we use

i = Ii/(hc), where Ii is the= 766.7 nm is the wavelength. For the= 3 2 ⌘/(2⇡), where ⌘ = 0.044 is the branching ratio backto the initial state.
Attenuation of the hollow light beam as it propagates through the atom cloud is observable in
the long direction of the cloud (along z), as seen in the inset of Figure 2.9. To include this e↵ect, weconsider the two hollow light beam pulses sequentially, and we assume that the number of photons
absorbed locally equals the number of optically pumped atoms. Interestingly, the model predicts
that the attenuation results in a smaller density variance in the probed gas when compared to a

fraction probed

20

1.0
0.5
0

-1

0

z / zF

1

Figure 2.9: Modeling the spatially selective optical pumping. We compare the normalized momentum distribution of the central 38% of the atoms to three di↵erent models (dotted, solid, anddashed lines; see text). The data (circles) are obtained from an average of four images. We findthat the attenuation of the hollow light beams (inset) does not strongly a↵ect the predicted final
momentum distribution when probing a small fraction of the gas. (Inset) We take images of thecloud after a short (1.3 ms) expansion and compare data with the horizontal hollow light beam andwithout any q
optical pumping in order to measure the fraction of atoms probed (circles) vs z/zF ,2E

F,trapwhere zF = m(2⇡⌫2 . For this data, the fraction probed is 71%. The prediction of our modelz)(solid line), which includes attenuation of the hollow light beam as it propagates through the cloud,agrees well with the data.

21
model that ignores attenuation but where we adjust the beam powers to probe the same fractionof the atoms. This e↵ect is relatively small and decreases as one probes a smaller fraction of thegas. This can be seen in Figure 2.9 where we show the measured momentum distribution for thecentral 38% of the atoms compared to three di↵erent models, each of which is adjusted to givethe same probed fraction. The solid line is the model explained above, which includes attenuation,
while the dotted line show the result when we ignore the depletion of the hollow light beams. For
comparison, the dashed line shows the expected distribution if one selects atoms in a cylindricalvolume with sharp boundaries.

2.7

Conclusion
In conclusion, we have directly observed the Fermi surface in the momentum distribution ofa weakly interacting Fermi gas. To do this, we probe the central region of a harmonically trapped
gas. A concern with this approach is that one might be left with very little signal after selecting
a small enough region to approximate a homogeneous gas. However, for a gas at T /TF,trap = 0.12
and our typical measurement precision, we find that probing the central 40% (or less) of the gas
is sufficient to approximate a homogeneous gas. In the next chapter, I will discuss combining thisdonut beam technique with RF spectroscopy to measure Tan’s contact and the spectral function
for a strongly interacting Fermi gas.

Chapter 3
Probing a homogeneous strongly interacting Fermi gas

The collective behavior of an ensemble of strongly interacting fermions is central to many
physical systems including liquid 3 He, high-Tc superconductors, quark-gluon plasma, neutron stars,
and ultracold Fermi gases. However, theoretical understanding of strongly interacting fermions ischallenging due to the many-body nature of the problem and the fact that there is no obvious small
parameter for a perturbative analysis. Therefore, in order to establish the validity and applicabilityof theoretical approaches, it is essential to compare them against experimental results. Ultracold
atomic Fermi gases are ideal for this purpose, as they provide excellent controllability, reproducibility, and unique detection methods [16, 13]. In particular, changing the magnetic field in the vicinityof a Feshbach resonance enables precise control of the interactions, which are characterized by thes-wave scattering length [18]. On resonance, the scattering length diverges and the behavior of theunitary gas no longer depends on it. Testing theories in this regime is especially desirable.
An outstanding issue for the unitary Fermi gas is the nature of the normal state just above thetransition temperature, Tc , for a superfluid of paired fermions. Some theories of strongly interacting Fermi gases (BCS-BEC crossover theories) predict that the normal state is not the ubiquitous
Fermi liquid but instead involves incoherent fermion pairing (preformed pairs) in what has beentermed the pseudogap state [32]. It has been suggested that the pseudogap state a↵ects the temperature dependence of a quantity called Tan’s contact [6]. The contact, which is a measure of theshort-range correlation function, has been shown to be an essential thermodynamic parameter for
ensembles with short-range interactions [33, 34, 35, 36, 37, 9]. The contact connects many seemingly

23
unrelated quantities through a set of universal relations that are valid for any temperature, anyinteraction strength, and any phase of the system. While the value of the contact, as well as manyof these relations, were tested successfully at low temperature [38, 39, 40, 41], there are significant
discrepancies among theories on how the contact of a unitary homogeneous Fermi gas depends on
temperature, especially around Tc [42, 6, 43, 5, 7]. The temperature dependence of the contact
was recently measured for a trapped unitary Fermi gas [44]. However, for the trapped gas, averaging over the inhomogeneous density distribution washes out any temperature-dependent features,and the measurement was unable to di↵erentiate between theoretical models. Here we present ameasurement of the homogeneous contact, which can be directly compared to the predictions ofdi↵erent many-body theories.
The behavior of strongly interacting quantum gases can be probed in many di↵erent ways.
In the early days, the momentum distribution after a time-of-flight was measured after ramping
into the molecular side of the resonance to either measure the molecular fraction or the condensate
fraction [45]. In-situ images provide information on density distribution, which can be used to
calculate the pressure of the gas, and thus the thermodynamics of the gas can be studied [21, 30].
Time-of-flight information can be used to study the energy, hydrodynamic and collective behaviorof the gas. Bragg spectroscopy [44] and Raman spectroscopy [46] have also allowed to learn the
structure factor and the dispersions in the Fermi gas. One other technique to study the stronglyinteracting Fermi gas is RF spectroscopy [16, 47, 38]. The RF spectroscopy technique allows us to
study quantities such as the contact and spectral function of the gas.
In this chapter, I will describe the basics of RF spectroscopy. Then I will describe ourmeasurement of homogeneous contact as a function of temperature at unitarity using this technique.
I will then describe another experiment where we performed a momentum resolved RF spectroscopyof strongly interacting Fermi gas across BCS-BEC crossover to measure the spectral function. Mostof the content in this chapter was published in Ref. [48] and Ref. [49].

24

3.1

Tan’s Contact
Tan’s contact is the measure of the short-range two-body correlation in an interacting quan-tum gas. In 2008, Shina Tan derived a number of universal relations for an interacting Fermi gas
with short-range, or contact, interactions [33, 34, 35, 36, 37, 9]. These universals relations connect
microscopic quantities, such as the momentum distribution of the fermions, to macroscopic quantities, such as the total energy of the system. These relations are quite powerful because they areapplicable whether the system is many-body and few-body, whether the gas is normal or superfluid
and whether the system is homogeneous or trapped.
One of the most intuitive ways to understand the contact is by looking at the gas microscopically [36, 9]. Let’s consider a Fermi gas atoms of atoms with spin-up and spin-down, and withs-wave scattering length a between the two spin states, and with ar0 , the physical range ofinteractions. We would like to count the number spin-up and spin-down atoms inside a volume
V = 43 ⇡l3 with center at the location of spin-up atom. Naively, one would expect the number ofpairs in this volume to be n" ⇥ n# ⇥ ( 43 ⇡l3 )2 , where n"(#) is the density of spin-up(down) atoms.
For short-range pairs with l ⌧ n1/3 , where n is the total density, the number of pairs scales as l4
instead of l6 and this enhancement factor corresponds to the Contact.

Npair !

l4
C
4

(3.1)

Because contact measures the short-range correlations, because of the nature of the wavefunction, it appears as the amplitude of the k 4 tail in momentum distribution [34]. That is,

C = lim k 4 n(k),

(3.2)

k!1

where k is the momentum in units of Fermi momentum, kF =R1
1
3
that (2⇡)
3 0 n(k)d k = 0.5.

p2mE
~ , and n(k) is normalized such

Another relations that was later derived comes from the high-frequency tail of RF spec-

troscopy. The high frequency tail of the rf line shape is predicted to scale as ⌫ 3/2 , with the25contact connecting the amplitude of the high frequency tail through (see Ref. [9] and references
therein):(⌫)
C/(N kF )=p
0
02⇡ 2 ⌫ 3/2
1 (⌫ )d⌫R1

for (EF /h) ⌧ ⌫ ⌧

h
2mr20

(3.3)

where N is the total number of atoms, and ~kF is the Fermi momentum, and ⌫ is the rf detuning
in units of the Fermi energy, EF /h, with h being the Planck constant (2⇡~ ⌘ h).
3.1.1

RF Spectroscopy

RF spectroscopy relies on the manipulation of Zeeman states of the atoms. Around the fieldnear Feshbach resonance, i.e. 202 G, mF =9/2, mF =7/2 and mF =5/2 are separated byabout 45 MHz in energy. Hence, we can use RF photons to increase or decrease mF by one.
RF spectroscopy can be understood using a semi-classical picture of light atom interaction[50]. Let the energy di↵erence between the two spin states |ei and |gi be ~!, Ee

Eg = ~!0 . The

electric field of the RF can be written as E(t) = E0 cos(!L t), where E0 is the strength of the field
and !L is the frequency of the RF. The Hamiltonian for this system is H = Hatom + Vint , where
0
1
0
1
0
0
0
dE
B
C
B
C
Hatom = @A , Vint = @
A.
0 ~!0
dE 0

Here, d = he|ˆ⇡ ·~r|gi is the dipole matrix element. We can use the time-dependent Schrodinger
equation to find the population of excited states,i~
The wave function

can be written as ,

@
=H .
@t

(3.4)

= c1 (t)|gi + c2 (t)|eiei!0 t , where c1 and c2 are probability

amplitude such that c21 + c22 = 1. Using RWA approximation,
0 1 0
10 1
i
t
⌦e C Bc1 Cd Bc1 C B 0
i @ A=@
A@ A.
dt
i
t
c2
⌦e
0
c2
Here, ⌦ = dE0 /~ is the Rabi frequency and

= !L

(3.5)

!0 is the RF detuning. We can solve

this coupled di↵erential equation with an initial condition of c1 (0) = 1 and c2 (0) = 0, to get the26
population in the excited state as a function of time,|c2 (t)|2 =When atoms in mF =

p
⌦
2
sin
(
⌦2 +
⌦2 + 2 /4

9/2 and mF =to move the atoms from mF =7/2 to mF =2 /4t).

(3.6)

7/2 are strongly interacting, using RF photons
5/2 is especially useful because the atoms in

mF = 5/2 are weakly interacting with the other two spin states. Hence, the information we get byscanning the frequency of RF and counting the number of atoms out coupled is easy to interpret.

3.1.2

Experimental Sequence

We perform the experiments with an optically trapped ultracold gas of 40 K atoms in an equal
mixture of the |F, mF i = |9/2, 9/2i and |9/2, 7/2i spin states. We determine the contact by
combining rf spectroscopy with the donut beam technique to probe the local properties of a trappedgas as described in Chapter 2. The experimental sequence is depicted in Figure 3.1. The magnetic
field is ramped adiabatically to the Feshbach resonance and kept at this value for 2 ms before
abruptly shutting o↵ the trapping potential. Before the potential is shut o↵, the hollow beams arepulsed on, followed by the rf pulse, which transfers a small fraction of the atoms in the occupied|9/2, 7/2i state to the initially unoccupied |9/2, 5/2i state (which is weakly interacting with theother two spin states). We detect these atoms using absorption imaging after 3 ms of expansion.
The temperature of the gas is varied by changing the final depth of the optical dipole trap in theevaporation process [31]. The number of atoms per spin state after the evaporation ranges from50, 000 to 220, 000. For the data presented, the radial trapping frequency, !r , ranges from 2⇡ ⇥ 200
Hz to 2⇡ ⇥ 410 Hz, while the axial trapping frequency, !z , ranges from 2⇡ ⇥ 19 Hz to 2⇡ ⇥ 25 Hz.
The contact is extracted from a measurement of the rf line shape(⌫) [38], where(⌫) isthe rate of atoms transferred from one of the two interacting spin states to a third state, by an rf
pulse centered at a frequency detuning ⌫. A representative data set, where the hollow light beams
were used to select the central 30% of the atom cloud, is shown in figure 3.2. For each line shape,
we take data at 30 di↵erent detunings between

16 kHz and +116 kHz, where ⌫ = 0 is defined as

27

(a)
initialcloud(b)203.4 G

50 ms

y

x

hollow beams (50 ms)zabsorption
imaging,
after 3 ms

atoms
probedtrap releaseFeshbach resonance(202.2 G)

rf pulse (100 ms)

Figure 3.1: Time sequence of the experiment. The magnetic field is ramped from 203.4 G, where
the atoms are initially prepared, to the Feshbach resonance. The hollow light beams are turned on
280 µs before trap release; initially, the beam that propagates perpendicular to the long axis of thecloud is pulsed on for 10 µs followed by 40 µs of the second beam. The line shape is measured using
an rf pulse with a total duration of 100 µs and a gaussian field envelope with = 17 µs, centered
180 µs before trap release. The cloud expands for 3 ms before being detected by absorption imaging.
To improve the signal-to-noise ratio, we remove the remaining atoms from the |9/2, 9/2i and|9/2, 7/2i states and then transfer the outcoupled atoms in the |9/2, 5/2i state to the |9/2, 9/2i
state, where we image on the cycling transition [3].

28
the single-particle transition frequency between the |9/2, 7/2i and |9/2, 5/2i states (measured
for a spin polarized gas in the |9/2, 7/2i state).
0.2
2 3/2π 2 ν 3/2Γ [NkF ]10

Γ [EF /h]

0.15
0.1

8
6
4
2
0

0

5

10

15

ν [EF /h]

0.05
0
0

5

ν [EF /h]10

15

Figure 3.2: An rf line shape for the unitary Fermi gas at T /TF = 0.25 Rwith 30% of the atoms
1
probed. The solid (red) line is a fit to Eq.(3.3) with the normalization
1 (⌫)d⌫ = 0.5, due
to the 50% 50% spin mixture. The inset shows the same data multiplied by 23/2 ⇡ 2 ⌫ 3/2 . We
make sure the rf pulse induces only a small perturbation, by setting its power to well below thevalue where we see the onset of saturation of the number of outcoupled atoms. The measurement
at di↵erent frequencies is done with di↵erent rf powers, and when analyzing the data, we linearly
scale the measured number of atoms outcoupled at each frequency to correspond to a common rf
power.

The inset of Figure 3.2 shows(⌫) multiplied by 23/2 ⇡ 2 ⌫ 3/2 , where we observe a plateau

for frequencies higher than 5 EF /h. We extract the contact by fitting the measured(⌫) for⌫ > 5 EF /h to Eq.(3.3) (solid line in figure 3.2). For the normalization, we integrate the line shape,
2 , where r
including the tail, up to ⌫ = ~/mre↵e↵ is the e↵ective range of the interaction [18] (which

is approximately 300EF /~).

