Photoemission Spectroscopy of a Strongly Interacting
Fermi Gas
byJohn Pagnucci Gaebler
B.S., Rice University, 2004

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
2010This thesis entitled:
Photoemission Spectroscopy of a Strongly Interacting Fermi Gas
written by John Pagnucci Gaebler
has been approved for the Department of Physics

Deborah Jin

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
Gaebler, John Pagnucci (Ph.D., Physics)
Photoemission Spectroscopy of a Strongly Interacting Fermi Gas
Thesis directed by Prof. Deborah Jin

The ability to study ultracold atomic Fermi gases holds the promise of signiﬁcant advancesin testing fundamental theories of many-body quantum physics. Of particular interest are stronglyinteracting Fermi gases in the BCS to BEC crossover that exhibit a transition to a superﬂuid state
at temperatures near 0.2TF , where TF is the Fermi temperature. This transition, as a fraction of
TF , is extremely high compared to any known superﬂuid or superconductor. These gases are also
in a universal regime where the physics is independent of the details of the atomic interactions
and is therefore relevant to ﬁelds as diverse as condensed matter, nuclear physics and astrophysics.
In this thesis, I present an experimental probe of atomic gases that uses momentum-resolved RF
spectroscopy to realize an analog of angle-resolved photoemission spectroscopy (ARPES) in materials. This measurement reveals the energy and momentum of single-particle states in the stronglyinteracting Fermi gas. In condensed matter, ARPES has proved to be one of the most powerful experimental techniques for studying the electronic structure of strongly correlated electron materials.
The ability to perform analogous measurements in ultracold Fermi gases constitutes a signiﬁcantadvance in our ability to directly connect ultracold atomic gases to strongly correlated electron
systems. Taking advantage of this new measurement technique, I investigate a long-standing problem in the ﬁeld of strongly interacting fermions, namely whether a pseudogap state consisting ofincoherent fermion pairs exists at temperatures above the critical temperature for superﬂuidity.
The photoemission data I present provide strong evidence for this state and have implications for
fundamental theories of strongly interacting Fermi gases and strongly correlated electron materials.
I also discuss the experimental conﬁrmation of recently predicted universal relations for stronglyinteracting Fermi gases, as well as some of the ﬁrst experiments involving atomic Fermi gases with
p-wave pairing.

Acknowledgements

This thesis represents the work of nearly six years of my life, and as it is with most things,
it would simply not have been possible without all of the people around me, not even close. First,
there are the people without whom I would never have come here in the ﬁrst place. I should
thank my parents ﬁrst. They have encouraged me nearly every day of my entire life and helped me
through so many problems and challenges. I want to thank my mom for encouraging me, helping
me stay organized, and always ﬁghting for the best for me. Sometimes having an Italian motherwilling to go all the way for her children comes in handy. Grazie mille mama. I want to thank my
dad for being my chess coach, for building model airplanes and science projects in the garage withme, for helping me with my homework despite my ADD, for encouraging me in math and science,and for being a great dad. I know you became a lawyer (I blame Harvard), but I’m sure you could
easily have been a great scientist in another life. I’d like to thank my Gramps for being a wonderful
grandfather and for always being, at least to me, a model intellectual with great compassion andhumanity, and an extraordinary mind. It meant a lot to see you at my thesis defense. I’d also liketo thank my siblings, it is so nice to have you both as good friends now that we are ﬁnally grownup.
I should also thank all of my teachers. In particular, my high school physics teacher Mr.
Pettus introduced me to the subject that I’ve never stopped studying. He challenged me and thathas kept my interest ever since. The way I feel about physics can be summed up by a quote fromJohn F. Kennedy who, in a now famous speech in Rice stadium, said “Why go to the moon? They
may well ask...why does Rice play Texas? Not because [it is] easy, but because [it is] hard.” I mayv
still get confused by basic quantum mechanics, and Rice may loose to Texas by 30 points and callit a good game, but at least it’s not easy.
I had a number of great professors at Rice who helped me along the way. I’d like to thank Paul
Padley for advising me and giving me the opportunity to work at Fermi Lab, and Paul Stevenson
for being such an excellent teacher of quantum mechanics. Most importantly, I’d like to thankRandy Hulet. I can still remember the day in our modern physics class when he described anexperiment in which electrons pass through a double slit mask and form an interference pattern.
It was at that moment when it struck me how amazing quantum mechanics is and that feeling has
never really left me since. He also told us about laser cooling of atoms, which later inspired me tocome to him and ask for the opportunity to work in his lab, which he gladly aﬀorded me. It wasthat experience that ultimately led me to apply to the University of Colorado and seek a research
assistantship with Debbie Jin. I should also thank Randy for contacting Alain Aspect and helping
me set up my year abroad in France after college. That was a great year where I made many
good friends from all over the world. What I learned working in Alain Aspect’s lab really made a
diﬀerence when I started at JILA. I must also thank the French department at Rice and the Clyde
Bull Ferguson travel fellowship they oﬀered me that made that year possible.
The work in this thesis is the research I performed in Debbie Jin’s group at JILA and I thankher for being such a wonderful advisor. She is an outstanding scientist and an incredibly clear
thinker. However, I would especially like to thank her for treating me as a colleague and taking myideas seriously, even at the beginning of graduate school when I barely knew what I was talking
about half of the time (still the case occasionally). This gave me the conﬁdence to keep trying out
my own ideas, and eventually function as a real scientist (most of the time). One of the things I
enjoyed most in graduate school was sitting around with Debbie and brainstorming ideas or coming
up with research strategies. I especially need to thank Jayson Stewart, who worked along side me
for nearly all of what is in this thesis. Jayson is a very talented experimentalist and certainly
deserves equal credit for the majority of the work in this thesis. I think we made a really great
research team and I had a lot of fun, whether it was messing around in the lab, mountain biking

vi
before work, ﬁshing for trout, or skiing. I look forward to keeping up contact with Jayson and am
sure he will be a very successful scientist. I’d also like to thank Tara Drake and Rabin Paudel, themore recent additions to our group. They have been very helpful, and will undoubtedly do a great
job carrying on this research.
I’d like to thank Brian DeMarco, Debbie Jin, and Cindy Regal for building such a great
experiment and laying the groundwork for the research in this thesis. As many scientists havenoted at one time or another, we only see farther than others by standing on the shoulders of the
giants before us.
I’d also like to thank all of the Jin and Cornell group members who have become my goodfriends and helped me so many times. I’d like to especially thank Juan Pino, Josh Zirbel, Kang-Kuen
Ni, Michele Olsen, Russ Stutz, Laura Sinclair, Aaron Leanhardt, Rob Wild, Tyler Cumby, Marcio
Miranda, Brian Neyenhuis, Silke Ospelkaus, Huanqian Loh, Ruth Shewmon, and Phil Makotyn.
I also need to acknowledge all the help I have gotten from theorists, here at JILA andelsewhere. John Bohn is an amazing resource for atomic physics knowledge and his calculations
were particularly useful for the p-wave Feshbach resonance paper. Leo Radzihovsky, Victor Gurarie,
and Jim Shepard helped me better understand many-body theory and condensed matter ideas. I
would also like to thank the Strinati group in Camerino, Italy and especially Andrea Perali. It wasa pleasure to see Andrea here in Boulder and collaborating with him on the pseudogap work has
been very helpful and has led to many interesting results.
I really have to thank all the people at JILA, NIST, and CU that help create what is probably
the world’s most concentrated center of knowledge in atomic physics and experimental techniques.
I especially acknowledge help from Eric Cornell, Jun Ye, and the JILA electronics and machine
shops. I think the collaborative environment at JILA combined with the great expertise of themembers is vitally important to all of the success we have here and I can only hope to ﬁnd a similar
environment in my future work.
Finally, I would like to thank Katie. Thanks for all of your support, and for lifting my spirits,
and making me happy. The last few years of my life here with you in Boulder have been amazing,

vii
and I’m so happy we will be staying here together.

viii

Contents

Chapter
1 Introduction

1

2 Creating a strongly interacting quantum degenerate atomic Fermi gas with a tabletop experiment

8

2.1

8

What is a quantum degenerate Fermi gas? . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1

2.2

A harmonically trapped Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . 12

Cooling an atomic gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1

Measuring temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Feshbach resonances

18

3.1

Interactions in an ultracold atomic gas . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2

Wide vs narrow resonances and universality . . . . . . . . . . . . . . . . . . . . . . . 22

3.3

Measurement of molecule binding energies and resonance centers . . . . . . . . . . . 25
3.3.1

Determination of s-wave Feshbach resonance parameters with rf molecule
dissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.2

Determination of p-wave molecule binding energies with magneto-association

29

3.4

Measurement of molecule lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5

Creating p-wave molecules near the Feshbach resonance . . . . . . . . . . . . . . . . 38

3.6

p-wave superﬂuids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

ix
4 Fermi superﬂuids and the BCS-BEC crossover
4.1

4.2

44

The pairing theory of Fermi superﬂuidity (BCS Theory) . . . . . . . . . . . . . . . . 44
4.1.1

Cooper pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1.2

Properties of the superﬂuid state . . . . . . . . . . . . . . . . . . . . . . . . . 46

Turning up interactions: The BCS-BEC crossover . . . . . . . . . . . . . . . . . . . . 51

5 Fermi liquids, spectral functions, and photoemission spectroscopy

57

5.1

Landau’s Fermi liquid theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2

Green’s function and many-body quantum theory . . . . . . . . . . . . . . . . . . . . 62

5.3

Photoemission spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.4

Photoemission spectroscopy experiments with a strongly interacting Fermi gas . . . 71

5.5

Improvements to the signal-to-noise ratio . . . . . . . . . . . . . . . . . . . . . . . . 75

6 The Pseudogap state of a strongly interacting Fermi gas

81

6.1

The pseudogap phase in the BCS-BEC crossover . . . . . . . . . . . . . . . . . . . . 81

6.2

(Some) History of the pseudogap phase and relation to high temperature superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.3

Atom photoemission spectroscopy and the pseudogap phase in the BCS-BEC crossover 88

6.4

Pseudogap to molecular gas crossover . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.5

Comparisons to a theory of the pseudogap . . . . . . . . . . . . . . . . . . . . . . . . 106

6.6

Extracting information from BCS-ﬁts to the spectral functions . . . . . . . . . . . . 111

6.7

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7 Towards atom-photoemission spectroscopy of a homogeneous gas
7.1

117

Focusing on the center of the cloud . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.1.1

Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.2

Application to a weakly interacting Fermi gas . . . . . . . . . . . . . . . . . . . . . . 123

7.3

Studying the eﬀect of the trapping potential on atom-photoemission spectroscopy . . 128

x
8 Universal relations and the contact

137

8.1

The contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

8.2

Measuring the contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

8.3

Testing the Tan relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

9 Conclusion and Outlook

151

9.1

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

9.2

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Bibliography

155

Appendix
A 40 K Transitions

163

B Subtracting the high-momentum background signal from optical pumping

168

C Accounting for interactions with the third spin-state in measurements of the contact

