Measuring local properties of a Fermi gas in the BCS-BEC
crossoverby
Tara E. Drake
B.S. Physics, University of Delaware, 2008
B.A. Music, University of Delaware, 2008

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
2015This thesis entitled:
Measuring local properties of a Fermi gas in the BCS-BEC crossover
written by Tara E. Drakehas been approved for the Department of Physics

Prof. Deborah Jin

Prof. Eric Cornell

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
Drake, Tara E. (PhD, Physics)
Measuring local properties of a Fermi gas in the BCS-BEC crossover
Thesis directed by Prof. Deborah Jin

This thesis presents experiments probing the physics of strongly interacting fermionic atoms
in the BCS-BEC crossover. Ultracold atom experiments bring the ability to arbitrarily tune interatomic interactions, which allows for unprecedented access to the regime of strongly interacting
physics. The majority of cold atom experiments, however, are carried out in an atom trap that imprints an inhomogeneous density on the cloud of atoms. Many phenomena, especially the signaturesof phase transitions, are significantly modified by this non-uniform density. In this thesis, I present
a novel imaging technique that allows us to probe a region of nearly homogeneous density within alarger, inhomogenenous cloud. Using this technique, I present new results for strongly interacting
fermionic atoms, including the first measurements of the contact and the occupied spectral function of a homogeneous Fermi gas, and the first direct observation of the “textbook” momentum
distribution of an ideal Fermi gas.

Dedication

For James, Marty, and Joe.

v

Acknowledgements

Nearly all of the work presented in this thesis was carried out by myself and two amazing
physicists, graduate student Rabin Paudel and post doc Yoav Sagi. Rabin is possibly the mosthardworking and the most patient person I know, and he has been an ideal co-worker. I truly could
ask for no one better, and it will be difficult to begin on another project without him. Yoav brought
his intelligence and inspiration to our group and kept us motivated through many difficulties. It
was his vision that guided much of the work presented here, and we could not have accomplishedwhat we did without him. I was incredibly lucky to have these two as labmates.
There have been several other graduate students with whom I have worked during my time
at JILA. John Gaebler and Jayson Stewart were the graduate students on my project when I
joined, and it was their idea that lead to our spatially selective imaging technique presented in thisthesis. Roman Chapurin has been working with us for the last three years and will be taking thefirst experiments with the new apparatus. Ruth Bloom and Phil Makotyn, who both joined the
Jin-Cornell group when I did, gave me their support and friendship during our mutual time here.
Debbie Jin has taught me a lot about being a scientist—how to get to the heart of a problem,
when data has promise and when to change your approach, how to communicate my ideas and how
to get others to communicate theirs. She has always pushed for the best I can give her, and I feelthat all aspects of my work and career are better for her guidance.
Eric Cornell often acts as a second advisor for Debbie’s students, and I am incredibly gratefulfor his presence. His understanding of complex phenomena in terms of simple physical pictures andhis unshakable kindness make him the kind of physicist and person I would like to be.

viMatt DeCamp was my advisor during undergrad and was the person to introduce me toexperimental physics. I owe him thanks for his patience and infectious enthusiasm and for greatly
influencing the decision to make physics my career.
Without Krista Beck, many of the graduate students in the Jin-Cornell group would be lost(or in some cases, stranded at DIA). She consistently lends us her help, from taking care of traveland scheduling to keeping us fed and caffeinated.
Lastly and never least, the electronic shop, computer, and machine shop staff at JILA are thelife support system of the experiments here. I feel they deserve particular credit in the case of mylab, which seemed to need emergency resuscitation more than most. They saw us through a disasterinvolving the complete reconstruction of our electronics systems (special thanks to James Fung-AFat!), a mess of computer issues (thanks JR Raith and Jim McKown!), and finally the disassemblyof our old apparatus and the construction of a new one (thanks especially to Hans Green, whohelped us make our new sources, set up the new experiment, and coordinate our move!). Thanks
also go to Beth Kroger, who, regardless of whether or not it was her responsibility, was always
ready to solve our problems and go to bat for us.

Contents

Chapter
1 Introduction
1.1

1.2

1.3

1

Strongly correlated fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.1

Context: the nature of matter . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.2

Interacting fermions in the wild . . . . . . . . . . . . . . . . . . . . . . . . . .

3Studying strong correlations in Fermi gases . . . . . . . . . . . . . . . . . . . . . . .

4

1.2.1

The BCS-BEC crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2.2

Our system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.2.3

Density inhomogeneity: an advantage or a disadvantage? . . . . . . . . . . .

9

1.2.4

Imaging from a uniform density sample . . . . . . . . . . . . . . . . . . . . . 12

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Spatially selective imaging

14

2.1

Laguerre-Gaussian mode beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2

Experimental setup of LG mode beams

2.3

Optical pumping transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4

Fitting the LG mode beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5

Aligning the LG beams onto the atom cloud . . . . . . . . . . . . . . . . . . . . . . . 22

2.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

. . . . . . . . . . . . . . . . . . . . . . . . . 15

viii
3 The ideal homogeneous gas: uncovering the Fermi step function

26

3.1

The momentum distribution of a homogeneous Fermi gas . . . . . . . . . . . . . . . 26

3.2

Experiment sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3

Seeing the Fermi step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4

3.5

3.3.1

Approximating our results as “homogeneous” . . . . . . . . . . . . . . . . . . 30

3.3.2

How homogeneous is “homogeneous enough”? . . . . . . . . . . . . . . . . . . 32

Modeling the signal removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4.1

The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4.2

Collisions in time of flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4.3

Effect of finite branching ratio: the return of hot atoms . . . . . . . . . . . . 40

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Contact of a homogeneous gas
4.1

4.2

4.3

What is the contact? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1.1

Tan’s definition: the high momentum tail and the Tan relations . . . . . . . . 46

4.1.2

An “intuitive interpretation” . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

The contact of a homogeneous gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.1

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.2

Extracting the contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Local contact vs. (local) T/TF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.1

4.4

4.5

Obtaining the in-situ density distribution . . . . . . . . . . . . . . . . . . . . 58

Modeling our results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4.1

The effect of the remaining density inhomogeneity . . . . . . . . . . . . . . . 63

4.4.2

The optical pumping model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5 The occupied spectral function of a homogeneous gas in the normal phase
5.1

45

67

The normal state of the BCS-BEC crossover . . . . . . . . . . . . . . . . . . . . . . . 68

ix
5.2

5.3

5.4

Photoemission spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2.1

The experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2.2

The PES signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

The two component fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3.1

The quasiparticle signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3.2

The “incoherent background” . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Results of the fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4.1

Comparison to the Fermi polaron case . . . . . . . . . . . . . . . . . . . . . . 86

5.5

The homogeneous contact vs. (kF a)−1 . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.6

Understanding the “incoherent background” . . . . . . . . . . . . . . . . . . . . . . . 89

5.7

5.6.1

Iincoherent as large pairs in the crossover . . . . . . . . . . . . . . . . . . . . . 90

5.6.2

Iincoherent as part of a Fermi liquid . . . . . . . . . . . . . . . . . . . . . . . . 92

5.6.3

Iincoherent and the Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6 Photoemission spectroscopy as a function of temperature

94

6.1

Temperature in the normal phase of atoms in the BCS-BEC crossover . . . . . . . . 94

6.2

The contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.3

The occupied spectral function as a function of temperature . . . . . . . . . . . . . . 97

6.4

Fitting to our two-component function . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7 Conclusions

106

7.1

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.2

Future Possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

xBibliography

109

Appendix
A Thermometry117

B Scaling the number of atoms in PES data120

C Comparing PES of a homogeneous sample to trap-averaged data

123

C.1 Back-bending of EDCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
D Derivation of the single-particle spectral function for atoms bound in thermal pairs

126Chapter 1

Introduction

This thesis tells the story of recent work at JILA in the field of ultracold Fermi gases. The
ability to create and study degenerate Fermi gases of atoms is less than two decades old, andin that time research in this area has grown from a small offshoot of the larger field of BoseEinstein condensates to a huge community comprised of many groups around the world. Studying
these gases gives us important insights into the basic quantum nature of matter, and the precisionand controllability offered by ultracold gas experiments makes them very well suited for studying
fundamental questions. In particular, the ability to arbitrarily tune interactions between particles
allows unprecedented access to the regime of strongly correlated physics.
Even after a decade of research using ultracold strongly interacting Fermi gases and nearly acentury of theoretical and experimental investigations of strongly correlated matter, there are basic
questions about the nature of systems of strongly interacting particles that remain unresolved.
This is due in part to the incredible richness and complexity of strongly correlated materials. In
addition, many strongly correlated systems are notoriously difficult to control or access; a neutronstar, for example, has this problem. Moreover, theoretical description of such complicated systems
is a very challenging undertaking.
The community of ultracold Fermi gas experiments attempting to answer these questions hasgrown exponentially in the last few years and now includes more than 50 experiments worldwide.
When I joined the JILA group in 2008, the technique of rf photoemission spectroscopy in ultracold
atoms had just been developed [1]. The next year, Tan’s Contact was introduced in the context of

2
ultracold atoms [2, 3], and in the year after, the equation of state of a strongly interacting Fermi
gas was first measured [4, 5]. In the intervening time, new experiments around the world starting
probing Fermi gases in novel geometries and using exotic atomic species [6, 7, 8]. This thesis covers
a very small part of the progress in understanding the fundamental physics of strongly interacting
fermions.

1.1

Strongly correlated fermions

1.1.1

Context: the nature of matterAll known particles can be divided into two groups based on their intrinsic total spin. Particles
with integer spin (including zero spin) are classified as bosons and are generally the carriers of forces;
they include the photon, which carries the electromagnetic force, the phonon, which quantizes
motion, and the Higgs boson, which is linked to mass and the gravitational force. Particles with halfinteger spin are classified as fermions. Electrons, protons, and neutrons, which are the constituentsof visible matter, are all fermions.
The basic behavioral difference between fermions and bosons comes from the symmetry of
their respective wavefunctions and is most easily seen in the character of their respective groundstates. Identical bosons have wavefunctions that are symmetric under two-particle exchange, meaning that the total wavefunction will not change if any two bosons swap states. This has the effectthat, at very low temperature, all bosons in a system of identical bosons will occupy the same energy state, which results in a “macroscopic wavefunction” of many particles called a Bose-Einsteincondensate (BEC) (shown in figure 1.1a). Identical fermions, however, must form wavefunctions
that are antisymmetric under exchange, and therefore no two identical fermions can share an energy state. At zero temperature, N identical fermions will occupy the N lowest energy states of thesystem, one fermion per state. The energy of the highest occupied state is called the Fermi energy(EF ) (shown in figure 1.1b). This fundamental difference between bosons and fermions underlies
and explains many familiar physical phenomena—for instance, why electrons (fermions) in atoms

3(a) Bosons(b) Fermions

Figure 1.1: The zero temperature ground states of bosons and fermions in a onedimensional harmonic potential. (a)The ground state of identical bosons is a macroscopicallyoccupied single-particle wavefunction—a BEC. (b)N identical fermions in their ground state will
occupy the N lowest energy states, one per state. Only distinguishable fermions can occupy thesame state at the same time.

occupy multiple energy orbitals, resulting in the diversity of elements found in the periodic table,
and why photons (bosons) can form massively occupied coherent states, as seen in lasers.

1.1.2

Interacting fermions in the wild

While the physics of non-interacting (ideal) fermions is well understood, “exotic” matterwith interesting and useful properties often consists of fermions with very strong interparticle
interactions. Strongly interacting fermions are crucial to our understanding of such phenomena
as high temperature superconductivity, Landau Fermi liquids, and certain novel photovoltaics [9].
Fermions with very strong interactions are also the constituents of astrophysical phenomena suchas neutron stars, the high density remnants of supernovae, and the quark-gluon plasma, which isthe predicted state of matter in the universe after the Big Bang and just before the formation of
protons and neutrons.
The direct study of many of these exotic systems is very challenging. Neutron stars are
remote and not subject to scientific manipulation. Quark-gluon plasmas have been achieved in
the Brookhaven Relativistic Heavy Ion Collider (RHIC) [10], but the short lifetime of the plasma
and the lack of many available probes present challenges to understanding these plasmas. Evenhigh–TC superconductors, for which a large pool of experimental data has been collected since their

4
discovery in 1986, are very complex materials with multiple and possibly competing phases andnon-trivial geometry, which makes identifying the underlying physics difficult [11, 12].
Theoretical description of fermions with very strong interactions also faces many challenges.
While perturbative approaches have been shown to work very well for weak interparticle interactions, sufficiently strong interactions (as in the above phenomena) require a non-perturbative
many-body approach. In some cases, a phenomenological theory that maps the complex, stronglyinteracting system onto a simpler system can be successful, such as Landau’s Fermi liquid theory. However, in many cases, even a phenomenological understanding is elusive. For this reason,
experimental measurements are very important to benchmark and guide theoretical progress.
The challenges and limitations listed above explain the need for a simple, controllable experimental system in which to investigate the behavior of strongly interacting fermions. At first,
the study of cold, dilute gases of atoms may seem unrelated to the big, unanswered questions ofhigh-TC superconductivity and quark-gluon plasmas. It is true in many respects that relatively
simple atomic physics experiments are not sufficient for gaining a complete understanding of thesemuch more complex phenomena. However, the basic question underpinning these systems—how
does an ensemble of fermions with very strong interactions behave?—remains open. Despite thesimplicity and fundamental importance of the question, the answer is still a hotly contested area of
debate. Experimental studies of ultracold Fermi gases are well positioned to answer this question.

1.2

Studying strong correlations in Fermi gases
The experimental realization of an ultracold Fermi gas occurred in 1999 [13], six years after

the first BECs were realized in ultracold Bose gases [14, 15, 16]. Around the same time, thediscovery of Fano-Feshbach resonances in atomic gases [17] added the ability to arbitrarily tune
the strength and sign of the interactions between atoms. For a history of these developments inthe field and particularly at JILA, one should read the PhD theses of Brian DeMarco and Cindy
Regal [18, 19]. These two achievements (cooling to degeneracy and controlling interactions) were
the two necessary ingredients for creating a strongly correlated, superfluid Fermi gas. The theory

5
that describes such a system is called the BCS-BEC crossover [20, 21, 22].

1.2.1

The BCS-BEC crossover

The ground state of interacting fermions is very different from the non-interacting case. Imagine an ensemble of fermions with an equal population of two spin states and with some interaction
between fermions in the two spin states. In 1957, John Bardeen, Leon Cooper, and Bob Schrieffer(“BCS”) showed that for such a system at zero temperature, even very weak attractive interactions
would lead to pairing of the fermions about the Fermi surface [23]. These pairs, called, “Cooper
pairs”, are generally very large and loosely bound. However, their total spin is an integer value,
and, being effective bosons, the pairs will condense to form a superfluid. This is the basic idea
behind conventional, or BCS, superfluidity.
Starting with a BCS superfluid, imagine increasing the strength of the interactions between
the two spin states. The Fermi surface will become broader, and more fermions will participate in
the pairing. The pairs become more and more tightly bound, such that the pairs are more localized.
For very strong interactions, the pair will become a tightly bound bosonic molecule, and the groundstate of the system will be a superfluid Bose-Einstein condensate (BEC).
The BCS-BEC crossover unites these two types of superfluidity. As expressed above, theimportant variable for moving from a BCS superfluid of Cooper pairs to a molecular BEC is thestrength of the interparticle interactions, which can be parameterized by a two-particle scattering
length, a. Figure 1.2 shows the phase diagram of the BCS-BEC crossover. For all interaction
strengths, the zero temperature ground state of the system is a superfluid of paired fermions. In
essence, the crossover tells us that the ground state of interacting fermions is always a superfluid
condensate—just like bosons!
The phase transition to a superfluid in the BCS-BEC crossover was first experimentallydemonstrated in 2004 by Debbie Jin’s group at JILA and again in the same year by Wolfgang
Ketterle’s group at MIT [25, 26]. Scientists working with 40 K at JILA and 6 Li at MIT were able tocool mixtures of two spins states of their atoms while simultaneously turning on strong interactions

6

Figure 1.2: Phase diagram of the BCS-BEC crossover, reproduced from Ref. [24]. In
the leftmost (BCS) limit, the particles have a weak attractive interaction and form a superfluid of
Cooper pairs below the critical temperature, TC . In the rightmost (BEC) limit, bosonic dimers
with large binding energies form above TC and form a Bose-Einstein condensate below TC . Tpair isa theoretical boundary delineating a crossover from a region of the normal phase that has pairs toa region without. The dashed µ = 0 line shows the expected point at which the chemical potential
crosses from positive (on the BCS side) to negative (as it is in the BEC limit).

between atoms in different spin states. An essential step forward for accessing the crossover regime
was the discovery of magnetic Fano-Feshbach resonances, which allow experimentalists to use an
external magnetic field to tune the scattering length that characterizes interparticle interactions toarbitrarily large values. Using these resonances, both groups were able to measure the expectedsuperfluid phase diagram of the crossover (figure 1.3), and researchers at MIT further demonstrated
superfluidity by rotating the gas and imaging vortices (figure 1.4).
The physics of the BCS-BEC crossover is universal, and should be observed in any groupof strongly interacting fermions, regardless of the type of atom, or even whether the fermion issubatomic, as in the case of a neutron star, a quark-gluon plasma, or an electron in a crystal

7

Figure 1.3: Condensate fraction throughout the BCS-BEC crossover. The contour plotshows the fraction of condensed atoms at various interaction strengths and temperatures. In thisplot, the BCS limit is on the far right and the BEC limit is on the far left. (T /TF )0 and kF0 weretaken from the initial conditions of the weakly interacting gas. Figure reproduced from Ref. [19].

Figure 1.4: Superfluidity seen throughout the crossover. Reproduced from Ref. [27].

8lattice 1 . The central region of the phase diagram, where −1 & (kF a)−1 & 1, has the property
that the two-particle scattering length, a, is as large as or larger than the average interparticle
spacing, (kF )−1 . (kF is the Fermi wavevector and is related to the Fermi energy by EF =

2
~2 kF
2m ,

where m is the atomic mass and ~ is Planck’s constant.) In this region, the gas behaves much
more like a liquid, exhibiting hydrodynamic flow and other collective effects of strong interactions.
Gases at these scattering lengths are said to be in the “crossover regime.” In this regime, the twobody scattering length is no longer sufficient to characterize the particles’ behavior, and as the gas
approaches a = ∞ (“unitarity”) and zero temperature, the only relevant length scale remainingis the interparticle spacing. This regime of interactions is the most interesting for us to access,not only due to the universal behavior, but because these are the interaction strengths at which
perturbation theory breaks down, and the behavior of the gas is difficult to predict. The crossover
regime provides us the greatest chance to observe new physics and guide new theoretical description.

1.2.2

Our system

While the advent of hydrodynamic Fermi gases in the unitary regime has caused the originalFermi gas community to grow to include more exotic geometries and conditions, our lab has mainly
stuck to the recipe originally set by the first JILA crossover studies—a gas of 40 K atoms in an equal
mixture of two spin states with a Fano-Feshbach resonance controlling interactions. The clouds
generally consist of 5–30 ∗ 104 atoms at 0.1–0.5TF . The atoms are trapped in a 3D geometry usinga crossed-beam optical dipole trap, and the final cloud has a cigar shape with a roughly 1:10 aspectratio.
In order to cool the atoms to quantum degeneracy, we use a series of trapping potentials andcooling techniques. We first gather 40 K atoms in a magneto-optical trap using the LVIS loading
technique developed at JILA [28]. We then load the atoms into a Ioffe-Pritchard magnetic trap in
a cloverleaf configuration. We evaporatively cool in this trap by using a single microwave frequency
1

The key assumption for this universality is that the scattering length, a, is much larger than the range of thephysical interaction. For atoms, this requirement means that a should be much larger than the van der Waals length,
r0 . For 40 K, r0 = 60 Bohr radii, which is much smaller than a at the interaction strengths we investigate.

9
to remove the hottest atoms from the trap and letting the remaining atoms re-thermalize. Whenthe atom gas temperature has reached about 6 µK, we load the atoms into the optical dipole trap,
where the final evaporation occurs. The final temperatures reached are generally at or below 100
nK. Further details about the construction of the apparatus and about the magnetic trap are foundin Brian DeMarcos thesis [18].
During the final stages of cooling in the dipole trap, we apply a uniform bias magnetic
field to the atoms to bring them close to a Fano-Feshbach resonance. This both allows us to
improve evaporation at the coldest temperatures and to set the interactions between the atoms
when we perform our science, which gives us access to the crossover regime. The resonance for 40 K
occurs at 202.2 G. It enhances interactions between two Zeeman states, |F, mF i = |9/2, −9/2i and|F, mF i = |9/2, −/2i, where F is the total atomic spin, F = 9/2 is the lowest hyperfine state, andmF is the projection onto the axis of the magnetic bias field. Loading into an optical dipole trap
(as opposed to staying in a magnetic trap) allows us to prepare our atoms in these two spin states,
which are not magnetically trappable. For more details about our optical dipole trap and the 40 K
Fano-Feshbach resonance, see the theses of Cindy Regal and Jayson Stewart [19, 29].

1.2.3

Density inhomogeneity: an advantage or a disadvantage?