3.1.3

Contact versus Temperature

The main result of this experiment, namely the homogeneous contact versus the temperature,is presented in figure 3.3. The contact is normalized to the average kF of the probed sample, andtemperature is given in terms of T /TF , with TF being the average Fermi temperature of the probed
sample. The data show a monotonic decrease of the contact with increasing temperature from

29

Tc

4
3

4

C/(NkF)

2
1
0.0

0.8

1.6

2.4

3

Data
G0G0,

2
0.0

Virial 2,

0.2

T/TF

GPF,
Virial 3,

0.4

GG
QMC,

ENS

0.6

Figure 3.3: The contact of a nearly homogeneous sample (about 30% of the trapped atoms probed),
versus T /TF at unitarity (black circles). The shaded area marks the superfluid phase transition,with some uncertainty in its exact position (Tc /TF = 0.16 0.23) [4]. As a comparison, we plot
the gaussian pair-fluctuation NSR model (GPF) [5], the self-consistent t-matrix model (GG) [4],the non-self-consistent t-matrix model (G0 G0 ) [6], the 2nd and 3rd order virial expansion [5],a quantum Monte-Carlo calculation (QMC) [7], and the contact extracted from a thermodynamic
measurement done at ENS [8]. The error bars represent one standard deviation. The inset shows thehigh temperature behavior of the contact, where we find good agreement with the virial expansion.

30
a maximum value of 3.3 N kF . For T /TF = 0.16, at the edge of our experimentally attainabletemperatures, we observe a sharp decrease of the contact to about 2.6 N kF . We compare our datawith several theoretical models [5] and a quantum Monte-Carlo (QMC) simulation [7]. The manybody theories are in the framework of the t-matrix approximation [5], di↵ering by their choice ofthe diagrammatic expansion, the particle-particle propagator, and the self-energy. For T /TF > 0.4,
the di↵erences between the theoretical models are small, and the predictions all lie within theuncertainty of the data. As expected, at higher temperatures (T /TF > 1), we find good agreementwith the virial expansions [5] (see inset of figure 3.3). For T /TF < 0.4, our data do not agree
fully with any of the many-body theories. It is worth noting, however, that two of the theories(GPF and G0 G0 ) predict a higher value for the contact above the superfluid phase transition thanbelow, which may be consistent with observed sharp decrease near T /TF = 0.16. We note thatthe predicted Tc /TF has some uncertainty, as indicated by the shaded region in figure 3.3. The
non-self-consistent t-matrix model (G0 G0 ) predicts an enhancement of about 50% in the value ofthe contact around Tc [6], which the data do not show. We also do not observe an increasing trend
in the contact for T > Tc , in contrast to a recent QMC simulation [7].

3.1.4

Obtaining the in-situ density distribution

As can be seen from Eq.(3.3), the contact is naturally normalized by N kF , and the detuning by the Fermi energy. However, a question which arises is how to define EF in our experiment. For a harmonically trapped gas, EF is defined in terms of the trap parameters EF,trap =~(!r2 !z )1/3 (6N )1/3 . On the other hand, the Fermi energy of a homogeneous gas is given in terms~of its density (in one spin state), n: EF,hom = 2m
(6⇡ 2 n)2/3 . In our experiment, as we increase2

the power of the hollow light beams, we probe a smaller portion of the gas that is more homogeneous. The relevant Fermi energy, which we use in figures 3.2 and 3.3, is therefore the averageR~2of the local (homogeneous) Fermi energy: EF,avg = 2mN
P (r)n(r)[6⇡ 2 n(r)]2/3 d3 r, where P (r) ispR
the detection probability after optical pumping, and Np = P (r)n(r)d3 r is the number of atoms
probed.

31
We use the in-situ density distribution, n(r), in order to calculate the average kF and EF of theprobed atoms. To get n(r), we use the fact that at unitarity the cloud expands hydrodynamically,
and the dynamics are governed by the continuity equation. The solution for the continuity equation
with harmonic confinement with a time-dependent trapping frequency !(t) is self-similar with thefollowing scaling transformation: ri (t) = bi (t)ri (0), where ri is the spatial coordinate (i = x, y, z),
and bi (t) obeys the equation [51, 13]:b̈i (t) =!i (t)2 bi (t) +!i (0)2bi (t) [bx (t)by (t)bz (t)]with the initial conditions bi (0) = 1 and ḃi (0) = 0. The constant

,

(3.7)

is the characteristic exponent

in the equation of state µ(n) / n , where µ is the chemical potential and= 2/3 at unitarity. Fora sudden turn o↵ of the trap, !i (0) is the trapping frequency along the i axis, and !i (t > 0) = 0.
In figure 3.4, we plot the measured width of the cloud in the axial and radial directions as
a function of the expansion time. The data show a rapid increase in the size in the radial (tight)
direction of the cloud and almost no increase in the axial direction–a characteristic of hydrodynamic
expansion. The solid lines show the numerical solution of Eq.(3.7) with !r = 2⇡ ⇥ 226 Hz and!z = 2⇡ ⇥ 19 Hz (which were measured independently), which agrees very well with the data. We
find that after 4 ms of expansion the finite resolution of the optical system does not a↵ect theextracted parameters, and therefore we choose this expansion time for the density measurements.
We fit the measured density profiles at 4 ms with a Thomas-Fermi distribution, which we find tobe general enough for this purpose.
We have tested our density determination method by looking at the ratio of the peak density
at unitarity to the peak density of a weakly interacting gas, at low temperatures. The density
distribution of the weakly interacting gas is measured using a fit to a Thomas-Fermi distribution
after ballistic expansion. For T = 0, this ratio is nU /n0 = ⇠ 3/4 , where ⇠ at unitarity is a universal
constant that relates the chemical potential to the Fermi energy: µ = ⇠✏F . From the measured
density ratio at a temperature of T /TF,trap = 0.15, we extract a value of ⇠ = 0.40 ± 0.05, which isconsistent with recent determinations of this universal constant by other groups [30, 8].

32

12

radialaxial

10

σ(t)/σ(0)

8
6
4
2
0
0

1

2

3

4

Time of flight [ms]

5

6

7

Figure 3.4: Hydrodynamic expansion at unitarity. We start with a weakly interacting gas with⇠ 90, 000 atoms per spin state at T /TF = 0.12 and ramp adiabatically to the Feshbach resonancefield. We fit the cloud with a Thomas-Fermi distribution after a variable expansion time and extract
the rms widths, (t), in the radial and axial directions. For the data, we de-convolve the measured
width with a gaussian point spread function with an rms width of 2.9 µm, to account for the finite
resolution of the optical system. The data is normalized by the initial cloud size, which is 33.4
µm and 2.8 µm in the axial and radial directions, respectively. The solid lines are the numericalsolution of the hydrodynamic equation.

33
The average local Fermi energy can be obtained from the density distribution of the atoms,n(r), and the detection probability, P (r). We measure n(r) by turning the trap o↵, without
applying the hollow light beams, and imaging the cloud after 4 ms of expansion at the resonance.
To determine the density distribution in trap, we fit the distribution measured after expansion andrescale the dimensions back to t = 0, assuming hydrodynamic expansion [51]. For the fit, we use
the Thomas-Fermi distribution, which is known to fit the data well [13].
We obtain P (r) using a model of the optical pumping by the hollow light beams [27]. In themodel, we assume that atoms that scatter a single photon are transferred to the dark state, and we
account for the attenuation of the hollow light beams as they propagate through the cloud. For agiven n(r), the propagation model gives us P (r) after the consecutive application of the two hollow
light beams. We note that the results presented in figure 3.3 are not sensitive to the details of themodel.

3.1.5

How homogeneous is homogeneous enough?

In figure 3.5, we show the contact at T /TF = 0.46 as a function of the fraction of atoms
probed, which is varied by changing the intensity of the hollow light beams. The main part offigure 3.5 shows the contact per particle in units of kF,trap in order to show the change in themeasured signal. We find that the signal increases as we probe fewer atoms near the center of thetrapped gas. We compare our results with several theoretical models, where the model lines areR
1model =
model [T /T (r)]k (r)d3 r, with C model (T /T ) being the
calculated by Ctrap
P (r)n(r)Chom
F
F
F
hom
Np kF,trap
model prediction for a homogeneous contact (normalized to N kF ), TF (r) = EF (r)/kB is the local
Fermi temperature, and kB is the Boltzmann constant. We find good agreement of the data withthe models.
In the inset of figure 3.5, we plot the contact divided by the average local kF , defined in thesame way as in figure 3.3. For comparison, we also plot theory predictions for the homogeneousmodel (hT /T i), where the notation hi stands for density-weighted
contact at the average T /TF , ChomF
model (hT /T i) ⇡ hC model (T /T )i.
averaging. A reasonable criterion for homogeneity is when ChomF
F
hom

C/(NpkF, avg )34

C/(NpkF,trap )

2.8
2.4

2.8
2.4
2.0
1.6
0.0

0.5

1.0

2.0
1.6
0.0

Data
GPF
GG
G0 G0

0.2

0.4

0.6

0.8

1.0

fraction probed

Figure 3.5: Contact versus the fraction of atoms probed for a gas with T /TF = 0.46 at the centerof the cloud. In the main plot, the measured contact (squares) is normalized in respect to the trap
kF , and is compared to the predictions of several theoretical models (lines) using the local density
approximation. The measured contact increases as we probe fewer atoms at the cloud center, where
the local density is largest. The inset shows the contact normalized by the average kF of the probed
atoms (squares), compared to theoretical predictions of the homogeneous contact at the average
T /TF (lines).

35
When the fraction of the atoms probed is less than 30% we find that this approximation holds tobetter than 2%. When probing 30% of the atoms, we calculate that the rms spread in the local TF
has been reduced to about 20%. We find that the data for T /TF = 0.46 and fractions lower than
30% agree with theory predictions for a homogeneous gas (see inset of figure 3.5).

3.1.6

The e↵ect of the remaining density inhomogeneity

In figure 3.6, we compare our data to the predictions of several theoretical models. Here weshow that the e↵ect of the remaining density inhomogeneity of the probed sample on the theory
predictions is negligible. We use the detection probability, P (r), and the density distribution, n(r),for each of the data points, to calculate the average contact predicted by each theoretical modelaccording to:hCi =1
Np hkF iZ

model
P (r)n(r)Chom[T /TF (r)]kF (r)d3 r ,(3.8)

model is the prediction for the contact of a homogeneous gas theory (normalized to N k ),where ChomF

Np is the number of probed atoms, and hkF i is the average kF . The comparison of the averagecontact, hCi, and the homogeneous contact for three di↵erent models is shown in figure 3.6. The
graph clearly demonstrates that (with 30% of the atoms probed) the e↵ect of the remaining density
inhomogeneity on the contact is negligible, and hence theories for the homogeneous contact can becompared directly to the data.

3.1.7

Thermometry of a unitary Fermi gas

Thermometry of a strongly interacting gas is not trivial, and di↵erent groups have usedvarious techniques, including thermometry with a minority component [8], measurement of theenergy versus entropy relation [52], and an empirical temperature extracted from fitting the cloud
to a Thomas-Fermi distribution [44]. We base our thermometry on a measurement of the releaseenergy of the gas and the recently reported equation of state [30]. We determine the release energy
by taking an image of cloud after 4 ms of expansion at unitarity. Knowing our trapping potential,
the equation of state, and the generalized virial theorem at unitarity [52], we are left with only

36

4

C/(NkF)

3

2

G0 G0

1

0

GPF
GG
0.0

0.2

T/TF

0.4

0.6

Figure 3.6: Comparison of the homogeneous contact calculated by several theoretical models (solidlines) and the contact averaged over the remaining density inhomogeneity when probing the central
30% of the cloud for the same models (open symbols). The excellent agreement of the points andthe lines shows that the e↵ect of the remaining density inhomogeneity on the contact data can beneglected. See text for the explanation of the theories.

37
the temperature, T , as a free parameter in the calculation of the release energy. We find T bymatching the calculated energy to the measured one. We estimate that the one sigma uncertainty
in the temperature is 5%. When reporting T /TF in figure 3.3, we use TF = EF,avg /kB .
Our thermometry assumes a knowledge of the trapping potential V (r) and the equation ofstate n(µ, T ), where T is the temperature and µ is the chemical potential. For a non-interacting
gas the equation of state is known, and for the unitary gas, we use the equation of state recently
measured at MIT [30]. The trapping potential is calibrated from the known optical trap beam
waists and the measured oscillation frequencies in all three directions. We adopt a local density
approximation approach; the local chemical potential is given by µ(r) = µ0
T and number of atoms, N , µ0 is set by the normalization requirement N =V (r). For a givenRn[µ(r), T ] d3 r. The

equation of state then determines the complete density profile n(r), from which we can calculate
other quantities such as the entropy, total energy, release energy, and shape of the cloud. Since
we measure N , we get a one-to-one correspondence between T and these quantities, and therefore
any of them can serve as a thermometer. With the unitary gas, we have chosen to use the releaseenergy as a thermometer.
The release energy per particle is calculated from the measured density profile of the expanded
gas using
Erel =

X

i=x,y,zZ
1 X
m ⇣ ri ⌘ 2
Ei,rel =[nt (r)N
2 t

n0 (r)]d3 r ,(3.9)

i=x,y,zwhere ri is the corresponding spatial coordinate (i = x, y, z), t is the expansion time, and nt (r) is
the density distribution at time t. In the experiment, we use t = 4 ms. We have verified that therelease energy measured at t = 4 ms is the same as that measured after 12 ms of expansion. For agiven potential V (r), the release energy is given by [52]:1Erel = hr · rV (r)i ,2where the symbol hi stands for the density-weighted average: hg(r)i = N1
the calculated Erel (T ) to the measured Erel , we determine T .

(3.10)
Rg(r)n(r)d3 r. By equating

As a comparison, we have used two other techniques to extract the temperature of the unitary

38
gas. The first technique we compare to is based on the widely used practice of fitting the stronglyinteracting gas to a Thomas-Fermi distribution and extracting an empirical temperature, T̃ , fromthe fitted fugacity [44]. At T = 0, the empirical temperature is connected to the real temperaturepby T = T̃ ⇠, where ⇠ is the universal constant defined above [13]. Albeit without a complete
theoretical justification, one can then extend this to finite temperatures and extract T [44]. In thefollowing analysis, shown as the blue triangles in figure 3.7, we used ⇠ = 0.376 [30].
The second thermometry method we compare to is based on the entropy of the weaklyinteracting gas before the ramp to unitarity. We calculate the entropy of the weakly interacting
gas from the measured temperature and the trapping potential. In the experiment, we start fromthe weakly interacting gas and slowly ramp to the Feshbach resonance field. By performing thisramp there and back and comparing the entropy before and after the ramp for a gas initially at
T /TF = 0.12 and T /TF = 0.22, we have determined that the entropy increases by about 6% whengoing to the Feshbach resonance field. Assuming this increase, we use the entropy of the unitarygas together with the equation of state as the thermometer. In figure 3.7, the T /TF we obtainfrom these two additional techniques are plotted against the release energy thermometry. We finda good agreement between all the three techniques up to T /TF = 0.4. Above that temperature, theempirical temperature technique becomes unreliable since the e↵ect of quantum degeneracy of the
shape of the cloud diminishes. The entropy technique starts to show a small systematic deviation
upwards above T /TF = 0.4. The close agreement of the three techniques, which are based on
independent observables, up to T /TF = 0.4 gives us confidence in our thermometry. To estimate
the errors in T we look at the di↵erence between the entropy and release energy thermometry
techniques.
In summary, we have presented a measurement of the homogeneous contact of a unitary
Fermi gas versus temperature. Our measurement is based on a novel technique that allows us toprobe local properties of the cloud. Our data show good agreement with theory predictions for
T /TF > 0.4, but at lower temperatures no single prediction fully agrees with the data. Furthermore,
the data do not show an enhanced narrow peak around Tc , which was predicted to exist due to

39

1.0

T/TF, other methods

0.8

0.6

0.4

0.2
entropy techniqueempirical temperature technique
0.0
0.0

0.2

0.4

0.6

0.8

1.0

T/TF from release energy

Figure 3.7: Comparison of di↵erent thermometry methods. The x-axis is the temperature, T /TF ,where TF is the trap Fermi temperature, as extracted from the release energy. The y-axis is thetemperature we get from the two other thermometry methods (see text for more details). The
dashed line is y = x.