171

D Simulating photoemission spectroscopy of molecules

174

Chapter 1

Introduction

The realization of Bose Einstein condensation in 1995 was a crowning achievement, marking
a major shift and a new direction for the ﬁeld atomic physics [23, 24, 25]. To understand the signiﬁcance of this shift, we can brieﬂy examine the history of atomic physics. Modern atomic physics
started at the beginning of the 20th century with, as the name implies, an eﬀort to understandthe mechanics of an atom, the basic building block of all materials. It was at this time that the
failure of classical physics to explain the stability of the atom led to the revolutionary developmentof quantum mechanics, and a golden age for atomic physics that spanned the 1910s and 1920s.
The new theory of quantum mechanics led to such rapid progress that, beginning with Planck in
1918, twelve of the next ﬁfteen Nobel prizes were awarded for work in atomic physics. However,at this point, the wild success of atomic physics became its downfall as physicists took the newtheoretical concepts developed to understand atoms and used them to begin the ﬁelds of nuclear
and particle physics, condensed matter and astrophysics. Indeed from 1933 to 1996, there were
only nine more Nobel prizes awarded in ﬁelds of traditional atomic physics. However, this drought
doesn’t tell the story of a major change that was slowly taking place in atomic physics, which wasmoving away from developing theories of the mechanics of an atom, and towards controlling andusing atoms as ideal quantum systems. Much of this progress was enabled by the invention of
the laser in 1960, which gave scientists the ability to coherently and precisely control the quantum
states of atoms. Developments in the 1980s to control and entangle the quantum states of atoms
and photons would lead to dramatic demonstrations of non-locality in quantum mechanics, and2eventually the ﬁeld of quantum information, which seeks to use precise control of quantum states
to perform computation. At the same time in the 1980s, there were ongoing developments in theuse of lasers to control the motion of atoms, including the demonstration of how to slow atoms,
and achieve laser cooling of atomic gases. This eventually led to the achievement of Bose-Einsteincondensation of atomic gases in the miraculous summer for atomic physics of 1995.
Bose-Einstein condensation (BEC) is a phase transition to a new state of matter where allof the particles in the system occupy the same quantum state. It can occur only at very low
temperatures or high densities, regimes where the quantum nature of the individual atoms in thegas cannot be ignored. In the dilute alkali gas experiments that reached BEC in 1995, the atoms
were cooled to nanoKelvin temperatures, which is (as far as we know) the coldest temperatures everreached in the universe. This dramatic achievement led to an explosion in the new ﬁeld of ultracold
atomic gases, with publication rates increasing exponentially to nearly a thousand academic papers
a year by 2000 [26]. Nobel prizes have been awarded in 1997, 2001, and 2005 in atomic physics
with surely many more to come to recognize this new and exciting research ﬁeld.
But beyond the “wow” factor of these ultracold gases, exactly what has attracted so muchattention? While ultracold gases have enabled new progress in precision measurement, leading
to ultra-precise interferometers and atomic clocks, much of the interest comes from the potential
to use cold atomic gases as models to test fundamental theories of quantum matter. Indeed, thework presented in this thesis regards the use of ultracold gases to test fundamental theories ofstrongly interacting fermions. It is because of our ability to control and study these systems at thequantum level, using the tools of atomic physics, that they are so useful in modeling and studying
quantum matter. Over the last 15 years, this high level of control has allowed experiments withquantum gases to realize physics previously believed to only exist as textbook or gedanken (thought)
experiments. Indeed, until 1995 a weakly interacting Bose-Einstein condensate was widely assumed
to be a gedanken experiment, with no relevance to any physically realizable system. Yet, if all we
could do with atomic gases is realize experiments for which we already understand the physics,
this would be a shallow ﬁeld. Fortunately, the development of Feshbach resonances and optical3
lattices to control atomic interactions and create strongly correlated gases means that we can use
ultracold atomic gases to investigate some of the most pressing scientiﬁc issues to date, such as
high temperature superconductivity and quantum phase transitions. Indeed, developing theories tounderstand strongly correlated systems is a central challenge in nearly all modern ﬁelds of physics
including nuclear physics, astrophysics, and condensed matter. The great importance of developing
theories of strongly correlated systems, and the high level of precision and control aﬀorded byultracold gases to study these regimes is what motivates the work in this thesis.
Phillip Anderson laid out the basic premise for why we study large and complicated systems
like materials or quantum gases in his oft cited paper “More is diﬀerent [27]”. He writes that even
if “the elementary entities of science X obey the laws of science Y ... this hierarchy does not imply
that science X is “just applied Y.” At each stage entirely new laws, concepts, and generalizations
are necessary, requiring inspiration and creativity to just as great a degree as the previous one.
Psychology is not applied biology, nor is biology applied chemistry.” In other words, if we takesome atoms, which we understand perfectly well on an individual basis, but then let them interact
with each other to form a larger system, we may know little to nothing of use about the behavior ofthat system based only on our understanding of those individual atoms. Instead, we must develop
new laws and concepts to understand this larger and more complicated system. In the extreme
case, this is quite obvious, our knowledge of the quantum mechanics of atoms does not imply we
now have all the basic tools to understand life, despite the fact that life is actually made of abunch of atoms obeying the laws of quantum mechanics. Amazingly, it turns out that systemsof multiple atoms become complicated so fast that simply trying to solve the basic equations ofquantum mechanics becomes impossible even for only a few tens of atoms, even with the worlds
most powerful supercomputers. This is why we need to develop new ways of understanding thesesystems. At a simpliﬁed level, one can divide physics into ﬁelds that are still trying to develop a
more fundamental understanding of the basic constituents of matter and energy in the universe,such as ﬁelds like high energy particle physics, and ﬁelds that are tackling systems for which weunderstand the properties of the basic constituents but are unable to formulate complete theories

4of the whole. Our study of ultracold atomic gases falls into the latter category.
So far, physicists have been quite successful at explaining large systems of particles if thoseparticles are weakly interacting. This is because the behavior of the system as whole may not deviatethat far from what would be expected of a single particle, but occurring many times over in each of
the particles constituting the system. Thus, as a ﬁrst approximation, one can simply try to describe
large systems by ignoring the presence of interactions. The next level of approximation, mean-ﬁeld
theory, is to take into account the average interaction between particles, but ignore ﬂuctuations
around the average. The electrons in most metals and semiconductors, as well as weakly interacting
ultracold atomic gases, are good examples of systems that can be largely understood in terms ofmean-ﬁeld theory. However, a separate class of systems exist where the constituent particles arestrongly interacting with each other and deviations from mean-ﬁeld theory are important. It
is in these systems that we should expect new behaviors and properties to emerge. In materials,
some examples of strongly correlated systems are high-temperature superconductors, heavy-fermion
materials, giant magneto-resistors, Mott insulators, and materials near quantum phase transitions[28]. Many of these materials have signiﬁcant potential applications, such as high-temperature
superconductors that could be used for eﬃcient transport of electricity. Developing theories tounderstand strongly correlated materials could unlock new and important technologies.
While ultracold atomic gases and strongly correlated electron materials may be worlds apartin many ways, including their density (dense solids vs dilute gases) and temperatures (Kelvin vs
nanoKelvin), their behaviors are underpinned by many of the same theoretical concepts of stronglyinteracting quantum systems. The connections to certain strongly interacting systems is especiallystrong. Indeed, when the short-range atomic interactions are increased to their maximum strength
using a Feshbach resonance, the system enters a universal regime where its properties no longer
depend on the details of the atomic species or interactions[29, 30]. Any other strongly correlated
system with short-range interactions in this universal regime will behave identically, after a simple
rescaling of the parameters depending on the density. It is thought that dense nuclear matter may
in fact be in this same regime, displaying identical behavior to an ultracold atomic gas [31, 32].

5
The real interest in studying ultracold atomic gases to learn about strongly interacting systems comes from the fact that the control and detailed information experimenters have with atomic
systems is so great. Ultracold atomic gases allow the experimenter to start with a simple system,
like a gas of weakly interacting particles, and incrementally add complexity by, for example, turning
up interactions with a Feshbach resonance, adding a periodic potential, or adding disorder. Thisability to start from what you understand and slowly go towards something you don’t understandis far more diﬃcult to achieve in materials, where the system often has many inseparable andcomplicated interactions going on at once. Furthermore, atomic gases oﬀer experimenters many
new measurement techniques that are not available in other systems. For example, a common
measurement in atomic gases is to measure the momentum distribution of the constituent particles, which is something far more diﬃcult to achieve for electrons in materials. Almost always inphysics, the ability to make new kinds of measurements drives the development of theories andleads to advances in basic understanding. Atomic gases thus oﬀer the opportunity to develop aground-up understanding of strongly interacting systems, which can ultimately impact our ability
to understand other complicated systems, whether in condensed matter, nuclear, or astrophysics.
A central challenge in realizing the great promise of ultracold atomic gases is to develop experimental measurements that can directly connect to the theories of strongly interacting systems
and enable sensitive searches for new phenomena. In this thesis, I will discuss a new measurement
technique for strongly interacting atomic gases that I helped to develop. This technique is called
atom photoemission spectroscopy, and it has allowed us to make a number of important measurements of fundamental properties of strongly interacting Fermi gases. We have used this techniqueto address the existence of a pseudogap state in the BCS-BEC crossover, as well as verify a numberof recently derived universal relations known as the Tan relations. In the next section, I will givea more detailed outline of the contents of this thesis.
Outline of Thesis Contents
Much of my graduate career was alongside John (Jayson) Stewart who wrote a Ph.D. thesis
in 2009 and, underlying my organization of this thesis, is a competing desire to give both a complete

6
description of the work I have done with adequate background information, while not repeating too
much of what is already contained in Jayson’s thesis. Furthermore, background information on theexperimental apparatus is contained in Brian Demarco’s thesis, and Cindy Regal’s thesis contains
excellent introductions to Feshbach resonances and the BCS-BEC crossover. As such I will be briefin my discussions of those topics and refer the reader to appropriate sources.
In the second chapter, I introduce and review basic concepts for quantum degenerate Fermi
gases. I also give a brief description of the experimental apparatus, explaining how we cool an
atomic 40 K gas to ultracold temperatures and make measurements.
In the third chapter, I explain how we realize a strongly interacting Fermi gas using a magnetically tunable Feshbach resonance. I discuss the physics of Feshbach resonances in atomic gases
including the concepts of narrow versus wide resonances and s-wave versus p-wave resonances.
While most of the work in this thesis regards s-wave resonances, we conducted some experiments
using a p-wave Feshbach resonance in order to explore the possibility of creating a p-wave superﬂuid. I discuss the results of these experiments, which include forming p-wave pairs and observing
their lifetimes and non-isotropic momentum distributions. I also review some prospects for further
research of atomic gases with p-wave interactions.
In the fourth chapter, I review the results of conventional BCS theory, which describes how
fermions pair up to form a superﬂuid or superconductor in the presence of weak attractive interactions. I then discuss what happens as the strength of interactions is increased to the limit of
the BCS-BEC crossover and, eventually, molecular pairing. The BCS-BEC crossover occurs as theattractive interaction between fermions is increased to the limit where the typical pair size is onorder of the interparticle spacing, and thus too strongly interacting to be described by BCS theory,
but not yet to the deeply-bound molecular limit where the fermionic properties of the constituentsplay no role. It is in this crossover that the gas is the most strongly correlated, and therefore where
the most interesting physics takes place. I review past experiments in the BCS-BEC crossover anddiscuss some of the goals of ongoing work.
In the ﬁfth chapter, I begin with a review of the concepts of Fermi-liquid theory and spectral

7functions, which serve as a language for discussing and understanding interacting Fermi gases. I
then present the technique of atom photoemission spectroscopy to measure the spectral function.
Atom photoemission spectroscopy measures the single-particle spectral function and is analogous
to angle resolved photoemission spectroscopy (ARPES) in materials. I present our ﬁrst atom
photoemission spectroscopy measurements in the BCS-BEC crossover. Finally, I explain how wehave improved on the techniques presented in that paper to signiﬁcantly increase the signal-to-noise
ratio of the data.
In the sixth chapter, I discuss the idea of a pseudogap state in the BCS-BEC crossover. Thisis a state in which the Fermi gas contains incoherent pairs of fermions existing at temperatures
above the superﬂuid phase transition. Then, I present our atom photoemission data that searches
for evidence of this state by looking at the properties of the single-particle spectral function aboveand below the superﬂuid phase transition. Indeed, we observe a spectral function that demonstrates
backbending both above and below the superﬂuid transition, consistent with the predictions of apseudogap state. I also present data that maps out the pseudogap regime as a function of interaction
strength and shows how the pseudogap state evolves into the molecular regime where the physics
is described by deeply bound molecules. Finally, I show comparisons to a theory of the pseudogap
state that captures many essential elements of the data.
In the seventh chapter, I present a new technique for measuring the momentum distributionof atoms at the center of a trapped gas using intersecting hollow light beams. This technique allows
us to obtain atom photoemission spectroscopy signal from the center of the atom cloud and thereby
reduce some of the eﬀects of density inhomogeneity. This allows us to better understand the eﬀectof the trapping potential on our atom photoemission experiments.
In the eighth chapter, I discuss experiments to verify the universal Tan relations for quantum gases with short range interactions. These relations predict connections between numerous
measurable quantities in ultracold gases. We take advantage of multiple measurement techniques,
including atom photoemission spectroscopy, to verify these powerful relations.
In the end, I present a summary of results and a brief outlook for future research.