Confining atoms in a potential well is a necessary step to carrying out BCS-BEC crossoverexperiments (and many others types of atomic physics experiments as well). The most commonlyused atom traps, whether magnetic, optical, or both, create a potential that approximates that ofa harmonic oscillator. These bowl-shaped potentials allow for the high densities (1012 atoms per
cm3 —“high” for a quantum gas) needed for the quantum degenerate regime. In such a potential,
the spatial extent of the trapped gas depends on its total energy, with more energetic atoms
traveling farther from the center of the trap. For a Fermi gas, even a T=0 cloud will have as manyoccupied energy states as there are atoms, due to Fermi statistics. This means that the atoms will
be distributed non-uniformly, with the highest density of atoms at the trap center and the lowest
densities at the edges of the bowl. This spread of densities is always present in trapped atom

10
experiments to some extent.2
This density inhomogeneity has important consequences. The energy scale of Fermi gases is
EF , the Fermi energy. This energy is directly determined by the density:

EF =

~2
(6π 2 n)2m

(1.1)

where m is the mass of the atom, ~ is Planck’s constant, and n is the number density ofatoms in a single spin state. Several other quantities depend on the Fermi energy in turn:

kF =

√

TF =

2mEF
~2

(1.2)

EF
kB

(1.3)

where kF and TF are the Fermi wavevector and temperature, respectively, and kB is theBoltzmann constant.
Since all of these quantities depend upon the local density, they vary across the atom cloud.
In a harmonic trap, for example, the condition T /TF = TC (where TC is the critical temperature of
the superfluid phase transition) will only ever be true for a single ellipsoidal, equipotential surface.
The atoms enclosed by that surface will be below the transition temperature, and the atoms outside
will be above.
In some cases, the spread of densities in a harmonically trapped gas can be an advantage.
From one spatially resolved image of the atom cloud, one can extract measurements at many
densities. This has been particularly useful in recent measurements of the equation of state [4, 35,
5, 36].
Some measurements, however, cannot differentiate between signal that comes from the more
dense center of the trap or the relatively dilute edges. Measurements done after the trapping
2
In 2013, a group in Cambridge trapped a BEC in a uniform optical potential, or “box trap”. This techniquesolved the problem of density inhomogeneity and has enabled the group to measure the dynamics of the BEC phasetransition with great accuracy. A box trap for fermionic Li6 atoms has very recently been demonstrated at MIT in
the group of Martin Zwierlein. [30, 31, 32, 33, 34]

11

Figure 1.5: Momentum distributions of weakly interacting fermions. Images of fermionic
atoms taken after a time of flight expansion from a harmonic trap reveal the momentum distribution.
The distribution of momenta can be fit to determine T /TF . The dashed line is the expectedBoltzmann distribution for the same number of atoms. Note the lack of a sharp Fermi surface,even for the coldest clouds. Figure reproduced from Ref. [37].

potential has been turned off and the atoms have expanded for some time of flight are of this type.
In this case, the quantity measured will reflect a sum over all the densities of the trap. If a sharp
signal is expected at the transition temperature or at the Fermi surface, this “trap averaging” will
broaden that signal, and even completely wash it out in some cases.
This effect of trap averaging has been present since the first measurements of a degenerate
Fermi gas. Figure 1.5 shows the first evidence of Fermi degeneracy in 40 K. The expected momentum
distribution for a non-interacting Fermi gas in a harmonic trap can be analytically determined asa function of T /TF , and the data can be fit to determine the temperature. However, the spreadin kF throughout the cloud means that, instead of a sharp step the Fermi momentum, ~kF , thatgets sharper with decreasing temperature, even very cold clouds look nearly Gaussian in shape.
In addition, while the momentum distribution of a non-interacting gas can be calculated, themomentum distribution of a Fermi gas with interactions is still an open question. In the case of
the strongly interacting Fermi gas, recovering features washed out by trap averaging can prove
impossible.

12
1.2.4

Imaging from a uniform density sample

The experiments within this thesis all have one thing in common. In each, we have taken
pains to eliminate the effects of trap averaging and extract a signal that, to a good approximation,
comes from a single density.
We do this not by creating a cloud with a single density, but by altering our measurement suchthat we only measure a small portion of the cloud at a time. This spatial selectivity is typically
performed immediately after the atoms are released from the trap and before they have had achance to expand in time of flight. In this way, our technique allows us to perform measurementsof the momentum distribution of the atoms while retaining the knowledge of where in the trap thesignal came from.

1.3

Contents
In this thesis, I will explain the work our group has done since developing spatially selective

imaging. In some sense, the experiments presented here have largely already been carried out
without the benefit of spatial selectivity. For each experiment, however, I will attempt to show how
measuring a single density has enhanced our understanding of the physics of interacting fermionsand uncovered new behavior that had been buried by trap-averaging.
In chapter two, I will present the spatially selective imaging technique. In chapter three, I
will present the first experiments using spatially selective imaging, in which we probed an ideal
Fermi gas and saw, for the first time, a sharp step in the momentum distribution of the gas. This
chapter will also contain much of the work the we did to characterize this technique, which relied
upon our knowledge of the form of the momentum distribution of a non-interacting gas. Chapter
four presents measurements of the contact of a homogeneous Fermi gas at unitarity, including a
few interpretations of the contact and why probing a single density is of particular importance forthis measurement.
The next two chapters will focus on measurements of the occupied spectral function of a13strongly interacting Fermi gas using atom photoemission spectroscopy, which has been the largest
focus of my PhD work. In chapter five, I will present the work we’ve done investigating the normal
phase of the gas in the BCS-BEC crossover by preparing a gas just above TC and varying theinteraction strength. This chapter will also include a small introduction to crossover physics and a
description of the interest and controversy surrounding the character of the normal phase. Chapter
six will discuss our preliminary investigations focusing on how the spectral function of a normal
phase gas changes with temperature.

Chapter 2

Spatially selective imaging

In this chapter, I present the details of our spatially selective imaging technique. This technique uses two Laguerre-Gaussian (LG) mode beams created from light tuned to an optical pumping
transition. These ring-shaped beams are focused on our atoms along the z (axial) and y (vertical)
directions and are carefully aligned such that they optically pump atoms from the low density
regions at the edge of the cloud. The signal in our images comes from the atoms that were notoptically pumped. These were at higher and more uniform density and can be approximated as
coming from a homogeneous density gas. This technique is used in all the experiments that makeup the main body of this thesis.

2.1

Laguerre-Gaussian mode beams
Laguerre-Gaussian spatial mode beams have recently been of interest to the ultracold atom

community for their unique intensity and phase patterns. These beams have an intensity of zero attheir center and are often called “donut” mode beams for their ring-shaped cross section. (Aroundthe lab, we refer to our spatially selective imaging as the “donut beam technique”, and in this thesis,
I’ll use these interchangeably.) This shape comes from a non-zero orbital angular momentum l~,which is separate from the spin (polarization) of the photons and which results in a phase windingaround the donut.
At the focus, the light intensity of a Laguerre-Gaussian mode beam with angular momentum

15
index, l, is given by
2 P
I(r) =
πl! w2



2r2
w2

l

2r 2

e− w2

(2.1)

where P is the total optical power, r is the radial coordinate transverse to the direction of beam
propagation, and w is the waist. (LG beams can also have more than one radial node, which results
in a spatial mode with several concentric rings. The number of radial nodes is given by a radialindex, p. In equation 2.1, p = 0, and the beam profile has only one ring.) LG mode beams can begenerated a variety of ways, depending upon the desired efficiency and mode purity [38, 39, 40, 41,30].
We found that printing an absorptive diffraction pattern in chromium onto a glass slide was acheap and quick solution that produced acceptable beams for our purpose. Figure 2.1 shows 3 suchgratings and images of the resulting beams. The gratings show 1, 2, or 3 forked dislocations in thecenter of the pattern, which produce beams with an angular index of l = 1, 2, or 3, respectively.
After printing, the width of a black line in each grating is 75 µm, and the total pattern extends overa 1 inch circular slide. A collimated laser beam, which is much smaller than the slide, is centeredon the dislocation, and the diffracted order is imaged (bottom row in figure 2.1). 1

2.2

Experimental setup of LG mode beams
Our spatially selective imaging technique involves the use of two donut-mode beams carefully

focused and aligned onto our atoms. These beams are each created with a grating with angularindex of l = 2. One beam is sent along the vertical (y) axis and another along the horizontal
axis along the long direction of the optical trap (z-axis) (see figure 2.2). Both donut-mode beams
propagate along the same axes as the 1064 nm optical trapping beams, and they are focused onto
the atoms using the same lenses that focus the trapping beams. Figure 2.3 shows the path of thehorizontal donut beam as it is first collimated to a 5.5 mm diameter waist, sent through the grating,1
It is important to note that, for our purposes, we care only about the intensity of the LG mode beam nearits center—specifically that the intensity in the center reaches zero and that the “inner walls” of the beam follow
equation 2.1. We do not care much about the phase properties or the mode purity beyond this, and one can see in
figure 2.1 that we generate a small amount of intensity in p 6= 0 modes, leading to faint outer rings around the main
ring.

16

l=1

l=2

l=3

Figure 2.1: Forked diffraction gratings. These gratings have central dislocations that determine
the angular index, l, of the Laguerre-Gaussian beam they create. The patterns on the top row wereprinted on glass slides, and the bottom row shows the resulting far field diffraction for each. Notethat the LG beams with higher l have larger central holes with sharper inner edges.

reflected off a mirror with motorized control, and combined with the horizontal optical trap pathusing a dichroic slide. This beam is focused onto our atoms using a 200 mm lens. The vertical
donut beam is similarly collimated, sent through a grating, focused on the atoms using a 250 mm
lens, and then combined with the vertical optical trapping beam via a polarizing beamsplitter cube.
The vertical donut beam is aligned onto the atoms by hand rather than by motorized micrometers,
but we have found that the alignment of this beam is stable for months, while the horizontal donut
beam must have its alignment checked before and after each data set. (A typical data set takes 2.5hours.)

17

vertical
beam

atoms
y

horizontal
beam

x
z

Figure 2.2: Illustration of the two intersecting hollow light beams used to opticallypump atoms at the edges of the trapped atom cloud.

2.3

Optical pumping transitions
The spatially selective imaging technique works by optically pumping the edges of the atom

cloud into a state invisible to our imaging such that only the central atoms at relatively uniform
density are imaged.2

The atoms that are optically pumped are shelved in the F = 7/2 hyperfine

manifold during imaging, and only the atoms that remain in their original state in the F = 9/2
manifold are imaged. We have two optical pumping transitions that we generally use, dependingon the initial state of the atoms and the choice of experiment.
Figure 2.4 shows the two optical pumping transitions utilized in this thesis. Transition a,used for the Fermi surface and homogeneous contact experiments (chapters 3 and 4), pumps atoms
from the |F, mF i = |9/2, −7/2i hyperfine ground state to the |F 0 , mF i = |5/2, −5/2i excited state.
Transition b is used for photoemission spectroscopy experiments (chapters 5 and 6) and pumps
2

The technique of using shaped optical pumping beams to select atoms for imaging has been used previously. For
an example, see Ref. [42].

18

atoms

2w = 5.5 mm

New Focus
Picomotor
mirror mountFigure 2.3: Setup of the horizontal donut beam. The horizontal donut-mode beam (propagating along the z-direction) is collimated to a 5.5 mm diameter before it is sent through a forked
diffraction grating that gives the beam an l = 2 Laguerre-Gaussian mode in the far field.

F

0

= 25

mF 0 = − 25

optical pumping
transition a

mF 0 = − 23

42 P3/2

optical pumping
transition b

F = 27
F = 92mF = − 92

mF = − 27

mF = − 52

42 S1/2

Figure 2.4: Optical pumping transitions. The straight lines indicate the two optical pumping transitions used in the donut beam technique. The squiggly lines indicate the most likely
spontaneous decay path of an optically pumped atom.

19
atoms from the |F, mF i = |9/2, −5/2i hyperfine ground state to the |F 0 , mF i = |5/2, −3/2i excitedstate. Further characterization of the details these two transitions, including the chance of an atom
to return to the imaging state and the possibility of colliding with an optically pumped atom, areinvestigated at the end of chapter 3.

2.4

Fitting the LG mode beams

intensity(arb. units)
1

0.5
0
90 µm
Figure 2.5: Image of the horizontal donut beam at the atoms.

As part of characterizing our LG beams, we image the beams at their focus, where they
overlap with the atom cloud. The horizontal LG beam propagates along an imaging axis, so it iseasy for us to image the beam at the atoms using the camera and imaging system already in place.
Figure 2.5 shows a 2D cross section of the horizontal LG beam at the atoms. By fitting to find thecenter of the beam and finding the distance of each pixel from that center, we construct a profileof the intensity, I(r), which can be fit to equation 2.1 (see figure 2.6). From this fit, we see thatthe horizontal LG beam is well approximated by an l = 2 mode beam with a waist of w = 16.8µm.
We also found the waist of the donut beam propagating in the vertical direction. Unlike
the horizontal donut beam, the vertical donut beam removes signal along only the one axis (thez-axis, along which the cloud is longest). In other words, the vertical donut can be approximated20

Intensity (arb. units)

·104
1

0

0

10
20
30
radial distance (µm)

Figure 2.6: Fitting the horizontal donut beam.

by a one dimensional intensity profile, I(z), with two peaks of high intensity and a center going tozero intensity. (The horizontal beam is, in contrast, modeled as I(x,y).) We imaged the focus of
the vertical donut beam using a beam profiling camera from DataRay Inc. (figure 2.7). Noticingthat the beam was slightly elliptical and wanting to know the I(z) seen by the atoms (which wasdifficult to determine from our setup), we found the principal axes of the beam and plotted theperpendicular cross sections. Figure 2.8 shows the two perpendicular cross sections of the vertical
donut beam. Both cross sections were fit to an l = 2 mode beam, and they returned waists of
w = 186µm and w = 151µm, giving us lower and upper limits for the size of the vertical donut at
the atoms.
We then decided to use our atoms themselves to measure the vertical donut size. Looking
along the x-direction (perpendicular to the long direction of the trap), we took an image of theatoms in the trap with and without removing signal using the vertical donut (figure 2.9a). Summingalong the y-direction, we calculated the removal as a function of z:
Fraction of atoms not optically pumped(z) =Atoms imaged after pumping(z)Atoms imaged before pumping(z)

(2.2)

21

Figure 2.7: Image of the vertical donut beam. The cross hairs indicate the principal axes of
the image and correspond to the two cross sections shown in figure 2.8. The line which is closer tohorizontal corresponds to figure 2.8b, which we found the oriented along the z axis of the trap. The
other cross hair is along the x axis. As z is the elongated axis of our cigar-shaped trap, and as thevertical donut beam has a waist of 186 µm, this beam only removes signal along the z-direction.
This image was captured using a DataRay beam profiling camera and software.

The fraction of atoms left in the signal by the vertical donut beam is shown in figure 2.9b.
To determine the size from figure 2.9, we used a very simple model of optical pumping by the
donut beam. In this model, we assumed that the probability for an atom to be optically pumped
was proportional to exp[−I(z)], where I(z) is the intensity of the vertical donut along z, given byequation 2.1. The solid black line in figure 2.9b shows the expected fraction of atoms that remainin the imaging state based on this model. For this calculation, we use the measured profile of theatoms without optical pumping (figure 2.9) and subject them to removal by an l = 2 LG beamwith a 186 µm waist. We see that this simple model reproduces the probability of the atoms toremain in the imaging state quite well, apart from a slight asymmetry on the right side of figure
2.9b. We believe that this asymmetry comes from a similar asymmetry in the vertical donut beam,
seen in the inset of figure 2.8b. We do not expect a feature of this size to have much effect on the
final measurements of the gas using the donut technique.

22Int. (a.u.)

Intensity (arb. unit)

a
8,000
6,000

0
−100 0100

z (µm)

4,000
2,000
0
−800

−400

0

400

800

z (µm)

Int. (a.u.)

Intensity (arb. unit)

b
8,000

0
−100 06,000

100

z (µm)

4,000
2,000
0
−800

−400

0

400

800

z (µm)

Figure 2.8: Cross sections of the vertical donut beam. These cross sections correspond to thecrosshairs in figure 2.7. The insets zoom in on the zero intensity center of the beam. The inset of
a was fit to an l = 2 LG beam with waist w = 151µm, and the inset of b fits to an l = 2 LG beamwith waist w = 186µm. From our measured removal of signal in our atom cloud, we find that theatoms see an intensity profile closer to b (see figure2.9).

2.5

Aligning the LG beams onto the atom cloud
Properly aligning the donut-mode beams onto the atom cloud has two stages. For roughalignment, we image the cloud of atoms after optically pumping away signal with only one beam tomake sure that the center of the cloud is in the same position and has the same optical depth (OD)
as it does before signal is removed. As the donut beam is moved across the cloud, the number ofwithout removalwith 27% removal

800
600
400
200
0

0

100

200
z (µm)b
Fraction of atoms imagedaIntegrated density (arb. units)

23

300

1

0.5

0

0

100

200
z (µm)

300

Figure 2.9: The removal of signal by the vertical donut beam. a. Density profiles before andafter application of the vertical donut beam. Note that the number of atoms imaged in the center
is unchanged, with signal only being removed from the cloud’s edges. This shows good alignmentof the donut beam onto the atoms. b. The fraction atoms remaining in the signal is shown as afunction of z (dots). This is compared to a very simple model predicting the amount of removalfrom an l = 2 LG beam with a waist of 186 µm (black solid line). This model fits the data well.
The slight asymmetry in the removal seen on the upper right side of the data comes from a slightasymmetry in the donut beam profile itself, just visible in figure 2.8b. This is further evidence
that the cross section fit by a 186 µm waist corresponds to the correct intensity profile seen by ouratoms.

Number

80,00040,000

0

−150

0

150

knob position (arb.)

Figure 2.10: Moving the donut beam across the atoms. Here, we shift the position of thehorizontal donut beam with respect to the cloud and measure how many atoms remain in theimaging state. We see two dips in atom number, corresponding to positions when either side of
the “donut” crosses through the center of the trap. The beam is best aligned somewhere between
these two dips, when the center of the atom cloud sits in the center of the LG beam.

24
atoms not optically pumped should have a “W” pattern, where the best alignment is found in thecenter of the W (figure 2.10).
Once we have established a rough alignment, we use the pendulum-like motion of atoms in
the harmonic trap for fine alignment. By optically pumping at some time, t, before the trap isturned off, we can look for a “slosh” in the atoms that remain. This is not truly a slosh, as wehave not given the atoms a directional kick; instead, it comes from the usual motion of atoms in aharmonic trap—atoms away from the center of the trap feel a restoring force towards the center.
If the beam is selecting atoms at the edge of the trap, we measure a large oscillation in their center
positions, but if the beam is perfectly centered, the center of the selected atom cloud should notchange. (In either case, we also see a “breathe” in the atoms’ width at twice the frequency.) We
have found that aligning the horizontal (vertical) beam to minimize the center-of-mass motion in
the radial (axial) direction results in very precise alignment. In addition, we often use this technique
to measure the frequencies of the trap, as it is the most non-perturbative way we have found to doso. Figure 2.11 shows one such frequency measurement taken during donut alignment.

2.6

Summary
In terms of new experimental techniques, the spatially selective imaging was fairly simple toset up. A minimal number of optics were added to the system, the optical pumping frequencies
were chosen, the shapes of the donut-mode beams were characterized, and the beams were carefullyaligned onto the atoms.
In the next chapter, I present our first experiment with spatially selective imaging—thedirect observation of the momentum distribution of a non-interacting Fermi gas. The rest of thecharacterization of the “donut beam technique” is discussed in the context of this experiment,
including the possibility of collisions with optically pumped atoms, the validity of assuming ourmeasurements are taken at a single density, and the development of a robust model for the removalof signal using this technique.

25

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 2.11: Donut alignment technique. a. When a hollow light beam is slightly misaligned
from the center of the atoms, we observe a small center of mass oscillation in the visible sample as
a function of time held in the trap after optical pumping. As a final alignment step, we minimize
this oscillation. We have also found this to be a good way to measure the trapping frequency. In
these data, we find the trapping frequency to be 266(2) Hz in the x-direction. b. We measure
the width of the atom cloud in time of flight after some time held in the trap. We observe thein-trap pendulum-like oscillations of the atoms as they exchange potential and kinetic energy. This“breathe” oscillation occurs at twice the trapping frequency and is present even for a perfectlyaligned beam.

Chapter 3

The ideal homogeneous gas: uncovering the Fermi step function

The first experiment carried out with our new spatially selective imaging technique was ameasurement of the momentum distribution of an ideal (non-interacting) Fermi gas. This experiment revealed the iconic step in the momentum distribution of a degenerate Fermi gas. To ourknowledge, this was the first time that the “textbook” momentum distribution of a Fermi gas was
directly observed. In addition, the momentum distribution of an ideal Fermi gas is well understood,
and our results allowed us to fully characterize the donut beam technique. Much of the content of
this chapter was published in Ref. [43].

3.1

The momentum distribution of a homogeneous 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
for an ideal Fermi gas is given by the Fermi-Dirac distribution:
1n(k) =

 2 2

e


/kB T

~ k
−µ
2m

,

(3.1)

+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 , has not previously been directly observed in experiments. One reason forthis is that the vast majority of Fermi systems, such as electrons in materials, valence electrons in27
atoms, and protons and/or neutrons in nuclear matter, are interacting. For ultracold atom gases,
the interactions can be made very strong or very weak, and the momentum distribution can bedirectly probed.