40pair fluctuation in a pseudogap phase.
To provide additional insight into the nature of the normal state of the unitary Fermi gas, I
will describe the momentum-resolved RF spectroscopy technique combined with the donut beamin the next section.

3.2

The spectral function of a strongly interacting Fermi gas
The spectral function provides the information about the modification of single particle states

due to interaction. A free particle, which shows a quadratic dispersion, is modified to a quasiparticle with an e↵ective mass as interaction increases. When the system is strongly interacting,due to pairing, the single-particle dispersion can also be negative. The measurement of such singleparticle excitation spectrum provides us with an insight on the many-body behavior of the system[53].
In condensed matter experiments, the spectral function can be measured by a techniquecalled Angle Resolved Photoemission Spectroscopy (ARPES). ARPES involves shining a laser on
the surface of a material and measuring the momentum (k) and the energy (✏k = ~k 2 /2m) of theejected electrons. From this information and using the energy and momentum conservation, thespectral intensity can be calculated. The energy of the electrons on the surface is given by,Es (k) = ✏k +
where

+ h⌫

(3.11)

is the work-function of the system and h⌫ is the photon energy of the incident beam. Theenergy Es gives the dispersion of the system. The spectral function thus measured is the imaginary
part of the Green’s function in many-body theories. The spectral intensity probed in an ARPES
experiment is given by [53],

IARP ES (k, !) = f (!)M0 (k, !)A(k, !),(3.12)

where f (!) is the Fermi function, M0 (k, !) is a matrix element which comes from the selection

41
rule and A(k, !) is the spectral function. Because of the Fermi function, the spectral intensity
measured with ARPES only contains the information about the occupied spectral function. The
spectral function is related to the Green’s function G(k, !) [54], as,
1
Im(G(k, !).
⇡A(k, !) =(3.13)The Green’s function for an interacting fermion is well-known and given by,
G(k, !) =1!

✏k⌃(k, !).

(3.14)

Here, ⌃(k, !) is called the self-energy. The real part of this term contains the information about
the e↵ective mass and the imaginary part contains the information about the lifetime of the quasiparticles. Thus, measuring the single particle spectral function of the interacting fermionic systemgives us an insight on the most important properties of the system.

3.2.1

Momentum-resolved RF spectroscopy

Our group developed a technique analogous to ARPES to measure the single particle spectral function of an interacting Fermi gas. This technique that is called momentum-resolved RF
spectroscopy (or atom photoemission spectroscopy) is an extension of RF spectroscopy work as described in the previous section. The atoms outcoupled by RF photons, in our case, are equivalent
to electrons ejected from the material in ARPES. In addition to measuring the outcoupled atoms
as a function of RF frequency, we also resolve their momentum after a TOF thus allowing us to
get the spectral function. Since, the momentum imparted by the RF photons is negligible, we canget the spectral function by simply using the conservation of energy. The matrix element M0 (k, !)in our case is 1. The only requirement for momentum-resolved RF spectroscopy is that the outcoupled atoms do not interact with the remaining two-spin states. In our case, for 40 K atoms, thescattering length between the outcoupled atoms in mF =states mF =9/2 and mF =5/2 and the remaining atoms in spin

7/2 is around 200a0 . The interaction e↵ect is negligible enough tomeasure the momentum distribution after a ballistic TOF expansion. Also, the final-state e↵ect
after RF spectroscopy is negligible at this interaction strength.

42
3.2.2

Experimental Sequence

The experiments are performed with an ultracold gas of 40 K atoms in an optical dipole trap.
The trapping frequencies are 243 Hz and 21.3 Hz in the radial and axial directions, respectively,
with the axial direction of the trap oriented horizontally. The final stage of evaporative cooling is
performed at a magnetic field of 203.3 G, after which the field is swept linearly in 50 ms to the final
value where we carry out the PES measurement. The initial temperature of the weakly interacting
gas, before the adiabatic sweep, is T0 = (0.16 ± 0.02)TF [an exception is the farthest point on the
BCS side, where T0 = (0.13 ± 0.02)TF ]. From our previous measurements of the release energy
at unitarity (described in the previous section), we estimate that the temperature at unitarity is(0.18 ± 0.02)TF . The gas has 80, 000 to 120, 000 atoms per spin state and is cooled such that thetemperature is just above Tc after a sweep to the Fano-Feshbach resonance at 202.20 G [12]. We
verified that for all interaction strengths the condensate fraction is less than 1%.

Figure 3.8: Momentum-resolved RF spectroscopy. The atoms in one of the strongly interacting
spin states are outcoupled to a third spin state. Since the outcoupled atoms have weak interaction
with the atoms in remaining spin states, the momentum distribution and energy of these atoms
can be measured after a balistic TOF expansion. The donut beam technique is employed on theoutcoupled atoms right before TOF to select the homogeneous sample.

The atom PES measurement uses an rf pulse with a Gaussian field envelope that has an rms

43
width of 51 µs (17 µs), and a total duration of 300 µs (100 µs), for ⌫ < 34 kHz (⌫34 kHz).

We use shorter pulses at higher ⌫ to minimize e↵ects due to the motion of the spin-flipped atoms
during the rf pulse. The rf detuning ⌫ is given with respect to the resonant transition frequency⌫0 , which is measured for a spin polarized gas. For each PES measurement, we take data for thesame 25 values of ⌫, each of which is repeated three times. The rf power is increased for larger ⌫,while keeping the fraction of spin-flipped atoms less than 40%. As in the RF spectroscopy work tomeasure Tan’s contact, we measure the dependence of the number of spin-flipped atoms on the rf
power, and scale the measurements done at di↵erent rf powers to correspond to a common level.
Immediately after the rf pulse, and exactly at trap release, we pulse on the hollow-core lightbeams, which propagate perpendicularly to each other and intersect at the center of the cloud[20,21]. The frequency of these beams is chosen to optically pump atoms from the |9/2, 5/2i state
into the upper hyperfine manifold (F = 7/2), where they are invisible to our imaging. Both the
rf pulse and the spatially selective optical pumping, which is pulsed for 40 µs, are completed in atimescale that is short compared to motion of atoms in the trap. We typically probe the |9/2, 5/2i
atoms that came from the central 30% of the cloud; we find this fraction to be a good compromise
between spatial selectivity and signal-to-noise ratio [20,21]. As described in Ref. [3], just before
imaging the cloud, we remove the remaining atoms from the |9/2, 9/2i and |9/2, 7/2i states
and then transfer the outcoupled atoms in the |9/2, 5/2i state to the |9/2, 9/2i state with two
short rf ⇡-pulses. This procedure enables us to image the atoms on the cycling transition, which
improves the signal-to-noise ratio.

3.2.3

Two-mode fitting function

Similar to the analysis done in electron systems, we use a two-mode function to describe the
PES signal [55]:I(k, E) = ZIcoherent (k, E) + (1Z)Iincoherent (k, E) ,(3.15)

where the first part describes quasiparticles with a positive dispersion, the second part accommodates an “incoherent background” that exhibits negative dispersion, and Z is the quasiparticle

44

b

c

T /TF

E

k2
hollow beams
selection

h 0
EF

Tc

h(

superfluidBCS

1ballistic expansion
absorption
imaging beam

BEC

0
(kF a) 1

d

0+ )
EF

cameraa10n(k , )

k

2

2

2

2

2

2

2

1

1

1

1

1

1

1

0

0

0

0

0

0

0

−1

−1

−1

−1

−1

−1

−1

−2

−2

−2

−2

−2

−2

−2

−3

−3

−3

−3

−3

−3

−3

100

E (EF)

10−1

−2

10

−3

10

(kF a)−1 = −0.88
−4

0

1

k (kF)
BCS

2

−0.43
−4

0

−0.25
1

k (kF)

2

−4

0

0.26

−0.08
1

k (kF)

2

−4

0

1

k (kF)

2

−4

0

0.42
1

k (kF)

2

−4

0

0.82
1

k (kF)

2

−4

0

1

2

10−4

k (kF)
BEC

Figure 3.9: Momentum-resolved RF spectroscopy data taken at T > Tc (see text) as a function ofinteraction strength. The color represents the probability distribution of atoms at a given E and k
in the strongly interacting gas. The white line is the quadratic dispersion of free particles.

45
spectral weight. When these two parts (defined below) are combined, the resulting dispersion can
exhibit back-bending.
The quasiparticles in Fermi liquid theory are long-lived and therefore give rise to narrowenergy peaks, which, in principle, could be directly observed. However, such peaks would bebroadened by our experimental resolution of about 0.25EF . This resolution is set by the numberof atoms (with EF scaling only weakly with increasing N ) and the rf pulse duration, which must
be short compared to the harmonic trap period in order to probe momentum states. We convolve
Eqn. 3.15 with a Gaussian function that accounts for our energy resolution before fitting to thedata in order to determine the spectral weight of the quasiparticles (Fig. 1d, upper panel).
To describe quasiparticles, we use
Icoherent (k, E) = 4⇡k 2 · (E

k2
m⇤

E0 )

h

⇣
⇣
(⇡m⇤ T )3/2 Li3/2
exp⇣⌘
exp ET µ + 1

E0 +µ
T

⌘⌘i 1

,

(3.16)

which consists of a quadratic dispersion of sharp quasiparticles multiplied by a normalized Fermi
distribution ( is the Dirac delta function, and Li is the polylogarithm function). We include as fit
parameters, a Hartree shift E0 , e↵ective mass m⇤ , chemical potential µ, and temperature T . Here,energies are given in units of EF and m⇤ in units of m, the mass of a 40 K atom. This description of
Fermi liquid quasiparticles is typically only used very near kF and for T approaching zero, whereas
we fit to data for a larger range in k and with temperatures near 0.2 TF (just above Tc ). The latteris necessitated by the unusually large interaction energy compared to EF , and we note that 0.2 TF
is still sufficiently cold that one can observe a sharp Fermi surface in momentum, as described in
Chapter 2. Any increase in quasiparticle widths away from kF will have little e↵ect on the data aslong as the quasiparticles have an energy width less than our energy resolution, which should be
the case for long-lived quasiparticles.
The second part in Eq. 3.15 needs to accommodate the remainder of the signal, which isoften referred to as an “incoherent background” in a Fermi liquid description. For fermions withcontact interactions, one expects an incoherent background at high momentum due to short-range
pair correlations [34, 9, 56]. Motivated by this and by the normal state in the BEC limit, we use

46
for Iincoherent a function that describes a thermal gas of pairs. The pairs have a wave function thatdecays as exp ( r/ R), where r is the relative distance and R is the pair size [57], and a Gaussian
distribution of center-of-mass kinetic energies characterized by an e↵ective temperature Tp . This
gives
8k
Iincoherent (k, E) = ⇥

Ep

E+k

2

q

Ep
Tp e

Ep +E 3k2
Tpsinh

⇡ 3/2 (E

✓ p p
2 2k

Ep E+k2Tpk 2 )2where ⇥ is the Heaviside step function, Ep is a pairing energy defined by kF R =p◆(3.17)

2/Ep , and both

Ep and Tp are dimensionless fitting parameters. While this description of the incoherent piece may
not fully capture the microscopic behavior except in the BEC limit, we find nonetheless that Eq.
3.15, after convolution with a Gaussian function that accounts for our energy resolution, fits thedata very well throughout the crossover.
In Figure 3.10, we show Z as a function of (kF a) 1 . For our lowest (kF a) 1 , Z ⇡ 0.8;
however, Z decreases rapidly going from the BCS side of the crossover (negative a) to the BEC side
(positive a), reaching Z ⇡ 0.3 at unitarity. Beyond (kF a) 1 = 0.28 ± 0.02, Z vanishes, signallingthe breakdown of a Fermi liquid description. Restricting the fitting to a smaller region around kF
gives results for Z that are consistent with the fits to k  1.5 (see Fig. 3a). We note that theinteraction strength where Z vanishes, as well as the sharpness with which Z goes to zero, arelikely to be temperature dependent [58]. The best fit values for the e↵ective mass, m⇤ are shownin Fig. 3b, where m⇤ increases with increasing interaction strength as expected for a Fermi liquid.
A linear fit gives m⇤ = 1.21 ± 0.03 at unitarity, which is somewhat higher than m⇤ = 1.13 ± 0.03
measured in Ref. [21], but close to the T = 0 prediction of m⇤ = 1.19 from Ref. [10].
3.2.4Energy Distribution Curves

Energy Distribution Curves, commonly known as EDCs, are the cut of the spectral function at
a given momentum. The width of these curves give the information on lifetime of the quasiparticles
and the peak gives the dispersion. An example of an energy distribution curve is shown in Figure
3.11, which is taken at (1/kF a =0.08). The circles are the data points and the lines come from

47

1

a

Z

0.8
0.6
0.4
0.2
0

1

0.5

0

0.5

b

1.4

m⇤ (m)

1

1.2
1

1

0.5

0
(kF a)

0.5

1

1

Figure 3.10: The quasiparticle residue Z and the e↵ective mass m⇤ as a function of the interaction
strength. Both of them are the fit parameters in Equation 3.15.

48
a single surface fit using the two-mode fitting function. In this case, the fit parameters are Z =0.37(3), m⇤ = 1.22(3), T = 0.25(2), E0 =The reduced

3.2.5

0.33(2), µ = 0.19(4), Ep = 0.23(4), and Tp = 1.09(8).

2 of this fit is 1.2.

Contact vs interaction strength

As (kF a) 1 increases, short-range correlations are expected to increase. This gives rise toincreased weight in the high-k part of the spectral function [56], which is quantified by a parameter
called the contact [34, 9, 38, 48]. In a Fermi liquid description, the contact must be accounted forby the incoherent part of the spectral function [56].
We note that our particular choice for Iincoherent has the expected form of a 1/k 4 high-k tailin the momentum distribution [34] and a 1/⌫ 3/2 large-⌫ tail in the rf line shape [9], where ⌫ isthe rf detuning. Remarkably, we find that the contact can be accurately extracted from the fits to
the PES data even though we restrict the fits to k  1.5. For comparison, 1/k 4 behavior in the
momentum distribution was observed for k > 1.5kF [38].
In Figure 3.12, we plot the measured contact per particle, C/N , in units of kF , as a functionof (kF a) 1 . The data extend previous measurements of the contact at unitarity [48, 44] and agree
well with several theoretical predictions [6, 10].