Chapter 2

Creating a strongly interacting quantum degenerate atomic Fermi gas with atabletop experiment

2.1

What is a quantum degenerate Fermi gas?
In quantum mechanics all identical particles must be considered to be completely indis-

tinguishable from each other, which implies important restrictions on the behavior of systems ofidentical particles. All particles can be classiﬁed as following one of two sets of quantum mechanicalrules, Fermi-Dirac statistics or Bose statistics, and the corresponding particles are called fermionsand bosons respectively. Whether a particle is a fermion or boson is determined by a quantum
mechanical phenomena called spin, which can take on any integer or half integer value (anythingelse is called an anyon). Particles with either no spin or integer values of spin are bosons, while
particles with half integer values of spin are fermions. The basic building blocs of matter, electrons,
protons, and neutrons, are all spin 12 particles and thus fermions. However, atoms are made up of
many of those particles and can be either bosons or fermions depending on whether they have aneven (bosons) or odd (fermions) number of constituents. An important result of Fermi statistics
is that no two identical fermions can occupy the same quantum state; this is referred to as thePauli exclusion principle. This arises from the requirement that the many-particle wave-function
describing a system of identical fermions must be antisymmetric under the exchange of any two of
those particles.
Let’s consider the case of two non-interacting identical fermions. To begin with, let’s take two
fermions with no internal degree of freedom so that they are completely described by their position-

9
space wave-functions. We will assume there is a ground state wave-function ψ0 (r) and a ﬁrst excitedstate ψ1 (r). Intuitively, one would think that the lowest energy state for the two fermions would beψ(r1 , r2 ) = ψ0 (r1 )ψ0 (r2 ) where r1 (r2 ) is the coordinate for particle 1 (2). This wave-function puts
both fermions in the ground state. However, this wave-function is symmetric under the exchangeof particles 1 and 2, in other words ψ(r1 , r2 ) = ψ(r2 , r1 ), which is not allowed for fermions. The
lowest energy wave-function that obeys the antisymmetry principal, ψ(r1 , r2 ) = −ψ(r2 , r1 ), isψ(r1 , r2 ) = √12 ψ0 (r1 )ψ1 (r2 ) − √12 ψ1 (r1 )ψ0 (r2 ). This wave-function has one particle in the groundstate and one particle in the ﬁrst excited state. Thus, by restricting the two-particle wave-function
to be antisymmetric, the ground state energy is raised because we are forced to put one particle
in an excited state. Note that the two fermions can never be found at the same location sinceψ(r1 = r2 ) = 0.
Now, let’s consider a case where the fermions also have an internal degree of freedom, which
may be spin in the case of an electron or internal atomic state in the case of an atom. We willconsider the simple case where there are two degenerate internal states, labeled up | ↑⟩, and down
| ↓⟩. The spin wave-function of the two identical fermions can be in one of four states: the spin()(
triplet states | ↑⟩1 | ↑⟩2 , √12 | ↑⟩1 | ↓⟩2 + | ↓⟩1 | ↑⟩2 , | ↓⟩1 | ↓⟩2 , and the spin singlet state √12 | ↑⟩1 | ↓)⟩2 −| ↓⟩1 | ↑⟩2 . The spin triplet states are all symmetric under exchange of particle 1 and 2, whereas
the spin singlet state is antisymmetric under exchange of particles. Since the overall wave-function
for the two fermions must be antisymmetric, if the fermions are in a spin triplet state, their position
space wave-function must be antisymmetric just as in the case where we had no internal degree of
freedom, but if they are in the spin singlet state, then their position space wave-function must be()
symmetric. Thus, the lowest energy state is now ψ(r1 , r2 ) = √12 ψ0 (r1 )ψ0 (r2 ) | ↑⟩1 | ↓⟩2 − | ↓⟩1 | ↑⟩2 ,which has one spin up particle and one spin down particle, both in the ground position-space state.
In this way, even if the internal degree of freedom is not explicitly contained in the Hamiltonian, it
can aﬀect the energy of the system by determining which wave-functions are allowed.
If there are many fermions, the antisymmetry principal says that the overall wave-function
must be antisymmetric under exchange of any two fermions. Again assuming two degenerate

10aEnergyb

EF

kBT

Figure 2.1: Fermions in a square well The quantized motional states are shown as horizontal
lines with the ground state at the bottom and energy increasing upwards. Blue circles are spin up
fermions and red circles are spin down fermions. a) The ground state of the fermion system has afermion of each spin into each motional level, ﬁlling up all the quantum states until there are no
more fermions. The energy of the highest occupied level is the Fermi energy. b) Finite temperaturecauses excitation of fermions above the Fermi energy by energies of order kB T .

1

Occupation

0.8
0.6
0.4
0.2
0
0

0.5

1

1.5

2

Energy/EF
Figure 2.2: The fermion distribution function The average occupation number of quantum
states for fermions of a single spin state are plotted as a function of the energy of the state fordiﬀerent temperatures. Reduced temperatures of T /TF = 0, 0.1,and 0.6 are shown as solid, dashed,
and dot-dashed lines. At zero temperature (solid line), the occupation is 1 for all states below
the Fermi energy and 0 for all states above. At T /TF = 0.1 (dashed line) the Fermi surface isbroadened by roughly 0.1EF . At T /TF = 0.6 (dot-dashed) there is no recognizable Fermi surface
and the distribution approaches the classical limit of an exponential.

11
internal states, the ground state wave-function will have one fermion of each spin in the groundposition-space state, one of each spin in the ﬁrst excited state, and so on with two particles ineach state until there are no more particles, as shown in Fig. 2.1 a. The energy of the highest
occupied state is called the Fermi energy, denoted EF . If we take the example of a 3-d square wellpotential, the states are labeled by their momentum ~k and the density of states in k-space, ρ(k),V
2is given by ρ(k)dk = 2π
2 k dk, where V is the volume of the system. The total number of fermions∫k V 2is N = 2 0 F 2π
2 k dk. Here the factor of two is because we can put one fermion of each internal

state in each k state, and kF is the Fermi wave vector, which is the value of k that corresponds to√the highest occupied state, and is given by kF =2mEFh, where m is the fermion mass and ~ = 2π,~

where h is Plank’s constant. This gives
kF = (3π 2 n)1/3(2.1)

and correspondingly
EF =~2(3π 2 n)2/3
2m

(2.2)

where n = N/V is the total fermion density.
At ﬁnite temperature there will be excitations above this ground state, with some particles
near the Fermi surface being excited to states above it, as shown in Fig 2.1 b. A system attemperature T will have excitations going up above the Fermi energy by an extra energy on order ofkB T where kB is the Boltzmann constant, Fig. 2.1. We can deﬁne a unitless reduced temperature of
the system as T /TF where TF is the Fermi temperature corresponding to EF /kB . A gas at T /TF ≪1 will have a particle distribution that looks mostly like the ground-state particle distribution, butwith a smearing around the Fermi surface of width kB T , see Fig. 2.2. A Fermi gas at TTF ≪ 1 iscalled a quantum degenerate Fermi gas and is strongly aﬀected by quantum statistics. As T /TF is
increased, the smearing will grow larger and larger until at T /TF ≫ 1 the particle occupation will
drop to much less than one per state, and the gas will essentially behave as a classical gas. The
particle distribution for a ﬁnite temperature Fermi gas is given by the Fermi-Dirac distribution

12function
1f (E) =
e

E
kb T

(2.3)

/ζ + 1

where ζ = eµ/kb T is called the fugacity and µ is the chemical potential, which is approximately EF
for a weakly interacting low-temperature Fermi gas.
Let’s consider an example. The valence electrons inside a metal can be considered to be aweakly-interacting Fermi gas. If we take copper, which has a density of 9 grams/cm3 , an atomic
mass of 63.5 and one valence electron, we get an electron density of 8.4 × 1028 m−3 . Using theelectron mass, this gives TF = 81, 000 K. Thus, even at room temperature, copper has a reduced
temperature of approximately T /TF = 0.004, which is far into the quantum degenerate regime.
This thesis is about dilute atomic Fermi gases made up of potassium-40 (40 K) atoms. A typicalatomic density in the gas is 1019 m−3 which gives TF = 270 nK, a full twelve orders of magnitude
lower than for electrons in copper. Thus, to study a degenerate atomic Fermi gas in this regime,
we will need to cool our atomic vapor to nano-Kelvin temperatures.

2.1.1

A harmonically trapped Fermi gas

Eqs. 2.2 and 2.1 give the Fermi energy and Fermi momentum for a homogenous Fermi
gas, however, in ultracold atom experiments the gas is typically conﬁned by a harmonic potential,
which alters the density of states. For a cylindrically symmetric trap, parameterized by a radial
trap frequency ωr = 2πfr and an axial trap frequency ωz = 2πfz , which gives a trap aspect ratio
2Eof λ = ωz /ωr , the density of states becomes ρ(E) = 2λ(~ω
3 . The Fermi energy is thenr)

EF = ~ωr (6λN )1/3

(2.4)

where N is the number of atoms in a single spin-state. A corresponding Fermi momentum can be√deﬁned as kF = 2mEF /~. A typical conﬁnement trap has ωr = 2π × 250 Hz and λ = 0.1, which
gives EF = h × 10 kHz or TF = 450 nK for N = 105 atoms.
The equation for the Fermi energy of a homogenous gas and a trapped gas can be linked
by noting that the Fermi energy obtained by plugging the peak density of an ideal, harmonically13
trapped gas at zero temperature into Eq. 2.2 gives the same Fermi energy as Eq. 2.4. In fact,
one way to think about a trapped gas is using local density approximation (LDA), which says thateach concentric shell of constant density n in the gas can be thought of as a homogenous system,with a local Fermi energy given by Eq. 2.2. Any quantity for the trapped gas can be obtained by
calculating the same quantity for a homogenous gas with density ranging from n = 0 to n = n0 ,where n0 is the peak density of the trapped gas, and then performing a density weighted averageof that quantity using the density proﬁle of the trapped gas.
In the rest of this thesis, the Fermi energy will typically refer to Eq. 2.4; however, at times
we will also use Eqs. 2.2 and 2.1, and I will try to clearly specify when this is the case.