3.2

Experiment sequence
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, where F is the quantum number denoting thetotal atomic spin and mF is its projection. The atoms are confined in an approximately cylindricallysymmetric trap, created by two orthogonal 1075 nm beams, one with a 30 µm waist and one witha waist of 200 µm. We measure a radial trap frequency νr of 214 Hz and an axial trap frequency
νz of 16 Hz. We take data at B = 208.2 G where the scattering length a between atoms in the|9/2, −9/2i and |9/2, −7/2i states is approximately −30 a0 [1], where a0 is the Bohr radius. Here,the gas is very weakly interacting, with a dimensionless interaction strength of kF a = −0.011.
Once we create an ideal Fermi gas, the sequence for spatial selection and probing the momentum distribution is as follows. We first turn off the trap suddenly and illuminate the atoms
with the vertical donut beam, followed immediately by pulsing on the horizontal beam. The powerin the beams is on the order of 10s to 100s of nW and is varied to control the fraction of atoms
that are optically pumped out of the imaging state (|9/2, −7/2i). Each beam is pulsed on for 10 to40 µs, with the pulse durations chosen such that the fraction of atoms optically pumped by eachof the two beams is roughly equal (within a factor of two). We then image the remaining atoms in
the imaging state, usually after some time of flight.
In this experiment, we probe the |9/2, −7/2i spin component. The hollow light beams areresonant with the transition from the |9/2, −7/2i state to the electronically excited |5/2, −5/2istate (see figure 3.1). Atoms in this excited state decay by spontaneous emission with a branchingratio of 0.955 to the |7/2, −7/2i ground state and 0.044 to the original |9/2, −7/2i state. Afterreleasing the trap and applying the hollow light beams, we allow the cloud to expand for 10 or 12
ms time of flight. To optimize the signal-to-noise ratio in the image, we use an rf π-pulse to transfer

28

4P3/2

|5/2, -5/2>

F = 7/2
F = 9/2
mF = -9/2

-7/2

4S1/2

Figure 3.1: Optical pumping transition driven by the LG mode beams. Schematic level
diagram showing the optical pumping transitions. The relevant states at B = 208.2 G are labeledwith the hyperfine quantum numbers of the B = 0 states to which they adiabatically connect.

the remaining |9/2, −7/2i atoms to the |9/2, −9/2i state and then image with light resonant withthe cycling transition. Just prior to this rf pulse, we clean out the |9/2, −9/2i state by opticallypumping these atoms to the upper hyperfine ground state (f =7/2). The imaging light propagatesalong the z direction, and we apply an inverse Abel transform to the 2D image (assuming sphericalsymmetry in k-space) to obtain the 3D momentum distribution, n(k).

3.3Seeing the Fermi step
In figure 3.2, we show momentum distributions measured with and without using the hollow

light beams. As shown in figure 1.5 and Ref. [13], the trap-averaged momentum distribution for the
Fermi gas is only modestly distorted from the Gaussian distribution of a classical gas. The dashed29an (k )

10

0

1
k / kF2

0

1
k / kF2bn (k )

1

0

Figure 3.2: Measured momentum distribution for a weakly interacting Fermi gas. a.
The momentum distribution of the central 16% of a harmonically trapped gas is obtained from theaverage of twelve images. This distribution has been normalized such that the area under the curveis equal to 1. The solid line shows a fit to a homogeneous momentum distribution, where T is fixed
to the value obtained from the trap-averaged distribution and kF is the only fit parameter. b. For
comparison, I show the trap-averaged distribution, taken from an average of six images, with fits
to the expected momentum distribution for an ideal gas in a harmonic trap (dashed line) and tothe homogeneous momentum distribution (solid line).

line in figure 3.2b shows a fit to the expected momentum distribution for a harmonically trapped
ideal Fermi gas, from which we determine the temperature of the gas to be T /TF,trap = 0.12 ± 0.02.

30
Here, the Fermi temperature for the trapped gas is given by TF,trap = EF,trap /kB , where EF,trap =h(νr2 νz )1/3 (6N )1/3 is the Fermi energy for the trapped gas. After optical pumping with the hollow
light beams so that we probe the central 16% of the atoms, the measured momentum distribution(part a of figure 3.2) has a clear step, as expected for a homogeneous Fermi gas described byequation 3.1.
For a sufficiently small density inhomogeneity, the momentum distribution should look likethat of 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), normalized in terms of
kF and TF :n(k/kF ) =kF−3(k/kF )2

1 T /TF
ζe

,

(3.2)

+1

where ζ is the fugacity of a homogeneous Fermi gas, which is a function of T /TF and given by,−4
Li3/2 (−ζ) = √3 π(T /TF )3/2

(3.3)

To obtain the solid lines in figure 3.2a and b, we fix T /TF to the value we measure for the trappedgas, which leaves only a single fit parameter, kF , which 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.

3.3.1

Approximating our results as “homogeneous”

In order to quantify how well the measured momentum distribution is described by that of ahomogeneous gas, we look at the reduced χ2 statistic in figure 3.3a. The reduced χ2 is much larger
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, χ2 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 only those atoms near the center of the trap. Figure 3.3b displays the fit value kF , in units

31

Reduced χ2

a 15
10

5

0

kF /(kF ,tr ap )

b

0

0.5
Fraction Probed

1

0.5
Fraction Probed

1

1

0.8

0.6
0

Figure 3.3: Fit results as a function of the fraction of atoms probed. (a) As the fraction of
atoms probed is decreased, the reduced χ2 for the fit to a homogeneous gas momentum distribution
decreases dramatically and approaches a value of 1.6 for fractions less than 40%. The minimum
χ2 is limited by systematic noise in the image (i.e. fringes). (b) The best fit value of kF increases
as we increasingly probe only the atoms in the central (highest density) part of our trap. A modelof the optical pumping by the hollow light beams yields an average local kF indicated by the solidline. As an indication of the density inhomogeneity, the shaded region shows the spread (standard
deviation) in kF from the model.

of kF,trap =p
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 I discuss in section 3.4. The model result for the average local kF , hkF i, of the probed gas is32

T / TF

0.4

0.2

0

0

0.5
fraction probed

1

Figure 3.4: Measured T /TF vs the fraction of atoms probed. We fit the measured momentum
distribution to a homogeneous gas distribution with two free parameters, T /TF and kF . The plotshows the best fit value for T /TF as a function of the fraction of atoms probed near the center of thecloud. The density inhomogeneity of the probed gas can result in a T /TF that is much larger thanthat 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 with the data.

shown with the solid line in figure 3.3b, 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 of the local kF , δ 2 = kF2 − hkF i2 , and the shaded region infigure3.3b shows hkF i ± δ. In the region where the reduced χ2 indicates that the measured n(k)fits well that for a homogeneous gas (fraction probed < 40%), δ/ hkF i < 0.08.
3.3.2

How homogeneous is “homogeneous enough”?

An important question when we use our selective imaging technique is how selective we need
to be for the results to be considered that of a “homogeneous” gas. This, of course, depends on thewidth of the features that one wishes to see. Since all of the signal originates from a gas confined ina smooth, harmonic potential, the only way to probe a perfectly homogeneous density is to probe
only one atom. Instead, we hope to find a “sweet spot” for which we can reproduce sharp features
but still have decent signal to noise.

33
Ideally, we’d like to measure features whose widths are determined by temperature or interactions and not significantly modified by any remaining spread in densities. For the ideal Fermi
gas, this would mean that the width of the step in the momentum distribution will reflect the true
T /TF of the gas.
We can check this by fitting the results to a homogeneous gas distribution, with both kF
and T /TF as fit parameters. For a gas with a completely homogeneous density, this fit returns theaverage temperature of the atoms. On the other hand, a large density inhomogeneity that washes
out the Fermi surface because of a spread in the local values of kF will result in an artificially highfit value for T /TF . This can be seen in figure 3.4. The solid line shows the average T / hTF i for theprobed gas calculated using our model. Here, T is fixed and the dependence on the fraction probed
comes from the fact that the average density, and therefore the average local TF , increases as weprobe a smaller fraction of atoms. 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 (data shown in figure 3.2), the best fit
value is T /TF = 0.14 ± 0.02. As a check of the model, we can also calculate n(k) for the probed
gas and fit this to the homogeneous gas distribution; the results (dashed line in figure 3.4) agree
well with the data.
From this data, we find that when probing 40% or less of the atoms, the observed momentum
distribution is consistent with that of a homogeneous gas, where the width and position of the Fermi
surface reflect the average temperature and density of the probed portion of the gas.

3.4

Modeling the signal removal
Even though the emergence of a sharp Fermi surface proves the effectiveness of our donut

beam technique, a model describing the optical pumping is useful for understanding the remaining
density inhomogeneity. While it might have been simpler if our donut beams had a perfectly sharp
cut, eliminating all signal past a certain radius, we know that the inner walls of the LG beams havesome finite width, and we can use the model to see how much this effects our final signal.

34
3.4.1

The model

In modeling the effect 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 at a particular location in the cloud is taken to be Pi (~r) = exp(−γi (~r)τi σ), where τi is the pulse
duration and the subscripts i = 1, 2 denote the two hollow light beams. The photon flux is given
by γi (~r) = Ii (~r)λ/(hc), where Ii (~r) is the position-dependent intensity, c is the speed of light, andλ = 766.7 nm is the wavelength. For the optical absorption cross section, we use σ = 3λ2 η/(2π),
where η is the branching ratio between the initial and final states. For the intensities, we use
equation 2.1 with l = 2 and make the approximation that w is constant across the cloud.
Attenuation of the hollow light beam as it propagates through the atom cloud is observablein the long direction of the cloud (along z), and the effect of this can be seen in figure 3.5b. To
include this effect, we consider 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 model that ignores attenuation but where we adjust the beam powers toprobe the same fraction of the atoms. This effect is relatively small and decreases as one probes asmaller fraction of the gas. This can be seen in figure 3.5 where we show the measured momentum
distribution for the central 38% of the atoms compared to three different models, each of whichis adjusted to give the same probed fraction. The solid line is the model explained above, which
includes attenuation, while the dotted line shows the result when we ignore the depletion of thehollow light beams. For comparison, the dashed line shows the expected distribution if one selects
atoms in a cylindrical volume with sharp boundaries. All three of the models give similar results,
which is good—it shows us that our technique is independent of any particular details of the opticalpumping.
Although we see that our results are model independent, for the rest of this thesis, we choose
to use the model that includes attenuation of both beams by the atoms, as we believe this provides

35

a

3

n (k )

2

1

0

0

1

k / kFfraction probed

b
1

0.5

0
−100

0

100

z / zF

Figure 3.5: Modeling the spatially selective optical pumping. a. Comparison of the normalized momentum distribution of the central 38% of the atoms with three different models (dotted,
solid, and dashed lines; see text). The data (circles) are obtained from an average of four images.
We find that the attenuation of the hollow light beams (included in solid line model) does notstrongly affect the predicted final momentum distribution when probing a small fraction of thegas. b. An image of the cloud is taken with a probe beam that is perpendicular to the long (z-)
direction. The image is taken after only a short (1.3 ms) expansion so that we probe the spatial
dependence of the density. We compared images with the horizontal hollow light beam and without any
q optical pumping in order to measure the fraction of atoms probed (circles) vs z/zF , where
2E

F,trapzF = m(2πν
2 . For this data, the total fraction of atoms probed is 71%. The prediction of ourz)model (solid line), which includes attenuation of the hollow light beam as it propagates throughthe cloud, agrees well with the data.

36

2
1
0

0

2
1
0

0.5
1
k /kF ,tr ap

0

0.5
1
k /kF ,tr ap

1.5

0.5
1
k /kF ,tr ap

1.5

35% probed

1
0

3
n(k )

0

0

2

0

21% probed

1

1

3

1.5

2

66% probed

2

0

1.5

56% probed

3
n(k )

0.5
1
k /kF ,tr ap

n(k )

n(k )

3

0

3

81% probed
n(k )

n(k )

3

0.5
1
k /kF ,tr ap

1.5

16% probed

2
1
0

0

0.5
1
k /kF ,tr ap

1.5

Figure 3.6: Model of spatially selective optical pumping technique compared to data at
six different donut√ beam powers. The x-axis of these graphs are in units of the kF of the trap,

given by kF,trap =2mEF,trap, instead of being determined by fitting, as in figure 3.2.
~

37
the most correct physical description. Using the well-known phase space distribution of an ideal
Fermi gas in a harmonic trap, we can make predictions for the momentum distributions of the Fermi
gas given a certain amount of signal probed. Figure 3.6 shows data from six removals compared to
the results from the model. The agreement is very good.

3.4.2

Collisions in time of flight

While we find our model of the donut beam technique reproduces our data very well, it does
not take into account any collisions that may occur after the donut beam optical pumping. In themodel, atoms that have been removed from the signal have no effect on the atoms that remainin the imaging state. This is how we hoped our technique worked, but it was important to verify
whether this was indeed the case.
One of the main reasons for developing the donut beam technique was to be able to takemeasurements of a homogeneous sample after a time of flight expansion such that we get momentum
information from a uniform density. As such, we tried to be extra careful to characterize the waysthat the donut beam technique might interfere with accurately measuring the momentum of thegas. One such way would be if the atoms in the imaging state underwent collisions with atoms
optically pumped to the dark state during time of flight expansion. This process could potentiallydestroy the momentum information we are trying to extract.
When imaging the momentum distribution of a non-interacting cloud, we use a magnetic
Fano-Feshbach resonance to set the s-wave scattering length that parametrizes interactions between
our two main spin states, |9/2, −9/2i and |9/2, −7/2i, to −30 a0 . The number of collisions during
expansion is negligible. (The collision rate is roughly three collisions a second for typical densities
in the optical trap, while the expansion time is no more than 12 ms.) However, the optically
pumped atoms in the F = 7/2 manifold could potentially interact with our |9/2, −7/2i atoms andscramble their momenta. To investigate this effect, we performed a Monte Carlo (MC) simulationof collisions on an atom cloud during a 10 ms expansion.
There are potentially two types of collisions that can change our signal, inelastic (or spin

38changing) and elastic. The elastic scattering rate can be determined by the s-wave scattering lengthand the atomic velocity; for a = 200a0 and vF = 0.0128 m/s (where a0 is the Bohr radius andvF = ~kmF is the Fermi velocity), the elastic scattering rate can be estimated to be Γel = Kelastic ∗ n,
where Kelastic = (4πa)2 ∗ vF = 1.8 × 10−11 cm3 /s. The inelastic scattering rate constant, calculated
by John Bohn (personal communication, March 2012), is smaller but comparable—Kinelastic =8.5 × 10−12 cm3 /s for |9/2, −7/2i colliding with |7/2, −5/2i. (The optically pumped atoms can fall
into other hyperfine states, but we expect all of the scattering rates to be similar.)In the simulation, we calculate the effects of both elastic and inelastic collisions in a cloud withtypical initial conditions and donut removal. First, we create an ensemble of roughly 100,000 atoms
at T = 0.12TF . Using a Monte Carlo technique, we assign three position and three momentum
coordinates to each atom (x, y, z, kx , ky , kz ). Then, using our model of donut removal, we put 77%of the atoms at the cloud edges into the higher energy F = 7/2 hyperfine manifold. 23% of theatoms remain in the |9/2, −7/2i imaging state.
Once the distributions of positions, momenta, and spin states are set, we look for atoms in
the imaging state that are close enough to an F = 7/2 atom to undergo a collision. If the atoms areclose enough (we check all particles that are closer than 40% of the cloud’s Thomas-Fermi radius),
we calculate their probability to collide (either elastically or inelastically) based on their relative
positions and velocities. Still using Monte Carlo, we decide whether the atoms collide based onthat probability. If the collision is elastic, we recalculate the momenta of the two atoms and allow
them to remain in the simulation; if the collision is inelastic, they are removed from the signal.
After calculating the collisions, we let the atoms move in free space (no confining potential) for asmall amount of time compared to the collision rate. After that motion, we use the new momenta
and positions of the atoms to move them for another time step and recalculate the probability tocollide. This process continues for 10 ms of time of flight expansion.
In the simulation, we find that 7% of those atoms left in the imaging state will undergo anelastic collision during expansion and roughly 2% will undergo spin-changing collisions (and be lostfrom the signal). Figure 3.7 shows a single simulated momentum distribution of our cloud extracted

39

calculated n(k )2typical shot noisen(k )collisions and noise

1

0

0

0.5

1

1.5

k / kF
Figure 3.7: Momentum distribution with and without collisions. Using the model of opticalpumping and the phase space distribution of an ideal Fermi gas, we calculate what the momentum
distribution would like if only 23% of the signal is probed (39% removed by the vertical beam).
The black line shows this calculation, while the blue points show this calculation plus a typicalamount of noise. The red points show how the (blue) momentum distribution changes after a 10
ms expansion, during which the atoms may collide with the optically pumped atoms. We conclude
from this that the effect of collisions on our signal is negligible.

from the spatial distribution of atoms after 10 ms time of flight expansion. We find that the collisions
during expansion have the effect of slightly scrambling our signal and slightly increasing the weightabout k = kF , but we also see that the effect is small. We fit the final momentum distributions to
the Fermi distribution for a homogeneous gas, allowing kF and T /TF as fit parameters. We findthat the fitted T /TF (from fitting to a homogeneous momentum distribution) increases from 0.141
to 0.155 (which is generally within our uncertainty) with no appreciable change to kF . From thiswe conclude that the effect of collisions during expansion is below what we can observe.

40
3.4.3

Effect of finite branching ratio: the return of hot atoms

While our model of spatially selective imaging relies on the assumption that we are only
measuring atoms that have not scattered a photon, this is not necessarily the case. The opticalpumping transitions that we use have a small but finite branching ratio, which gives the probability
for an atom to spontaneously decay back to initial state after scattering a photon. If this occurs,
the atom will remain to be imaged (barring the possibility of scattering another photon), and willretain two extra momentum ’kicks’ from first absorbing and then re-emitting a photon. This extra
momentum would be on the order of the Fermi momentum, so this process could potentially havea significant effect on the measurement of the Fermi step.
In chapter 2, I presented two optical pumping transitions that we utilize in the donut
beam technique. The transition used in this chapter (and the next) is |F, mF i = |9/2, −7/2i →|F 0 , mF 0 i=|5/2, −5/2i. This has a 0.044 branching ratio, so 4.4% of the atoms that absorb a photon
will return to the |9/2, −7/2i imaging state. The transition used with photoemission spectroscopy(chapters 5 and 6) is |F, mF i = |9/2, −5/2i → |F 0 , mF 0 i=|5/2, −3/2i, which has a 0.096 branchingratio and thus an enhanced effect. We spent a good deal of time worrying about and looking for thisbackground of hot atoms, particularly for the higher branching ratio transition. It was importantfor us to understand how much these unwanted hot atoms could distort the momentum informationof our gas.
Generally, when using the technique of “optical pumping”, one does not need to worry aboutatoms falling back into the initial state, even for transitions with a particularly high branchingratio. This is because eventually all atoms will fall into another state, called the “dark state.” Ifthey are not pumped there at first, increasing the power or duration of the optical pumping should
continue to drive the transition until no atoms are left in the initial state. Our problem is that
our optical pumping beams have a shape, with inner walls that are not infinitely sharp. Even as
the power is increased, there will be a small part of the beam with low enough intensity that it
will only ever drive one transition at most. On the other hand, some of the atoms (those closer

41
to the cloud edges and sitting in the middle of the donut “ring”) experience a much higher opticalpumping intensity, and will almost certainly be pumped again if they happen to fall back to theinitial state.

3.4.3.1

Estimating the size of the effectIf atoms are returning to our imaging state after being pumped, the signal we image will
be composed of atoms that were not pumped (“good signal”) and a small amount of atoms which
return from the excited state to their initial state with extra momentum (“bad signal”). We canmake a simple calculation to estimate how large the effect of atoms falling back into the imaging
state would be. A simple picture of this process is as follows. There is some probability, P , for
an atom to be optically pumped, which depends on the intensity of the donut beam. There isalso some probability, η, for an optically pumped atom to decay back to the imaging state, whichis equal to the branching ratio between the initial and excited states. For the purposes of thiscalculation, let’s assume that the intensity of the optical pumping light is uniform across the cloudand unchanging in time. Thus, P is also uniform and unchanging. We know that this is not true;
the optical pumping beam has an LG spatial mode and loses photons as it scatters off the atoms.
However, this simple calculation should give us a rough estimate of the amount of the “bad signal”
we might expect.
This picture can be described by the following equation:
Nimaged = Ninitial −NXeventsiP i ∗ Ninitial − η i ∗ P i ∗ Ninitial ,(3.4)

where Ninitial is the number of atoms before optical pumping, Nevents is the number of times anatom is allowed to scatter a photon, and Nimaged is the total number of atoms in the imaging state
after optical pumping. The terms in the sum correspond to the effects of optical pumping by atmost i photons; for i = 1, we calculate that P ∗ Ninitial atoms scatter one photon, removing themfrom the signal, and then η ∗ P ∗ Ninitial of those atoms return with extra momentum. In this case,the “good signal” (i.e. atoms with their original momenta) is equal to (1 − P ) ∗ Ninitial , while the42fraction of probed signal
coming from returning atoms(a) chance to return is 4.4 %
0.3

0.2

0.1

0

0

0.25
0.5
0.75
fraction of atoms probed

1

Scattering events allowed:
one
two
three

fraction of probed signal
coming from returning atoms(b) chance to return is 9.6 %
0.5
0.4
0.3
0.2
0.1
0

0

0.25
0.5
0.75
fraction of atoms probed

1

Figure 3.8: The size of the “bad signal” for two branching ratios. Using equation 3.4,
I can estimate the fraction of the signal that comes from atoms with an extra momentum kickfrom having scattered a photon. The blue line is the estimated fraction of bad signal if the atoms
can scatter at most one photon. As we account for more scattering events, the atoms which havefallen back into the imaging state after being pumped have another chance to be removed, and thefraction of bad signal begins to converge to a lower value. The two graphs correspond to the two
optical pumping transitions we use, found in figure 2.4. From these calculations, we estimate theerror from “hot atoms” to be below 5% for transition a and below 10% for transition b.