3.2.6

Conclusion

The results presented here can explain how di↵erent observations lead to di↵erent conclusions
regarding the nature of the normal state of the unitary Fermi gas. Although the data here taken
just above Tc show that a Fermi liquid description breaks down for (kF a) 1

0.28±0.02, Z remains

finite at unitary. Fermionic quasiparticles may play a key role in thermodynamics, while PES data
reveal back-bending and significant spectral weight in an “incoherent” part that is consistent withpairing.

49

1
0.5

k/kF = 0.24

Spectral weight (arb. units)

0
0.5

k/kF = 0.47

0
0.5

k/kF = 0.71

0
0.5

k/kF = 1.03

0
0.5

k/kF = 1.26

0
0.5
0

k/kF = 1.5

2

1

0

1

2

E (EF )
Figure 3.11: Energy Distribution Curves at 1/kF a = 0.08. Each panel is the cut of the spectral
function at the given k/kF . The circles are data and the lines are the fits.

50

C /N (kF )

12
9
6
3
0

1

0.5

0
(kF a)

0.5

1

1

Figure 3.12: The contact density per particle (in units of kF ) for a homogeneous Fermi gas above Tc
is shown versus (kF a) 1 . The contact measured from the tail of the rf lineshape (blue circles) agrees
well with the contact extrapolated from the fits of the PES data (red squares). For comparison, we
also plot the BCS (dashed black line) and BEC (dashed magenta line) limits, given by 4(kF a)2 /3
and 4⇡(kF a) 1 , respectively [9], the non-self-consistent t-matrix at T = 0 (dotted blue line) and
its Popov version at Tc (dash-dotted red line) [6], and the self-consistent t-matrix model at T = 0(double-dotted green line) [10]. We find that the contact measured above Tc agrees well with the
T = 0 theories.

Chapter 4
New Generation Fermi Gas Apparatus Design

In 2012, we decided that the first generation potassium Fermi gas experiment apparatus was
becoming outdated. We were running the last (out of four) potassium dispensers, and expected
to exhaust the source within a few years. We started planning for a new generation apparatus in2012 and started early work of assembling optics, electronics and vacuum chambers in 2013. In
December 2014, the very first Fermi gas apparatus was disassembled, and we put all our e↵orts tothe new generation apparatus starting in January 2015.
The new Fermi gas apparatus (Figure 4.2) consists of three chambers. In the first chamber,
we put potassium dispensers and make a MOT. The atoms collected in the first MOT are pushed
to the second chamber through a di↵erential pumping tube to the second chamber where we makeanother MOT. The second chamber has a much longer vacuum lifetime, so we can trap more atoms
and perform sub-Doppler gray molasses cooling. Atoms are then loaded into a purely magnetic
quadrupole trap and transferred to the science cell with a “cart”. We do subsequent cooling of theatoms in magnetic and optical dipole traps in this cell down to quantum degeneracy.

4.1

Vacuum chambers

4.1.1

Dispenser Chamber

The dispenser chamber (Figure 4.1) is a 6-way cross glass cell made by Ron Bihler at Precision
Glassblowing, Denver. It is made up of pyrex and has six 1.5”(3.81 cm) diameter windows for MOT
light. Additionally, there are three 0.5” (1.25 cm) diameter windows, which will be used for push

52
beam and MOT fluorescence detection. The chamber has two arms for atomic sources. We put anenriched source of potassium chloride (bought from Trace Sciences International at $240 per mg in2011) in one of them (see the description of how to make the potassium dispensers in Appendix),
and we put Rb getters in the second arm. We later noticed that the enriched potassium salt pricehas increased by four fold starting in 2013. The arms have a diameter of 1.1” (2.8 cm) where thesource-boats are placed and are reduced to 0.868” (2.2 cm) tubes that fuse to the main chamber.

Figure 4.1: Six-way cross glass chamber which contains the atomic source for the system. The
potassium atoms are collected in a 3D MOT in the first stage of the experiment.

A transfer tube is fused to the glass chamber with a 2.73” (6.93 cm) long glass tube with adiameter of 0.75 in (1.91 cm). This tube has a glass-to-metal transition to a stainless steel with a1.33” (mini) CF. A longer stainless steel transfer tube, custom made by MDC Vacuum, is attachedto the mini flange. The transfer tube has a length of 10” (25.4 cm) and an inner diameter of 1.1 cm.
The diameter is chosen to allow for a good di↵erential pumping between the dispenser chamber
and the main MOT-chamber. The other end of the transfer tube has a 2.75” regular CF, which isconnected to the main MOT-chamber.

53
4.1.2

Second MOT-chamber

To save time from glass-chamber manufacturing work, we bought a commercially available
stainless steel chamber for our main MOT-chamber. It’s a 6-inch Spherical Octagon Chamber from
Kimball Physics (MCF600-SphOct-F2C8). The chamber consists of eight 2.75” (6.99 cm) ports
and two 6” (15.24 cm) ports. Four of the 2.75” ports and the 6” ports are used for MOT beams.
One of the 2.75” port is connected to the transfer tube coming from the dispenser chamber sideand the port orthogonal to it is connected to the transfer tube to the science cell. One of the 2.75”
port is used for pumping.
The viewports needed for MOT and optical pumping are also bought from Kimball Physics(MCF275-MtgFlg-C1VP and MCF600-MtgFlg-F1VP). The viewports are made up of Corning type7056 glass.
A pump arm is attached to the spherical octagon chamber. The arm is primarily used forpumping out the dispenser chamber and the octagon chamber. The arm consists of a titaniumsublimation pump (Gamma Vacuum TSP,3 FILAMENT,2.75 CF,MS,STD) and a 75 L/s ion-pump(Gamma Vacuum 75S-CV-4V-SC-N-N). A reducer (MDC ConicalReducer, 4.5”x 2.75”) is used toconnect the spherical octagon chamber to a four-way cross (MDC 4-WayCross, 4.5”). The TSP is
attached to the upper port with another reducer (MDC ConicalReducer, 4.5”x 2.75”); the ion pumpis attached to the side and the lower port is used for the initial pumping with a turbo-molecularpump. We use a zero-length reducer (MDC ZeroLengthReducer, 3.38”x2.75”Tapped) to connect
the four-way cross with a angle-valve (Lesker, Manual SS All-Metal Angle Valve CF flanged). A
turbo-molecular pumping station is attached to the valve during the initial pumping stage and later
removed after closing the valve. The port is then sealed o↵ with a CF blank.
The second transfer tube, which connects the second MOT chamber with the science cell,consists of a reducer, an all-metal-gate-valve, a four-way cross and a glass-to-metal transition section
to the science chamber. The atoms move 20.87” (53 cm) from the center of the MOT cell to thecenter of the science cell.

54

Figure 4.2: CAD Drawings of the apparatus. Left: Top view, Right: Side View

55
The all-metal-gate-valve (VAT Inc. All-metal gate valve DN 16(5/8”)) serves the purpose of
separating the MOT section from the science chamber. We put it there such that we can easilychange the source or make changes to the science chamber without having to rebake the entire
apparatus.
There is another pump arm attached to the science cell that is similar to the first pumping
arm except we use a 40 L/s pump (Gamma Vacuum 45S-2V) instead of a 75 L/s pump because thesurface area of the science cell is much smaller than that of the Kimball octagon cell. A zero-length
reducer (MDC Vacuum ZeroLengthReducer, 2.75”x1.33”) is attached to the cross, which is thenattached to another 4-way cross (MDC Vacuum 4-WayCross, 2.75”). Like in the pumping arm #1,
a TSP (Gamma Vacuum TSP,3 FILAMENT,2.75 CF,MS) is attached to the upper port and theion-pump is attached to the side. The lower port is used for pumping with a turbo-molecular pumpduring the initial stage of vacuum pump-down.

4.1.3

Science Cell

The science chamber (Figure 4.3) is designed by Jay Meikrantz from Precision Glassblowing,
Denver and is based on the cell designed by Jay for Adam Kaufman (Regal lab) . The cell isdesigned to have a large optical access enabling us to have beams for a 3D-lattice and also have ahigh resolution imaging. It is an all-glass spherical octagon with seven 0.625” (1.59 cm) windows.
The top and the bottom of the cell consists of a 1.7” (4.32 cm) windows. The chamber is attached
to a 1.33” (mini) flange with a glass-to-metal transition tube with a diameter of 1.9 cm.
The cell is coated inside and out such that it has a good transmission of both 767 nm and1064 nm beams. Figure 4.4 shows the transmission data we measured for a large and a small
window before they were assembled.

56

Figure 4.3: The Science Cell: the cell is manufactured by Precision Glassblowing in Denver andhas seven 0.625” windows and two 1.7” that are double AR-coated at around 767 nm and 1064 nm.

120[a.u.]
Transmission [%]

100
80
60
40
Large Window
Small Window
Baseline measurement

20
0
200

400

600

800

1000

1200

1400

Wavelength [nm]Figure 4.4: Transmission data for the science cell windows. The windows are AR coated for 767
nm and 1064 nm light on both sides. The baseline measurement was done without any windows inplace.

57

4.2

Pumping Speed Calculations

4.2.1

BasicsConsidering the nature of the gas and the relative quantity of gas through a tube, the gas
flow can be either in the continuum flow or in the molecular flow regime [59]. In the high pressure
regime, the gas flow can be either viscous or turbulent, and is termed continuum flow. All of thepressures we work at (< 10 9 Torr) are described by the molecular flow regime in which the mean
free path is long compared to the tube size making the flow dynamics entirely governed by gas-wallcollisions.
The quantity that describes the flow in a tube is called conductance (C), which is measured
in L/s. The conductance of a cylindrical tube in the continuum flow regime is given by [59],C[L/s] = 78

d3 [in3 ]
l[in]

(4.1)

where d is the tube diameter in inches and l is the length of the tube in inches. The coefficient
includes all the factors for units conversion.
When there are multiple tubes connected in either series or parallel, the total conductancefollows the following equations:1
Ctotal
Ctotal =

=

X 1

X
i

! Series Conductance(4.2)

Ci ! Parallel Conductance,

(4.3)

i

Ci

where the symbols used are obvious.
Similarly, when there is a 90 degree elbow in a tube, Monte Carlo simulations have shownthat the conductance is reduced to 22% of the original conductance.
Another important quantity in the calculation is the pumping speed. The chamber surfaces outgas with a certain rate depending upon the material. These outgassed materials need
to be pumped out to maintain a desired vacuum pressure. The pumping speed depends on the

58
conductance of the tube to which it is connected. The e↵ective pumping speed is given by,1
Pe↵ective

=

1
Ppump

+

1
,
C

(4.4)

where Pe↵ective is the e↵ective pumping speed at the chamber, Ppump is the pumping speed of thevacuum pump and C is the conductance of the tube which connects the vacuum chamber with thevacuum pump.
We will usually have an idea of the e↵ective pumping speed we want based on the following
equation,
Pumping speed =4.2.2

outgassing rate ⇥ surface area.
Desired Pressure(4.5)

Conductance in our vacuum system setup and pressure estimates

The outgassing rate depends upon the materials and temperature used. For a baked 316L
stainless steel, it is 2 ⇥ 10 13 Torr L/s cm2 . Because we have an idea of the e↵ective pumping
speed, we use Eq. 4.4 to calculate the required pumping speed of the vacuum pump.
The conductance of the tube that connect the dispenser chamber and the main MOT chamber
is given by,
C12 = 78 ⇥0.433
= 0.43 L/s.
14.54

(4.6)

Now, let’s calculate the conductance of the pumping arm. First, let’s estimate the conductance of the cone,
Ccone =

1
1.3753 + 2.3753⇥ 78 ⇥= 211 L/s.
2
2.95

(4.7)

The conductance of the cross is,
Ccross = 78 ⇥2.3753= 155 L/s
6.73

(4.8)

59
Hence, the net conductance of the pumping arm is,
1
1
Cpump arm = (Ccone+ Ccross) 1 = 89 L/s(4.9)

The total conductance from the dispenser MOT to the ion pump is,
C1P = (C121 + (0.3 ⇥ Cpump arm ) 1 ) 1 = 0.43 L/s(4.10)We have a 75 L/s pump in this arm but the e↵ective pumping speed in the dispenser chamberis dominated by the transfer tube which is 0.43 L/s. Considering the outgassing rate of Pyrex fromthe glass cell [59], the estimated pressure in the dispenser chamber is given by,
surface area ⇥ outgassing ratepumping speed
2
10(195 cm ⇥ 1.5 ⇥ 10
Torr L/s/cm2 )== 7 ⇥ 10 8 Torr
0.43 L/s
P =(4.11)(4.12)

Next, the main MOT chamber. The e↵ective pumping speed is the same as the conductanceof the pump arm, which is 41 L/s. The surface area is 687 cm2 and the outgassing rate of stainlesssteel is 10 13 Torr L/s/cm2 . Hence, the estimated pressure is 1.7 ⇥ 10 12 Torr.
Next, let’s calculate the e↵ective pressure in the science cell. The conductance of the tube
from the cross to the cell is 2.93 L/s. The conductance of the pump arm is 13 L/s. We have a
40 L/s ion pump in this pumping arm. The net conductance is 1.85 L/s. The outgassing rate of
the glass is 10 12 Torr L/s/cm2 . And the surface area of the cell is 56 cm2 . Hence, the estimatedpressure is,
P =

4.3

56 ⇥ 10 12= 3 ⇥ 10 11 Torr
1.85

(4.13)

Pumping down to ultrahigh vacuum
Since the science cell was not ready, we assembled the apparatus in two phases. In the firstphase we assembled upto the gate valve, then a year later we assembled the science cell. During
the construction we made sure that all the fasteners we use are 316L stainless steel to avoid any
magnetization in the future.

60
Reaching an ultrahigh vacuum requires a thorough cleaning of the vacuum parts. We started
by cleaning all the stainless steel parts in an ultrasonic bath of soap-water (alconex) solution for 10
minutes. Then, we rinsed them with deionized water. The next step of cleaning process involved
ultrasonic bath of acetone, methanol and isopropanol (in that order) for 2 minutes each.
We found the hard way that glass chambers cannot be cleaned in an ultrasonic cleaner. Our
first MOT cell got shattered beyond mending while cleaning in an ultrasonic cleaner.
After the vacuum chambers parts are cleaned and assembled, the first stage in pumping down
is with a turbo-pump (Agilent Technology Turbo-V-301-AG). We reached a pressure of about 10 8
Torr with just a turbo-pump. To go below this value, we constructed a brick oven around our
vacuum chamber (Figure 4.5). We used heating plates underneath the chamber to uniformly heat
the oven. We slowly increased the oven temperature to 300 C over a course of the day and kept theentire system at that temperature for about 10 days. We also fired all our Ti-Sub pump filaments
during the bake to clean them out. We took out the magnet from the ion-pump while the oven was
hot. After over a week of bakeout and cooling down, we reached a pressure of 2 ⇥ 10 11 Torr. The
pressure log during the first stage of baking is shown in Figure 4.6.