2.2

Cooling an atomic gas
The experimental techniques we use to create ultracold 40 K gases are now more than a decadeold, and they vary little from the techniques presented in the original BEC papers of 1995, withthe exception of the ﬁnal stage of evaporation being done in a far-detuned optical trap (FORT),
and that two spin states are now occupied during the evaporation. I will provide a brief descriptionof the experimental apparatus below; for a more detailed presentation I would recommend Brian
Demarco’s thesis [33] for everything concerning the MOT and magnetic trap, and Cindy Regal’s[6] and Jayson Stewart’s thesis [34] for the optical trap.
The ﬁrst stage of cooling is laser cooling in a magneto-optical trap (MOT). A MOT uses
counterpropagating laser beams on three orthogonal axes as well as a quadruple magnetic ﬁeld toslow down and trap atoms using the radiation pressure of light. With the MOT we are able tocreate a 40 K gas of a few 109 atoms cooled to a few hundred mK. The second stage of cooling
involves trapping two spin states of 40 K in a cloverleaf magnetic trap and performing microwave
evaporation to selectively remove atoms with the highest energy. In the ground state, 40 K is a spin
9/2 atom, and the spin states we use for evaporation are the |f, mf ⟩ = |9/2, 9/2⟩ and |9/2, 7/2⟩
states, where f is the total atomic spin and mf is the projection along the magnetic-ﬁeld axis. Afterevaporating for approximately 45 s in the magnetic trap, we typically achieve conditions of 25 × 107

14
atoms at 7 µK. At this point the gas has a reduced temperature close to T /TF = 3. For the ﬁnal
stage of evaporation, we use a crossed-beam FORT with lasers near 1070 nm, which is far enoughdetuned from the atomic transitions near 770 nm that resonant scattering is essentially negligible.
A FORT works through the AC atomic Stark shift, and for red detuned lasers, the Stark shift will
always be negative. Thus, a focused laser beam creates a trapping potential because atoms will
be attracted to the high intensity at the focus. The laser beam creates a gaussian conﬁnement
potential for the atoms, which can be approximated as harmonic near the center. We use a FORT
consisting of two crossed beams to achieve a higher axial trap frequency and thus a lower trap
aspect ratio. Forced evaporation in a FORT is achieved by slowly lowering the laser beam intensity
to allow the highest energy atoms to fall out of the trap. After about 10 s of evaporation in the
FORT, we reach a degenerate Fermi gas with 2 × 105 atoms at or just below T /TF = 0.1.
For the evaporation stages in both the magnetic trap and the FORT, it is important to have
two spin states of 40 K occupied in order to have collisions, and thus allow rethermalization of thegas as high energy atoms are removed. In scattering theory, collisions can be understood using
partial-wave analysis on the two-particle wave functions. The wave-function of the two colliding
particles can be broken into angular and radial parts, with the angular part being described by
the spherical harmonics. An l = 0, or s-wave collision, involves two atoms interacting with awave-function in the lowest spherical harmonic, while an l = 1, or p-wave collision, has atoms
interacting in the ﬁrst spherical harmonic. For l > 0, the collisions involve angular momenta
and there is a centrifugal barrier that can prevent the atoms from getting close, and this reduces
their chances of colliding. For interacting atoms with kinetic energy greater than the height ofthe centrifugal barrier, this is not an impediment to colliding; however, for kinetic energies lower
than the barrier height, the centrifugal barrier suppresses scattering. This is known as the Wigner
threshold law. As the temperature of a gas is reduced, the average collision energy decreases, andbelow a certain temperature collisions with l > 0 will be “frozen out” (except in the case of a l ̸= 0
scattering resonance as described in the next chapter). For 40 K, the barrier height or thresholdfor p-wave collisions is approximately 1 mK, and only s-wave collisions are energetically allowed

15
in the evaporation stages of the experiment [33]. However, fermions in a single spin state cannot
interact via an s-wave channel due to the antisymmetry principal, as discussed above, and thus it
is important to occupy at least two spin states to allow collisions in the gas to take place.
The experimental apparatus consists of two glass cells connected by a meter long tube. Eachcell has a MOT. The collection cell contains potassium getters, enriched for 40 K, for a source.
The getters are small samples of potassium-chloride salt and calcium that emit potassium vapor
when heated by running current through them. We can control the background vapor pressure ofpotassium in the collection cell by changing the current we are running through getter. However,the background pressure in the collection cell is too large for eﬃcient evaporation because a typicalatom will be hit by a high-energy background atom once every second or so. This is why we havea second cell, that we call the science cell, for evaporation where the typical time for a collision
with a background atom is approximately 100s. We move atoms from the collection cell MOT toa second MOT in the science cell with a “push beam.”
The bias coils used for the magnetic trap in the science cell can also be used to create abias magnetic-ﬁeld for atoms trapped in the FORT. We can create ﬁelds up to 250 G. In order toincrease interactions in the gas, we take advantage of a Feshbach resonance near 200 G that canincrease collisions in the gas by orders of magnitude. This both allows the evaporation in the FORT
to be more eﬃcient, by decreasing the rethermalization time scale, and also allows us to create astrongly interacting Fermi gas at the end of the evaporation. I will explain Feshbach resonances
in detail in the next chapter, including the physics behind them as well as some experiments tomeasure them and use them to create diatomic molecules.

2.2.1

Measuring temperature

You might be wondering: How do we measure the temperature of a dilute microscopic gasthat is only a few nano-Kelvin? There are rather signiﬁcant challenges: the gas has a size of justa few tens of microns, it exists in an ultrahigh vacuum, it is a million times more dilute than air,it will only scatter light that is precisely tuned to the atomic transition, and it is the coldest thing

16
in the universe (assuming intelligent life somewhere else in the universe haven’t created somethingeven colder without us knowing); hwoever, it turns out it is not that diﬃcult to measure thetemperature. First, we already have lasers precisely tuned to the atomic transition that we use
to make the MOT. Also, even though the gas is dilute compared to air, the trapped cloud can bevery optically dense for resonant light and absorb more than 99% of light that travels through it.
This allows us to take a picture of the cloud, using lenses to magnify and focus the image onto a
low noise CCD camera. Speciﬁcally, the atom gas absorbs photons and scatters them to create ashadow image. By taking a second image of the light without any atoms present, we can determine
how many photons were absorbed at each position and back out the column density of the atoms.
One minor problem is that when the atoms absorb and scatter photons during an image acquisition,
they are heated. In fact, a single photon imparts an energy to an atom that is roughly equal to theaverage energy of an atom in the ultracold gas. To take a high quality image, we need to scatter
hundreds of photons oﬀ of each atom, which easily imparts enough energy to the heat the gas outof the trap. Fortunately, we are able to scatter hundreds of photons oﬀ of each atom in just a few
tens of microseconds such that they are essentially frozen in place during the picture. However,this means that after each picture we will need to prepare another ultracold Fermi gas from scratch
in order to make another measurement.
There are two kinds of measurements we can make by taking a picture of an ultracold Fermi
gas. The ﬁrst is to take a picture of the cloud while it is being held in the conﬁnement trap. This
gives the density proﬁle n(r) of the trapped gas. Actually, it gives the projection of n(r) onto a
2d surface. From this, the 3d density distribution can be extracted assuming symmetry and using
an inverse Abel transform. One can compare the measured n(r) to the theoretical prediction fora non-interacting Fermi gas held in a harmonic trapping potential and extract a temperature. In
reality, this measurement turns out to be rather diﬃcult with our apparatus due to the diﬃcultyof measuring a very small and optically dense sample with our imaging system. A second kind of
measurement is to take a picture after a time of ﬂight (TOF). This involves quickly turning oﬀ thetrapping potential, after which the atom cloud begins to fall due to gravity, as well as expand as the

17gas ﬂies apart. If the atom cloud is weakly interacting, we can assume the expansion is ballistic, orin other words, that no momentum changing collisions take place during the expansion. Once thecloud has expanded to much larger than its original size, the density distribution of the cloud will
reﬂect the velocity distribution of the gas just before we turned the trap oﬀ. A typical expansion
time is 10 ms. With a TOF measurement, we can compare the measured velocity distribution to
the theoretically predicted velocity distribution for a harmonically trapped Fermi gas to extract atemperature.
For a non-degenerate Fermi gas, we can use classical thermodynamics, which predicts a√gaussian distribution for the velocities, with the rms velocity ⟨v 2 ⟩ being related to temperatureby the equipartition theorem as 21 m⟨v 2 ⟩ = 32 kb T . Once the gas is cooled below approximately
T /TF = 0.6 it becomes necessary to account for quantum eﬀects and use Fermi-Dirac statistics tounderstand velocity distributions. Below T /TF = 0.1, the velocity distribution approaches the zerotemperature limit of a trapped Fermi gas and it becomes diﬃcult to measure the temperature tohigh accuracy due to imaging resolution and a limited signal-to-noise ratio. The lowest temperaturegases we have measured are T /TF = 0.07 ± 0.01.
For more detailed discussions of measuring the temperature of a trapped Fermi gas, I would
recommend the discussions in the theses of Brian Demarco, Cindy Regal, and Jayson Stewart[33, 6, 34].

Chapter 3

Feshbach resonances

In this chapter, I will discuss how atoms in an ultracold gas interact with each other and,speciﬁcally, how these interactions can be tuned using a Feshbach resonance to make the systemstrongly interacting. Feshbach resonances can enhance scattering in any angular momentum channel, and I will discuss both experiments near an s-wave and near a p-wave Feshbach resonance. In
particular, I will present a new measurement of the location of an s-wave Feshbach resonance for
40 K atoms as well as experiments creating and observing p-wave molecules near a p-wave Feshbachresonance. The theoretical discussion of Feshbach resonances follows largely from Ref. [35] although for the s-wave case similar treatments can be found elsewhere, for example Ref. [4]. Cindy
Regal’s thesis [6] is also a good reference for the particular case of 40 K Feshbach resonances. The
experiments involving the p-wave Feshbach resonance were published in references [2, 3].

3.1

Interactions in an ultracold atomic gas
One of the great features of using ultracold atomic gases to study the physics of stronglyinteracting systems is that the interactions at the two-body level, between two isolated atoms, canbe understood in a rather simple framework. This is due to the fact that we are usually in a situation
where the dominant interaction, the Van der Waals interaction, which falls oﬀ as r−6 where r is the
distance between atoms, is a short-range interaction and can be mapped onto a pseudopotentialconsisting of a delta-function interaction. The interaction can then be characterized by single
parameter, a, the s-wave scattering length. This amounts to an approximation for scattering at low

19energy with a short-range potential. Speciﬁcally, the s-wave scattering amplitude from a short-range
potential can be expanded for low k as

f (k) = −1a1
2

+ reﬀ k2 − ik

(3.1)

where ~k is the collision momentum, and reﬀ is the eﬀective range of the potential [36, 35]. In thelimit ka → 0, the scattering amplitude asymptotes to a constant value, −a, which gives a scatteringcross section σ = 4πa2 . For ka ≫ 1 and reﬀ k ≪ 1, we ﬁnd f = i/k giving the unitarity limited
cross section of σ = 4π
. In the limit where reﬀ k ≪ 1, the interaction can be modeled by a deltak2function pseudopotential and the atom-atom interaction energy becomes [35]U (r) = λδ(r)(3.2)

where λ is adjusted to produce the proper scattering length viaa=m
λ
4π~2 1 − λλ

(3.3)

c

where m is the atom mass and λc is the critical value of λ where a bound state appears in theattractive delta-function potential (λc < 0) [35]. It can be seen that as λ → λc the scattering
length diverges. It is in this limit of large scattering length where the many-body system becomesstrongly interacting.
This delta function pseudopotential reproduces all of the essential physics of the more complicated atom-atom interaction [35] (note: it is important to impose a proper cutoﬀ at large momenta
for the delta function potential to obtain a non-zero scattering length [35]).
In principle, the scattering length a depends on the details of the full atom-atom potential,
but in practice a is almost completely determined by the position of the single bound state closest
to the threshold energy for two atoms scattering with zero collision energy (as in the case above
for the delta function potential). If there is a bound state at an energy slightly below threshold,the scattering length is positive. If there is a bound state exactly at threshold, there will be azero energy scattering resonance and |a| → ∞. In a situation where there is no bound state below

20
threshold, but if the potential were to be made a little bit deeper one would appear, the scattering
length is negative [36].
This picture gives an idea of how one could tune the scattering length and thus the strengthof the pseudopotential. If the interatomic potential could be tweaked in such a way as to move theposition of the last bound state around threshold then the scattering length, and thus the strengthof the pseudopotential, could be tuned to any value. This is exactly how a magnetic-ﬁeld Feshbachresonance works. It turns out that the position of the bound states relative to the thresholdenergy of the atoms can be tuned using the diﬀerential Zeeman energy shifts of the various spinchannels[6, 35]. The magnetic-ﬁeld strength, B0 , where a new bound state appears at thresholdcorresponds to the peak of a Feshbach resonance, and near B0 the scattering length tunes with themagnetic ﬁeld strength B as [4, 6]
a = abg (1 −w)
B − B0(3.4)

where abg is the background scattering length, w is the width of the resonance, and B0 is the center
position of the Feshbach resonance, see Fig. 3.1. Feshbach resonances are an amazing tool forstudying strongly interacting quantum gases because they allow us to tune the scattering length toany value simply by varying the strength of an applied external magnetic ﬁeld.
The bound state that is brought into resonance can be either an s-wave (ground-state rotational) or p-wave (1st excited rotational) molecular state and the corresponding scattering resonancethen occurs in the s-wave or p-wave collision channel. Away from any scattering resonance, the low
collision energies in ultracold gases mean that p-wave scattering is usually “frozen out” in accordance with the Wigner threshold law. However, in the presence of a p-wave scattering resonance,a spin-polarized Fermi gas can be dominated by p-wave interactions. This raises the interesting
possibility of creating a p-wave paired superﬂuid. Before we discuss this possibility further, it will
be helpful to go over a few more concepts and basic measurements. In particular, an understandingof the concepts of broad and narrow resonances is essential.