43“bad signal” which comes from returning atoms is given by η ∗ P ∗ Ninitial . There is some chancethat these “bad atoms” will scatter more photons, which gives them more chances to leave theimaging state. The terms for which i > 1 describe this process.
Figure 3.8 shows what fraction of the imaged signal will be composed of atoms with extra
momentum kicks, given by equation 3.4. This quantity is found by solving for a particular fractionof atoms imaged after pumping, Nimaged /Ninitial , and taking the difference between that and thefraction of atoms pumped by one photon, (1−P ) (the “good signal”). The three lines correspond to
calculations including the first one, two, or three terms in equation 3.4. Figure 3.8a was performed
for a branching ratio of η = 0.044, which corresponds to the optical pumping transition used in thischapter and in chapter 4. Figure 3.8b includes a branching ratio of η = 0.096, which corresponds
to the transition used in this chapters 5 and 6.
The fraction of “bad atoms” contributing to the signal increases as more atoms are optically
pumped, which is expected. However, allowing for more than one photon scattering event forsituations when the optical pumping intensities are high (low fraction of atoms probed) causes thecalculation to converge to a smaller “bad signal.” This result makes sense—in the case of highoptical pumping intensity, the atoms would be more likely to scatter multiple photons, and any
returning atoms would have another chance to leave the signal. For most of the data taken inthis thesis, the fraction of atoms imaged after the donut beam technique is 25–30%. Thus, we can
expect a 3% effect on the signal for transition a and a 7% or less effect for transition b. Both ofthese are quite small effects, which, as with the effects of collisions with our dark atoms, we expect
will be beneath the noise in our system.

3.5

Conclusions
Our first experiments with the donut beam technique gave us confidence that we could

recover sharp features, such as the Fermi step, that would otherwise be washed out of our data.
We were initially concerned that this approach might leave too little signal when selecting a smallenough region to approximate a homogeneous gas. However, for a gas at T /TF,trap = 0.12 and44
our typical measurement precision, we find that probing the central 40% (or less) of the gas issufficient to approximate a homogeneous gas. We take advantage of the “sweet” spot in the following
experiments; in general we probe 25 − 30% of the gas and see that we are sufficiently homogeneous.
Direct observation of the Fermi surface was pretty satisfying as far as experiments go; it is notevery day that your results look like they came from a statistical mechanics textbook! Additionally,
the momentum distribution of an ideal Fermi gas has a well-known form, and comparing our results
to our expectations allowed us to test our understanding of what the donut beam technique does.
Armed with this, we stepped away from the known and into the unknown—in this case, the regimeof strong interactions, where the properties of the Fermi gas are still a subject of research. The
rest of this thesis presents experiments on gases in the BCS-BEC crossover, where we use spatiallyselective imaging to probe the local properties of our gases.

Chapter 4

Contact of a homogeneous gas

This chapter presents our measurement of the local, or homogeneous, contact of a unitary
Fermi gas as a function of temperature. The contact is a measure of short-range correlations
between atoms with a short-range interaction. First defined by Tan in 2005 [2], the contact is auniversal quantity that is valid for a very wide range of circumstances—including zero and finite
temperature gases, superfluid and normal phases, few and many particle systems, and balanced
or imbalanced spin populations. Recent work has shown that the contact can be extended tobosons [44, 45], three-body interactions [46], and nucleons in atoms [47].
Theoretical predictions for the temperature dependence of the contact of a Fermi gas differ
substantially, especially near the superfluid transition temperature, TC /TF . However, since theFermi temperature, TF , is density dependent, even large features near the phase transition can belost when the measurement is taken from atoms in a harmonic trap. Using our spatially selective
imaging, we measure the contact for atoms at the center of our trap where the density is nearlyhomogeneous. In this data, we observe a sharp feature not distinguishable in previous contact
measurements [48]. This feature may be associated with the superfluid transition. The work inthis chapter has been published in Ref. [49].

4.1

What is the contact?
In 2005, Shina Tan derived a number of universal relations for an interacting Fermi gas

with short-range, or contact, interactions [2, 50, 51]. These powerful relations connect microscopic

46
quantities, such as the momentum distribution of the fermions, to macroscopic quantities, such as
the total energy of the system. Furthermore, the relations are universal in that they do not dependon the details of the interparticle potential, nor do they depend on the state of the system, which
could be an exotic Fermi superfluid, a normal Fermi liquid, or even a simple two-body state such asa diatomic molecule. At the heart of the universal relations is a single quantity, which Tan termed
the contact.

4.1.1

Tan’s definition: the high momentum tail and the Tan relations

The contact was first introduced as a quantity in Ref. [2]. It was defined as the coefficient of
the high momentum tail of the momentum distribution, n(k):C ≡ lim k 4 n(k).

(4.1)

k→∞

The contact is an integration of the contact density, C(R); C =R

d3 R C(R). C has units ofinverse length and is often normalized by kF .
Equation 4.1 is the first of the so-called “Tan relations”. It is has been known for some timethat the momentum distribution of particles interacting via a contact interaction has a tail thatscales as k −4 [52]. However, the real advance came in the universal relations that Tan presented [2,50, 51], which connected the contact to other extensive properties of the gas, such as energy andpressure. The other relations connect the contact to the spatial correlation atoms in the two spinstates, n1 and n2 , for short distances [51],rr1hn1 (R + )n2 (R − )i −−−→
r→0
2
2
16π 2



1
2
−
2
r
ar



C(R);

(4.2)

the sum of the kinetic and interaction energies (T and I, respectively) [51],T +I =XZσ

d3 k ~2 k 2(2π)3 2mC
~2nσ (k) − 4 +
C;
k
4πma

(4.3)

the virial theorem [50],T +I −V =−

~2
C,
8πma

(4.4)

47
where V is the potential energy of the gas; the pressure, P, of the gas [50],
2
~2
P= E+
C,
3
12πma

(4.5)

where E is the energy density; the change in total energy, E, for an adiabatic (constant entropy, S)change in scattering length [2],dEd(a−1 )


=−

S

~2
C;
4πm

(4.6)

and the change in total energy for an instantaneous jump in scattering length from ainitial to
af inal [2],~2∆E = −
4πm

1
af inal

−



1

C.

ainitial(4.7)

Incorporating Tan’s definitions into their own work, others have shown that the contact canalso be related to the total integral of the rf line shape [53, 54, 55],Z∞dωΓ(ω) =

~Ω2
4m



−∞

1
1
−
a12 a13


C12 ,

(4.8)

and its high frequency tail [56],Γ(ω) −−−→
ω→∞−1 2Ω2 (a−1
12 − a13 )C12 ,4πω(mω/~)1/2 (a−1
13 + mω/~)(4.9)

where ω is the frequency of the rf pulse, Ω is the Rabi frequency. In taking such a line shape, atoms
are outcoupled from state 2, where their interactions with atoms in state 1 is given by a12 , to state
3, where the interaction is given by a13 . These relations depend on the contact of the atoms in
the initial two states, C12 . Other work has related the contact to the number of closed channelmolecules measured by photoassociation [57, 58],
Nmol =R ∗ ∆2C,
4π[∆ − (B − B0 )]2(4.10)

where B0 and ∆ are the center and the width of the magnetic Fano-Feshbach resonance, and R∗ isa length equal to half the effective range at the center of the resonance [55]; and the static structure
factor, S(q), measured using Bragg spectroscopy [59],1
S12 (q) −−−→
ω→∞ 8



1
4
−
q πaq 2

C,

(4.11)

48
where q is the momentum selected by the Bragg pulses. (For all of the above equations, m is theatomic mass, a is the s-wave scattering length between the two Fano-Feshbach resonance states,and ~/2π is Planck’s constant.)
The contact and its relations are valid for all scattering lengths and phases of the system andencapsulate all of the many-body physics of the gas [57]. Many of these relations have been verified
in experiment, both by our group and by others [60, 3, 45, 48, 61, 62].

4.1.2

An “intuitive interpretation”

In Ref. [55], Eric Braaten presented a physical picture of the contact, using equation 4.2 listed
above. It is the explanation I find most intuitive, and I have paraphrased it below.
Suppose you have a gas with two types of fermionic atoms, atom A and atom B. A and B
interact with each other via a low energy (s-wave) scattering length, a, that is much larger thanthe effective range of the interactions. (This type of interaction is called a contact interaction, andit can be modeled using potential wells with depths of finite size [63].) Being fermions, A doesn’tinteract with A nor B with B.
Now, let’s calculate the probability to find a B atom near a particular A atom. To be
concrete, we are looking in a volume, V = 43 πl3 , surrounding the A atom. This probability will be
determine by the average number density of B atoms, nB . Indeed, for a = 0, this is the only factor.
Additionally, as the magnitude of a increases, nB may adjust, but for l as large or larger than the
average distance between particles, nB still completely determines the probability of finding a B
atom near the A atom. However, for l much smaller than the average inter-atomic spacing, thechance to find a B atom close to an A is higher than naı̈vely expected from nB . The increased
magnitude of the contact interactions make the atoms “sticky” and more likely to be found close
together.
Now, let’s find the probability to find any two atoms, one of type A and one of type B, neareach other. Inside large volumes, the number of such pairs is roughly nA ∗ nB ∗ ( 43 π)2 ∗ l6 . However,for l << n1/3 , (where n = nA + nB ) the number of pairs can be shown to scale as l4 , which is an49
increase from the naı̈ve expectation of l6 scaling:
Npair (l) →l4
C4(4.12)

So, the contact is directly proportional to this increased probability of short-range pairs. From
Fourier transforms (or simple dimensional analysis), it is straightforward to see how the increased
probability to find atoms at short distances leads to increased weight in the high momentum tailof the momentum distribution.

4.2

The contact of a homogeneous gas
The contact can be measured as a sum over the entirety of the system, or it can be presented

as a local quantity, C(R), the contact density. The first measurements of the contact were theformer type, where the contact had been integrated over a distribution of densities in the trappedgas (for example, see Ref. [3]). It would be years before the contact for a single density wasmeasured.
Figures 4.1 and 4.2 illustrates the need for a measurement of the local contact. Both figures
show three theoretical predictions of the contact of a gas at unitarity as a function of T /TF . These
three theories predict very different behavior, particularly around the phase transition. However,
one can use the measured density distribution of atoms at unitarity in a harmonic trap to calculate
what each theory predicts would be measured in such a system. In this case, the three theories
become indistinguishable, and they are each shown to agree well with the results of the measurementof the contact from Ref. [61]. (Note that, for Refs. [66, 61] and figures 4.1 and 4.2, I is the total
contact, which we call C and which they define in terms of the contact density, I ≡ V ∗ C(R).)
Clearly, a measurement of the contact from a single density would be helpful for distinguishingamong theories and guiding future theoretical progress.

50

a
GG

4

G0 G0
GPF

I/(NkF )

3
2
1
0

b

0

0.5
T /TF

1

4
GG
G0 G0
GPF

I/(NkF )

3
2
1
0

0

0.5
T /TF

1

Figure 4.1: Theories for the contact as a function of temperature. As a comparison, weplot a self-consistent t-matrix model (GG) [64], a non-self-consistent t-matrix model (G0 G0 ) [65],and a Gaussian pair-fluctuation NSR model (GPF) [66]. a. The theories presented for a single
density. b. The theories after being averaged over the densities in a harmonic trap. Despite being
qualitatively and quantitatively very different, after trap-averaging the three theories look similar.

51

Figure 4.2: First measurement of the contact as a function of temperature. The measurement of the trap-averaged contact agrees very well with the theories presented in figure 4.1, as
well as the virial expansion. This figure has been reproduced from Kuhnle, et al, Ref. [61].

4.2.1

ExperimentWith this in mind, we combine our spatially selective imaging with rf spectroscopy to determine the local contact at the center of our trap. In previous work, we measured the (trap-averaged)
contact in three different ways: by looking at the tail of the momentum distribution, through atom
photoemission spectroscopy, and by looking at the tail of the rf line shape [3]. We chose to look at
the tail of the rf line shape for this data, although in chapters 5 and 6 of this thesis, we extract the
local contact from both the line shape and atom photoemission spectroscopy.
We begin with an equal mixture of 40 K atoms in our two Fano-Feshbach resonance states,|f, mf i = |9/2, −9/2i and |9/2, −7/2i. The temperature of the gas is varied by changing the final
depth of the optical dipole trap in the evaporation process. The number of atoms per spin state
after the evaporation ranges from 50, 000 to 220, 000. In these measurements, the radial trapping

52
frequency, ωr , ranges from 2π × 200 Hz to 2π × 410 Hz, while the axial trapping frequency, ωz ,ranges from 2π × 19 Hz to 2π × 25 Hz.

203.4 G

50 ms

absorption
imaging,
after 3 ms

hollow beams (50 ms)

trap releaseFeshbach resonance(202.2 G)

rf pulse (100 ms)

Figure 4.3: Schematic of the experimental sequence. The magnetic field is ramped from
203.4 G, where the atoms are initially prepared, to the Fano-Feshbach resonance. The hollow lightbeams are turned on 280 µs before trap release; the beam that propagates perpendicular to thelong axis of the cloud is pulsed on for 10 µs followed by 40 µs of the second beam. The line shapeis measured using an rf pulse with a total duration of 100 µs and a Gaussian field envelope withσ = 17 µs, centered 180 µs before trap release. The cloud expands for 3 ms before being probed byabsorption imaging.

The experimental sequence is depicted in figure 4.3. The magnetic field is ramped adiabatically to the Fano-Feshbach resonance and kept at this value for 2 ms before the trapping potential
is abruptly shut off. Prior to turning off the potential, the hollow beams are sequentially pulsed
on for 50 µs. This is followed by an rf pulse, which transfers a small fraction of the atoms in theoccupied |9/2, −7/2i state to the initially unoccupied |9/2, −5/2i state. The |9/2, −5/2i state hasa background interaction of only 180a0 with the other two states in the left in the trap, whichallows us to approximate atoms in this state as non-interacting. The total duration of this pulse is
100 µs. We then turn off the trapping potential and let the cloud expand for 3 ms. We detect the
number of transferred atoms using absorption imaging.
Absorption imaging of a low number of atoms can be difficult. To guarantee our best signal
to noise, we remove all atoms not in the |9/2, −5/2i state using optical pumping light and use two
π/2 pulses to move the |9/2, −5/2i atoms to the |9/2, −9/2i just before imaging. This state can be53
imaged using the |9/2, −9/2i to electronically excited |11/2, −11/2i closed (“cycling”) transition.
This imaging procedure is standard in our lab, and it is particularly important for experiments
incorporating our donut beam technique, as this technique leaves us with only around 30% of thesignal we would otherwise have.

4.2.2

Extracting the contact

0.2
2 3/2π 2 ν 3/2Γ [NkF ]10

Γ [EF /h]

0.15

0.1

8
6
4
2
0

0

0.05

5

10

ν [EF /h]

15

0
0

5

ν [EF /h]10

15

Figure 4.4: An rf line shape for the unitary Fermi gas at T /TF = 0.25 withR 30% of the atoms∞
probed. The solid (red) line is a fit to equation 4.13 with the normalization −∞ Γ(ν)dν = 1/2,
due to the 50% − 50% spin mixture. The inset shows the same data multiplied by 23/2 π 2 ν 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 different frequencies is done with different 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.

54
The contact is extracted from a measurement of the rf line shape Γ(ν) [3], where Γ(ν) is therate of atoms transferred from |9/2, −7/2i state to the |9/2, −5/2i state by an rf pulse centered ata frequency detuning ν. Equation 4.9 of the Tan relations predicts that the high frequency tail ofthe rf line shape scales as ν −3/2 , with the contact determining the amplitude. This equation canbe rewritten:
Γ(ν)
C/(N kF )= √00
2π 2 ν 3/2−∞ Γ(ν )dνR∞for ν → ∞(4.13)

where N is the total number of atoms, and ~kF is the Fermi momentum, and ν is the rf detuning
in units of the Fermi energy, EF /h, with h being the Planck constant (2π~ ≡ h).
A representative data set, where the hollow light beams were used to select the central 30%of the atom cloud, is shown in figure 4.4. For each line shape, we take data at 30 different detunings
between −16 kHz and +116 kHz, where ν = 0 is defined as 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, −7/2istate). The highest frequency typically corresponds to approximately 13 EF /h. The inset of figure
4.4 shows Γ(ν) multiplied by 23/2 π 2 ν 3/2 , where we observe a plateau for frequencies higher than
5 EF /h. We extract the contact by fitting the measured Γ(ν) for ν > 5 EF /h to equation 4.13 (solidline in figure 4.4). For the normalization, we integrate the line shape, including the extrapolated
2 , where rtail, up to ν = h/mreff
eff is the effective range of the interaction [67]. We find this cutoff

to be ν = 320EF /h at the Fano-Feshbach resonance.

4.2.2.1

Setting the rf power for the line shape measurements

When taking the rf line shape, it is important outcouple only a small fraction of the atoms
with the rf pulse so that the data can be compared with the theoretical predictions made for the
linear response regime. On the other hand, our measurements depend on having good signal tonoise in the tail of the rf line shape, which requires increasing the power of the rf pulse to outcouple
more atoms for larger detunings. In order to be able to compare the number of atoms taken using
different powers, we have measured the number of atoms transferred by the rf pulse as a functionof the rf power for different frequencies. For this measurement, the data was taken without the

55
0.6

fraction of atoms transferred

(a)

(b)

0.5
0.4
0.3
0.2
0.1
0
0

1

2

3

0

0.5

(rf power)1/2 [arb]1
rf power [arb]1.5

2

fraction of atoms transferredFigure 4.5: a. Coherent oscillations with increasing rf power at a detuning of 5 MHz relative to thesingle-particle transition frequency. To show the coherent oscillations, the x-axis is the square rootof the rf power (which is proportional to the Rabi frequency). b. Zooming in on the initial rise infigure 4.5a, this time plotted as function of the rf power. The rf power is shown in experimental
units and varied by changing the output of a function generator. The solid line is a fit to the
exponential model introduced in the text.

0.4

0.3

0.2

0.1

0
0

10

20

30

40

50

60

rf power [arb]Figure 4.6: Saturation with increasing rf power at a detuning of 45 MHz. Note the absence ofcoherent oscillations for larger detunings. The solid line is a fit to the exponential model introducedin the text.

56
donut-mode beams, for a gas at unitarity with ∼ 80000 atoms per spin state and EF /h = 7700 Hz,
with h being the Planck constant. The temperature of the gas, T /TF = 0.13, was measured before
ramping to the resonance field.
For small detunings, we observe oscillatory behavior reminiscent of coherent Rabi oscillations(figure 4.5a). For large detunings, the transfer is incoherent (see figure 4.6), and the number ofoutcoupled atoms saturates to a value that is between a quarter and a half of the total numberof atoms (depending on the rf detuning). We fit the data to Noutcoupled = A(1 − e−P/P0 ) to findthe saturation power P0 , where Noutcoupled is the number of atoms outcoupled by the rf pulse withpower, P . The results are given in figure 4.7. For the lowest detuning, where coherent effects
exist, we fit the signal up to the first peak (figure 4.5b). We see that P0 increases with increasing
detuning, allowing us to continually increase the rf power as we measure further out on the tail ofthe line shape.
When measuring the rf line shape we keep the power less than P0 /5, where the number ofatoms transferred by the rf pulse is small compared to the total number of atoms. Below thispower, the linear approximation, Noutcoupled ≈ A PP0 , is correct to within 10%. By keeping our rf
power in the “linear regime,” we stay on the linear rising edge of the curves in figures 4.5 and 4.6.
In this regime, we outcouple only about 20% of the total atoms, and the number outcoupled is not
yet affected by the depletion of the atoms in the original state. As we increase the rf detuning, we
also increase the power to always outcouple about 20% of the atoms. Afterwards, we linearly scale
down the number of atoms at each detuning to match the number that would have been outcoupledby the power used for the central part of the line shape.

4.3

Local contact vs. (local) T/TF
The main result of these experiments, namely the contact of a homogeneous gas as a functionof temperature, is presented in figure 4.8. The contact is normalized to the average kF of theprobed sample, and temperature is given in terms of T /TF , with TF being the average Fermi
temperature of the probed sample. (These quantities are extracted from the average density,

57

20

P0 [arb]
0.33
2.5
8.4
12.7
17.1

15

P0 (arb)

detuning [kHz]
5
25
45
65
85

10
5
0

0

20

40
60
80
detuning (kHz)100

Figure 4.7: The measured saturation power, P0 , in arbitrary units, for different rf
frequency detunings. This power is extracted from fitting to curves of Noutcoupled versus rf
power, as seen in figures 4.5 and 4.6. We choose this power in order to stay in the “linear regime”
and outcouple about 20% of the atoms.

which I’ll explain how we obtain in section 4.3.1.) The data show a monotonic decrease of thecontact with increasing temperature from a maximum value of 3.3 N kF . For T /TF = 0.16, at theedge of our experimentally attainable temperatures, we observe a sharp decrease of the contact
to about 2.6 N kF . In figure 4.8, we compare our data with several theoretical models [66] and aquantum Monte-Carlo (QMC) simulation [68]. The many-body theories are in the framework of
the t-matrix approximation, differing by their choice of the diagrammatic expansion, the particleparticle propagator, and the self-energy. For T /TF > 0.4, the differences between the theoreticalmodels are small, and the predictions all lie within the uncertainty of the data. As expected, at
higher temperatures (T /TF > 1), we also find good agreement with the virial expansions [66] (seeinset of figure 4.8). For T /TF < 0.4, our data do not agree fully with any of the many-bodytheories. 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 than below, which may be consistent
with observed sharp decrease near T /TF = 0.16. We note that the predicted Tc /TF has someuncertainty, as indicated by the shaded region in figure 4.8. The non-self-consistent t-matrix model(G0 G0 ) predicts an enhancement of about 50% in the value of the contact around Tc [65], which
the data do not show. We also do not observe an increasing trend in the contact for T > Tc , in

58
contrast to a recent QMC simulation [68].
Thermometry of a strongly interacting gas is not trivial, and different groups have used various techniques, including thermometry with a minority component [4], measurement of the energy
versus entropy relation [69], and an empirical temperature extracted from fitting the cloud to a
Thomas-Fermi distribution [48]. We base our thermometry on a measurement of the release energyof the gas and the recently reported equation of state measured by the MIT group [36]. 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 [69],we are left with only the temperature, T , as a free parameter in the calculation of the release energy.
We estimate that the one sigma uncertainty in the temperature is 5%. When reporting T /TF in
figure 4.8, we use TF = EF,avg /kB . Appendix A includes the details of calculating the temperatureusing this technique.