4.4

LasersWe decided to switch to DBR lasers (PH767DBR080T8) manufactured by Photodigm fromthe usual homemade External Cavity Diode lasers (ECDLs) that we used in the previous generation
machine. The main motivation for the switch was the stability of the DBR lasers. The linewidth
for these lasers is slightly wider (⇠ 1 MHz) than typical ECDLs (few hundred kHz). Since thetransition linewidths for potassium atoms are ⇠ 6 MHz, we didn’t expect any problems. These
diodes also come with an in-built thermisor/TEC and are quite compact.
We use two DBR lasers with maximum power of 80 mW for the D2 MOT setup. One ofthem is locked to the absorption signal of 39 K crossover line (Figure 4.9) using Pound-Drevor-Hall(PDH) technique. The transition is shifted to the correct frequency of 40 K using AOMs. The
remaining light is used for repumping. The second laser is beatnote locked to this laser with an

61

Figure 4.5: Oven construction for vacuum bakeout. Heating plates are placed underneath theapparatus. We make an oven around the chamber using bricks and cover with thermally insulatingblankets.

Figure 4.6: Pressure log during the first vacuum bakeout.

62

Asph. lensCyl. lenses
F= 3.1

4Photodigm
DBR
80mW

OpticalIsolators
30dB +30 dB

F =100 F =-50

2

PBS

2

PBS
To TA

Glan Thomson
Pol.

PBS

PBS
To beatnote

K-Vapor cell

EOM

2

2

2

Repump

F =10

New Focus 1601
PhotodiodeAsph. lensCyl. lenses
F= 3.1
Photodigm
DBR
80mW
F =100 F =-50

Trap
OpticalIsolators
30dB +30 dB

2

PBS

2

PBS
To TA#1

To TA#2
From repump
PBS to combine
trap and repump

Figure 4.7: D2 Laser setup. We set up two DBR lasers for repump and trap light. The repump
laser is locked to the crossover transition of 39 K atoms using a spectroscopy in a vapor cell. The
trap laser is o↵set locked to the repump laser using a photodetector.

63o↵set frequency of 666 MHz. The laser setup for trap and repump beams is shown in Figure 4.7.
Because we have two MOTs and require large intensity, 80 mW light from the DBR lasers is
not enough. We use three 1.5W Tapered Amplifiers, two for trap beams and one for repump, from
Eagleyard(EYP-TPA-0765-01500-3006-CMT03-0000). The setup is shown in Figure 4.8 After anappropriate frequency shifts using AOMs, we split the repump light for the first and second MOT.
The first MOT repump is combined with the first MOT trap beam with a PBS and fiber coupled
together. The second MOT repump is fiber coupled by itself.
The trap beam from MOT# 2 TA is split into two beams using the 0th and 1st order beam ofan 80MHz AOM. The -1st order beam is used for trapping. The 0th order beam is sent to a second
AOM to be used for pushing (-82 MHz), optical pumping (-62 MHz) and probing (-47 MHz). Since
these processes don’t occur at the same time, we use a single AOM and three D-shaped mirrors to
pick-up three deflections, corresponding to each frequencies, and fiber couple the beams. Opticalpumping beam is sent through a +69 MHz AOM before fiber coupling. During optical pumping
and probing stages, the o↵set lock is shifted to get to the right frequency.
The trapping frequencies and intensities of the trap, repump, push and pump beam are listed
below:
Beam

Frequency

Power

Trap MOT 1-3

120 mW

Repump MOT 1

-3

40 mW

Trap MOT 2-3

120 mW

Repump MOT 2

-3

10 mW

Push

-3

15 mW

Pump

+3

100 uW

64

TaperedPhotodigm
DBR
ampliﬁer
80mW
Trap#1Cyl. lensF = 50F =175F =-50

OpticalIsolator
30dB

2
From repump TA

F = 250

Beam dump

AOM-80 MHzTo MOT#1

To MOT#2

TaperedPhotodigm
ampliﬁer
DBR
80mW
Trap#2Cyl. lensF = 502x telescope

OpticalIsolator
30dB

2

AOM
80 MHz

To science probe
To push/probe/pump

Figure 4.8: Tapered Amplifier setup.

65

2P
3/2

F’ = 5/2

55.2

F’ = 7/2

31.0

F’ = 9/2-2.3

F’ = 11/2

Op@cal Pump
Trap

766.7 nm
D2

2S

1/2

-46.4

Push

RepumpF = 7/2

714.3

F = 9/2-571.5Figure 4.9: 40 K D2 transition lines. The frequencies on the right side are in MHz.

66

4.5

Cart
We transfer atoms from the second MOT chamber to the science cell, using a pair ofquadrupole coils which are mounted on a linear stage (cart). The atoms travel a distance of
20.87” (53 cm) from the center of the second MOT chamber to the center of the science cell. The
quadrupole coils have 30 turns each and are wound with copper square hollow tubings which allow
us to water cool these coils. They serve the purpose of providing quadrupole confinement during
MOT stage and the transfer stage.
The coils are mounted on a custom made Parker XR-404 with 700mm of travel. We use
Parker BE233GL-NPSN servo motor and GV6K-U6E servo controller. The motion of the linear
stage is programmed on a Motion Control software also provided by Parker. We have seen in some
JILA labs that this servo/motor can be noisy because they use switching power supply, so they
had to use an Aerotech motor. This is not a problem for us.

4.6

Magnetic Trap Description
After discussing various magnetic trap designs, we decided to build a QUadrupole Io↵e Con-

figuration (QUIC) trap around the science cell. Io↵e-Pritchard traps are quite common in JILA,
but they have a poor optical access. Plugged quadrupole trap would have been another option, butthey require a separate blue-detuned laser and are also quite sensitive to pointing stability. Hence a
QUIC trap is an attractive option as it is relatively simple to design design, provides a large opticalaccess and is “quick” to build.
The QUIC trap primarily consists of three coils: a pair of quadrupole coils and a Io↵e coil
(Figure 4.10). The Io↵e coil shifts the zero of the magnetic field from the quadrupole pair, thus
atoms avoid undergoing Majorana spin-flips. However, because the Io↵e coil also breaks the axial
symmetry, the field profile is not symmetric. The field minima is shifted in the direction of theIo↵e coil (Figure 4.11.
We use the quadrupole coils of the QUIC trap as a bias-coil-pair for Feshbach field. Hence,

67
the quadrupole trap includes an H-bridge circuit. The electronics of the switches is shown in Figure
4.13.
The coils are wound with a square hollow copper tubing, which are good for water cooling.
The quadrupole coil pairs have 4x4 turns and have the inner diameter 3.75 in and the outer diameterof 5.12 in. The coils are separated by 1.53 in (inner separation). The Io↵e coil has 4x2 turns andhas an inner diameter of 0.72 in and an outer diameter of 1.36 in. The coil is placed 1.18 in fromthe quadrupole coils center.
With the measured coil resistitivity of 1.88 mOhms/m, we expect the power dissipation on
the coils is about 2900 W at the current required for the QUIC trap. Hence, we use JILA-chilledwater to cool these coils.
An important part of the magnetic trap design is the field stability. To maintain a 3 mG
trap-bottom stability, we need 30 ppm current stability in the Io↵e-coil. We use the circuit shownin Figure 4.12 to stabilize the current in these coils.

4.7Electronics for the magnetic trap
We use two Keysight 6690A power supplies (15V, 440A) to drive cart coils and the QUIC

trap. The cart coils and the Io↵e coil are connected to one power supply and the science quadrupole
trap is connected to the other power supply. To switch between cart coils and Io↵e coils, we use amechanical relay switch from Kissling(29.511.11). These switches can handle up to 500A of current
and have a switching time of ⇠ 60 ms. We also use four of these switches to make an H-bridge toswitch between quadrupole and Feshbach configuration (Figure 4.13.
The QUIC trap needs to be stable to about 10 ppm in current noise to avoid heating or losses
during the evaporation stage. We use analog servo electronics to maintain the current stability.
The electronics are home-built and an improved version of the one designed by Brian DeMarco. We
use an IGBT (SKM400GA12V) to regulate the current and Danfysik IT 700-S as a current sensor.

68

Figure 4.10: The top view of the QUIC trap. The quadrupole pair (orange), the fast-B coil (yellow),Io↵e (light yellow) are visible.

69

B@GaussD

B@GaussD

200

200

150

150

100

100

Ioﬀe = 0A

50

-0.04

-0.02

0.00

0.02

0.04

50

y@mmD

-0.04

-0.02

B@GaussD

B@GaussD

200

200

150

150

100

-0.04

-0.02

-0.02

0.00

0.02

0.04

y@mmD

-0.04

-0.02

0.00

0.02

200

200

150

150

100

100

0.00

0.02

0.04

y@mmD

y@mmD

Ioﬀe = 200 A

B@GaussD

Ioﬀe = 250 A

0.04

50

B@GaussD

50

0.02

100

Ioﬀe = 150 A

50

-0.04

0.00

Ioﬀe = 100A

-0.04

-0.02

0.04

y@mmD

50

Ioﬀe = 337 A

0.00

0.02

Figure 4.11: B-field profile along the direction of Io↵e-coil while increasing the Io↵e coil current.
Quadrupole current = 380 A.

0.04

y@mmD

70

Figure 4.12: Current Servo electronics. This servo is used to control the gate voltage of MOSFETs/IGBTs for MOT coils, shim coils, quadrupole coils and the Io↵e coil.

H

H

Monitor

Servo

IGBT

Relay
IGBT

Relay 4

H
H

H

Servo

Relay 3

Relay

Ioﬀe

Coil 2Servo Monitor

Relay 2

Keysight
6690A

Keysight
6690A

Coil 1

Relay 1

Cart Quadrupole coils

71

IGBT

Figure 4.13: Circuit diagram for cart quadrupole trap and QUIC trap. Left: H-bridge setup to
switch between science quadrupole trap and Feshbach coils. Right: Switching circuit between theIo↵e coil and the cart quadrupole coil. The element H are the Danfysik IT 700-S, used for in-loop
and out-of-loop current sensors for servoing and monitoring purpose respectively. Relays are theKissling (29.511.11) switches. IGBTs used in these circuits are SKM400GA12V.

72

4.8

Dipole Trap
The accessible Feshbach resonances in 40 K are in the negative Zeeman levels, which are highfield seeking. Hence, we load the atoms at the end of magnetic trap evaporation to a far-o↵-resonant
dipole trap (FORT). We use an IPG 10W multimode fiber laser (IPG YLR-10-1075-LP) at 1075
nm for this purpose. We split the beam into two arms and fiber couple them to make a crossed
dipole trap. The two arms the beam have orthogonal polarization to avoid potential issue withfluctuating interferences. The schematic of the setup is show in Figure 4.15. The horizontal beam
has a beam waist of 25 µm and the vertical beam has a beam waist of 150 µm. The trapping
potential in the horizontal direction is given by,
3⇡c2
Udipole (x, y, z) =
2!0

where

✓

1
!

1
+
!0 ! + !0◆(Ihorizontal (x, y, z) + Ivertical (x, y, z))(4.14)+mgy,(4.15)

is the excited state linewidth of the P3/2 level, !0 is the resonant D2-line frequency, ! isthe frequency of the dipole trap laser, and Ihorizontal (x, y, z) and Ivertical (x, y, z) are the intensityof horizontal and vertical beams respectively. The last term in the equation is potential due togravity, which becomes important when the trap depth is lowered.

GLC 003

Pol

dichroic
mirrorF = 250
doubletF = 150
doubletdichroic
mirrorFigure 4.14: Dipole trap setup. Left: Setup for horizontal dipole trap to produce 25 µm beam
waist. Right: Setup for vertical dipole trap to produce 150 µm beam waist.

73

4.9

Computer Control and Data Acquisition
We use two Xilinx FPGA cards to control the experiment. Each of these cards provide

48 digital channels and 8 analog channels. The card is programmed in LabView. The front-end
computer control interface is written in MATLAB to be more user friendly.. The sequence is sent
as 64-bit information from MATLAB to LabView using TCP/IP communication. This enables thecontrol of the experiment from any computer. Since 16 analog channels are not enough for us,
we use two 16-bit 8-channel analog cards (EVAL-AD5360) from Analog Devices. These cards areprogrammed using 6 digital channels from the FPGA.
In MATLAB, channel numbering starts at 0. Analog channels from FPGA1 are labelled as
100 for AO0, 101 for AO1 etc. and those from FPGA2 are labelled 200 for AO0, 201 for AO1 etc.
Digital lines are labelled similarly– 100 for DO0, 101 for DO1 and so on.
All the timings are in seconds and analog voltages are in volts. For digital channels, 0 or
OFF is logical low and 1 or ON is logical high. We use an optical isolator board between the FPGA
digital lines and the instruments. This helps to prevent potential ground loops and accidental fryingof the FPGA board.
Commands that can be used in MATLAB are as follows:• Aout(channel, time, value) - set the value of an analog channel at the given time
• Aout ramp(channel, time, ramp duration, start value, end value) - ramp an analog channel
from start value to end value at the given time during the ramp duration• Dout(channel, time, value) - set the value (ON/OFF) of a digital line
• Dout pulse(channel, time, pulse duration) - generate a pulse in a digital channel• GPIBout(time, address, command) - send a command to the GPIB address.
An example code to switch the relay from Cart configuration to Io↵e configuration is shown
below. This set of codes also sets the voltage and current limit on the power supply.

74
GPIBout(1, 5, ‘VOLT 14’) % GPIB channel 5 at 1 second
GPIBout(1, 5, ‘CURR 440’)% GPIB channel 5 at 1 secondDout(cart or ioffe voltage multiplexer, ti coils, OFF) %multiplexer
Dout(cart relay switch, ti coils, ON) %relay
Aout ramp(cart coil servo control, ti coils, 100*ms, 0, -1) %current
Aout ramp(cart coil servo control, 5, 100*ms, -1, 0) %current

We use a PixelFly camera and a Princeton Instrument (PI) ProEM CCD camera to acquire
images of the atoms. The PixelFly camera is used to image atoms in the second MOT. Princeton
Instrument camera is used to acquire images of atoms in the science cell. Even though an EM CCD
is an overkill, we found out recently that this is the only camera currently available from PI thathas kinetics mode imaging.

75

Breakout box

External Analog

Opto-isolators

Digital lines

VFG

Parker Motor
Controller
Keysight 33220A
Dipole trap

GPIB

HP E4420

FPGA Analog

FPGA#2
FPGA#1

PixelﬂycameraPI camera

K-40

K-41

K-39

K-38

Figure 4.15: Layout of the computer control. We use one main computer for FPGA and GPIB
control. Three other workstations are used to acquire data from PixelFly/Princeton Instrument
cameras, and to send commands to Versatile Frequency Generator (VFG), Parker Motor and dipole
trap waveform. The computers are named after potassium isotopes.

Chapter 5
Cooling to degeneracy

Cooling of the atoms down to degeneracy begins in the dispenser MOT cell. Here, atoms
from potassium vapor are trapped in a magneto-optical trap (MOT) and cooled to the Doppler
limited temperature. We collect about 108 atoms in the first MOT. Since the vacuum lifetime inthis chamber is only around one second, we push the atoms to the second chamber, which a better
lifetime. Better vacuum lifetime and use of the dark-SPOT repump allow us to trap 5 ⇥ 108 atoms
in the second MOT. Atoms are further cooled below the sub-Doppler limit using gray molasses
cooling technique. At this stage, they are optically pumped into the correct Zeeman level andloaded into a purely magnetic quadrupole trap. The atoms are transferred to the science cell using
a translation stage with a unity efficiency. Then, they are loaded to the stationary quadrupole
trap. Io↵e coil is adiabatically ramped on to create a harmonic potential. We use a microwave
transition (|F = 9/2i ! |F = 7/2i) to evaporate hotter atoms out of the trap. About 2 ⇥ 107
atoms are cooled to about 20µK in this trap. They are then transferred to an optical dipole trap
where further evaporation down to degeneracy occurs. We get about 105 atoms at T /TF = 0.2 atthe end of optical trap evaporation. By putting them in the right Zeeman states, we can access the
Feshbach resonance near 202 Gauss to create a strongly interacting ultracold Fermi gas.