21

Figure 3.1: A Feshbach resonance The scattering length near an s-wave Feshbach resonance for
40 K atoms is plotted as a function of magnetic ﬁeld. The solid lines are the theoretical prediction
given by Eq. 3.4 where the parameters B0 and w were adjusted to ﬁt the data and abg was
known from previous measurements. The data points (circles and squares) are measurements ofthe scattering length from rf spectroscopy of mean-ﬁeld energy shifts. The data shows that the
scattering length, and therefore the strength of the atom-atom interactions, can be tuned over alarge range. It is important to note that one would not expect to be able to measure scattering
lengths above a few thousand Bohr radii (a0 ) in this experiment due to the unitarity limit. Thisﬁgure is taken from Ref. [1]
.

22

3.2

Wide vs narrow resonances and universality
The situation I described in the last section, where the interactions can be parameterized bya single parameter, the scattering length a, and understood in terms of a delta function pseudopotential, is the situation of a universal or broad resonance [6, 35]. To understand what is meant by
this, consider Eq. 3.1, which gives the scattering amplitude for s-wave collisions, and take ka → ∞.
In order for the scattering amplitude to be approximately what it would be in the case wherereﬀ = 0, we require reﬀ k ≪ 1 for all relevant k in the system [35]. We can see this by rewriting eq.
i −1
3.1 as f (k) = ki (1 + 2i reﬀ k − ka
) . For a degenerate Fermi gas, this means we require reﬀ kF ≪ 1,

where kF is the Fermi wavevector, for a “broad” or “universal” Feshbach resonance. The resonanceis universal in the sense that the scattering only depends on the scattering length a, as opposed toa situation where higher order terms would need to be taken into account and interactions would
depend on the details of the full atom-atom potential. It is “broad” in the sense that when ka → ∞all collisions in the gas are resonant; in other words, they all have the unitarity limited value forthe scattering amplitude. An equivalent deﬁnition of a broad resonance is that the energy width
2

4~of the resonance, given by Γ = mr
2 , be larger than the Fermi energy [35].
eﬀ

If a resonance is narrow, then only a small fraction of collisions in a band of energies of widthΓ (or a band of momenta of width ~/reﬀ ) are resonant. The width of that band depends on reﬀ , andtherefore the physics is not universal. Interestingly, when the resonance is narrow the many-bodyproblem can be treated perturbatively, even at the center of the resonance [35].
Studying the scattering problem in a two-channel model allows one to relate reﬀ to theparameters of the Feshbach resonance, which is fundamentally a multi-channel scattering problem.
It can be shown thatreﬀ = −

2~2mµco abg w(3.5)

where µco is the diﬀerence in magnetic moment between the closed and open channels and istypically close to 2µB where µB is the Bohr magneton [35, 4, 37]. For the 40 K resonance, near 202
G, we ﬁnd reﬀ ≈ −60a0 . Using a typical value of kF we have kF reﬀ ≈ −0.03, so the resonance is23well into the universal regime.
The discussion above centered on s-wave resonances, but it turns out that the situation for
p-wave resonances is a bit diﬀerent due to the existence of a centrifugal barrier. In particular, thelong-range nature of the centrifugal barrier (the potential falls of as r12 ) ensures that a quasi-boundresonance state with a well deﬁned energy persists down to asymptotically low scattering energies.
In contrast, for s-wave scattering resonances, a quasi-bound state only exists at high energies and is
not accessed when one is in the limit of a broad resonance [35]. For p-wave scattering, the expansion
for the scattering amplitude becomesfp (k) =k2−ν −1 + 12 k0 k 2 − ik 3

(3.6)

where ν, the scattering volume, is the p-wave analog of the scattering length and k0 is theanalog of the eﬀective range [38, 35]. By expanding the scattering amplitude in the low energy
22~
limit it can be shown there is a resonance at energy E ≈ mνk, which scales approximately linearly0with magnetic ﬁeld [2, 35], and has an energy width given byΓp ≈4m1/2 E 3/2
.
k0 ~

(3.7)

Unlike the energy width for a broad s-wave resonance, where there is no quasi-bound state
and the resonance is at zero collision energy, the width of the p-wave resonance depends on a ﬁniteΓresonant scattering energy and scales as E 3/2 . This means that as E → 0, then Ep ∝ E 1/2 → 0.
In other words, in the low energy limit the p-wave resonance is always a narrow resonance. The
physical reason for this is that the energy width Γp of the p-wave resonance can be thought of interms of the inverse tunneling time into a quasi-bound state at energy E, which scales as E 3/2 [35]due to the centrifugal barrier (see Fig. 3.2). At low energies, the tunneling time becomes inﬁnite
and thus the energy width goes to zero.

24

V(R)

0

R

Figure 3.2: Tunneling through a p-wave barrier The schematic curve V (R) shows the atom~2
atom potential with the p-wave centrifugal barrier equal to mR
2 where R is the internuclear separation. The curve is not drawn to scale as the height of the barrier is h × 5.8 MHz for 40 K atoms
whereas the depth of the full potential is in the THz regime. The red line is a sketch of the wave
function for a low energy quasi-bound state which oscillates quickly inside the potential, decays
exponentially underneath the centrifugal barrier, and then oscillates slowly in free space at large
R. For E much smaller than the barrier height, the tunneling time through the barrier scales as
E −3/2 where E is the scattering energy.

25

3.3

Measurement of molecule binding energies and resonance centers
In this section I describe measurements of the molecule binding energies near an s-wave

resonance and near a p-wave Feshbach resonance. These binding energies can be used to determine
Feshbach resonance parameters including the center of the resonance. While previous work in ourgroup made determinations of the positions and widths of these resonances, the measurements I
present here are more precise and also give some important insight into these resonances. The
measurement of the p-wave molecule binding energies was published in Ref. [2].

3.3.1

Determination of s-wave Feshbach resonance parameters with rf molecule
dissociation

One way to eﬃciently create Feshbach molecules is to adiabatically convert atom pairs intomolecules by slowly ramping the magnetic ﬁeld from above the resonance (where there is no boundstate near threshold) to below the resonance where a bound state exists [39]. This method wasﬁrst demonstrated in Ref. [39] for a resonance between the |9/2, −9/2⟩ and |9/2, −5/2⟩ states of
40 K near 220 G. The s-wave resonance we are interested for the purpose of this thesis is between

the |9/2, −9/2⟩ and |9/2, −7/2⟩ states of 40 K and occurs near 202 G [6]. Here, we create molecules
in an atom gas consisting of an equal mixture of atoms in the |9/2, −9/2⟩ and |9/2, −7/2⟩ states
at T /TF = 0.1 by ramping the magnetic ﬁeld from 203.5 G to a value B below the resonance. The
ramps are done at rates much slower than 40µs/G to be well within the adiabatic limit for creating
molecules [39].
In order to measure the molecule binding energy, we perform rf spectroscopy on the gas.
Speciﬁcally, we apply an rf ﬁeld at a frequency near the energy splitting of the |9/2, −7/2⟩ and|9/2, −5/2⟩ states, which is approximately 47 MHz for ﬁelds near 200 G. Then, we selectively
image any atoms in the |9/2, −5/2⟩ state and record their number as a function of the rf frequency.
To make sure the rf lineshape is unaﬀected by many-body eﬀects, we ﬁrst turn oﬀ the trapping
potential and let the cloud expand for 5 − 15 ms to lower the atom density before applying the rf

26

3x104

Natom

-5/2
4

2x10

-7/2
-9/2

1x104

0
47.05

47.10

47.15

RF Frequency (MHz)Figure 3.3: RF molecule dissociation (Main ﬁgure) An rf lineshape taken at 201.25 G at frequencies near the |9/2, −7/2⟩ and |9/2, −5/2⟩ energy splitting. The sharp feature at lower energy
corresponds to the bare transition for an unpaired atom with a width determined by the energy
resolution. This energy splitting can be used to calibrate the magnetic ﬁeld. The broad feature athigher energy is the molecule dissociation feature. The solid line is the combination of a gaussian forthe bare atom transition and a theoretical molecule dissociation curve (eq. 3.8) for the higher frequency feature. By ﬁtting the the molecule dissociation curve, the binding energy for the moleculeat this magnetic ﬁeld is obtained. In this case, we obtain a binding energy of h × (46.4 ± 1.5) kHz.
(Inset) A schematic showing the transitions corresponding to the two observed features. Threesolid lines mark the energies (vertical axis) of the three lowest mF states. Initially, only the lower
two states are occupied. The shorter arrow marks the bare atom transition of the rf lineshape. If
the atom is bound in a molecular state its energy is lowered by EB (dotted line) and the transitionis shifted to higher frequency. In addition, the atoms can be dissociated into a continuum statewith ﬁnite kinetic energy above the bare |9/2, −5/2⟩ line, which accounts for the high frequency
tail of the molecule dissociation feature.

27
pulse. The power of the rf pulse is modulated by a gaussian envelope with a full width 1/e2 max
duration of 300 µs, such that the Fourier limited energy resolution is a Gaussian of full width 1/e2
max equal to 4.3 kHz. A sample rf lineshape is shown in Figure 3.3.
Two features can be observed in the lineshape. A narrow feature at low frequency corresponds
to ﬂipping the spin of unpaired or free atoms from the |9/2, −7/2⟩ state to the |9/2, −5/2⟩ state. The
center frequency of this feature can be used to calibrate the experimental value of the magnetic ﬁeld.
A second broad feature at higher energy corresponds to the dissociation of bound molecules. The
shape of this feature is determined simply from the wavefunction overlap between the molecularstate and scattering states of two atoms and Fermi’s golden rule. An analytic formula for thenumber of atoms transferred as a function of rf frequency is given in Ref. [39, 40] to be√I(νrf ) ∝hνrf − EB(hνrf )2

(3.8)

where νrf is the rf frequency minus the frequency of the single-atom transition and EB is themolecule binding energy. Here we have assumed that interactions between the initial and ﬁnal
states are negligible.
By ﬁtting the rf lineshape to Eq. 3.8 we can determine EB . This measurement was repeated
for a number of magnetic-ﬁeld values with the results plotted in Figure 3.4. The binding energy of
the Feshbach molecules near the resonance is well approximated by [6]EB =~2m(a − r0 )2(3.9)

where a is given by Eq. 3.4 and r0 ≈ 60a0 is the range of the Van der Waals potential [6]. Fitting
the binding energy data to this formula with r0 = 60a0 and abg = 174a0 yields B0 = 202.20±0.02 G
and w = 7.04±0.1 G. The parameters r0 and abg are constrained by theory and other measurements[6, 41], however, prior to this measurement B0 was only known to within an error of 0.07 G [6].
In this experiment, rf spectroscopy simply served as a tool to measure the binding energy of alow density and, hence, weakly interacting gas of Feshbach molecules. This was possible because we−r/ahave an analytic formula for the Fesbach molecule wavefunction (Φ(r) ∝ e r[4]) and can predict

Binding Energy (kHz)

28

0
30
60
90
120
200.5

201.0

201.5

202.0

Magnetic Field (G)Figure 3.4: s-Wave molecule binding energy Measurements of molecule binding energies near
the s-wave Feshbach resonance between the |9/2, −7/2⟩ and |9/2, −9/2⟩ states of 40 K atoms (blackdots) are plotted as a function of magnetic ﬁeld. The black line is a ﬁt to eq. 3.9. The ﬁt values
are given in the text.