4.3.1

Obtaining the in-situ density distribution

The contact is an extensive property with units of inverse length; it is naturally normalized
by N kF , as seen in equation 4.13. The rf detuning is often given in units of EF . However, aquestion which arises is how to define kF and EF in our experiment. For a harmonically trappedgas, EF is defined in terms of the trap parameters EF,trap = ~(ωr2 ωz )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),2~n: EF,hom = 2m
(6π 2 n)2/3 . In our experiment, as we increase 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 4.4 and 4.8, is therefore the average of the local (homogeneous) Fermi
R~2
energy: EF,avg = 2mN
P (r)n(r)[6π 2 n(r)]2/3 d3 r, where P (r) is the detection probability afterpR
optical pumping, and Np = P (r)n(r)d3 r is the number of atoms probed.
We measure the in-situ density distribution, n(r), in order to calculate the average kF and
EF of the probed 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 [70, 27]. The solution for

59ao u r d a ta ( r f s lin e s h a p e s )G 0G 0,G P F ,G G

4

V ir ia l 2 ,V ir ia l 3 ,Q M C ,E N S

C /(N k F )

3

2

1

0

0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

T /T FbT co u r d a ta ( r f s lin e s h a p e s )G 0G 0,G P F ,G G

4C /(N k F )Q M C ,

E N S

2

3

0 .0

0 .2

0 .4

0 .6

T /T F
Figure 4.8: The contact of a nearly homogeneous sample, versus T /TF at unitarity. We
take the rf line shapes for clouds at unitarity and various temperatures and extract the contactfrom the high momentum part. We use our spatially selective imaging to look at only 30% of thespin flipped atoms, which come from the center of our trap. The contact is normalized by the
total number of atoms and kF , which we find by measuring the in situ density distribution andusing our optical pumping model to calculate the average density of the probed atoms. We find
T /TF using the thermometry technique described in Appendix A. We present our data as black
points and compare them to the three theories from figure 4.1 [66, 64, 65]. We also include the
2nd and 3rd order virial expansions [66], a quantum Monte-Carlo calculation (QMC) [68], and thecontact extracted from a thermodynamic measurement done at ENS [4]. The error bars on our data
represent one standard deviation. a. Our data show good agreement with the virial expansion athigh temperatures, where the virial expansion is expected to be valid. b. The shaded area marks
the superfluid phase transition, with some uncertainty in its exact position (Tc /TF = 0.16 − 0.23)[64]. The differences in the theories are most obvious near the superfluid phase transition, and in
this region, it is clear that the data do not fully support any one theory presented here.

60
the continuity equation with harmonic confinement with a time-dependent trapping frequency ω(t)is self-similar with the following scaling transformation: ri (t) = bi (t)ri (0), where ri is the spatial
coordinate (i = x, y, z), and bi (t) obeys the equation:b̈i (t) = −ωi (t)2 bi (t) +ωi (0)2,bi (t) [bx (t)by (t)bz (t)]γ(4.14)

with the initial conditions bi (0) = 1 and ḃi (0) = 0. The constant γ is the characteristic exponent
in the equation of state µ(n) ∝ nγ , where µ is the chemical potential and γ = 2/3 at unitarity. Fora sudden turn off of the trap, ωi (0) is the trapping frequency along the i axis, and ωi (t > 0) = 0.
In figure 4.9, 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 equation 4.14 with ωr = 2π × 226 Hz
and ωz = 2π × 19 Hz (which were measured independently), which agrees very well with thedata. We find that after 4 ms of expansion the finite resolution of the optical imaging system does
not affect the extracted 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 to be 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 = ξ −3/4 , where ξ at unitarity is a universal
constant that relates the chemical potential to the Fermi energy: µ = ξF [71]. From the measured
density ratio at a temperature of T /TF,trap = 0.15, we extract a value of ξ = 0.40 ± 0.05, which isconsistent with recent determinations of this universal constant by other groups [36, 4].

61
12
radialaxial
10

σ(t)/σ(0)

8
6
4
2
0
0

1

2

3

4

5

6

7

Time of flight [ms]Figure 4.9: Hydrodynamic expansion at unitarity. We start with a weakly interacting gas with
∼ 90000 atoms per spin state at T /TF = 0.12 and ramp adiabatically to the Fano-Feshbachresonance field. We fit the cloud with a Thomas-Fermi distribution after a variable expansion timeand extract the rms widths, σ(t), in the radial and axial directions. For the data, we deconvolve themeasured width with a Gaussian point spread function with an rms width of 2.9 µm, to accountfor 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 thenumerical solution of the hydrodynamic equation.

4.4

Modeling our results
We obtain P (r), the probability to detect an atom after applying spatial selectivity, using

the model of the optical pumping presented in section 3.4. We assume that atoms that scatter asingle 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 a given n(r), the propagation model gives
us P (r) after the consecutive application of the two hollow light beams.
In figure 4.10, 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 4.10 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 are62a

D a taG P F
G G
G 0G 0

2 .5

2 .0

C /(N

p

k F ,tra p )

3 .0

1 .5

1 .0

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

0 .8

1 .0

fr a c tio n p r o b e d

b

2 .5

2 .0

C /(N

p

k F,avg )3 .0

D a taG P F
G G
G 0G 0

1 .5

1 .0

0 .0

0 .2

0 .4

0 .6

fr a c tio n p r o b e d
Figure 4.10: Contact versus the fraction of atoms probed for a gas with T /TF = 0.46 at thecenter of the cloud. a The measured contact (squares), normalized by the trap kF , is comparedto 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 islargest. b Here, the contact is normalized by the average kF of the probed atoms (squares) and iscompared to theoretical predictions of the homogeneous contact at the average T /TF (lines).

1
model =
calculated by Ctrap
Np kF,trap

Rmodel [T /T (r)]k (r)d3 r, with C model (T /T ) being the
P (r)n(r)Chom
F
F
F
hom

model prediction for a homogeneous contact (normalized to N kF ), TF (r) = EF (r)/kB is the local

63
Fermi temperature, and kB is the Boltzmann constant. We find good agreement of the data withthe models.
In figure 4.10 b, we plot the contact divided by the average local kF , defined in the same way
as in figure 4.8. For comparison, we also plot theory predictions for the homogeneous contact atmodel (hT /T i), where the notation hi stands for density-weighted averaging.
the average T /TF , ChomF
model (hT /T i) ≈ hC model (T /T )i. When the
A reasonable criterion for homogeneity is when ChomFF
homfraction of the atoms probed is less than 30% we find that this approximation holds to better than2%. 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% agreewith theory predictions for a homogeneous gas.

4.4.1

The effect of the remaining density inhomogeneity

In figure 3 in the paper, we compare our data to the predictions of several theoretical models.
Here we show that the effect of the remaining density inhomogeneity of the probed sample on thetheory 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[T /TF (r)]kF (r)d3 r ,P (r)n(r)Chom(4.15)

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 different models is shown in figure 4.11.
The graph demonstrates that (with 30% of the atoms probed) the effect of the remaining density
inhomogeneity on the contact is negligible, and hence theories for the homogeneous contact can becompared directly to the data.

64

4

C /(N k F )

3

2

G
1

0

G
0

G P F
G G
0

0 .0

0 .2

0 .4

0 .6

T /T F
Figure 4.11: 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 effect of the remaining density inhomogeneity on the contact data can beneglected.

4.4.2

The optical pumping model

To calculate the spatially dependent probability that atoms are probed, P (r), we use amodel for the optical pumping by the hollow light beams. In Ref. [43] we have introduced such amodel, which was also tested experimentally. The calculated P (r) along with n(r) is used in our
determination of the average quantities (density, kF , TF , EF and the contact) of the probed atoms.
We show here that the results presented in this paper are not sensitive to the details of the opticalpumping model.
To show this, we introduce a second, much more simplified model, which we refer to as the“sharp cut” model. In the sharp cut model, we assume that atoms are left for probing only if theirposition (x, y, z) satisfy x2 + y 2 < R12 and z 2 + y 2 < R22 , where R1 and R2 are set to reproduce thefraction of atoms probed after the application of one or both of the hollow beams. In figure 4.12,
we present the homogeneous contact, similar to figure 3 in the paper, but using the two different
models. As can be clearly seen in the figure, the results of the two models are essentially the same,

65
4 .0
f u ll r e m o v a l m o d e ls h a rp c u t m o d e l

3 .5[N k F ]

2 .5

C

3 .0

2 .0
1 .5
1 .0

0 .0

0 .5

1 .0

T /T F
Figure 4.12: The homogeneous contact at unitarity as function of the temperature, using the full
removal model (squares) and the sharp cut model (triangles). The good agreement of the twomodels shows that the exact shape of the detection probability, P (r), does not affect the contact
data. This is because the central part of the trapped gas is nearly homogeneous, and the data wastaken probing the central 30% of the atoms.

1 0 5 0 0
1 0 0 0 0

E F /h [H z ]9 5 0 09 0 0 0
8 5 0 08 0 0 0f u ll r e m o v a l m o d e ls h a rp c u t m o d e l

7 5 0 07 0 0 00 .0 00 .2 5

0 .5 00 .7 5

T /T F
Figure 4.13: The average Fermi energy with ∼ 30% of the atoms probed, using the full removal
model (squares) and the sharp cut model (triangles).

which demonstrates that the details of the calculation of P (r) are not important when the fractionof atoms probed is as small as 30%. In figure 4.13, we show the average Fermi energy for the data,

66
where again the two models give consistent results.

4.5

Conclusions
The results presented here are the first measurement of the temperature dependence of thecontact for a nearly homogeneous Fermi gas. The data show good agreement with theory at“high” temperatures—T /TF > 0.4 or so—but also have a sharp feature near where the superfluid
transition occurs. This feature was not seen in previous contact versus temperature measurements
presumably because averaging the signal over a spread of densities within the trap smoothed it out.
Furthermore, near the phase transition no single prediction fully agrees with the data.
It is our hope that this data will prove useful for theorists attempting to understand theunitary Fermi gas. The contact is a useful parameter for many reasons; not only is it straightforward
to measure in experiment, the Tan relations make it fairly straightforward to calculate from theory,
too. In particular, equation 4.6 shows that the contact can be directly taken from the dependence
of energy on scattering length, which is often calculated in theory.
Since no theory fully predicts the behavior that we see, we cannot say that our data supports
one picture of the unitary Fermi gas over another. While an important and useful thermodynamic
parameter, it is also not clear how the contact can connects the state of the system, particularlyaround the superfluid phase transition. In particular, it would be interesting to know more about
the so-called “pseudogap regime” in the normal state in the crossover, just above the criticaltemperature. In the next chapter, I will present our efforts to directly probe the nature of the normal
state using momentum-resolved rf spectroscopy. In that work, we develop a more detailed pictureof a Fermi gas in the crossover, and we find a surprising connection between our photoemission
spectroscopy results and the contact.

Chapter 5

The occupied spectral function of a homogeneous gas in the normal phase

This chapter describes our work measuring the local properties of a strongly interacting Fermi
gas in the normal phase of the BCS-BEC crossover. While the ground state of the Fermi gas in the
crossover is well established and studied, the nature of the normal state—whether it can be thoughtof as a Fermi liquid with well-defined quasiparticle excitations or whether is exhibits signatures of
uncondensed pairs, as expected for the pseudogap regime—is still a topic of ongoing research.
We measure the spectral function of our atoms using atom photoemission spectroscopy (PES)(realized with momentum resolved rf spectroscopy), which is a technique developed by our groupin 2008 [1, 72]. This technique is closely analogous to a condensed matter technique used to probehigh-Tc superconductors. With PES, we can measure the probability to find atoms at specificmomenta and energies; this is called the spectral function. The energy and momenta dispersion
we measure can tell us about the nature of the gas and the state of individual atoms (i.e. paired,unpaired, and so forth). Previously, our lab has measured spectral functions for a gas in thenormal phase, which showed evidence of pairing [73, 74]. However, these measurements came froman inhomogeneous sample, and the results were therefore difficult to quantify without relying on
theory. This consideration, and the simultaneous measurements of the equation of state in Parisand Tokyo that agreed with a Fermi liquid description of the normal phase [5, 35] convinced usthat PES on a homogeneous sample was necessary to understand our system.
In this chapter, I discuss PES measurements that used our spatially selective imaging technique. This turns out to be essential to our ability to understand and quantify our results. The

68
work in this chapter has been published in Ref. [75].

5.1

The normal state of the BCS-BEC crossover
In section 1.2.1 of this thesis, I introduced the BCS-BEC crossover as the theory describing

the ground state of a system of interacting fermions. For weak attractive interactions, the gas atzero temperature is a superfluid of paired fermions that behave like Cooper pairs. For increased interactions, the gas remains a paired superfluid at low temperatures, and for very strong interactions,those pairs resemble bosonic dimers in a molecular BEC. [19, 27, 76, 77, 78]
But what happens to the normal state (i.e. the gas in the non-superfluid regime at finite
temperature) is also a critical part of the BCS-BEC crossover. On the BCS side of the crossover(weak attractive interactions), the gas should resemble a nearly non-interacting Fermi sea. If theinteraction strength is adiabatically swept across the Fano-Feshbach resonance, on the far BEC
side of the crossover the atoms will form uncondensed, bosonic Fano-Feshbach molecules. How thegas evolves from an ensemble of fermions to an ensemble of bosons is extremely interesting and is
the subject of ongoing research.
A well-established and powerful paradigm for describing systems of interacting fermions is
the theory of Fermi liquids, which was developed by Landau in 1956 [79, 80]. It postulates thateven in the presence of strong interactions, the system retains a Fermi surface and has low energy
excitations that are long-lived, fermionic, and nearly non-interacting. The effect of interactions isincorporated into renormalized properties of these quasiparticle excitations, such as an effective
mass, m∗ , that is larger than the bare fermion mass, m, and a spectral weight, or quasiparticle
residue, that is between zero and one [80].
While Fermi liquid theory is extremely successful in describing a wide range of materials, it
fails in systems exhibiting strong fluctuations or spatial correlations. Understanding the breakdown
of a Fermi liquid description is an outstanding challenge in strongly correlated electron physics [81].
If the normal state in the crossover can be described as a Fermi liquid for some range of interaction
strengths, there must be an interaction strength at which it fails, either before or as the atoms form

69
localized dimers.
The question of how the Fermi liquid paradigm breaks down in the normal state of the Fermi
gas in the crossover is related to the prediction of a “pseudogap” phase, where incoherent manybody pairing occurs for strong interactions and just above the transition temperature TC [77].
This pseudogap phase has bosonic (pair) excitations, in contrast to the fermionic excitations of aconventional Fermi liquid. However, unlike true, tightly bound bosonic molecules that would befound in the BEC limit, these bosonic pairs have a binding energy that is on the order of or lessthan the Fermi energy, and the pairs are as large as or larger than the average interparticle spacing.
In the pseudogap region near unitarity, pairing occurs between atoms with similar momenta close
to the Fermi momentum, ~kF , and the pairing is facilitated by the presence of a Fermi surface andFermi statistics, a very different picture from the two-body dimer.
While much experimental investigation into the nature of atoms in the normal phase of thecrossover had been carried out prior to the work presented in this chapter, no consensus had beenreached. In experiments that probed the strongly interacting gas in the middle of the crossover,Fermi-liquid-like behavior was observed in thermodynamic [35, 4] and spin transport properties[82]. Meanwhile, evidence for pairing above TC was reported in photoemission spectroscopy (PES)
measurements [73], which reveal the distribution of single-particle energies and momenta in a manybody system [1, 83]. Interpretation of these data has been controversial, with a Fermi liquid theory
and a pseudogap theory each separately argued to agree with the observations [84, 74, 85]. Issues
raised include the fact that the PES measurements probed a trapped gas, where averaging over the
inhomogeneous density can obscure the intrinsic physics [84, 86], and that thermodynamics measurements are relatively insensitive to a pseudogap compared to spectroscopy. Thus, the questionof how a Fermi liquid evolves into a Bose gas of paired fermions in the BCS-BEC crossover, andhow the Fermi liquid description breaks down, remains open.

70

5.2

Photoemission spectroscopy
We approach this question of the nature of the normal state by probing the energies andmomenta of the atoms in our system above TC and at various interaction strengths throughout thecrossover. We combine our new technique of imaging a uniform density portion of our gas with thetechnique of atom photoemission spectroscopy (PES) to measure the occupied part of the spectral
function of a homogeneous gas.
Atom photoemission spectroscopy was developed in our lab as an analog to angle-resolved
photoemission spectroscopy (ARPES) used to probe electronic systems. ARPES uses the photoelectric effect, in which photons incident on a sample can kick out electrons from a material. If theenergy and velocity of both the initial photon and the ejected electron are known, conservation ofenergy and momentum will reveal the energy and momentum of the electron in the sample. By
varying the initial photon angle with respect to the sample, ARPES can be used to measure theprobability that an electron will be found in a particular energy and momentum state, which yields
the occupied part of the electrons’ spectral function. A diagram of the ARPES technique is shownin figure 5.1.
Similarly, our goal is to take our strongly interacting atoms and measure the probability
to find an atom at a particular energy and momentum. As with electrons in a solid, the shapeof this dispersion will tell us about the microscopic details of the system—paired, unpaired, noninteracting, liquid-like, and so on. In atom PES, the “photo-ejection” occurs when a pulse of rf
light outcouples atoms from one of the interacting spin states to a third spin state that is only
weakly interacting with the other two. Simultaneously, the confining optical potential is switched
off. These steps allow us to project some of our interacting atoms very quickly onto non-interacting
states in free space, just as the photo-ejection of electrons does in ARPES. By allowing the noninteracting atoms to expand for some time of flight (TOF), we are able to measure the distributionof velocities/momenta in the ejected atoms. One difference between atom PES and ARPES is that,since there is no underlying lattice and the inter-atomic interactions are engineered to be s-wave,

71

Figure 5.1: Angle resolved photoemission spectroscopy (ARPES). In ARPES, a photon
incident on a surface kicks out an electron (right), which is then collected by a detector after sometime of flight (left). The velocity of the ejected electron and its angle with respect to the incident
beam are recorded. Taking these and the energy of the incident photons into account, researchers
can solve for the energy and transverse momentum of the electrons in the crystal. This diagram isreproduced from Ref. [11] and the Stanford Shen Laboratory website [87].

the distribution of velocities for the atoms is spherically symmetric. Figure 5.2 shows a diagram of
the atom PES technique. 1

5.2.1

The experiment

We begin this experiment by preparing a gas of 40 K atoms in an equal mixture of two spinstates, |9/2, −9/2i and |9/2, −7/2i. The temperature of the atoms after evaporation is T0 =(0.16 ± 0.02)TF with 80, 000 to 120, 000 atoms per spin state. The atoms are held in an optical
dipole trap, and the trapping frequencies at the end of evaporation are 243 Hz and 21.3 Hz in theradial and axial directions, respectively.
We use the Fano-Feshbach resonance between these two spin states to control interactions[19, 73]. The final stage of evaporative cooling is performed at a magnetic field of 203.3 G wherea ≈ 1100a0 (a0 is the Bohr radius), after which the field is swept linearly in 50 ms to the final value
where we carry out the PES measurement (see figure 5.3).
1

An alternate spectroscopy technique to measure occupied spectral functions was proposed prior to the development of the “atom PES” presented here [88]. This technique uses two-photon Raman transitions instead of single
photons of MHz frequencies to transfer atoms to an unoccupied Zeeman state.

72a|-9/2i|-7/2istronginteractions

hollow beams
selection|-5/2irf
photonE = h̄νbhν0EFballistic expansion after
release from trap

CCD camera

absorptionbeamEn (k , E )k2h(ν0 +ν)EF

0k

Figure 5.2: Atom photoemission spectroscopy (PES). a. In our PES experiments, we startwith an equal mixture of atoms in the |9/2, −9/2i and |9/2, −7/2i states, with a Fano-Feshbachresonance controlling their interactions. An rf photon moves some atoms from the |9/2, −7/2i state
to the |9/2, −5/2i state, which does not interact with the first two states. Nearly simultaneouslywith the rf pulse, the hollow light beams are pulsed and the trapping potential is turned off. The
atoms expand and fall during time of flight, and we take an image of only those atoms selected by
the hollow beams. b. The signal that we measure from PES is constructed through energy andmomentum conservation. By measuring the momentum of the outcoupled atoms (green line), we
know the momentum of the atoms in the trap, and we can use the rf photon energy to also get theenergy of the atoms in the trap (red line). The shape of the measured dispersion (red line) tells us
something about the state of the interacting system (the back-bending seen here can be indicativeof BCS-like pairing).

203.3 G

50 ms
hollow beams(50 μs)absorption
imaging, after
5 ms

rf pulse (300 μs)

trap release202.7 G ≥ Bfinal ≥ 201.8 G

Figure 5.3: PES sequence timing.