5.1

MOT Basics
The first stage of cooling starts with a magneto-optical trap (MOT). Magneto-optical trap

works in the principle of Doppler cooling and an inhomogeneous magnetic field [60]. The basic

77
principle of optical pumping in a MOT is shown in Figure 5.1. The atoms in a quadrupole magnetic
field experience a position dependent Zeeman shift in the excited state. When we introduce twocounter-propagating circularly polarized red-detuned beams on the atoms, they scatter photons
depending upon their position and are driven towards the center of the trap where the magnetic
field is zero [60].
For a small Zeeman shift compared to the detuning ( ), the force on the atoms with velocity~v and displacement (~r) is given by [60],
F~ =
where

~v

~r,

(5.1)

= (8~k 2 s0 )/( (1 + s0 + (2 / )2 )2 ), and  = (µB 0 )/(~k) . Here, ~k is the momentum of

the photons, s0 is the I/Isat ,is the transition linewidth, µ is the magnetic moment, and B’ isthe magnetic field gradient. For a typical MOT parameter of the field gradient of ⇠ 10 G/cm anddetuning of a few , the oscillation frequency is a few kHz and the damping rate is a few hundred
kHz, making it an over damped system.

b.

a.
σ-

σ+

σ-

σ-

σ+

IMOT

me= +1
me= 0
me= -1

σ+

σ+

IMOT

σmg= 0posi6onFigure 5.1: a. A sketch of a magneto-optical trap (MOT) setup. Three orthogonal beams are
passed through the chamber and are retro-reflected in combination with quarter wave plates toform a +setup. We run current in opposite direction in the two coils to form a quadrupole
magnetic field. b. The basic energy level diagram for a simple MOT transition, which shows shift
in energy level as a function of position.

The cooling limit in a MOT depends on the atomic transition linewidth, and is given by,
TDoppler =

~
.
2kB

(5.2)

78
The natural linewidth of the D2 transition in 40 K is 2⇡ ⇥ 6 MHz, hence the Doppler temperature
limit in the MOT is 144 µK.

5.2First MOTThe collection MOT is formed with three retroreflected beams. The MOT beams consist of

120 mW of trap and 40 mW of repump power coming out of a single mode fiber. The light is split
into three beams and sent through the three orthogonal windows of the collection MOT cell withappropriate polarizations. The beams have a Gaussian waist of 1.4 cm. The quadrupole magnetic
field in for trapping is produced by a pair of coils with 110 turns run in quadrupole configuration
at 3A.
We collect potassium vapor from following reaction which occurs at around 400 C:
2KCl + Ca ! 2K + CaCl2 .

(5.3)

To avoid any deposition of potassium in the collection cell windows, we use heating tapes
around each windows to heat the cell at 70 C. When we did not use the heating tapes, we noticed
formation of silver potassium layers on MOT windows that a↵ected beam transmission.

F = 150

Photodiode

MOT Cell#1

Security camera

Figure 5.2: Fluorescence measurement of the first MOT. A 2f ! 2f imaging system is setup tocollect the fluorescence of the first MOT. By knowing the solid angle of the lens subtained on
the atoms, and intensity and detuning of the MOT beams, we can calculate the number of atoms
collected in the MOT. There’s also a security camera, which is sensitive to near IR light, to view
the MOT for troubleshooting purpose.

MOT fluorescence is observed through the half-inch windows. We setup a security camera(forgood sensitivity in the IR regime) on one side and a photodiode with one lens to focus the MOT
fluorescence on the second side.

79
The yield of potassium and the fill tau as a function of current is shown in Figure 5.3. We
noticed that with a fill time of 1.2 s, 3.8 A is ideal for us. We started to see potassium deposits
above 4A.
We scanned the frequency of the trap laser to optimize the filling rate of the MOT. Figure
5.4 shows the results of the scan.

5.2.1

Calculating MOT Number from fluorescence

We setup a single lens imaging system to collect the fluorescence onto a photodiode. The
number of atoms in the MOT is given by,
Natoms =VP D,gain ⇥ QE ⇥ solid angle ⇥ ~! ⇥ pexcited /⌧(5.4)

where gain and QE are the photodiode specifications, solid angle is the measured quantity of theimaging setup, ! is the frequency that corresponds to the 767 nm light, pexcited is the excited state
population and ⌧ is the lifetime of the excited state. The excited-state population is given by,
pexcited =I/Isat,1 + I/Isat + 2( / )2

where I is the intensity of the trap beam, Isat = 2.2 mW/cm2 and(5.5)
is the trap detuning. Using

this model, we typically get the atom number in the first MOT to be around 3 ⇥ 107 . However, thissimple model is not entirely correct. In the next section, I will discuss the more accurate six-level
model to get the excite-state fraction.

80

Figure 5.3: Potassium yield vs dispenser current. We raised the dispenser current and waited
for a few hours to measure the fill level and the half fill time. The measurement is taken with aphotodiode.

Figure 5.4: MOT fill level vs beat frequency. The y-axis is normalized such that it is proportional
to the atom number. The resonance is at 682 MHz.

81
5.2.2

Six-level MOT

A simple two-level model is not a good model to account for the number of atoms in a 40 K
MOT because of the complicated energy levels. To get a more precise number, we need to consider
the full six-level model which correspond to: F = 9/2 ! F 0 = 11/2, F 0 = 9/2 and F 0 = 7/2 andF = 7/2 ! F 0 = 9/2, F 0 = 7/2 and F 0 = 5/2 and find the total scattering rate.
From Ref. [60], we have Isat = 1.75 mW/cm2 . Using the branching ratios from ref [61],averaging over polarization (or Zeeman states) gives a factor of 2.5, or Isat = 4.4 mW/cm2 . To findthe scattering rate, we first need to find the transition strength and the branching ratio of groundstates and excited states respectively. The transition strength is proportional to the square of thematrix element µeg and is given by,
8
98
90
>
>
>
0
0
0
0
< L J S =< J F I >
= F
6p
B
0
0
D = 4 (2J + 1)(2J + 1)(2F + 1)(2F + 1)@>
: J L 1 >;>: F J 1 >
; mF
2

1 F0
q

m0F

132

C7
A5 ,(5.6)

where the curly brackets denote the Wigner-6j symbols, the parenthesis denotes the Wigner-3j
symbol, and q = ±1 for sigma transitions, q = 0 for ⇡-transitions. For the D2 transition in 40 K, theground state has S =1/2, L=0 and J= 1/2, and excited states has S = 1/2, L = 1, J = 3/2, and I=

4 for both ground and excited states. The matrix c gives the transition strength from F={9/2, 7/2}
to F 0 = {11/2, 9/2, 7/2, 5/2} relative to the total transition strength for that F (columns sum to1). The matrix b gives the transition strength from F = {9/2, 7/2} to F 0 = {11/2, 9/2, 7/2, 5/2}
relative to the total transition strength for that F’ (rows sum to 1).
0

1
0
1
0.6
0
1
0
B
C
B
C
B
C
B
C
B
C
B
C
0.296
0.255
0.592
0.408
B
C
B
C
B
C
B
C
c=B
C,b = BC
B 0.104 0.37 C
B 0.295 0.741 C
B
C
B
C
@
A
@
A
0
0.375
0
1

82
The rate equation can derived as follows:
dpF= RF 1 (p1dt

pF ) + RF 2 (p2

pF )

pF

(5.7)

p1 ) + bF1 pF(5.8)

p2 ) + bF2 pF ,

(5.9)

11

2
X
dp1
=
RF 1 (pF
dt
5

F=2
11

2
X
dp2
=
RF 2 (pFdt
5

F=2

where pF is the population in the excited state F, p1 and p2 are the populations in the groundstates, and RF f is the transition rate from f to F.
✓
cF f
If /Isat,Ff
RF f =
2
1 + If /Isat,Ff + 4(

Ff/

)2

◆

,

(5.10)

where Isat = Isat /cFf .
dp

In the steady state, we set dpdtF = 0 and dtf = 0. Hence, we get,p1 R F 1 + p2 R F 2
RF 1 + RF 2 +

pF =

p1 =

p2 =

P 11
2

F = 52P 11
2pF (RF 1 + bF 1 )(5.12)P 11
2

F = 52F = 52(5.11)RF 1

pF (RF 2 + bF 2 )

P 11
2

.

(5.13)R
F=5 F2
2

We can substitute Equation for pF into Equation for p2 . Then, we can solve for p2 in terms of p1 .
P 11
RF 2 +bF 2
2
F = 5 RF 1 +RF 2 +p2 = P 11 2
2

F = 52

RF 1 +bF 1
RF 1 +RF 2 +p1 .

(5.14)

We can then set p1 to 1 and calculate p2 . Finally, we can calculate all pF s. Then, the excited state
population is given by,
pe =

P

F
F p
P

1 + p2 +F pF.

(5.15)For the values of detuning and intensities we use in our MOT, the two-level model gives a factorof 4 less number than the six-level model. We have verified that the six-level model is consistent
with number extracted from absorption images.

83

5.3

Push to the second MOT
Atoms cooled in the first MOT are transferred to the second chamber with a near resonant

beam (|F = 9/2i ! |F 0 = 11/2i). The atoms collected in the first MOT are pushed in the directionof the second chamber. Once the atoms start to move out of the first MOT, they are o↵-resonantwith the push beam. Hence, they don’t get subsequent momentum kicks. We use a continuous
beam to push the atoms to the second chamber. We found out that the pulsed and the continuous
beam have a similar efficiency of transfer. We, however, noticed that the loading of the second
MOT is quite sensitive to the position of the push beam. If the push beam is near the atoms insecond MOT, it kills the second MOT instantly. Hence, we used two mirrors to walk the beam:
first mirror to efficiently push the atoms from the first MOT and the second mirror to miss thesecond MOT.

Figure 5.5: MOT fill level vs push frequency detuning.

84

Figure 5.6: MOT fill level vs push beam power.

85

5.4

Second MOT
Atoms pushed to the second chamber are collected in another MOT formed by three retro-reflected beams. The beams have a Gaussian waist of 1.4 cm. The gradient of 10 Gauss/cm for the
MOT is formed by water cooled coils, which also form the quadrupole trap in the subsequent stage.
The schematic of the trap, bright/dark repump and the D1 cooling beams are shown in Figure 5.7.
We use a dark-SPOT repump for this MOT in order to avoid density dependent losses. We load
the MOT for 60 s and collect 109 atoms. We have noticed that the dark-SPOT repump improves
the number by 10 folds but the loading ⌧ is increased by a factor of 4.

5.4.1

Lifetime

A good vacuum lifetime is desirable for an efficient cooling of atoms in a magnetic trap oran optical dipole trap. For the second chamber we can measure the lifetime of atoms in two ways.
The first way is to fully load the atoms into the MOT and then turn o↵ the push beam. We
can then monitor the decaying fluorescence signal of the MOT and fit it to an exponential to findthe lifetime. Using a bright repump beam, as opposed to a dark-SPOT repump, we observed twodi↵erent timescales as shown in Figure 5.9. The first lifetime corresponds to the fast decay due tohigh density of the MOT. The second, longer, timescale comes from the loss due to collisions withbackground atoms.
The second way to measure the lifetime of the atoms is to load the atoms into a magnetic
trap after optical pumping to the right spin states. Then, hold the atoms in the trap for variable
amount of time. At the end of the hold, either do an absorption imaging or load them to the MOT
followed by a recapture measurement. We found that the lifetime measured this way in the second
chamber is around 40s (Figure 5.10), which is shorter than measured using MOT fluorescence.
This is because the atoms in a magnetic trap are more sensitive to spin flips and losses due to thescattered light.
We also measured the lifetime in a QUIC trap in the science chamber by doing absorption

86

Beam#1Beam#2

D1
beam/2

Beam#3

F=100

x3
telescope

F=1000PBS
x4
telescope

Dark
repumpF=13.86x8
telescope
Trap

F=300
Dark
Spot

Brightrepumpx6
telescope

RepumpFigure 5.7: Schematic of the second MOT beams. The dark-repump is formed by imaging a 1.6mm aluminum dot on to the atoms by using F = 300 mm and F = 1000 mm lens setup as seenin the figure above. The bright repump is turned on only during optical pumping stage. D1 beam
consists of both trap and repump light out of the fiber.

87

Figure 5.8: MOT fill level vs push time. a. with dark SPOT repump b. without dark SPOT
repump. We notice a significant increase in atom number with the dark SPOT repump beams.

Figure 5.9: Lifetime of atoms in the second MOT. We load the atoms to the MOT and turn o↵ thepush beam and wait. We see a two clear timescales, a fast timescale of about 8s corresponding todensity dependent losses and a slower timescale corresponding to background losses

88

Figure 5.10: Lifetime of atoms in the quadrupole trap in the second MOT chamber. We load theatoms into a quadrupole trap and wait there for a certain time(dark time), then reload the atom
to a MOT and measure the recapture fraction. This lifetime is less than the MOT lifetime because
atoms are more sensitive to losses due to scattered light in a quadrupole trap. Also, the magnetic
trap is more shallow, thus vulnerable to grazing collisions with background gas.

89
imaging. We found out that the lifetime in the science cell is around 80 s. This is long enough todo evaporation in the magnetic and dipole trap.

90

5.5

Gray Molasses cooling
Doppler temperature, which is 145 µK for 40 K, is the limit to cool atoms in the MOT. Several

techniques have been tried to reach below this temperature. For example, molasses cooling with D2
light was tried in late 90s by the group at LENS [62]. The molasses cooling resulted in an inefficientcooling of the gas and worked only for a small sample (⇠ 107 atoms). This was mainly because theexcited state in the D2 transition is not well-resolved as in other species. The narrow line 4S-5P
cooling was tried by the Toronto group [63] but it is a little bit complicated to implement requiringdi↵erent set of optics which work in the near-UV regime.
Recently, the group of Christophe Salomon at ENS has implemented gray molasses coolingin potassium [64]. This cooling scheme works on the 4 S1/2 ! 4 P1/2 , more commonly known as theD1 transition. The cooling works in combination of velocity-selective coherent population trapping(VSCPT) [65] and Sisyphus cooling [66]. VSCPT works by optically pumping atoms into the darkstates whose lifetime depends on the square of the atomic velocity. This combines with of Sisyphus
cooling, in which atom climb optical potential, created by counter-propagating beams, thus losing
the kinetic energy.

|ψB>

|ψD>
Posi+onFigure 5.11: Gray molasses cooling. The cooling is a combination of velocity-selective coherent
population trapping and Sisyphus cooling.