29
the rf lineshape (Eq. 3.8) based on this and Fermi’s golden rule. However, we will see in Chapters
5,6 how rf spectroscopy can also be used as a tool for probing a strongly interacting system where
we do not know the form of the many-body wave function.

3.3.2

Determination of p-wave molecule binding energies with magneto-association

To measure the binding energy of p-wave molecules we took advantage of another spectroscopic technique pioneered at JILA and demonstrated for s-wave resonances in 40 K by our groupin Ref. [42]. The technique is a molecule association technique whereby one starts with a gas of
unpaired atoms and then oscillates the magnetic ﬁeld at a frequency close to the molecule binding
energy divided by h. The oscillating ﬁeld drives the atoms into bound molecules via a stimulated
emission process. When molecules are formed they can be lost due to inelastic collisions with otheratoms or molecules, which allow them to fall into deeply bound molecular states and gain enoughenergy to escape the conﬁnement trap. Thus, an atom loss feature can be a signature of molecule
association.
In these experiments, we cool a balanced spin mixture of atoms in the |9/2, −9/2⟩ and|9/2, −7/2⟩ states to T /TF = 0.2. We ramp the magnetic ﬁeld to a value near the p-wave Feshbachresonance, which occurs near 199 G. Then, we apply a small sinusoidal oscillation to the magnetic
ﬁeld at a frequency νmod for a duration of 36 ms. We record the number of atoms in the |9/2, −7/2⟩
as a function of νmod ; sample lineshapes are shown in Fig. 3.5.
In Fig. 3.5, we observe an asymmetric atom loss feature that has a width that increases
with the cloud energy EG = 12 m⟨v 2 ⟩, where the rms velocity is obtained from a Gaussian ﬁt
to the expanded cloud. EG is thus a measure of the average kinetic energy of the particles and is
proportional to the range of atom collision energies. The width and shape of the association featurearises from the distribution of collision energies in the atom cloud. To determine the binding energyof the molecule, we measure the peak energy of the loss feature as a function of EG and extrapolate
to EG = 0. We repeated this measurement at multiple ﬁelds around the resonance and the results
are shown in Fig. 3.6.

30

Scaled Natom

1.0
0.8
0.6
0.4
0.2
0

0

20

40

60

80

100

nmod (kHz)
Figure 3.5: p-Wave molecule magneto-association lineshapes Lineshapes for association toa bound p-wave molecule using a sinusoidally modulated magnetic ﬁeld. For this data, B = 198.12
G. We ﬁnd that the line shape is asymmetric and has a width that depends on the kinetic energy of
the atoms EG , which is deﬁned in the text. The data are for values of EG of h × 2.0 kHz (circles),h × 4.5 kHz (triangles), and h × 6.0 kHz (squares). To highlight the asymmetry and dependence
on EG , the total number of atoms, Natom , for each lineshape was scaled to one. This ﬁgure is from
Ref. [2].

31

200

E (kHz)

100
0
100

-100

0

-200

-100
0

-1.0

-0.5

0

0.5

0.5

1.0

1.0

1.5

DB (G)
Figure 3.6: p-Wave molecule energies Energy of the molecule as a function of magnetic ﬁeld forboth the ml = 0 (open circles) and ml = ±1 (closed circles) resonances. E < 0 corresponds to abound molecule; E > 0 corresponds to a quasi-bound state. ∆B is the detuning from the ml = ±1
resonance position, which we measure to be B = 198.30 ± .02 G. Linear ﬁts give a slope of 188 ± 2
kHz/G for the ml = 0 resonance and 193 ± 2 kHz/G for ml = ±1. The inset shows data for the
ml = 0 resonance that suggest some non-linearity near the resonance. These data were taken all in
one day to reduce the uncertainty in the magnetic ﬁeld. This ﬁgure is from Ref. [2].

32
As can be seen in Fig. 3.6, there are two features, which we identify as corresponding to
p-wave molecules with diﬀerent values of ml . We measure the magnetic-ﬁeld diﬀerence between theresonances for molecules with ml = ±1 and ml = 0 to be approximately 0.5 G. The splitting arises
from the magnetic dipole interaction and was ﬁrst observed in Ref. [43] and explained in Ref. [38].
For both molecule features, we observe a linear dependence of the pair energy, E, on magnetic-ﬁeld
detuning. We measure the center of the ml = ±1 resonance, the lower magnetic-ﬁeld feature, tobe 198.30 ± 0.02 G.
Interestingly, we can follow the molecule association feature to positive energies where notrue bound molecular state exists. The feature at positive energies is due to association to a longlived resonance state, or quasi-bound state, with the lifetime of that state being determined by thetunneling time through the centrifugal barrier (Fig. 3.2). The tunneling time causes the width ofthese features to be as much as three times larger than for bound states, for data taken with similar
initial cloud energies.
We can take advantage of this resonant state to directly observe the presence of p-wave
molecules and distinguish between the ml = 0 and ml = ±1 states. For these experiments wefound it advantageous to use a spin polarized gas of atoms in the |9/2, −7/2⟩ state to eliminate
non-resonant s-wave collisions. We start with a 95/5 mixture of |9/2, −7/2⟩ and |9/2, −9/2⟩ andthen obtain a nearly pure spin-polarized gas with a slow sweep through the s-wave resonance at
202.2 G where inelastic losses on the low side of the resonance ensure that nearly all |9/2, −9/2⟩
atoms are lost [44]. At this point, we run into the unfortunate situation whereby the gas cannot
rethermalize any further because s-wave collisions are forbidden by quantum statistics. We lower
the trap depth to reach 105 atoms, but the gas is no longer in thermal equilibrium.
While we can create molecules using magneto-association as discussed above, it turns out
to be far more eﬃcient to simply hold the magnetic ﬁeld near the Feshbach resonance for a few
ms. To detect the presence of molecules, we quickly increase the magnetic ﬁeld, in 10 µs, to avalue above the resonance where the quasi-bound molecules have a large positive energy. The
ramp adiabatically converts the bound molecule into the quasi-bound state. The paired atoms

33

Figure 3.7: Dissociated p-wave molecule clouds Images of dissociated p-wave Feshbach
molecules. A linear grayscale indicates the optical depth, with white corresponding to more absorption of the resonant probe light. The images are taken after a 4 ms ballistic expansion fromthe trap and therefore show the velocity distribution of the atoms resulting from the dissociation of
the Feshbach molecules. The left image corresponds to the ml = ±1 resonance and the right image
corresponds to the ml = 0 resonance. The image plane is transverse to the quantization axis, which
is deﬁned by the external magnetic ﬁeld. The images show the expected angular distributions forp-wave pairs. This ﬁgure is from Ref. [3].

34then tunnel out of the centrifugal barrier and this energy is converted to kinetic energy of atoms
ﬂying apart. We then immediately turn oﬀ the optical trap, expand for a variable time of 1.9 to
5 ms, and take an absorption image. If we wish to remove the unpaired atoms before imaging themolecules, we can use a pulse of laser light that is resonant with the free atoms while the magnetic
ﬁeld is still at a value on the bound state side of the resonance. Since the molecules do not absorb
this light they are unperturbed. The result is a large energetic cloud of atoms, as seen in Fig. 3.7.
This is similar to a detection scheme used in Ref. [45] for narrow s-wave resonances. The angular
distributions seen in Fig. 3.7 are consistent with p-wave pairing.

3.4

Measurement of molecule lifetimes
In the discussion of p-wave resonance, we saw that to parameterize the resonance we need tomeasure the lifetimes of the resonant states. We accomplish this using two methods. One method
takes advantage of the molecule detection technique discussed above. Here, we create molecules onresonance and then ramp to a ﬁeld where we wish to measure the lifetime. Then, we detect the
number of molecules by quickly jumping the ﬁeld to a value where the pair energy is much higher
and will be converted to kinetic energy of free atoms, as discussed above. These measurements areshown as circles in Fig. 3.8.
A second technique takes advantage of magneto-association. Here, we can use the measured
width of the magneto-association line shapes to deduce an energy width or lifetime of the quasibound molecules. It is important to remove the eﬀect of broadening due to the ﬁnite energy
distribution of the atoms in order to measure the intrinsic energy width we are interested in. These
points are shown as squares in Fig. 3.8.
Putting these two measurements together and ﬁtting a line through the data we ﬁnd that theinverse tunneling rate τ −1 = αE 3/2 with α =

1
3
h2

1

(2.2 ± 0.1)10−3 s 2 . This agrees well with a theory

prediction from John Bohn [2] shown as the solid line in Fig. 3.8. If we use the inverse tunnelingrate to calculate an energy width using Γp = τ −1 /(2π), and take E = h × 10 kHz, which is a typicalFermi energy for our system, we ﬁnd an energy width of h × 0.35 kHz, which is much smaller than

1/e Lifetime (ms)

35

100
10-1
10-2
10-3

-200

-100

0

10

100

E (kHz)Figure 3.8: p-Wave molecule lifetimes Lifetimes of p-wave molecules are plotted as a functionof the molecule energy. Negative energies (left) correspond to bound molecules, while positiveenergies (right) correspond to quasi-bound pairs. Data for the ml = 0 resonance are shown in open
symbols, while ml = ±1 are closed symbols. The two dotted lines on the left indicate averages ofthe measured bound state lifetimes. The solid line on the right is a theory curve for the quasi-boundlifetimes. This ﬁgure is from Ref. [2].