It is important to note that this field sweep is adiabatic, which means the gas ideally stays
at constant entropy but not necessarily constant temperature. T0 , the temperature of the atoms at

73
203.3 G before the final field sweep, was chosen such that the final temperature is just above TC
after sweeping to the Fano-Feshbach resonance at 202.20 G [73]. From our previous measurementsof the release energy at unitarity [49], we estimate that the temperature after a sweep to unitarity
increases from T0 = (0.16 ± 0.02)TF to (0.18 ± 0.02)TF . Using the pair projection technique from
Ref. [25], we verified that for all interaction strengths the condensate fraction is less than 1%. 2
The atom PES measurement uses an rf pulse with a Gaussian field envelope that has an rms
width of 51 µs (17 µs), and a total duration of 300 µs (100 µs), for ν < 34 kHz (ν ≥ 34 kHz)(see figure 5.3). We use shorter pulses at higher ν to minimize effects due to the motion of thespin-flipped atoms during the rf pulse. The rf detuning ν is given with respect to the resonanttransition frequency ν0 , which is measured for a spin-polarized gas. For each PES data set, we takedata for the same 25 values of ν, each of which is repeated three times. The rf power is increased forlarger ν while keeping the fraction of spin-flipped atoms less than 40%. As we did when measuring
the contact (chapter 4), we measure the dependence of the number of spin-flipped atoms and scale
the measurements done at different rf powers to correspond to a common level (see Appendix B
for more details).
Immediately after the rf pulse, and exactly at trap release, we pulse on the hollow-core
light beams, which propagate perpendicularly to each other and intersect at the center of thecloud [43, 49]. The frequency of these beams is chosen to optically pump atoms from the |9/2, −5/2istate into the upper hyperfine manifold (F = 7/2), where they are invisible to our imaging (seefigure 5.4). Both the rf pulse and the spatially selective optical pumping, which is pulsed for 40 µs,
are completed in a timescale that is short compared to motion of atoms in the trap. We typicallyprobe 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 [43, 49]. As describedin chapter 4 and Ref. [49], 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 state2

Measuring the temperature of an interacting Fermi gas away from unitarity is thus far beyond the limits ofwhat we know how to do. At unitarity, we can rely on other groups’ measurements of the equation of state[4, 5, 36].
However, this measurement is only valid if the gas is at a single (kF a)−1 , which only happens if a → ∞.

74
to the |9/2, −9/2i state with two short rf π-pulses. This procedure enables us to image the atoms
on the cycling transition, which improves the signal-to-noise ratio.
F = 52M = − 23

42 P3/2

hollow beams
transition

F = 72
F = 92M = − 92

M = − 72

M = − 52

42 S1/2

rf photon, νFigure 5.4: Schematic level diagram of 40 K energy states relevant to the experiment.
The gas is prepared in an equal mixture of two hyperfine states. These two states, that canbe made strongly interacting near a Fano-Feshbach resonance, are |9/2, −9/2i and |9/2, −7/2i,
where the first number is the total atomic spin F and the second number is its projection M .
For photoemission spectroscopy, an rf pulse drives the transition from the |9/2, −7/2i state to the|9/2, −5/2i state, which is weakly interacting with the two other spin states. Also shown are thetransition driven by the hollow-core light beams (straight red arrow) and the spontaneous emission
transition with the largest Clebsch-Gordan coefficient (curvy red arrow).

5.2.2

The PES signal

Each PES data set consists of 75 images of the atoms each taken after 5 ms time of flight.
One such image is displayed in figure 5.5b, which corresponds to the detuning shown in red on
the rf line shape (figure 5.5a). Images taken for the same detuning are averaged together, andthe radial momentum distribution is extracted by performing an azimuthal average and an inverse
Abel transform on the image. The inverse Abel transform maps the 2D projection onto a 3D imageof the full cloud, and the azimuthal average sums over the azimuthal direction to get the radial
momentum distribution, n(r).
A full PES dispersion plot is constructed from 25 momentum distributions, each at different

75

Γ(ν)

a

b

0.4

−
→
k

0.3
0.2
0.1
0
0

2

4

ν [EF/h]

6

8

many
atoms

−2

0

−2

−4
0

no
atoms

c

Single−particle energy (E/E F)

2

0.5

1

1.5

2

k/kF
Figure 5.5: Creating the PES plots.a. A line shape showing the Rrelative fraction of atoms∞
outcoupled at each rf detuning. The line shape is normalized such that −∞ Γ(ν)dν = 1/2, due tothe 50% − 50% spin mixture. b. An image of an atom cloud, taking for the detuning marked in red
on the line shape (ν ≈ 6.5 kHz). In our analysis, this image will be averaged with 2 others at thesame detuning, and the momentum distribution will be extracted from the atoms’ TOF positions.
c. The contribution of a single image to a full PES dispersion plot. The image in b is shown after
extracting the momentum distribution. The color corresponds to the fraction of the atoms having
a specific momentum. Each frequency on the line shape (a) will add a parabolic strip of dispersion
data to the plot. Each strip is shifted from the bare transition (the white line, also called thefree-particle line) by −~ν.

76
detunings. Using energy conservation, the detuning of the rf and an atom’s final momentum canbe used to find the energy of the atom before “photo-ejection.” Each detuning corresponds to aparabolic slice of the plot:
Eafter spin flip = Ebefore spin flip
~2 k 2= hν + E;
2m

(5.1)

where E is the energy of a single atom in the strongly interacting sample. The distribution of
momenta along that slice is shown by the color. (Note that positive detunings yield E = −~ν forzero momentum.) Figure 5.5c shows the single slice of the PES dispersion plot that comes fromthe image shown in 5.5b (ν ≈ 6.5 kHz). The final signal from PES is displayed in a 2D plot created
when data from all frequencies are combined.
The PES signal, I(k, E), is proportional to the probability to find an atom in the interacting
gas at a particular energy, E, and momentum, k. This signal is proportional to k 2 A(k, E)f (E),where A(k, E) is the atomic (single-particle) spectral function [1, 89], and f (E) is the Fermi function. As in chapter 4, we use the contact extracted from the rf line shapes when we normalize theR∞ R∞
PES data so that −∞ 0 I(k, E)dk dE = 1. Since the PES data span a finite range of E and k(typically, −4 < E < 3 and k < 4), knowing the behavior of Γ(ν) for large ν allows us to accountfor signal beyond the range of the data in our normalization of the PES data.
Since we eject, or spin flip, atoms from the interacting gas, we only measure the occupied partof the spectral function. If we wanted to measure unoccupied states, one would need a different
technique that somehow populated those higher energy states; this might involve the injection of
non-interacting atoms into the strongly interacting gas (inverse PES) or the forced excitation of
atoms into the unoccupied bands. 3
Figure 5.6 shows I(k, E) measured for 11 different values of (kF a)−1 in the BCS-BEC
crossover at a temperature above TC . The color encodes the likelihood that an atom will bein the corresponding energy/momentum state. The y-axis shows the energy in units of the Fermi
3

The development of a technique to measure the unoccupied higher energy states would be a great project for
the near future!

E (EF)

77
2

2

2

2

0

0

0

0

−2

−2

−2

−2

(kF a)−1 = −0.88 −0.68

−4

0

1

2

−4

0

1

k (k )−0.66
01

k (k )F

E (EF)

2

−4

2

−4

F

2

0

0

0

0

−2

−2

−2

−2

0

1

2

−0.08
0

1

2

−4

0.1
0

1

2

k (kF)

k (kF)

k (kF)22

2

2

k (k )

2

−4

1

F

2

−0.25

0

k (k )

F

2

−4

−0.43

−4

0.26
0

1

2

k (kF)
100

E (EF)

10−1
0

0

0
10−2

−2

−4

−2

0.42
0

1

k (kF)

2

−4

−2

0.61
0

1

k (kF)

2

−4

10−3

0.82
0

1

2

10−4

k (kF)

Figure 5.6: Atom photoemission spectroscopy (PES). In these plots of PES data, the color
encodes the probability distribution of atoms at a given E and k in the strongly interacting gas.
We estimate that the error bar of (kF a)−1 is 0.03. E = 0 is the energy of a free atom at rest andthe white line shows the free-particle dispersion E = k 2 . Here, E is in units of EF , k is in units of
kF .

78
energy, EF , and the x-axis shows the momentum in units of the Fermi wavevector, kF . The whiteparabolas show the dispersion of non-interacting atoms (the “free-particle line”). Zero energy was
found by measuring the transition energy of a spin polarized gas of |9/2, −7/2i atoms.
EF and kF , which scale the axes of the figure 5.6, are calculated from the average density ofthe probed gas. As when we measured the contact in chapter 4, the average density is determinedfrom the central part of the in-situ density distribution of the gas (see section 4.3.1). For thatexperiment, we took a picture of our gas after a small TOF expansion, keeping the gas at the FanoFeshbach resonance. Expansion of a gas at unitarity is hydrodynamic and well characterized byprevious measurements [70, 27]; this allows us to back out the density distribution in trap from the
expanded cloud image. Throughout the BCS-BEC crossover, the expansion is also hydrodynamic,
although the rate of expansion differs from that at unitarity. We deduce the in-situ density distribution by solving the Euler equation with a scaling solution that assumes the chemical potential is
proportional to nγ , where n is the density, and γ depends on the interaction strength (see section
4.3.1 and Ref. [27] for the equations describing this scaling). For the purpose of this work, we use
γ as approximated by the Leggett ansatz [27]. After finding the in-situ density distribution, we use
our model of donut beam removal (presented in section 3.4) to find the density distribution of theselected atoms and take the average density-weighted density.
The PES data in figure 5.6 show several trends as a function of (kF a)−1 . For the point at(kF a)−1 = −0.88, which is the furthest on the BCS side, I(k, E) shows a maximum centered on
the free particle line with a relatively narrow spread in energy for any given k. As interactions
increase, the maximum signal in I(k, E) shifts down to more negative energies, and the spread inenergy gets larger. By the time (kF a)−1 = 0.82, which is the data set furthest on the BEC side, thespread in energy is quite large, and the data appear very different from the free particle dispersion.
Previous trap-averaged atom PES data showed back-bending and large energy widths [73, 74].
These features are also apparent in the nearly homogeneous data (shown in Appendix C). However,these data are more amenable to quantitative analysis because kF (and EF ) are approximately
single-valued across the sample. The remainder of this chapter explains how we implement such

79
a quantitative analysis, the results from this analysis, and the greater understanding it brings us
about the nature of the normal state.

5.3

The two component fits
Similar to the analysis done in electron systems, we use a two-mode function to describe the

PES signal [11]:I(k, E) = ZIcoherent (k, E) + (1 − Z)Iincoherent (k, E) ,(5.2)

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
spectral weight. When these two parts (defined below) are combined, the resulting dispersion can
exhibit back-bending.

5.3.1

The quasiparticle signal

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 . In practice, this resolution is difficult
to improve. It is set by the number of 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 toprobe momentum states. We convolve equation 5.2 with a Gaussian function that accounts for ourenergy resolution before fitting to the data.
To describe quasiparticles, we use
i−1
h


− (πm∗ T )3/2 Li3/2 − exp −ET0 +µ
Icoherent (k, E) = 4πk 2 · δ(E − ∗ − E0 ),m
exp E−µ
+
1
T
k2

(5.3)

which consists of a quadratic dispersion of sharp quasiparticles multiplied by a normalized Fermi
distribution (δ is the Dirac delta function, and Li is the polylogarithm function). We include as
fit parameters, a Hartree shift E0 , effective mass m∗ , chemical potential µ, and temperature T .
Here, energies are given in units of EF and m∗ in units of m. This description of Fermi liquid

80
quasiparticles is typically only used very near kF and for T approaching zero, whereas we fit todata for a larger range in k and with temperatures near 0.2 TF (just above TC ). The latter isnecessitated 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 [43]. Any increasein quasiparticle widths away from kF will have little effect on the data as long as the quasiparticles
have an energy width less than our energy resolution, which should be the case for long-lived
quasiparticles. Finally, as discussed in figure 5.7, using a quadratic dispersion over a large range of

2αPeak position (EF )k is supported by the data.

2

−1

1

−0.5

0
−1

(kF a)

0
0

0.5

1

1.5

k (kF )

Figure 5.7: Observation of quadratic dispersion. An example for the peak dispersion, which
is determined by finding the peak location for each k, is shown for (kF a)−1 = −0.43 (circles). We
locate the peaks by fitting each EDC separately to a Gaussian plus an offset, where the Gaussian
width is fixed to the experimental resolution. The red line shows a fit to Ak α + B in the range
0.2 < k < 1.5. Inset, The exponent from fits to the peak dispersion (squares). The error bars
indicate ± 1 standard deviation. We find that α is consistent with 2.

In equation 5.3, we assume a quadratic dispersion for the quasiparticle peak, which agrees
well with the data but is only predicted to be valid for k ≈ kF . To empirically test the validity ofthis assumption, we locate the peak positions in the EDCs at each k and fit the dispersion E(k).
To focus on a quasiparticle-like peak, and to reduce the effect of the asymmetric tails, we use a

81Gaussian fit where the width is constrained to be equal to our resolution (h × 2400 Hz) and where
we also include a constant offset as a fit parameter. An example of the resulting peak dispersion for(kF a)−1 = −0.43 is shown in figure 5.7. We can then fit the peak dispersion to Ak α + B to extract
an exponent. The results are shown in the inset of figure 5.7 as a function of (kF a)−1 , where we
restrict this analysis to the region where Z > 0. (For higher (kF a)−1 , the EDCs are increasingly
asymmetric and the peak positions from the Gaussian fits are noisy.) We find that α is consistent
with 2, which matches the quadratic dispersion of Fermi liquid theory.

5.3.2

The “incoherent background”

The second part in equation 5.2 needs to accommodate the remainder of the signal, which
is often referred to as an “incoherent background” in a Fermi liquid description. However, to thebest of our knowledge the functional form of this background has not been determined. In Fermi
liquid theory, the fractional signal in the quasiparticle feature decreases as stronger interactions
cause greater distortion of the Fermi surface, so it is logical that the growth of the background isdirectly connected to the “breakdown” of a Fermi liquid picture.
For fermions with contact interactions, one expects signal at high momentum due to shortrange pair correlations [2, 55, 90]. In addition, the normal state in the BEC limit of the crossoveris a gas of bosonic dimers. Motivated by these two ideas, we use for Iincoherent a function that
describes a thermal gas of pairs. The pairs have a wave function that decays as exp (−r/ R), wherer is the relative distance and R is the pair size [91], and a Gaussian distribution of center-of-masskinetic energies characterized by an effective temperature Tp . This gives
 √ √

q
2 2k −Ep −E+k2
Ep
Ep +E−3k2
8k Tp exp(
) sinhTp
Tp

Iincoherent (k, E) = Θ −Ep − E + k 2, (5.4)2
π 3/2 (E − k 2 )pwhere Θ is the Heaviside step function, Ep is a pairing energy defined by kF R = 2/Ep , andboth Ep and Tp are dimensionless fitting parameters. The full derivation of this formula is given in
Appendix D. While this description of the incoherent piece may not fully capture the microscopic
behavior except in the BEC limit, we find nonetheless that equations 5.2, 5.3, and 5.4 fit the data

82
very well throughout the crossover, after convolution with a Gaussian function that accounts forour energy resolution. For each value of (kF a)−1 , we perform a surface fit to the roughly 300 points
that comprise the PES data I(k, E) for k ≤ 1.5 and E ≥ −3. The reduced chi-squared statistic, χ2 ,after accounting for the seven fit parameters, is between 0.75 and 1.3 (see figure 5.8b.). An example
fit is shown in figure 5.8a, where I show several traces at fixed k for PES data taken near unitarity.

a

1
0.5

k/kF = 0.24

0.5

k/kF = 0.47

0
0.5

k/kF = 0.71

0
0.5

2
1

k/kF = 1.030
−1 −0.5

0
0.5

b
χ2

Spectral weight (arb. units)

0k/kF = 1.26

0

0.5

1

(kF a)−1

0
0.5
0k/kF = 1.5

−2

−1

0

1

2

E (EF )

Figure 5.8: Two-component fits to the atom PES data. a Data (circles) and fits (lines) are
shown for several example traces at fixed k through the PES data at (kF a)−1 = −0.08. These
traces are often called energy distribution curves, or EDCs. Here, the fitting parameters are Z =0.37 ± 0.03, m∗ = 1.22 ± 0.03, T = 0.24 ± 0.02, E0 = −0.33 ± 0.02, µ = 0.19 ± 0.04, Ep = 0.23 ± 0.04,
Tp = 1.09 ± 0.08, where the error margins are for one standard deviation and also include a 5%
uncertainty in EF . bThe χ2 values from fitting our PES data to our two-component function range
from 0.75 to 1.3. For the EDCs shown on the right, the reduced χ2 is 1.2.

5.4

Results of the fits
In figure 5.10, 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E (EF)−4

0

k (kF)

1

2

−4

−3

−3

(kF a)−1 = −0.88

−2

−2

0

0

−1

1

1

−1

2

2

0

1

k (kF)−0.43

Z=0.65F

Z=0.831

k (k )

F

0

−0.43

k (k )

2

−3

−3

1

−2

−2

0

−1

−1

−4

0

0

−4

1

1

(kF a)−1 = −0.88

2

2

2

−4

−3

−2

−1

0

1

2

0

0

−4

−3

−2

−1

0

1

2

1

F

1

k (kF)

−0.25

Z=0.6

k (k )

−0.25

2

2

−4

−3

−2

−1

0

1

2

0

0

−4

−3

−2

−1

0

1

2

1

F

1

k (kF)−0.08

Z=0.37k (k )

−0.08

2

2

−4

−3

−2

−1

0

1

2

0

0

−4

−3

−2

−1

0

1

2

F

k (k )1

0.26
k (kF)

1

Z=0.022

0.26

2

2

−4

−3

−2

−1

0

1

2

0

0

−4

−3

−2

−1

0

1

2

F

k (k )

1

0.42
k (kF)

1

Z=0.036

0.42

2

2

−4

−3

−2

−1

0

1

2

0

0

−4

−3

−2

−1

0

1

2

F

k (k )1

0.82
k (kF)

1

Z=0.029

0.82

2

2

Figure 5.9: Comparing the PES data with the results of the two-component fits. The top row shows seven of the PES dispersion
plots from figure 5.6. The bottom row shows dispersion plots constructed from equations 5.2, 5.3, and 5.4 and the best fit parameters
corresponding to the data above. The fitted value of Z is shown between eE (EF)

2

10−4

10−3

10−2

10−1

0

10

10−4

10−3

10−2

10−1

100

83

84

1

Z

0.8
0.6
0.4
0.2
0
−1

−0.5

0

0.5

1

(kF a)−1

Figure 5.10: Z, the quasiparticle spectral weight The quasiparticle spectral weight decreases
as (kF a)−1 increases. Using a linear fit to the range −0.5 ≤ (kF a)−1 ≤ 0.3, we find that Z vanishesat (kF a)−1 = 0.28 ± 0.02 (dashed line); this marks the breakdown of a Fermi liquid description.
The blue circles come from fits of a large range of data, from 0 to 1.5 kF . The red squares show
the result of restricting the fit to 0.7 to 1.3 kF , and they show a similar trend and slightly larger
error bars.

(positive a), reaching Z ≈ 0.3 at unitarity. Beyond (kF a)−1 = 0.28 ± 0.02, Z vanishes, signaling
the 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 (red points in figure 5.10). We notethat the interaction strength where Z vanishes, as well as the sharpness with which Z goes to zero,are likely to be temperature dependent [92].
The best fit values for the effective mass, m∗ are shown in figure 5.11, where m∗ increases withincreasing 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. [35], but close to
the T = 0 prediction of m∗ = 1.19 from Ref. [86].
The fitted values of the remaining parameters are shown in figure 5.12. The temperaturescharacterizing the two parts of equation 5.2 T and Tp are shown in figure 5.12a. For the quasiparticle
component, T is close to, but somewhat above the estimated temperature of the gas, T = (0.18 ±0.02)TF . We attribute the discrepancy to the remaining density inhomogeneity of the probed

85m∗ (m)

1.4
1.2
1
−1

−0.5

0

0.5

1

(kF a)−1

Figure 5.11: m∗ , the quasiparticle effective mass.
The quasiparticle effective mass m∗ is∗shown for the region where Z > 0. Interactions increase m , and the data (circles) agree surprisingly
well with the theoretical prediction for the limiting case of the Fermi polaron (solid line) [93, 4].
Restricting the fit to EDCs close to the Fermi surface produces similar results with increased error
bars (red squares).

portion of the trapped gas, which can broaden sharp features. In particular, when applying our
PES measurement technique to a weakly interacting gas, we find that the resulting momentum
distribution has a step at the Fermi surface whose width typically corresponds to T ≈ 0.25TF .
For the incoherent component, Tp , which in the model characterizes the spread of center-of-massenergies of pairs, is around 0.75TF . This is much higher than the temperature of the gas andsuggests that the data cannot be simply interpreted as coming from weakly bound bosonic moleculesin thermal equilibrium with a Fermi gas of atoms.
2~
Figure 5.12b shows the second fit parameter in Iincoherent , which is a binding energy Ep = mR
2

that sets the size of the pairs R. For the data farthest on the BEC side of the crossover, Ep
2~
approaches the two-body molecule binding energy, ma
2 (solid line in figure 5.12b). In the crossover,

Ep is larger than the binding energy for two-body molecules, as one would expect for many-bodypairs.
The two remaining fit parameters, E0 and µ, come from the quasiparticle part of the fitting
function and are shown in figure 5.12c and d, respectively. The Hartree energy shift E0 is negative

86
and has a magnitude that increases as (kF a)−1 increases, reaching a value of −0.46 ± 0.03EF at
unitarity, in good agreement with several theoretical calculations [94, 86, 95, 96]. Figure 5.12cshows E0 for the full range of (kF a)−1 for our data. Here, one can clearly see the jump in the fit
values and the large uncertainties that appear in the region where the fits return a small Z. This
happens because the quasiparticle component that is fit for (kF a)−1 < 0.2 has vanished. For thisreason, when discussing the fit parameters for the quasiparticle part (m∗ ,T ,E0 , and µ), we focus
on the region (kF a)−1 < 0.2.
The chemical potential, µ, from the quasiparticle part of the fit is shown in figure 5.12d. In
general, one expects µ to decrease as (kF a)−1 increases. The data (blue circles) for (kF a)−1 > −0.5
follow this expected trend, but the points for our two lowest (kF a)−1 values are surprisingly low.
In addition, all the values of µ here are somewhat lower than the predicted chemical potential in
the BCS-BEC crossover [77]. We have found that there is some interdependence of the two fit
parameters µ and T for the quasiparticles. This is illustrated in figure 5.12d, where the red squares
show the result for µ when we fix T = 0.25 in the fits. (The effect of fixing T on the other fit
parameters is to reduce some scatter but is otherwise minimal.) This interdependence, and the factthat T is increased by technical issues (namely, the remaining density inhomogeneity that limits
our resolution in k), may play a role in explaining the lower than expected values for µ.