More specifically, atoms in the dark state do not change energy as a function of position.
They can be excited to the bright state. The bright state energy level does vary with position

91
depending on the intensity and polarization of light. An atom has the highest probably of getting
excited into the bright state at the bottom of the potential hill. Once the atom is in the brightstate, it starts to climb the optical potential. As it reaches the top, it is optically pumped back to
the dark state. As a consequence, excitation, climbing and pumping reduce the temperature of thesample below the Doppler limit.
The transitions for D1 cooling is shown in Figure 5.12. The ground state consists of |F = 9/2i
and |F = 7/2i hyperfine manifolds of 2 S1/2 state. The excited state is the |F 0 = 9/2i and |F 0 = 7/2i
hyperfine manifolds of the 2 P1/2 state. These are well separated by 155 MHz. The optical transitionsof 770 nm light are blue shifted for both the “trap” |F = 9/2i ! |F 0 = 7/2i and the “repump”|F = 7/2i ! |F 0 = 7/2i transitions. We denote the detuning for the trap light asrepump light as

and that for

repump . The di↵erence between the trap and the repump light is denoted by.

In the experimental sequence, once we load the atoms in the second MOT for 60 s, we change
the detuning of the trap beam to move it closer to the resonance (1.5 ) for 70 ms. This stage
is known as compressed MOT (CMOT) which creates a slightly colder and denser MOT suitable
for molasses cooling. Then, we turn o↵ the optical beam and the quadrupole field in 100µs. We
then shine the D1 beams at the atoms. In Figure 5.13. We vary the detuning of the trap laser
while keeping

to be 0. Note that, since our goal was to improve the condition of the atoms

after the transfer rather than optimize the gray molasses itself, we did the measurement of number
and temperature at the end of cart transfer in the science cell. We see that the transferred atom
number is maximum and the temperature is minimum at=2 .

We also vary , shown in Figure 5.14, and find some strange behavior in number and temperature which could be due to on-resonance Raman transitions. Next we vary the power (Figure
5.16 and duration (Figure 5.15) of the molasses beams and observe a clear dependence.
In conclusion, we have implemented the D1 molasses beam to improve the initial condition
for evaporation in the magnetic trap. Even though we did not fully characterize this cooling stage,
we benefited tremendously in both transfer fraction and an efficient evaporation because of thisstage of cooling, which I will discuss in the next two sections.

92δ
ΔrepumpF’ = 7/2

Δ
86.3

2P
1/2

F’ = 9/2

770.1 nm
D1

2S

1/2

Repump

-69.0

Trap

F = 7/2

714.3

F = 9/2-571.5Figure 5.12: D1 transition lines for gray molasses cooling in 40 K. The numbers on the rightcorrespond to the frequency shift of the hyperfine levels in MHz.

93

Figure 5.13: Number (black square) and temperature (green circle) of the atoms transferred to thescience cell as a function of D1 trap frequency while keeping = 0. Note that zero is suppressed
for temperature. The reason we don’t see a drastic change in temperature is explained in the main
text.

Figure 5.14: Number (black square) and temperature (green circle) of the atoms transferred to thescience cell as a function of frequency di↵erence between trap and repump beams .

94

Figure 5.15: Number(black square) and temperature (green circle) of the atoms transferred to thescience cell as a function of D1 molasses pulse duration.

Figure 5.16: Number (black square) and temperature (green circle) of the atoms transferred to thescience cell as a function of D1 molasses power (trap plus repump). Repump beam power is 1/10th
trap beam power.

95

5.6

Loading to the quadrupole trap and cart transfer

5.6.1

Trapping in a magnetic trap

Atoms can be trapped in a magnetic trap by using the magnetic dipole moment. Atoms feel
~ µ · B),~ where µ is the magnetic moment and rB~ is the magnetic field gradient.
a force of F~ = r(~~Di↵erent Zeeman spin states have di↵erent magnetic moments that results in F~ = gmF µ0 rB,
where g is the Lande-g-factor and µ0 is the Bohr magneton. If we use the low field seeking states,
such as |F = 9/2, mF = 9/2i, |F = 9/2, mF = 7/2i and |F = 9/2, mF = 5/2i with a positive
magnetic moment, these atoms can be trapped in a magnetic trap. mF = 9/2 has a magnetic
moment of µB , Bohr magneton. Similarly, at low field mF = 7/2 has a magnetic moment of 7/9µB and mF = 5/2 has a magnetic moment of 5/9 µB . The Zeeman energy shift of the magnetic
Zeeman sub-levels is shown in Figure 5.28.

5.6.2

Optical pumping and loading

Once the atoms are cooled to sub-Doppler temperatures after the D1 molasses cooling stage,
they are optically pumped to low field seeking zeeman states using a circularly polarized ( + ) pulse.
For potassium, the |F = 9/2, mF = 9/2i and |F = 9/2, mF = 7/2i are low field seeking states.
During optical pumping, we keep the bias field at about 4 Gauss.
We scanned the frequency of the transition to find that the most efficient transition is the|F 0 = 11/2, mF 0 = 9/2i (Figure 5.17). We also scanned the polarization of the light using aquarterwave plate (Figure 5.19). This way we determine the purely circularly polarized light. Oncethe atoms are optically pumped, we ramp up the magnetic field gradient to 100 G/cm in 10 ms.
The pair of quadrupole coils sit on a track, which allows us to move the atoms to the final sciencecell region.

96

Figure 5.17: Finding the right transition for optical pumping. We vary the optical pumping frequency, load the atoms to the quadrupole trap, hold for a short time and measure the MOT
recapture fraction. The four lines corresponds to the energy of the Zeeman levels.

Figure 5.18: We vary the optical pumping pulse duration and measure the recapture fraction. We
find out that the recapture fraction saturates beyond 80 µs

97Figure 5.19: Optical pumping transitions needs a + light. Here, we vary the polarization of lightusing a quarter-wave plate to find the right polarization.

Figure 5.20: We vary the optical pumping power to get the highest number of atoms into thequadrupole trap. We vary the voltage driving the AOM frequency driver. 0.15V corresponds to13.5 µW of optical power and the power varies linearly with voltage in this regime.

98
5.6.3

Pros and cons of using a cart

At the building stage of the apparatus, we explored di↵erent options for transporting atoms
to science cell. The three popular options are: optical transport, multi-coil transport and cart
transport. We quickly ruled out the option for optical transport because it requires a high power
far-detuned dipole trap. The power requirement is set by the MOT or gray molasses temperature,size of the cloud and the mass of the atoms. That means for potassium it amounts to around 80 Wof 1064 nm light. The second option was to do multi-coil transfer, which requires ramping up anddown the current in a series of quadrupole coils to guide the atoms smoothly to the science area.
Although initially this was our plan, we hesitated at the end because first it seemed like it willtake a long time to implement for an efficient transfer of atoms. The other reason is the separation
distance between the coils. Since our coils are separated by a large distance, we need to run around
300A of current to produce a field gradient of 100 G/cm in the strong direction. To run 300 A of
current, the coils have to be water cooled. It seemed like this will be a big hassle to water cool 8
or 9 pairs of coils.
Finally, we decided to adopt the cart transfer technology that was developed in JILA in late
90s [67]. Implementing cart transfer is relatively straight forward as it requires only a single pairof quadrupole coils. We bought a commercial track from Parker Motion in Denver, installed thecart coils, programmed the motion using the software provided by the company.
One problem of using a cart is the space requirement on the optics table. The other downsideis that it forces us to have a magnetic trap in our sequence. With a good gray molasses cooling,
we could have tried optical transport and performed all-optical cooling, which would have resulted
in a much faster cycle time.

5.6.4

Transfer efficiency

The transfer efficiency of the atoms to the science cell is shown in Figure 5.21. To perform
this measurement, we captured the fluorescence of the MOT on the PixelFly camera. Then we

99
loaded the atoms to the quadrupole trap, and moved them to a specific point. We then moved theatoms back and recaptured back to the second MOT and measured the fluorescence. This gave
us the fraction of the atoms captured back in the MOT after the transfer. The one-way transfer
efficiency is the square root of that fraction. All the measurements took the same amount of time.
We found out that the gray molasses cooling significantly improves the transfer efficiency to thepoint where it is limited only by the vacuum lifetime.

5.7

Evaporation in the QUIC trap
Once the atoms are transferred to the science cell, we ramp down the cart quadrupole current

and ramp up the stationary quadrupole current simultaneous until the field gradient reaches 200
G/cm. We cannot evaporate in the purely quadrupole trap because of the Majorana losses. In aquadrupole magnetic trap there is a zero field region at the center at which the atoms undergo spin
flip to untrapped states and are eventually lost from the trap. This phenomena is called Majorana
spin flip. Majorana spin flip is avoided as long as the Larmor period of an atom in the magnetic trap
remains small compared to the typical flipping time r/v of the field seen by the atom. Following
Ref [68], we get the Majorana loss rate is
⌧=

mR2
~

(5.16)

Using this condition, for a cloud of potassium atoms of size 0.5 mm , this means the timescale
for Majorana loss is 157 s. However, we spent a couple of months trying to evaporate in a purely
quadrupole trap but did not succeed.
We turned the quadrupole potential produced by the quadrupole coils into a harmonic potential by ramping up the current in the Io↵e coil in 500 ms. The field produced by the QUIC trap
is,
~ QUIC | = |B
~q + B~ Io↵e |
|B
!
2
3
rB
1
2
2
⇠ B0 + y 2 +
⇢2 ,
2
B0

(5.17)(5.18)

100

Figure 5.21: We measure the transfer efficiency of the atoms as a function of transfer distancethrough the transfer tube to the science cell. The red points are with D1-gray molasses cooling andthe green points are without molasses cooling. The blue line is the diameter of the transfer tube.
The center of the science cell is 54 cm away. We find that the transfer efficiency is only limited by
the vacuum lifetime in the case of D1-cooled atoms. Without D1-cooling, we see that most of theatoms are lost coming right out of the chamber going through the narrow part of the tube.

101
where B0 is the o↵set frequency,is the axial curvature and ⇢ =p

x2 + z 2 . We can then derive

the trapping frequency,
r
1
µ
fy =
2
2⇡ m
s ✓
◆
1
µ rB 2
fr =
2⇡ m
B0

(5.19)(5.20)The measurement of the trapping frequency as a function of the field gradient is shown in
Figure 5.23.

Figure 5.22: We measure the trapping frequency in the QUIC trap by sloshing the atoms with afield gradient pulse and measuring the center position in TOF as a function of hold time.

The background s-wave scattering length of mF = 9/2 and mF = 7/2 is 170a0 and is favorable
for the evaporation. Until we reach a cold temperature, the p-wave cross-section is also favorable asshown in Figure . We use the microwave transition between |F = 9/2i and |F 0 = 7/2i transition forevaporation. We sweep the frequency from 1190 MHz to around 1278 MHz in multiple exponentialstages. Each stage has its exponential parameters ↵ andf (t) = fb + (f0fb )e ↵tdescribed as follows:t2,(5.21)

102

Figure 5.23: We vary the current in the Io↵e coil and measure the trapping frequency.

Figure 5.24: We measure the bias field of the QUIC trap by measuring the temperature of thecloud at the end of the evaporation and linearly extrapolating to find the frequency where thetemperature goes to zero. Depending on this frequency, we can calculate the magnetic field.

103

Figure 5.25: We vary the current in the Io↵e coil and measure the bias field. For a good evaporation,
we would like to have a bias field of a few Gauss.

Figure 5.26: We measure the axial trapping frequency of the QUIC trap by sloshing the atoms
in the axial direction with a field gradient pulse and measuring the center position in TOF as afunction of hold time.

104
where f(t) is the frequency at time t during evaporation, fb is the bottom frequency and f0 is thestart frequency for each stage. We optimized the evaporation as follows using the parameters in
Table 5.1
Table 5.1: We use three exponential stages and optimized the evaporation in the QUIC trap.
Stage
1190 - 1250 MHz
1250 - 1270 MHz
1270 - 1278 MHz↵ (Hz)
0.07
0.05
0.05

(Hz2 )
00
0.001

From the evaporation trajectory, we find that the parameter ↵ = d Log T/ d Log N = 1.05
> 1. From phase space density, we get= d Log PSD/ d Log N = -2.2. The condition fora “runaway” evaporation is that the collision rate increases with evaporation [69]. The collision
rate is given by= n v, where n is the density of the particles in the trap,

is the collisioncross-section, which is roughly constant in the temperature of range of evaporation, and v is therms velocity. In a simple case of a harmonic trap, the density scales as ⇠ N/T 3/2 and v scales as
T 1/2 , hence the collision rate scales as N/T . Thus, for a Log T vs Log N plot, we expect a slope oflarger than one for a good evaporation.
We typically evaporate the atoms down to 10-20 µK with N ⇠ 107 . This temperature is low
enough for us to load atoms into an optical dipole trap. The typical shot-to-shot number noise(standard deviation/average) at the end of the evaporation is 7% and temperature noise is lessthan 5%.

5.8

Optical Dipole Trap
The last stage of evaporation occurs in an optical dipole trap (ODT). Since the useful Fes-hbach resonance to do RF spectroscopy is between |F = 9/2, mF =9/2i and |F = 9/2, mF =7/2i, which are high-field seeking spin states, we cannot continue using a magnetic trap. Atoms
are therefore transferred to an optical dipole trap, where trapping of both high and low magnetic

105

Figure 5.27: Elastic cross-section vs temperature. The plot is reproduced from [11]. The filled
circles are the p-wave scattering cross-section for spin polarized cloud and the open circles are thes-wave cross section for mixed spin states. We see that the p-wave cross-section is large down to
100 µK and plummets as the temperature is reduced. The s-wave cross-section is relatively flat asa function of temperature.

1000

E@MHzD500

F = 7ê20- 500

F = 9ê2- 1000
0.000

0.005

0.010

0.015

0.020

0.025

0.030

B@TD

Figure 5.28: Zeeman sublevels of the ground levels in potassium. The evaporation microwave
transfers the atoms in |F = 9/2, mF = 9/2i and |F = 9/2, mF = 7/2i to F = 7/2 hyperfine level,
where they become high-field seeking and leave the trap.

106

Figure 5.29: The evaporation trajectory of the atoms plotted as Log T vs Log N. For a good
evaporation which increases the collision rate, we need the slope of this to be greater than 1.

Figure 5.30: The evaporation trajectory of the atoms plotted as temperature vs final microwave
frequency.

107

Figure 5.31: The evaporation trajectory of the atoms plotted as Log PSD vs Log N.

108field seeking states is possible.
An optical trap works in the principle of atom interaction with a far o↵-resonant light [70].
The electric field of a laser E induces an atomic dipole moment p in an atom that oscillates withthe light frequency !, where E(r, t) = êE(r)exp( i!t) + c.c. and p = êp(r)exp( i!t) + c.c.. The
amplitude of the dipole moment p is given by,
p = ↵E,(5.22)

where ↵ is the complex polarizability. The interaction potential is given by,1
hpEi =
2

Udip (r) =1Re(↵)I(r),2✏0 c(5.23)/! 2

where is ↵ = 6⇡✏0 c3 !2 !2 i(!03 /!2 ) . Hence,00

Udip (r) =

3⇡c2
2!03

✓

!0

!

+

!0

!