36
EF , putting us well into the narrow resonance regime.
We can also use direct molecule detection to measure the lifetimes of bound molecules below
the resonance using the same method as for the quasi-bound molecules. Here, we measure lifetimesof 2.3 ± 0.2 ms for the ml = 0 molecules and 1.0 ± 0.1 ms for ml = ±1 molecules. These values
show no measurable dependence on binding energy or magnetic-ﬁeld value. Theory work from John
Bohn predicts molecule lifetimes set by dipolar relaxation rates of 8.7 ms and 6.8 ms for the ml = 0
and ml = ±1 molecules [2]. Dipolar relaxation is a process where two atoms fall into lower energy
spin states and ﬂy apart with the extra kinetic energy. In the dipolar relaxation process angularmomentum is conserved by an increase or decrease in the orbital angular momenta of the atoms.
Dipolar relaxation is possible for these p-wave Feshbach molecules because atoms in the |9/2, −7/2⟩
are not in the lowest energy spin state. The fact that we measured shorter lifetimes than predictedby theory points to other possible decay mechanisms such as collisional decay.
It is important to remember that Feshbach molecules are molecules in the least bound state of
the atom-atom potential and that many other deeply bound states exist. In the pseudo-potential
approximation, we ignore the presence of these states as they have no eﬀect on the two-bodyscattering and interactions. However, if three or more atoms come together, inelastic processes
that leave two atoms in a deeply bound state become possible. If two of the atoms are alreadyin the Feshbach molecule state, the wave-function overlap with the more deeply bound states issigniﬁcantly increased and the process can be strongly enhanced.
In order to test whether atom-molecule collisions were playing a role in the p-wave molecule
lifetimes, we measured the molecule lifetime after removing all remaining free atoms. Since theconversion eﬃciency of atoms to molecules was about 20%, the remaining free atoms were themost likely collision partners for the molecules. Free atoms were removed with a resonant pulse oflight that heated them out of the trap. Using this technique, we measured a molecule lifetime of
7 ± 1 ms for the ml = 0 molecules (Fig. 3.9), consistent with the theoretical prediction for dipolar
relaxation.
If a p-wave resonance existed in the lowest energy Zeeman state, it would be possible toN molecules

37

time
Figure 3.9: Eﬀect of atom removal p-wave Feshbach molecule lifetime after removal of free atoms.
The molecules were created at the mL = 0 resonance. The solid line is a ﬁt to an exponential decay
which gives a lifetime of 7 ± 1 ms. This ﬁgure is from Ref. [3].

38eliminate dipolar relaxation as a possible decay channel. However, no such resonance exists for40 K. Interestingly, 6 Li has a p-wave resonance in the lowest energy state and, following our work,groups in Australia and Japan set about creating and measuring p-wave molecule lifetimes in 6 Li
[46, 47]. The Japanese group was able to observe p-wave molecule lifetimes that were limited only
by inelastic collisional losses [46].
It is interesting to contrast the general features of the p-wave molecule lifetime measurementswith previous s-wave molecule lifetime measurements performed by our group [44]. First of all,measurements of quasi-bound state lifetimes were not possible for the broad s-wave resonance. For
the s-wave resonance, we expect the quasi-bound state lifetimes to be extremely short, and thus
the energy width of any feature to be very large, since there is no centrifugal barrier. In fact, fora broad s-wave resonance there is no quasi-bound state immediately above the resonance [35]. On
the bound-state side of the resonance, the s-wave molecule lifetimes are found to depend strongly
on magnetic-ﬁeld detuning from the resonance. This is because the size of the s-wave molecule is astrong function of binding energy and hence detuning from the resonance. Smaller molecules havebetter wave function overlap (Frank-Condon overlap) with deeper bound molecular states and thus
decay more rapidly into those states via inelastic collision processes. In contrast, the size of thep-wave molecules are not expected to vary with molecule binding energy since their size is largely
set by the presence of the centrifugal tunneling barrier.
Furthermore, the s-wave molecules are found to have long lifetimes at small binding energies,
which are much longer than what we measured in the case of p-wave molecules. This can be
attributed to the greater size of the s-wave molecules and hence smaller overlap with deeper boundstates as well as a stronger Pauli suppression of inelastic processes [48].

3.5Creating p-wave molecules near the Feshbach resonance
As noted above, we found that it was possible to produce large numbers of p-wave molecules

by setting the magnetic ﬁeld to a value near the resonance and waiting. With our method to see themolecules, we could dynamically probe this process. In these experiments, we ramp the magnetic

39

1.2x105Natom(a)8.0x104
4.0x1040B

Nmol

1.0x104

(b)

BRes
t

5.0x103

0

0

5

10

15

Time (ms)Figure 3.10: p-Wave molecule creation (a) Measured atom number as a function of time thatthe magnetic ﬁeld is held at the mL = ±1 resonance. (b) Measured molecule number for the same
hold time on resonance. The inset shows the timing sequence for this experiment. The numberof molecules is measured using the dissociation technique described in the text (solid line.) The
number of atoms is measured by ramping the ﬁeld below the resonance (dashed line) where theimaging does not detect molecules. This ﬁgure is from Ref. [2].

Nmolecules

40

Figure 3.11: p-Wave molecule creation vs. magnetic ﬁeld Number of molecules created for a
1 ms hold vs. magnetic-ﬁeld detuning from the resonance. The data suggest that molecule creation
occurs only when the p-wave resonant state has a positive energy that is less than the maximum
collision energy between atoms in the Fermi gas.

41
ﬁeld to a value near the Feshbach resonance and then hold for a variable amount of time. To detect
the presence of molecules we quickly jump the ﬁeld above the resonance and expand the cloud as
described in the previous section. Alternatively, the number of atoms can be measured by ramping
the ﬁeld below the resonance where the imaging light is not sensitive to the bound molecules. The
experimental sequence is shown in the inset of Fig. 3.10. In Fig. 3.10, we show the number ofatoms and molecules as a function of hold time near the resonance. The data in Fig. 3.10 was
taken for the magnetic-ﬁeld value where we observe the highest conversion eﬃciency to molecules.
The molecule population increases rapidly for approximately 1 ms and then decays with a time
constant on the order of a few ms. At long times both the atom and molecule populations decay
monotonically, indicating that inelastic loss processes are taking place.
In Fig. 3.11, we show the number of molecules created after a 1 ms hold time for diﬀerent
detunings from the Feshbach resonance and also for diﬀerent values of the Fermi energy. We seethat molecule creation occurs only over a small range of magnetic-ﬁeld values near the resonance.
The molecule creation feature is asymmetric and has a width that increases with increasing cloudenergy. This suggests the width of the feature arises from the atomic kinetic energy distribution in
the cloud rather than any intrinsic width. This is consistent with our determination that the gas
is in the narrow resonance limit.

3.6

p-wave superﬂuids
Ultimately, the goal of this research was to explore the possibility of creating a stronglyinteracting Fermi gas dominated by p-wave interactions where a p-wave paired superﬂuid statemight be accessed. Such an unconventional superﬂuid would be extremely interesting from thepoint of view of condensed matter physics due to the vector order parameter (compared to thes-wave case) and possibilities for quantum phase transitions [49, 50, 51, 52]. Furthermore, in two
dimensions it is predicted that such a system could exhibit non-Abelian topological excitations,which is a topic of wide interest with possible applications to decoherence-free quantum computing
[53].

42
In order to study many-body physics in a strongly interacting atomic gas, it is important
for the rate of inelastic processes in the gas to be small compared to the rate of elastic scattering.
One way to think about this is that the true ground state of the cold atom gas is actually for
all the atoms to bind together and crystalize. In this sense, the dilute gas is in a highly excitedmetastable state. However, as long as the relaxation rate to lower energy states is slow enough, it
can essentially be ignored. In this case, we can study the physics of the metastable state, whichis governed by elastic scattering processes between the atoms and can be mapped onto a pseudopotential problem that has no deeply bound states. In practice, one runs up against some inelastic
processes, and in particular, three-body collisions that lead to the formation of more deeply bounddimers. The amount of energy released in such a collision is so great that the atoms promptly ﬂyout of the trap and stick to the wall of the vacuum chamber. Thus, the presence of inelastic decay
to deeply bound states manifests itself as atom and molecule loss in the trapped gas. As long as
this loss process is slow compared to many-body timescales in the system, which are generally set
by the elastic collision rate, we can study the low energy many-body physics of the quantum gas.
In the case of Fermi gases near s-wave resonances, this criteria is met as long as one doesn’t
go too far onto the molecule side of the resonance [6]. However, it does not seem to be met in thecase of the narrow p-wave resonance. Atom and molecule lifetimes near the p-wave resonance inour system were too short for us to observe any elastic scattering processes. The Japanese groupconﬁrmed that for 6 Li near the p-wave resonance inelastic collision rates were too large to allow for
equilibration of the system [46].
Nevertheless, I am hopeful that this is not the end of the story. We were never able to develop
a satisfactory model of the molecule creation process that we observed for a cold gas held near thep-wave resonance [3]. It is possible that interesting many-body physics is occurring here at thesame time as the inelastic decay processes. If we knew what questions to ask and had a good way
to probe the gas, there might be interesting physics to be learned. I suspect atom photoemission
spectroscopy, the main topic of this thesis, would be an interesting tool to try and apply to thissystem and there is a group out there working towards this possibility [54].

43
Moreover, it is predicted that for atoms conﬁned to two-dimensions the ratio of the elastic toinelastic loss rates would be more favorable, though still less favorable than the s-wave case [48]. An
optical Feshbach resonance could perhaps be used to enhance p-wave collisions while minimizing
inelastic rates. Also, optical lattices may help in preventing three-body loses [55]. While theseroutes to p-wave superﬂuidity are uncertain, the reward of success would be high and it may be
a worthwhile endeavor to pursue them (assuming you already have tenure). However, for the restof this thesis, I will return to studies of strongly interacting gases via a broad s-wave Feshbachresonance.

Chapter 4

Fermi superﬂuids and the BCS-BEC crossoverIn this section I discuss the BCS theory of superﬂuidity (or superconductivity in metals)
and the concept of the BCS-BEC crossover. My discussion of BCS theory comes from the bookby J.R. Schrieﬀer [5], which was written back in 1964 but is still quite readable, as well as theclassic many-body theory book by Fetter and Walecka [56], and also the helpful review [4]. For
the BCS-BEC crossover, I will present some of the early results obtained in our group from Cindy
Regal’s thesis [6] and also use insights from the review [4].

4.1

The pairing theory of Fermi superﬂuidity (BCS Theory)

4.1.1

Cooper pairs

While a non-interacting Bose gas undergoes a phase transition at low temeperatures to a BoseEinstein condensate, non-interacting Fermi gases have no phase transitions even as the temperature
is lowered to zero. Furthermore, Fermi gases with short-range repulsive interactions remain a Fermi
liquid, a state similar to a non-interacting Fermi gas but with renormalized parameters, down tozero temperature (with the possible exception of a transition to a ferromagnetic state if interactions
are extremely strong [57]). However, if attractive interactions exist, an instability in the Fermi liquidstate is introduced and the system has a transition to a superﬂuid state at low temperatures. A
hand-wavy way to think about this transition is to say that the attractive interactions allow thefermions to form pairs, those pairs are bosons, and the bosons can undergo Bose condensation.
What is truly amazing about the superﬂuid transition however, is that it occurs even for arbitrarily

45
weak attractive interactions, even though those attractions would be far too weak to bind together
two fermions in free space. To get some insight into this conundrum, it is helpful to examine what
has been termed the Cooper instability.

kF
Occupied
Statesk-k

Figure 4.1: Cooper’s problem In Cooper’s problem, two fermions coexist with a non-interacting
Fermi sea (ﬁlled circle). The two fermions with zero center of mass momentum interact via a weak
attractive interaction. The Fermi sea aﬀects the problem only by Pauli blocking, which prevents
states with momentum less than ~kF from being occupied. Shown are possible wave vectors k and−k outside of the Fermi sea that the two fermions may occupy. In the absence of the Fermi sea,
the kinetic energy cost associated with forming a bound state would outweigh the energy decrease
from forming a paired state. However, in the presence of the Fermi sea it can be shown that a sucha pair will lower the energy of the two fermions below 2EF . This many-body pair is stabilized by
the Fermi sea and called a Cooper pair.