5.4.1

Comparison to the Fermi polaron case

It is interesting to compare our results with those of Fermi polaron, which is the quasiparticle
in the limit of a highly imbalanced Fermi gas. Schirotzek et al. measured Z = 0.39 ± 0.09 for the
Fermi polaron at unitarity [97], which is similar to our result for the balanced Fermi gas. For thepolaron case, Z also goes to zero in a similar fashion to our results, but farther on the BEC side of
the crossover [97]. This similarity is surprising because we expect a phase transition from polarons
to molecules in the extreme imbalance limit [98, 99], with Z acting as an order parameter [100],
while, in contrast, the balanced Fermi gas should exhibit a continuous crossover. For m∗ , we alsofind that our result is close to the measured effective mass of the Fermi polaron at unitarity [35],

87

1.5

a

Tp
T

1.5

1

EP (EF )T , TP (TF )

1.25

0.75
0.5

b

1
0.5

0.25
0
−1

−0.5

0
(kF a)

0.5

0
−1

1

−1

1

c

0

−0.25

(kF a)

0.5

1

0.5

1

−1

d

0

−0.5

−0.5
−0.75

0

0.5
µ (EF )

E0 (EF )

0.25

−0.5

−1

−0.5

0
(kF a)−1

0.5

1

−1
−1

−0.5

0
(kF a)−1

Figure 5.12: Parameters of the fits to the PES data. a, We extract two temperaturesfrom our fits to I(k, E). The quasiparticle parts of the fits to the PES data give T values (redsquares) that are comparable to, but higher than, our estimated temperature of the gas at unitarity(T = 0.18 ± 0.02). In contrast, the incoherent parts of the fit give Tp values (blue circles) around0.75. Tp can be thought of as an effective temperature that characterizes the energy width of theincoherent part of the spectral function. b, The pairing energy, Ep , comes from the incoherent
part of the fit. At the largest (kF a)−1 , the data (circles) approach the two-body molecular binding
energy (line). c, The Hartree energy shift, E0 , comes from the quasiparticle part (blue circles).
The error bars correspond to ± 1 standard deviation from the fits and 5% uncertainty in EF . Also
shown are the results of a quantum Monte-Carlo calculation [94] (black square) and three theories
by Haussmann et al. [86] (red asterisks), Kinnunen [95] (green triangle), and Bruun and Baym [96](orange diamond). d, The chemical potential µ is also found in the quasiparticle part of the fit. We
show the results of the fit with (red squares) and without (blue circles) the constraint T = 0.25.
The fact that constraining T affects the best fit value for µ reveals an interdependence of µ and Tthat we do not observe for the other fit parameters.

m∗ = 1.20 ± 0.02, and similar to the predicted polaron mass [93, 4] throughout our measurement
range (solid line in figure 5.11).

88

C /N (kF )

12
9
6
3
0

−1

−0.5

0

0.5

1

(kF a)−1

Figure 5.13: The contact parameter. From the PES data, we extract the contact per particle(in units of kF ) for a homogeneous Fermi gas above TC as a function of the interaction strength(kF a)−1 . The contact measured from the tail of the rf line shape, using data for hν ≥ 5EF , isshown in blue circles, while the contact extrapolated from the fits of the PES data is shown in red
squares. Remarkably, even though we limited the fits of the PES data to k ≤ 1.5 and E ≥ −3, aregion with a relatively small contribution of short-range correlations [2, 55, 3, 49], we find that thecontact from the PES fits is consistent with the contact measured from the tail of the momentumintegrated rf line shape. For comparison with the data, we also plot the BCS (dashed black line)
and BEC (dashed magenta line) limits, given by 4(kF a)2 /3 and 4π(kF a)−1 , respectively [55], thenon-self-consistent t-matrix at T = 0 (dotted blue line) and its Popov version at TC (dash-dotted
red line) [65], and the self-consistent t-matrix model at T = 0 (double-dotted green line) [86].
Interestingly, we find that the contact measured above TC agrees well with the T = 0 theories.

The homogeneous contact vs. (kF a)−1

5.5

In chapter 4, we measured the contact of a homogeneous gas at unitarity as a function oftemperature. In this chapter, we also measure the rf line shape, and we can extract the contact asa function of (kF a)−1 from these measurements. As (kF a)−1 increases, short-range correlations are
expected to increase, and so is the contact. Figure 5.13 shows the measured contact per particle,
C/N , in units of kF , as a function of (kF a)−1 . The blue points come from the weight in the rf line
shape tail.

89
We can also extract the contact in another, perhaps surprising, way. In a Fermi liquid
description of strongly interacting atoms, the contact is expected to be valid and finite, and it
must be accounted for by the incoherent part of the spectral function [90]. (The contact cannot
be accounted for by the quasiparticle part, which only includes narrow peaks that are symmetricin energy.) Our particular choice for Iincoherent has the expected form of a 1/k 4 high-k tail in the
momentum distribution [2] and a 1/ν 3/2 large-ν tail in the rf line shape [55], where ν is the rf
detuning (see equations 4.1 and 4.9 from chapter 4). Thus, we can extract the contact directlyfrom the fits the to PES data (figure 5.13, red points). To do this, we use the model and the bestR∞
fit parameters for each data set, to calculate the momentum distribution: n(k) = −∞ Ip (k, E)dE
out to k = 5. We then find the contact from the average value of k 4 n(k) in the region 3 ≤ k ≤ 5.
Remarkably, we find that the contact can be accurately extracted in this way even though we
restrict the fits to k ≤ 1.5. For comparison, 1/k 4 behavior in the momentum distribution wasobserved for k > 1.5kF [3], and the fits to the rf line shape tails are restricted to hν ≥ 5EF .
As a check of the fitting function, we have compared these results with a simple expression
derived from the contact for weakly bound molecules. For a dimer with a wave function φ(r) =q
2 −r/R, the contact [101] is Cdimer = 8π/R. Since the dimer consists of two atoms, the contactRe
per particle is C/N = 4π/R. Expressing this in terms of the dimensionless Ep and multiplying by
the amplitude (1 − Z), we haverEpC
= 4π(1 − Z)
.
N kF
2

(5.5)

We find that the result of equation (5.5) agrees very well with the contact extracted directly fromthe momentum tail of the fitted dispersion.

5.6

Understanding the “incoherent background”
The two-part function that we use to fit and analyzed our PES data is simply the sum of

the expected forms of the spectral function in the BCS and BEC limits. One can argue thatthe “quasiparticle” part has a generic form that should describe Fermi liquid-like quasiparticles

90
regardless of how strong the interactions, provided that the lifetimes of those quasiparticles giveenergy widths which are far below the energy resolution of our spectroscopy. However, the otherpart of the fitting function describes tightly bound 40 K2 molecules with a specific binding energy,
Ep , and a thermal distribution described by Boltzmann statistics. This is a good description fora gas of molecules far from resonance on the BEC side of the crossover. In the center of thecrossover, where any pairs are of the similar size to the interparticle spacing and Fermi statistics
presumably still play a role in pairing, it is unclear whether the choice of this function to describe
the non-quasiparticle signal is justified.
In this section, I attempt to lay out an intuitive understanding of the non-quasiparticle signal
and our choice of functional form for Iincoherent . I also present some arguments justifying our choice
as not only a good fit to our data, but a reasonable physical description.

5.6.1

Iincoherent as large pairs in the crossover

One interpretation of Iincoherent is that it describes many-body pairing in the crossover. The
functional form correctly describes molecules in the BEC limit, and one can think of those molecules
remaining but getting more weakly bound as interactions are decreased. Weakly bound pairs witha very large size are expected for many-body pairing in the superfluid phase in the center of the
BCS-BEC crossover, and this is also a way to describe pairing in the normal phase in a pseudogap
picture.
To understand Iincoherent as describing many-body pairs, we can look at the fit parameter Ep
as giving a characteristic pair size. As seen in figures 5.12 and 5.14a, fitted Ep values from thedata sets taken farthest on the BEC side are consistent with the prediction of dimers with binding~
energy determined by the scattering length, Ep = ma
2 . As the interaction strength decreases, thefitted pairing energy drops to less than half EF , indicating a pair size that is more than twice the~interparticle spacing. The pair size, R, given by Ep = mR
2 , is shown in figure 5.14b.

The divergence of the fit values for R from the two-body prediction could be thought ofas the gradual change from two-body dimers to main-body pairs. Certainly, pairs that are a few

kF R

EP (EF )

91

1.5 a
1
0.5
0
6
4

b

2
0
−1 −0.5

0

0.5

1(kF a)−1

Figure 5.14: Fit results for binding energy and pair size from Iincoherent . a, Fitting to our
two-part function yields results for Ep that are consistent with the two-body prediction in the BEC
limit and deviate from this prediction around unitarity. This is the same plot as presented in figure~
5.12b. b, Ep is directly related to a characteristic pair size through Ep = ma
2 . In the case of only
two atoms, the pair size is predicted to be equal to the scattering length, a. At unitarity, thisprediction diverges and is therefore unphysical. The fit returns pair sizes in the crossover which
are larger than the interparticle spacing, kF−1 .

times larger than the average interparticle spacing, kF−1 , can no longer be thought of as “traditional”
molecules. Molecules that are much smaller than kF−1 are predicted to exist outside of the crossover,for (kF a)−1 > 1. Seeing these small molecules would require data taken farther on the BEC side.
Evidence of a pseudogap and the associated many-body pairing encompasses more than seeing
large pairs. In addition, these pairs must form around the Fermi surface, causing a back-bending in
the dispersion near kF [73]. In Appendix C, it is shown that both our data and the two-part fitting

92function produce the qualitative back-bending effect. However, it is unclear how to quantify thisback-bending from our fit parameters. Additionally, the hallmark of a pseudo-gap is a suppression
in the density of states, which is almost, but not quite, an energy gap between occupied andunoccupied bands. Unfortunately, atom PES only measures the occupied spectral function, and so
we cannot see the pseudogap directly.

5.6.2

Iincoherent as part of a Fermi liquid

Descriptions of a Fermi liquid typically focus on the presence of coherent, long-lived quasiparticles. However, a Fermi liquid as first presented by Landau has another part—an incoherent
background signal with large energy widths that contrasts with the peaked dispersion of the quasiparticles [80].
For interaction strengths less than (kF a)−1 = 0.28, the PES data show some component thatis (relatively) narrow in energy and positively dispersing, which is well described by Icoherent , ourfunction based on quasiparticles. If the gas has some quasiparticles (the presence of which areconsistent with, if not completely proven by, our results), it can be thought of as a Fermi liquid,with the non-quasiparticle signal being the broad, incoherent background described by Landau.
The form of this background signal is not specified, but we know that its growth is linked to theshrinking of the quasiparticle residue and the eventual breakdown of the Fermi liquid. In the BCSBEC crossover, this motivates a background piece, Iincoherent , that describes uncondensed pairs.

5.6.3

Iincoherent and the Contact

Theoretical work done by Schneider and Randeria calculated the spectral function for a Fermi
liquid of atoms and showed that the “background” signal has a high momentum, negative energy
component [90]. This component has a 1/k 4 dependence on momentum and is related to Tan’scontact. We see that our form of Iincoherent has spectral weight in this high momentum, low energy
region and, moreover, has the same form predicted by Randeria.
Our choice for Iincoherent , moreover, as briefly discussed in section 5.5, not only has the93correct form of the tails of the momentum distribution and rf line shape for a gas with a contact(see equations 4.1 and 4.9), but our own best fit parameters when fitting a restricted region of
PES data near kF nevertheless give the correct value for the contact (when compared to the highfrequency tails in rf spectroscopy).

5.7

Conclusions
The results presented in this chapter show that PES data are consistent with a spectral

function that includes two parts, namely a coherent part that corresponds to long-lived fermionic
quasiparticles and an incoherent part that describes large, many-body pairs. These results explain how different observations lead to different 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 andsignificant spectral weight in an “incoherent” part that is consistent with pairing. With the nearlyhomogeneous PES data, we find that Z vanishes surprisingly abruptly and note some similarity toFermi polaron measurements. Comparing the PES data with various BCS-BEC crossover theories
may help elucidate these observations and advance quantitative understanding of the crossover.

Chapter 6

Photoemission spectroscopy as a function of temperature

In chapter 5, I presented our results regarding the evolution of the spectral function in thenormal phase of the BCS-BEC crossover. In addition to these studies, we wanted to see how
the spectral function of a homogeneous gas changes with temperature. This chapter presents ourpreliminary findings.
We took PES data for several temperatures at two interaction strengths, one on either sideof the Feshbach resonance. This data was taken using the sequence and procedures laid out inchapter 5. The data on the BCS side was taken for (kF a)−1 = −0.24(2), and the data on the BEC
side was at (kF a)−1 = 0.23(3). (The error reflects the small variations in (kF a)−1 in the data sets.)

6.1

Temperature in the normal phase of atoms in the BCS-BEC crossover
Looking back to the phase diagram of atoms in the BCS-BEC crossover (figure 1.2), one cansee that the presence of preformed pairs is expected for some interactions and temperatures above
TC , but not for all. There is another temperature, Tpair (often called T ∗ [24, 27]), which signifies acrossover from a region of the phase diagram with pairs to a region without pairs. Consistent withthe gradual increase of pair-like signal as a function of (kF a)−1 in the PES data in chapter 5, Tpair
does not mark a phase transition but instead corresponds to a gradual transition over some range
of interaction strengths and temperatures. We expect that the fraction of pair-like signal in PES
data should depend on the temperature of the gas, as well as on the interaction strength.
A Fermi liquid is also expected to change its character as a function of temperature. The

95
Fermi liquid is a description of gas of interacting fermions at a temperature much lower than the
Fermi temperature. At these temperatures, the deBroglie wavelength of the particles is comparable
to the interparticle spacing; this is the regime of a “quantum” liquid [80]. At higher temperatures,the gas moves out of the quantum regime, and the properties that depend on the quantum natureof the system go away. At this point, we would expect the effective mass of the Fermi liquid toapproach the bare mass and the dispersion to be close to that of an ideal (non-interacting) Fermi
gas. At finite temperature in the quantum regime, Fermi liquid behavior may persist, but thequasiparticle lifetime may decrease. In the PES data, a shorter quasiparticle lifetime should appear
as an increased energy width in the dispersion.
Investigating the dependence of PES signal on temperature is as interesting and challenging
an endeavor as investigating the dependence on interaction strength at T ≈ TC . One particularchallenge is the possible increase of the energy widths of the quasiparticle feature with temperature.
In the T ≈ TC data, we isolated the quasiparticle signal from the background by looking forresolution-limited features (chapter 5). If the width of the “sharp” quasiparticle feature were toincrease to the size of our resolution or above, this approach would no longer work. In order to see
this very interesting behavior, we would need to develop better resolution in our experiment or adifferent analysis technique. In the first part of this chapter, I analyze PES data vs. temperatureby extracting the contact from the rf line shape, as was done in chapter 4. This analysis is morestraightforward, as it is not affected by the energy widths in the data, and it produces the firstmeasurement of the homogenous contact as a function of temperature at interaction strengths otherthan unitarity. In the second part of this chapter, I analyze the PES signal using our two-part fitting
function, and I show how this yields some nice results, as well as some evidence that another way
to analyze this data is needed.

6.2

The contact
As in chapters 4 and 5, the contact of a homogeneous gas can be extracted by fitting the highfrequency tails of the rf line shapes (see equation 4.13). Figure 6.1 shows the results for our data

96

C /N (kF )

6
4
2
0

0

1

2
T /TC

3

4

Figure 6.1: The contact of a homogeneous gas at three interaction strengths.The redpoints were taken at (kF a)−1 = 0.23(3), the black points at (kF a)−1 = 0 (from figure 4.8), and theblue at (kF a)−1 = −0.24(2).

1.5Unitar ity

/



T
TF



0

2

1



T
TF



T ≈ TC

0.5
0

0

0.2

0.4

0.6

T /TF ,0Figure 6.2: Comparing initial temperature of the weakly interacting gas with the temperature at unitarity. We measure T /TF,0 of the gas at a ≈ 1100a0 , after finishing evaporation
and before a 50 ms ramp to unitarity. T /TF,U nitarity of the gas after this ramp is measured using
the release energy of the gas, as presented in Appendix A. The ratio of T /TF,U nitarity to T /TF,0
is roughly constant (1.4 ± 0.2) for T > TC . We expected this to also be true for data taken at(kF a)−1 = 0.23(3) and −0.24(2).

97
exploring temperature dependence at the different values of (kF a)−1 in the strongly interacting
regime.
As expected, the contact increases as the interaction strength, (kF a)−1 increases. The x-axis
in figure 6.1 is the temperature normalized by TC , which is the critical temperature for condensation
at the field in question. For the data points at unitarity, we were able to determine the temperatureof the atoms at the final interaction strength through methods outlined in Appendix A. However,
we do not have a reliable way to determine the temperature of the atoms at values of (kF a)−1 otherthan unitarity. For these data, we experimentally determine TC in terms of TF,0 using the pairprojection technique [25]. (T /TF,0 is the initial temperature of the gas at a ≈ 1100a0 before thefinal magnetic field sweep towards the Feshbach resonance, so TC /TF,0 is the initial temperature
that, after sweeping to the final interaction strength, yields a gas at the critical temperature.) To
estimate T /TC for the non-unitarity data, we take the ratio of T /TF,0 to TC /TF,0 , and we makethe assumption that the ratio of T /TF of the gas at the final field to its initial T /TF,0 is constantfor many temperatures. We find this assumption to be approximately true for the majority of theunitarity data above T = TC and below T = 0.6TF (see figure 6.2). (At unitarity, TF ≈ 6TC .)The data show a strong dependence of the contact on temperature for the BEC side, while
the data taken at unitarity and on the BCS side have a much weaker dependence. Neither the BEC
nor the BCS side data definitively show the non-monotonic behavior seen in the unitarity data at
low temperature, but one would need to take more data near and below TC to investigate this.

6.3

The occupied spectral function as a function of temperature
The PES data on either side of the resonance is shown in figures 6.3 and 6.4. There area few distinguishing features of this data. First of all, the PES data on the BCS side is narrow
with most of the signal contained in a positively dispersing peak. The data on the BEC side iscomparatively broader in energy. For both sets, the signal along the positively dispersing featurereaches to higher E/EF as temperature increases; this is expected as the higher energy broadens
the Fermi surface. Also, both sets show the signal shifting closer to the free particle line for higher

98
temperatures. In fact, the highest temperature sets on either side ([T /TC , (kF a)−1 ] = [3.8, −0.24]
and [3.6, 0.23]) look very similar. This might be understood as the temperature setting the energy
scale instead of the interactions.

6.4

Fitting to our two-component function
I fit our new PES data to the two-part function presented in chapter 5, equation 5.2. While

this analysis worked well for the data just above TC , whether it extends to higher temperatures or todata with significant superfluid fractions was uncertain. In particular, while we clearly saw for datanear T = TC that sharp features (i.e. the quasiparticle part) have widths limited by the resolution,
it is not clear that this should be the case at higher temperatures as well. According to Landau,
the energy width of the quasiparticle peak should scale with the square of the temperature [79, 80],so the higher temperature data might have a larger width. To look for this effect, I fit the data at(kF a)−1 = −0.24(2) to a version of equation 5.2 in which the delta function in Icoherent is replaced
by a Gaussian function with a width, σ, determined by the fit:
Icoherent (k, E) = 4πk 2 · √

1
2πσ

k2 2
−[E − (E0 + m
∗ )]
exp
2
2σ

h



i−1
− (πm∗ T )3/2 Li3/2 − exp −ET0 +µ,
exp E−µ
+
1
T
(6.1)

Figure 6.5 shows the results of fitting the data on the BCS side while allowing the width to float.
As before, we apply a Gaussian convolution to account for our experimental energy resolution. For
the lowest temperatures, σ is consistent with zero, which means that the width of the quasiparticle
peak is limited by our resolution. For the highest temperatures, we see that the best fit includes afinite σ, but the values are comparable to our resolution of 2.4 kHz.
How well the fitting function describes our data is shown by the reduced chi-squared statistic,
χ2 , in figure 6.6. For the data presented in chapter 5, χ2 is between 0.75 and 1.3, indicating a goodfit. Fits to the data at (kF a)−1 = −0.24(2) are similarly good, and χ2 is between 0.7 and 1.4 withno discernible trend versus temperature. Allowing the width to be an additional fitting parameter
does not change χ2 significantly. Fits to the data at (kF a)−1 = 0.23(3) are not as good, and χ2

2

2

2

2

1

1

1

1

0

0

0

0

−1

−1

−1

−1

−2

−2

−2

−2

−3

−3

−3

−3

F

E (E )

99

T /TC = 0.72
01

2

T /TC = 1
−40

1

T /TC = 1.1
2

−4

0

1

2

k (kF)

k (kF)

k (kF)

2

2

2

1

1

1

0

0

0

−1

−1

−1

−2

−2

−2

E (EF)−4

T /TC = 1.4
−4

0

1

2

k (kF)

100

10−1

10−2

10−3
−3

−3

−3

T /TC = 1.8
−4

0

1

k (kF)

2

T /TC = 2.2
−4

0

1

2

T /TC = 3.6−40

k (kF)

1

2

10−4

k (kF)

Figure 6.3: PES data at (kF a)−1 = −0.24(2).

ranges from 1 to 4. It is unclear why this is the case for these data.
These results suggest that an additional width parameter in the fits is not needed to fit ourE (EF)

100
2

2

2

1

1

1

0

0

0

−1

−1

−1

−2

−2

−2

−3

−3

−3

T /TC = 0.84
−4

0

1

T /TC = 12

−40

E (EF)

k (kF)

1

T /TC = 1.4
2

−4

0

1

k (kF)2

k (kF)

2

2

2

1

1

1

0

0

0

−1

−1

−1

−2

−2

−2

100

10−1

10−2

10−3
−3

−3

−3

T /TC = 2.2
−4

0

1

k (kF)

T /TC = 2.3
2

−4

0

1

k (kF)

T /TC = 3.8
2

−4

0

1

2

10−4

k (kF)

Figure 6.4: PES data at (kF a)−1 = 0.23(3).

data. In addition, if the width of Icoherent can vary, some of the incoherent background signal could
be fit as quasiparticles. We clearly see this effect for the data taken at (kF a)−1 = 0.23(3), where
the incoherent background makes up the majority of the signal. For the remainder of this section,
I fit the data at (kF a)−1 = −0.24(2) two ways, both including an energy width fit parameter in

101

σ(k Hz )

4
2
0

−2

0

0.2

0.4
T /TF ,00.6

0.8

Figure 6.5: The energy widths in Icoherent . The data sets shown here were taken at (kF a)−1 =−0.24(2).
2

4

χ2

χ2

3
1

2
1

0

0

0.2

0.4
T /TF ,0

0.6

0.8

0

0

0.2

0.4
T /TF ,00.6

0.8

Figure 6.6: The reduced chi-squared. On the left, we see the χ2 values for data on the near
BCS side. The red points are fits where the width of the quasiparticles was a fit parameter (sameas figure 6.5), and the black points were taken from fits where the width was constrained to bemuch smaller than the resolution. The plot on the right shows the χ2 values for fits of data takenon the near BEC side.