◆

I(r),

(5.24)

where !0 is the frequency of the resonance. And the scattering rate is given by,
scatt (r) =

3⇡c2
2~!03

✓

!
!0

◆3 ✓

!0

!

+

!0

!

◆

I(r).

(5.25)

In our case, ! ⇡ !0 , so,
3⇡c2
I(r),
2!03
✓ ◆2
3⇡c2
=
I(r),
scatt
2~!03
Udip (r) =

where

=!

(5.26)(5.27)!0 .

For a Gaussian laser beam, the intensity profile is given by,I(r) =

2P/(⇡w2 )2
2
E r /(1+z/zR ) ,2
1 + z/zR(5.28)

where P is the power of the beam, w is the Gaussian beam waist and zR = ⇡w2 / is the rayleighrange.
We setup a crossed dipole trap (horizontal and vertical) as described in Chapter 4, with a1064 nm light. The beam waist of the horizontal beam is 25 µm and that of the vertical beam

109
is 150 µm. We start with a 2 W beam in the horizontal direction and 150 mW in the verticaldirection. The trap depth is 302 µK and the trapping frequency is 3 kHz in the radial direction.
We measure the trapping frequency at the end of the evaporation by modulating the intensityof the beam. We modulate the intensity by using an external set point from a frequency generator
in the set point port of the intensity servo at 30% of the amplitude. We see a loss of atoms at twice
the trapping frequency (Figure 5.32).

Figure 5.32: a. Measurement of the radial trapping frequency by modulating the horizontal dipole
trap beam. We see a loss in number at twice the trapping frequency and a small feature at the trap
frequency. We have done a similar measurement to measure the trapping frequency in the axial
direction by modulating the vertical dipole trap beam. b. Radial trap frequency vs power of thedipole trap beam. The red line is the fit to P 1/2 and the black line is from calculation.

We load the atoms into the dipole trap by turning on the dipole trap beam to 2W while atthe same time ramping down the quadrupole trap current in 500 ms to half the value. After 400 ms
overlap between the quadrupole trap and the optical dipole trap, we shut down the magnetic trap
(both quadrupole and Io↵e coil) in 200 µs leaving the atoms trapped only by the optical potential.
Then, we do a Landau-Zener transition to move the atoms from positive mF states (mostly in
mF = 9/2 state) to negative mF states using an RF sweep from 7 MHz to 5 MHz in 10 ms at 20
Gauss. We transfer about 90% of the atoms to the negative spin states. Immediately following the

110
sweep, we ramp the magnetic field from 20 Gauss to 208 Gauss in 100 ms. At the end of the ramp,
we use another RF pulse of 100 µs at 47 MHz to create a 50/50 mixture of |F = 9/2, mF =and |F = 9/2, mF =9/2i

7/2i spin states.

We evaporate the atoms in the dipole trap by lowering the intensity of the horizontal trapping
beam. We exponentially lower the power of the beam from 2 W to 20 mW in 12 s.
The evaporation trajectory of a typical evaporation is shown in Figure 5.33. The slope↵ =Log T/ Log N = 1.8, which means the evaporation is efficient. At the end of the evaporation,
we reach about N = 4 ⇥ 104 and T = 100 nK, which corresponds to T /TF = 0.1. The thermometryof fermions at low temperature is described in the next section.

5.9

Thermometry
Thermometry of ultracold atoms can be done by imaging the atoms after a long time offlight. This measures the momentum distribution of the cloud and the average size(momentum)
corresponds to the temperature of the cloud. However, this technique only works for a thermal
cloud. Once the cloud is degenerate, the cloud is no longer Gaussian. For an in-depth thermometry
derivation for trapped Fermi gases, please see ref. [11].
The Hamiltonian for a single particle trapped in a harmonic potential is:
H=

1
m 2 2
p2x + p2y + p2z +!r x + !r2 y 2 + !z2 z 22m
2(5.29)

where !r and !z are the trap frequency in radial and axial direction respectively. The distribution
and the density of states of the particles isn(✏) =

1Z

1 exp(✏/k T ) + 1b✏2g(✏) =,(5.30),(5.31)

2(!z /!r )(~!r )3

where Z is the fugacity of the gas. The total number of atoms can be calculated by integrating thedensity of states to EF = kB TF , which gives
N=

EF3
.
6~3 !r2 !z(5.32)

111

Figure 5.33: Evaporation trajectory for the optical trap evaporation as we lower the horizontal
dipole trap intensity. We see that the slope of Log T/Log N is 1.8 which means the evaporationin the dipole trap is much better than that in the magnetic trap. At certain point, we see theevaporation saturates: we see a loss in number without seeing a loss in temperature. This isbecause for this evaporation we started with an imbalanced gas in mF = 9/2 and mF = 7/2. We
lost all of the minority mF = 7/2 atoms and only mF = 9/2 remain. To get to colder temperature,
we had to start with a 50/50 mixture of spin states.

112
Hence, the Fermi energy is
EF = ~ 6!r2 !z N
Now, we can integrate N =

R1
0

1/3

.

(5.33)

Li3 ( Z),

(5.34)g(✏)n(✏)d✏ to give

N=

1
2
!r !z

✓

kT T
~

◆3

where Li3 is the polylog function. By rearranging this equation and substituting TF for N, we get,Li3 ( Z) =1
6(T /TF )3(5.35)

Since we expand the cloud for time t and measure the column integrated profile. The 2D opticaldepth for a trapped Fermi gas is given by,
OD(y, z) =
Here,

⇣
⌘
m(kB T )2y 2 /(2 r2 )z 2 /(2 z2 )
p
p
Li
Ze
e
.
2
2 1 + (!r t)2 1 + (!z t)2 ⇡~3 !r

is the absorption cross-section, andkB T
2r = m!r2

1 + (!r t)2 andkB T
2z = m!z2(5.36)

1 + (!r t)2 are

the cloud size. For the fitting purpose, we reduce this equation to,
OD(y, z) = ODpk Li2

⇣

⌘
2
2
2
2
Ze y /(2 r ) e z /(2 z ) /Li2 ( Z)We fit a two dimensional image of atoms to this equation with fit parameters ODpk , Z,

(5.37)
y,z andbackground o↵sets. A typical fit of a degenerate Fermi gas to this distribution is shown in Figure
5.34.
By substituting Z to equation 5.35 and numerically solving, we can determine T /TF of thegas. We can then compare this T /TF to that found by finding T fromand TF from N and thetrapping frequencies. The comparison of these two fits at various temperature is shown in Figure
5.35.

5.10

Imaging
We can get the number of atoms by doing fluorescence imaging as described in Section 5.2.1.

However, this method does not allow us to measure the momentum distribution or temperature of

113

peakDepth
OD (mOD)
Op5cal(mOD)\\k−45\Data\2016\July\July 18\ScienceCell 29.spe
10
20
30
40
50
10

20

30

40

50

x cross−section
600
400
200
0
−200
0

20

40

60

x(pixel)y cross−section

Op5cal
Depth
(mOD)
peak
OD (mOD)

600
400

FD ﬁts:T/T = 0.15 (From fugacity)
T/TF =F 0.15
N = 7.49e+04 (From Gaussian fit)
N = 7.5 x 104

200
0
−200
0

20

40

60

y(pixel)Figure 5.34: An example of a 2D surface fit of a degenerate cloud to an FD fit in 5.37.

114

Figure 5.35: A comparison of the temperature of the gas measured in two di↵erent ways. The
x-axis is the T /TF extracted from the fugacity of the cloud using an FD fit. The y-axis is the
T /TF extracted by measuring the temperature from of the cloud (Equation 5.37) and the TF
from number and trapping frequency. The line has a slope of 1.

115
the gas. We thus have to perform absorption imaging after a long time of flight compared to thetrapping frequency. Absorption imaging works on the principle of Beer’s law. When we shine abeam on a cloud of atoms, the intensity of the beam as passes through the gas isI(z) = I0 e OD(z) ,(5.38)or the total optical depth of the cloud is,
OD = ln (I/I0 )(5.39)

Experimentally, we shine a 40µs pulse of the probe light on the atoms and take a picture of thebeam after it passes through the atoms on a CCD camera and label as shadow “S”. The intensityof the beam is kept to less than 10% of Isat to avoid further correction. We wait 200 ms and let
the atoms fall under gravity to the bottom of the cell. At that time, we pulse an identical pulse oflight and take another image and label as light “L”. We take a third picture without any probing
pulse and label as background “B”. The optical depth of the cloud is then,
OD(y, z) =

ln

✓

S
L

B
B

◆

(5.40)

We can thus do a surface Gaussian fit or a surface Fermi Dirac fit to find the temperature andnumber of the cloud. The imaging setup to do fluorescence and absorption imaging is show in Figure5.36. Because the probe beam has to go though a long transfer tube with a narrow diameter, wecannot do a long TOF in this chamber.
The imaging setup for the radial profile in the science cell is shown in Figure 5.37. The probetransition for the atoms are |F = 9/2, mF = 9/2i ! |F = 11/2, mF = 11/2i, and |F = 9/2, mF =7/2i ! |F = 11/2, mF = 9/2i for the positive spin states requiring astates, we use |F = 9/2, mF =|F =9/2i ! |F =+ light.

For negative spin

11/2, mF = 11/2i, and |F = 9/2, mF =7/2i !

9/2, mF = 7/2i.
We do spin resolved imaging by either using the Stern-Gerlach technique or by going tohigh-magnetic field such that the transition of the di↵erent spin states are much larger than thetransition linewidth. For the Stern-Gerlach method, we use a small coil placed underneath the cell

116

Pixelﬂycamera50/50
spliBer

F = 150

MOT Chamber #2

F = 150

Probe light

Security camera

Figure 5.36: Imaging setup for MOT # 2. We can do either absorption imaging or fluorescenceimaging with this setup. The setup has a magnification of 1. There’s also a 50/50 beam splitter
which allow us to look a the MOT fluorescence with a security camera.

Science Cell
Probe lightF = 125(AC254-125-B-ML)
F = 150(AC254-150-B-ML)

Princeton Instrument
ProEM 512
Figure 5.37: Imaging setup for imaging atoms in the science cell in the radial direction. The probe
light is sent in the same direction as the horizontal dipole trap.

117
which produces a gradient of 70 Gauss/cm. During the time of flight, we turn on this coil for 5 ms
such that atoms experience a force dependent on the magnetic moment. We do a usual absorption
imaging at low field to image various spin states which are spatially separated. An example ofsuch a measurement is shown in Figure 5.38. Note that we stitched two OD images together withdi↵erent probe polarizations as the transition for positive spin states require athat for negative requires a+ probe light and.

For high field imaging, we use the Feshbach coil as the bias field for imaging. Near 200G,
the transitions for mF = 9/2 and mF = 7/2 are separated by about 45 MHz. We send a linearly polarized beam to the atoms, thus the OD we measure for the atoms needs a factor of twocorrection(verified experimentally).

118

mF= 9/2
mF= 7/2
mF= 5/2
mF= 3/2
mF= 1/2
mF= -1/2
mF= -3/2
mF= -5/2
mF= -7/2
mF= -9/2

Figure 5.38: Stern Gerlach imaging of the atoms in di↵erent Zeeman state. Note that this image
consists of two OD images stitched together, one for positive spin states and another for negative
spin states.

Chapter 6
Conclusion and Future WorkIn this thesis, I presented the work on probing homogeneous Fermi gases using a LaguerreGaussian beam. We developed a technique to spatially select the center part of the cloud trappedin a harmonic trap by optically pumping the edge of the cloud to the states dark to the probe
light. As a proof-of-principle experiment, we applied this technique to a weakly interacting Fermi
gas and directly observed the Fermi surface in momentum distribution. Motivated by the successof this result, we combined the same technique with RF spectroscopy to study strongly interacting
Fermi gases. We locally measured the Tan’s contact as a function of temperature at unitarity andcompared to various many-body theories. We then measured the spectral function of the gas above
Tc across BCS-BEC crossover. This study was important to understand the pairing phenomena in
the normal phase of the strongly interacting Fermi gases as it is related to the pseudogap phasein high Tc superconductors. Because we were probing the homogeneous sample, we were able torigorously quantify our measurement and extract quantities like quasi-particle residue (Z), e↵ective
mass and Hartree energy shift.
For the second half of my thesis, I presented the work on the construction of the new generation Fermi gas apparatus. This apparatus was a significant upgrade from the old one. The
old apparatus, which was the first degenerate Fermi gas machine in the world, was undoubtedly
one of the best in the world. The amount of scientific results it produced has been remarkable.
However, after 17 years it was getting a little unstable and unreliable. With the advancement thathas happened in the last decade in optics and software, it surely needed a major upgrade. The

120main problem with the old apparatus was its poor optical access and a long cycle time of around 2.5
minutes. The lasers needed frequent relocking and realignment, some of the old electronics needed
to be replaced on weekly basis and the computer control and data acquisition software was runningon Windows XP. Keeping all these things in mind, we decided to build a new lab from scratch. We
designed a three-chamber setup to trap a large number of potassium atoms and at the same timeallow good optical access and a large numerical aperture in the science cell. The science cell in thisapparatus is AR-coated on both sides to allow lattice experiments. The double-MOT setup witha dark SPOT repump allowed us to have a large potassium atom numbers. We also implemented
the dark molasses cooling scheme with D1-transition of 40 K to cool the cloud below the Doppler
temperature before loading the magnetic quadrupole trap. This cooling stage enabled us to retain
a large potassium number while transferring atoms to the science cell. The QUIC trap, which has
only three coils, was constructed to provide a large optical access. The quadrupole coils used in
the QUIC trap also served as bias coil to reach large magnetic fields for Feshbach resonance. Afterevaporation in the QUIC trap and an optical dipole trap, we were able to get a degenerate Fermi
gas at a temperature of around 0.15TF with 50,000 atoms.
In terms of scientific goals, we planned to study strongly interacting quenched Fermi gases.
For example, we could have studied the dynamic Tan’s relation which connects the release energy of
Fermi gases before and after the quench to Tan’s contact. We could have also studied the dynamics
of Tan’s contact after the quench. This would have allowed us to get an insight into the timescaleof short-range correlation.
Further work on momentum-resolved RF spectroscopy is still needed to understand the phase
diagram completely. We had only measured the spectral as a function of interaction strength atfixed temperature. We had performed some initial experiments to look at the gas as a functionof temperature at various interaction strengths across the crossover. This would have also allowed
us to test our model over a wide range of the phase diagram. In another experiment, we had
also tried to develop a new technique to probe the “unoccupied” branch of the spectral function.
This technique, which required injecting atoms to strongly interacting states, was analogous to the121inverse-ARPES experiments in condensed matter and could have allowed us to measure the gap inthe superfluid phase. However, the signal-to-noise ratio we were getting was too poor to perform
any meaningful measurements and needed some improvement.
However, in the fall of 2016, due to the sad and unexpected demise of Debbie, some ofher projects needed to be consolidated. The Fermi gas experiment we built is now converted to astrongly interacting BEC experiment. The machine has a capability to study 39 K and 85 Rb stronglyinteracting BECs. The quenched 85 Rb BEC experiments have definitely excited both experimental and theoretical AMO communities, and the study of these systems seems quite important tounderstand various few-body and many-body phenomena. I look forward to seeing great scientificresearch from this machine in coming years.

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