Icoherent and not. For the most part, this only affects the highest three temperatures and then only
slightly. For the data at (kF a)−1 = 0.23(3), I do not include an additional width parameter in thefits.
Figure 6.7 shows Z, the best fit value for the quasiparticle fraction for (kF a)−1 = −0.24(2)and (kF a)−1 = 0.23(3). One might expect that the quasiparticle fraction would increase with

1

0.5

0.5

0

Z

Z

1

102

0
0

0.2

0.4
T /TF ,0

0.6

0.8

0

0.2

0.4
T /TF ,00.6

0.8

Figure 6.7: The quasiparticle fraction. The results on the left were taken at (kF a)−1 = −0.24(2),and those on the right were taken at (kF a)−1 = 0.23(3). As before, red points indicate that σ wasa fit parameter, and black points indicate that the width of Icoherent was fixed to a value muchsmaller than the resolution.

temperature and possibly gradually go to 1 as T crosses T ∗ . The value of T ∗ has never beenmeasured experimentally for atoms in the BCS-BEC crossover, but there have been a few predictions
from various theories. Perali et al predict T ∗ of a homogeneous gas at unitarity to be roughly 0.55
TF [102]. Sá De Melo et al predict that the pseudogap region closes at T = 0.45TF at unitarity[103]. By these predictions, only the data set at the highest temperature on either side of theresonance would be above T ∗ . In this case, we might not expect to see Z = 1 because we are never
hot enough. On the other hand, it is interesting to note that the highest temperature PES data ateither interaction strength look remarkably similar. This may suggest that both sets are above T ∗ .
While the data show an increase of Z from 0 to 0.44 at (kF a)−1 = 0.23(3), the trend isless clear at (kF a)−1 = −0.24(2). This could be for several reasons. One reason might relate to
the quasiparticle width; if the width is allowed to be fit to the values in figure 6.5 (red points infigure 6.7a), Z has a more monotonic trend. It is problematic, however, that this difference in thebest fit value of Z does not lead to much change in χ2 .
The quasiparticle parameters that we get from fitting, such as the effective mass and thetemperature, show the expected trends. (Since the effective mass is determined by the slope of thequasiparticle dispersion, and the temperature is determined by how quickly the quasiparticle signal

103m∗ /mbar e

1.2
1.1
1
0.9

0

0.2

0.4
T /TF ,00.6

0.8

Figure 6.8: The effective mass at (kF a)−1 = −0.24(2). Red points indicate that σ was a fit
parameter, and black points indicate that the width of Icoherent was fixed to σ << 2.4 kHz.

disappears with increasing energy, it may be that a Z value which underestimates the quasiparticle
fraction still returns the correct value for these other parameters.) Figure 6.8 shows the effective
mass extracted from Icoherent as a function of temperature. m∗ starts higher than the bare mass(by 20%) and drops as temperature increases until it becomes consistent with the bare mass near
T /TF,0 = 0.3. This decrease with increasing temperature is expected; quasiparticles in a Fermi
liquid are only well-defined for low temperatures, and at high enough temperature, we expect theexcitations to look like bare 40 K atoms with a mass equal to the bare mass. The transition from asystem with dressed quasiparticles to one of bare particles could be thought of as another breakdownof the Fermi liquid description—this time due to temperature, rather than interaction strength.
The temperature of the quasiparticles is also a parameter in the fitting function. Figure 6.9
shows this temperature. We find that the best fit value for the temperature follows the trend ofincreasing temperature nicely, with the fit T /TF slightly above T /TF,0 .

104

T /TF ,fit

0.8
0.6
0.4
0.2
0

0

0.2

0.4
0.6
T /TF ,00.8

Figure 6.9: The fitted T /TF from Icoherent . These data were taken at (kF a)−1 = −0.24(2). Red
points indicate that σ was a fit parameter, and black points indicate that the width of Icoherent
was fixed to a negligibly small value. The dashed line indicates T /TF,f it = T /TF,0 . The fit to thedata returns a slightly higher temperature than the initial temperature at a = 1100a0 , which isconsistent with results at unitarity (figure 5.12a).

6.5

Conclusions
The PES data that we took for different temperatures on either side of the crossover has

the potential to answer some interesting questions regarding the role of temperature in the Fermi
liquid description and in the formation of pairs. In this chapter, I have presented some limited
preliminary data that was analyzed using our two-part phenomenological model. I also used thedata to extract the first measurement of the homogeneous contact as a function of temperature atinteraction strengths other than unitarity.
There are several ways we might proceed. If we were to analyze data where we would expect
only coherent signal or only pair-like signal, we could look at the evolution of these signals as afunction of temperature separately. This would be easily done by taking data farther from unitarity
on either side of the crossover. We might also be able to separate out the quasiparticle signal by

105
looking directly at the coherence, as others have done in the past [104]. I believe that further
investigation into this regime would yield very interesting results.

Chapter 7

Conclusions

7.1

Summary
In this thesis, I have discussed experimental work on ultracold Fermi gases carried out at

JILA in the past four years. The main body of work in this thesis involves the use of a spatiallyselective imaging technique that allows us to take measurements from a region of the trapped gasthat has a nearly homogeneous density. I described how the use of beams with a Laguerre-Gaussian(LG) profile tuned to an optical pumping frequency achieve this end, and I described the opticalsetup and several procedures we employ for aligning and characterizing the LG beams.
Most of this thesis is taken from experiments that lead to the publication of Refs. [43], [49],and [75]. In the first, we used the spatially selective imaging technique to reveal the “textbook”
sharp step in the ideal Fermi momentum distribution. In the second, we measured the contact ofa homogeneous Fermi gas with strong interactions as a function of temperature. We saw a sharp
drop in the value of the contact near the superfluid transition; this feature would not be visible
in measurements of the contact of a gas with inhomogeneous density. In the last reference, we
combined atom photoemission spectroscopy with the spatially selective imaging to measure the occupied spectral function of a Fermi gas for different interaction strength in the BCS-BEC crossover.
To understand our results, we used a phenomenological model that splits the signal into two parts:a part resembling quasiparticles and a part based on thermal pairs. We found that this model
described our data well, and this led to interesting intuition about the nature of non-superfluidgas in the BCS-BEC crossover. Finally, I presented some photoemission spectroscopy data taken

107
for different temperatures in the crossover. This data is preliminary, and the higher temperature
behavior may not be fully captured by the phenomenological model previously employed. However,these first results are interesting and, this subject is worth pursuing with further research.

7.2

Future Possibilities
The future study of Fermi gases in the BCS-BEC crossover has many possible directions.

Many of these possibilities follow naturally from the work presented here. For instance, the temperature dependence of the contact of a homogeneous gas in the crossover has been studied in
depth only at unitarity. The preliminary results from chapter 6 indicate an increase in the steepestof the slope of the contact as a function of temperature as one crosses from the BCS side to the
BEC side that deserves more investigation. The preliminary data in chapter 6 also indicates that
extending the photoemission spectroscopy data to more temperatures (both higher and lower) in
the BCS-BEC crossover would be very interesting.
Besides temperature and interaction strength, an important variable in the study of stronglyinteracting Fermi gases is the relative population of the two atomic spin states. All of the work
presented here is carried out with 40 K in an equal mixture of two spins states, but there are systems
in nature composed of strongly interacting fermions with imbalanced populations; the mixture ofneutrons and protons in heavy atomic nuclei is an important example. The connection between
the polaron picture (maximum population imbalance) and the results for a balanced gas discussed
in section 5.4.1 is interesting and worth further research. In addition, it would be interesting tomeasure the contact of an imbalanced gas. The contact of an imbalanced gas is predicted to beequal for the minority and majority states. This prediction has interesting implications for the
structures of heavy nuclei and of neutron stars.
In chapter 5, I briefly mentioned the problem of thermometry of a gas away from unitarity.
The equation of state of a Fermi gas at different interaction strengths in the crossover has beenmeasured but only for zero temperature [4]. Measuring the temperature for any interaction strengthis an important piece still missing from our understanding of the BCS-BEC crossover.

108
In chapter 5, I also briefly mentioned that, while atom photoemission spectroscopy measures
the occupied part of the atoms’ spectral function, the unoccupied branches remain unknown. Researchers have had luck with injection spectroscopy (the inverse of photoemission spectroscopy,
where you couple atoms into the system instead of outcoupling them) for measuring short-lived
states in Rb-Cs mixtures [104]. This technique could possibly be adapted to measure the unoccupied spectral function. Being able to measure the unoccupied branch would immediately allow
us to extract the superfluid gap, as well as allow us to directly look for a pseudo-gap above thesuperfluid transition temperature.
The immediate future of Fermi gas research at JILA involves the construction of a newexperimental setup. In December of 2014, the 17-year-old apparatus in which the entirety of workfrom this thesis had been carried out was retired and disassembled for parts. The new apparatus
has been designed, assembled, and baked, and optimization is currently being carried out for thecreation an ultracold gas of 40 K atoms. The new apparatus should have faster experimental cycle
times and better optical access in the final chamber. I look forward to seeing which new avenuesof research will be pursued and where they will lead.

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Appendix A

Thermometry

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 [36]. 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 − V (r). For a givenR
T and number of atoms, N , µ0 is set by the normalization requirement N = n[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.
We determine the release energy by measuring the cloud after it expands at unitary for 4 ms.
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) − n0 (r)]d3 r ,
N
2 t

(A.1)

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 a

118
given potential V (r), the release energy is given by [69]:1Erel = hr · ∇V (r)i ,2where the symbol hi stands for the density-weighted average: hg(r)i = N1

(A.2)
R

g(r)n(r)d3 r. By equating

the calculated Erel (T ) to the measured Erel , we determine T .
As a comparison, we have used two other techniques to extract the temperature of the unitarygas. 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 [48]. At T = 0, the empirical temperature is connected to the real temperature√
by T = T̃ ξ, where ξ is the universal constant defined above [27]. Albeit without a complete
theoretical justification, one can then extend this to finite temperatures and extract T [48]. In thefollowing analysis, shown as the blue triangles in figure A.1, we used ξ = 0.376 [36].
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 Fano-Feshbach resonance field. By performing
this ramp there and back and comparing the entropy before and after the ramp for a gas initiallyat T /TF = 0.12 and T /TF = 0.22, we have determined that the entropy increases by about 6%
when going to the Fano-Feshbach resonance field. Assuming this increase, we use the entropy ofthe unitary gas together with the equation of state as the thermometer. In figure A.1, the T /TF we
obtain from these two additional techniques are plotted against the release energy thermometry. We
find a good agreement between all the three techniques up to T /TF = 0.4. Above that temperature,
the empirical temperature technique becomes unreliable since the effect of quantum degeneracyof the shape of the cloud diminishes. The entropy technique starts to show a small systematicdeviation 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 difference between the entropy and release energy

119
thermometry techniques.

1 .0

T /T F , o th e r m e th o d s

0 .8

0 .6

0 .4

0 .2
e n tr o p y te c h n iq u ee m p ir ic a l te m p e r a tu r e te c h n iq u e
0 .0

0 .0

0 .2

0 .4

T /T F fro m

0 .6

0 .8

1 .0

r e le a s e e n e r g y

Figure A.1: Comparison of different 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.

Appendix B

Scaling the number of atoms in PES data

When taking an rf line shape to extract the contact, we increase the power of our rf pulse
for large detunings in order to increase our signal to noise when there is very low signal (seesection 4.2.2.1). This technique is even more necessary for photoemission spectroscopy, where, in
addition to measuring number, we resolve the momentum distribution of the atoms outcoupled foreach detuning (chapter 5).
As described in chapter 4, section 4.2.2.1, we measure the number of atoms outcoupled as afunction of power for several detunings, ν, and use the results of this measurement to determine
how to scale the data at each ν for PES to correspond to the signal for a single rf power. We fit the
number outcoupled vs. power to an exponential and used the linear approximation to scale down
the number of atoms from shots with higher rf powers:
Nout = A(1 − e−P/P0 ) ≈ A

P
P0

(B.1)

However, for the contact data, we keep the power sufficiently low such that this approximationis correct to within 10%. For the PES data, in order to get more signal in the high-momentum,
negative energy portion of the plots, we use a higher rf pulse power and scale the number using thefull exponential. Below is a table of the calibrated P0 that we used for the PES data:detuning [kHz]
-25
-5
-0
5
20
40
60
80

P0 [arb]
6.3
0.70
0.15
0.092
1.4
8.0
19.8
32.3

121

Table B.1: The measured saturation power, P0 , in arbitrary units, for different rf frequency detunings. These results are consistent with figure 4.7. The slight differences come from slight changes inthe electronics over time. As before, we see that P0 increases with increasing detuning, and is lowest
near the center of the line shape, where the larger numbers of atoms available to be outcoupledcause the signal to saturate at lower powers.

Note that, for the PES data, we scaled the signal for frequencies lower than ν0 as well as
higher. We did this to be certain that the increased power wasn’t outcoupling signal from thecenter of the line shape; if it had been, extra signal would have appeared at negative frequencies
as well. This also allowed us to reduce the noise in the negative detuning region.
For PES data, we also vary the pulse duration as we change frequencies—a total duration of
300 µs for ν < 34 kHz and 100 µs for ν > 34 kHz. We find that the ratio of the saturation powers,
P0 , of two different pulse durations is equal to the ratio of the pulse durations (see figure B.1). In
the linear regime (away from saturation effects), this means that the number of atoms outcoupledis scaled by the ratio of pulse durations, as you would expect.

fraction of atoms outcoupled

122

0.4
300 µs pulse: P0 = 0.0058

0.3
0.2
0.1
0

100 µs pulse: P0 = 0.017

02
rf power (arb)

4
·10−2

Figure B.1: Fraction of atoms outcoupled vs rf power. At a detuning of ν = 80 kHz, wemeasured the number of atoms outcoupled by a 300 µs pulse and a 100 µs pulse. (Here, the quitedpulse duration corresponds to four times the 1/e2 amplitude half width.) For data taken at thisdetuning, we use the 100 µs pulse and, staying in the linear regime, increase the number of atoms
by a factor of three to compare it with data taken at small detunings using a 300 µs pulse.

Appendix C

Comparing PES of a homogeneous sample to trap-averaged data

C.1

Back-bending of EDCs
To compare the nearly homogeneous data with previously published trap-averaged data,

we use the analysis described in Ref. [73]. Specifically, independent Gaussian fits to fixed-k traces
through the PES data are used to find a dispersion E(k). Figure C.1a shows the nearly homogeneous
PES data for (kF a)−1 = 0.1, with white points marking the centers from the Gaussian fits. Figure
C.1b shows the fixed-k traces through the data, which are often called energy distribution curves, or
EDCs. These plots can be compared to figures 1 and 2 of Ref. [73], which show trap-averaged dataat (kF a)−1 ≈ 0.15 and T /TC = 1.24. We find that the PES results for a nearly homogeneous gas
are qualitatively similar to the trap-averaged results. Specifically, in both cases we see dispersions
that exhibit large energy widths and back-bending.
In the two-mode model, the combination of the positively dispersing coherent part and theincoherent part can produce a spectral function that exhibits back-bending. To demonstrate this,in figure C.2 we calculate the dispersion with the two-mode model using parameters similar to
those from our fits and find the peak using the same analysis technique as was in Ref. [73]. The
resulting dispersion clearly shows a BCS-like back-bending.

124

100
2

a

b
k = 0.37

10−1

1

0.52

E (EF)

0.66
0.81

0
10−2
−1

0.96
1.1

10−3

1.25

−2
1.4
1.55
−3
0

0.5

1

k (kF)

1.5

2

10−4

−2

−1
0
E (EF)

1

Figure C.1: Gaussian fits to PES data at (kF a)−1 = 0.1. a, The white circles indicate thecenters from weighted Gaussian fits to EDCs at fixed k. The white line shows the free-particle
dispersion, E = k 2 . b, Individual EDCs (blue points) are shown, along with the fitted Gaussian(red lines). Here, each EDC is individually normalized to have the same area, as in Ref. [73]. A
solid black line marks E = EF . Red stars show the center of each Gaussian, and the dashed redline is a guide to the eye.

125

Ec [EF ]

0.5
0
−0.5
−1
−1.5

0

0.5

1
k [kF ]

1.5

2

Figure C.2: Back-bending in the two-mode model. We generate a simulated spectral function
using the two-mode model with the following parameters: Z = 0.2, m∗ = 1.2, T = 0.25, E0 = −0.5,µ = −0.5, Ep = 0.5, Tp = 0.75. Each point in this figure was then obtained by fitting the spectral
22function at a given momentum k by the Gaussian function Ae−(E−Ec ) /2σ . In the fits, the valueof σ was constrained to be above the experimental resolution.

Appendix D

Derivation of the single-particle spectral function for atoms bound in thermalpairs

As the title of this section might suggest, understanding what we expect thermal pairs to look
like in our PES signal is not straightforward. PES measures the energies and momenta of single
particles, which means that in addition to outcoupling the atom to a weakly interacting state, therf photon must also overcome the binding energy of the pair.
To derive the functional form of our model, we first consider a single stationary pair that is
dissociated by absorbing an rf photon. For rf dissociation, energy conservation gives− Ep + hν = Erel ,where Erel =

(D.1)

2
~2 krelmis the relative kinetic energy of the two resulting atoms after dissociation andq~2
Ep = mR2 is the binding energy corresponding to a bound wave function φ(r) = R2 e−r/R . On
the other hand, in PES [1], the single-particle energy E is given by
E + hν =~2 k 2,2m(D.2)

where ~k is the momentum of the spin-flipped atom. If the pair is initially at rest, the two resulting
atoms will have momenta ~krel and −~krel , so that combining equations D.2 and D.1, we find2 2k
E = −Ep − ~2m. Here, we can see that pair dissociation yields a negative, quadratic single-particle

dispersion.
The amplitude of the signal as a function of k can be obtained from the normalized rf line

127
shape for the dissociation of a weakly bound molecule, which was derived by Chin and Julienne [91]:p
Erel Ep2,(D.3)
F (Erel ) =π (Erel + Ep )2
R∞
where 0 F (Erel )dErel = 1 and we have assumed that the final state is weakly interacting. Equation D.3 can be rewritten in terms of krel to givep
Ep
~3
G(krel ) = 2 3/2 2 2,
π m (~ krel /m + Ep )2

(D.4)

R∞

2 dk
G(krel )4πkrel
rel = 1. This line shape is highly asymmetric and has an extent in krelp
4 , which is consistentthat scales as Ep , or equivalently 1/R. At high krel , G(krel ) falls off as 1/krel

where

0

with Tan’s contact.
If the pair is moving, then combining equation D.2 and D.1 gives
E = −Ep −2~2 krel~2 k 2
+
.
m
2m

(D.5)

In PES, we apply rf with detuning ν, measure k for the spin-flipped atom, and extract E using
equation D.2. From equation D.5, we see that for the dissociation of non-stationary pairs, data with2 2ka particular k (but variable ν) can, in principle, yield E anywhere in the range −Ep − ~2m
≤E≤
2 2

k
−Ep + ~2m
, where the lower limit corresponds to a stationary pair and the upper limit corresponds

to a pair with the maximum possible center-of-mass momentum, ~Kcm = 2~k.
To model the center-of-mass momentum distribution of non-condensed pairs, we assume a
Maxwell-Boltzmann distribution with an effective temperature Tp :
P (Kcm ) =where kb is the Boltzmann constant and~2
4πkb mTp

R∞
0

3/2

2

e

cm
− ~4kKmT
b

2

p

,

(D.6)

2 dK
P (Krel )4πKcm
rel = 1. The product of the functions

in equations D.6 and D.4 gives the transition probability as a function of Kcm and krel . We change
variables to k and krel and integrate over the angles to find the transition probability as a functionof k and krel :4~4H(k, krel ) = 3/2 2
π m

s

−

Eb
kb Tp

kkrel e2 )
2~2 (k2 +krel
kb mTpe~2 (k+krel )2
kb mTp−e

2 /m + E )2(~2 krelb~2 (−k+krel )2
kb mTp

!
,

(D.7)

128
where

R∞R∞
0

0

H(k, krel )dkdkrel = 1. Finally, we change variables to E and k to derive the PES

signal (equation 5.4 in the main text):
8k
Ip (k, E) = Θ −Ep − E + k

2



q

Ep
Tp e

Ep +E−3k2
Tp

sinh

 √ √
2 2k

π 3/2 (E − k 2 )2−Ep −E+k2
Tp


.

(D.8)

In this last step, we also changed to dimensionless parameters (k, E, Ep , and Tp ) normalized by
kF and EF .