The thought experiment is as follows [58]: Consider a non-interacting Fermi sea and then
add two interacting fermions with zero center-of-mass momentum on top of that Fermi sea (Fig.
4.1). The Fermi sea aﬀects the problem only by Pauli blocking, which prevents the two fermions
from occupying any of the k states below the Fermi surface. If there were no interactions, the two
new fermions would have energy 2EF since each would have to sit on top of the Fermi sea. With
attractive interactions, it turns out they can lower their energy a small amount by forming a looselybound state called a Cooper pair. This problem is worked out in detail in a number of textbooks
and reviews. For the problem of fermions where the interactions can be characterized by scattering
length a, the binding energy would be EB = e82 EF e−π/kF |a| . This bound state could not exist in
the absence of the underlying Fermi sea and therefore the pair is really a many-body pair, although46
it is important to note that in this case the correlations with all the other atoms in the Fermi sea
are not caused by interactions but by quantum statistics.
Now, we have not solved the problem of many fermions interacting via a weak attractive
interactions, but we can use our insight from the Cooper problem to guess what the solution will
look like. In the fully interacting problem, many fermion pairs should form and condense leading
to a superﬂuid state. It turns out the pairing gap of the full many-body state in the BCS limit isactually [5, 56, 4]

∆0 =

4.1.2
4.1.2.1

8
EF e−π/2kF |a|
e2

(4.1)

Properties of the superﬂuid state
Critical temperature

The ﬁrst natural question about the superﬂuid state is what is the critical transition temperature. In order for the Cooper pairs to survive, one might guess that the temperature should beon the order of the pair binding energy. This turns out to be approximately correct with [5, 56, 4]

Tc = TFeγ 8eγ−π/2kF |a|
∆0
E
e
=
F
π e2
π

(4.2)

where γ is Euler’s constant and eγ ≈ 1.78.
For 40 K, the background scattering length is 174a0 and in our experiments typically kF−1 ≈2000a0 which would give TTFc ≈ 10−8 ; this is far below the temperature range that is experimentally
accessible. Without a Feshbach resonance to enhance interactions, atomic Fermi gas experiments
would not be able to access the superﬂuid regime.

47
4.1.2.2

The energy

The total energy of the superﬂuid is lowered from the non-interacting state by pairing andcan be written [5, 56, 4]
3∆0
EBCS = N EF −ρ(EF )∆0
5
2

(4.3)

where ρ(EF ) is the density of states at the Fermi energy. The ﬁrst term is simply the non-interacting
energy of the normal state. The second term is the pairing energy and can be interpreted as comingfrom ρ(EF )∆0 paired atoms each with pairing energy ∆20 . In other words, pairing occurs in a regionof energy width ∆0 about the Fermi surface.

4.1.2.3

The gap

Figure 4.2: The temperature dependence of the pairing gap The temperature dependent
pairing gap ∆(T ) as a fraction of the gap at zero temperature, ∆0 , is plotted as a function of thetemperature compared to the critical temperature T /Tc . The pairing gap vanishes at Tc . Thisﬁgure is taken from Ref. [4].

In the superﬂuid phase at ﬁnite temperature, the pairing gap decreases as T increases andabove Tc it vanishes. In the BCS limit, the pairing gap can be considered to be the order parameter

48
for the phase transition. In other words, the sudden appearance of pairing and the gap are thesame phenomenon as the transition to a superﬂuid. The temperature-dependent gap formula must
be solved numerically and is plotted in Fig. 4.2.

4.1.2.4

Pair-size

In the BCS limit, the pair wave function at large r can be approximated as sin(kF r)e−r/πξ
[4] where ξ can be written k2F E∆F . Since ∆ is much smaller than EF in the BCS limit, the pair size
is much greater than the average inter-particle spacing, which is of order k1F . This is the reason it
is not so easy to simply say the BCS superﬂuidity is a bunch of fermion pairs that Bose condense.
Rather, it is a system of many very large and overlapping pairs where the pair constituents are allstrongly correlated with each other through the Pauli exclusion principle.

4.1.2.5

Quasi-particles and the excitation spectrum

Another question is how does the gap manifest itself in the excitation spectrum and what
do the single-particle states look like in the paired superﬂuid. One way to answer this question
is to examine the solution of the BCS problem by canonical transformation. We can write themean-ﬁeld BCS Hamiltonian in second quantization in grand canonical (non-particle conserving)form as [56, 4]
HBCS =∑(ϵk − µ)a†k,σ ak,σ − ∆k,σ∑(a−k↓ ak↑ + a†k↑ a†−k↓ )

(4.4)

k

where a†k,σ (ak,σ ) are the fermion creation (destruction) operators for a fermion at momentumkk with spin σ and ϵk = ~2m
is the bare fermion energy. In order to diagonalize the Hamiltonian,
2 2

we introduce the quasi-particle operatorsγk,↑ = uk ak↑ − vk a†−k↓(4.5)

γk,↓ = uk ak↓ + vk a†−k↑

(4.6)

49
This transformation is canonical if the anticommutation relations {γk,σ γk†′ ,σ′ } = δk,k′ δσ,σ′ aresatisﬁed, which leads to u2k + vk2 = 1. If we plug eq. 4.5 into eq. 4.4 and do enough algebra, we ﬁnd[5, 56, 4]

HBCS =∑†
Ek γk,σ
γk,σ

(4.7)

(ϵk − µ)2 + ∆2

(4.8)

k,σ

where
Ek =

√

†
Thus, the elementary excitations of the system are given by the γk,σ’s and Ek is the energy due

to a quasi-particle at momentum k. The energy to add a particle at k is Ek (N +1)−E(N ) = µ+Ek ,where Ek (N ) is the energy of an N-particle system with a quasi-particle excitation at k. The energy
to remove a particle at k is E(N ) − Ek (N − 1) = µ − Ek . These excitation energies are plottedin the top part of Fig. 4.3. As can be seen, there is gap of 2∆ between the lower and the upper
branch. One way to think about this spectrum is to say the lower branch for the quasi-particle hole
excitations represents the energy of the single-particle states in the ground state and the upper
branch are the excitations.
It is also possible to obtain the values of uk and vk , which are also called the coherences. We
ﬁnd [5, 56, 4]1
ϵk − µu2k = (1 +)2Ek1
ϵk − µvk2 = (1 −
)
2
Ek

(4.9)
(4.10)

The coherences are plotted in the bottom of Fig. 4.3. We see that away from the Fermi
surface the quasi-particles are essentially the bare particle and hole operators. Only near the Fermi
surface where pairing occurs is there signiﬁcant particle-hole mixing between states with opposite
spin and momenta.

50

2.5

E(k)

2
1.5

2D

1
0.5
0
1

0.6

2

uk ,vk

2

0.8

0.4
0.2
0
0

0.5

k/kF

1

1.5

Figure 4.3: BCS quasi-particles Top) The dispersion relation for the BCS quasiparticles. The
top branch (red) is the energy of excitations on top of the BCS ground state (adding unpaired
fermions). The lower branch (blue) is the energy associated with removing particles in the BCS
ground state. The dashed line is the free-particle dispersion. The upper and lower branches of
the BCS dispersion are separated by twice the pairing gap ∆. Bottom) The coherences u2k (red)and vk2 (blue) are plotted vs k. Far from the Fermi surface, the coherences approach either 1 or 0
showing that in those limits the quasi-particle states are the same as the free-particle states. Near
the Fermi surface, the quasi-particles are superpositions of particle and hole states with opposite
momenta and spin. The coherence vk2 gives the occupied k states in the ground state and showsthat the momentum distribution is smeared out [5]. The plots here are for a gap equal to 0.2EF .
We have used µ =2~ 2 kF2m

for a weakly interacting Fermi gas.

51

4.2

Turning up interactions: The BCS-BEC crossover
From eq. 4.2 we know that reaching the superﬂuid regime in our 40 K gas will not be possible

with background interactions. But how high can we make Tc if we enhance interactions by turningup the scattering length using a Feshbach resonance? With the Feshbach resonance, we can tune
the scattering length all the way up to −∞ where a true bound state is on the verge of appearing
in the atom-atom potential. If we plug this into eq. 4.2, we ﬁnd Tc = 0.61TF . However, this isclearly outside the regime of validity for BCS theory, which requires weak interactions and ∆ ≪ EF[5, 56, 4] (here we would have ∆ = 1.1EF ). If we backtrack to an interaction strength where we
still expect BCS theory to be approximately correct, say kF a = −1, we ﬁnd Tc = 0.13TF and∆ = 0.23EF . This temperature is experimentally accessible.
It is instructive to look at what happens as the Feshbach resonance is crossed and a truebound state is introduced into the two-particle potential. This is the “BEC side” of the resonance.
The reason for the name is clear; if we go far enough so that we are again in the limit of weakinteractions ( kF1 a ≫ 1) the ground state is a Bose-Einstein condensate of dimer molecules where
the binding energy of each molecule is given in Eq. 3.9 to be ~2 /(ma2 ) for a broad Feshbachresonance. Here, the pair size is on order of the scattering length a [4] and thus much smaller than
the inter-particle spacing. In this limit, we can calculate Tc from the Bose-Einstein condensation
temperature with a boson density of n/2 bosons and mass 2m. This gives Tc /TF = 0.22 for the
BEC limit.
We can guess that in the range of interaction strengths of −1 < kF1 a < 1 the transition temperature is in the range of 0.1−0.2 TF , which is an extremely high superﬂuid transition temperature
for a Fermi gas. This is the region of the BCS-BEC crossover, named because we are going fromthe limit of small negative scattering length where the superﬂuid state is described by conventional
BCS theory to the limit of small positive scattering length where the ground state is a molecular
BEC. It was ﬁrst proposed by Leggett [59] that this crossover should be smooth with the groundstate being a Fermi superﬂuid all the way across. As a ﬁrst naive approximation, an adaptation

52

Figure 4.4: Pairs in the crossover The diﬀerent regimes of pairing are shown for a Fermi gas
with attractive interactions. In the BEC limit, the pairs are tightly bound molecules with a pairsize much smaller than the inter-particle spacing and the critical temperature is given by that for
Bose-Einstein condensation of weakly interacting bosons, which is approximately 0.22TF . In the
BCS limit, where the attractive interactions are weak, Cooper pairs can form and the Fermi gas
is a BCS superﬂuid. Here, the critical temperature is exponentially small, given by eq. 4.2. The
crossover takes places for interaction strengths of kF1 a = −1 to 1. In this region, the pair size is onorder of the inter-particle spacing and the gas is strongly interacting with a critical temperature in
the approximate range of 0.1 − 0.2TF . This ﬁgure is taken from [6].

53
to conventional BCS theory can be made by allowing the chemical potential, which equals EF on
the BCS side, to vary continuously to its limit of −Eb /2 on the BEC side where Eb is the moleculebinding energy [59, 60, 61]. This allows one to solve for quantities like the chemical potential,
the gap, and the pair wave function. The theory predicts that the pair size varies smoothly fromthe BCS limit of large Cooper pairs to a pair size on the order of the interparticle spacing in the
crossover and then to a pair size on the order of the scattering length a in the BEC limit [4], asshown in Fig. 4.4. The minimum fermion excitation energy varies smoothly from twice the BCS
gap, eq. 4.1, to a value on order of the Fermi energy in the crossover and then to the moleculebinding energy on the BEC side [6]. However, while this theory captures some of the physics ofthe crossover, it is not quantitatively accurate since it does not take into account all of the strong
correlations that occur in the BCS-BEC crossover. In fact, no controlled theory has been possible
in this regime due to the lack of a small parameter with which one can apply perturbation theory[35]. From the point of view of experiments, though, this is exactly the regime we want to be in,where many-body correlations are the strongest and the system is the hardest to understand. It ishere that we can hope to make measurements that contribute to fundamental many-body quantum
theories.
The ﬁrst major experimental goal was to demonstrate that a superﬂuid could be achievedin this strongly interaction regime and that a smooth crossover between the BCS and BEC limits
of interacting fermions did indeed exist. The ﬁrst experimental evidence for the strongly interacting Fermi superﬂuid came from our group in 2004 with the observation of a condensate in thecenter-of-mass pair momentum distribution [7]. This experiment is somewhat analogous to the observation of Bose-Einstein condensation by looking at the atom momentum distribution and seeing
a condensate peak at zero-momentum. The important diﬀerence is that with fermions, there isno condensate in the atom momentum distribution. The trick was to ﬁnd a way to measure thecenter-of-mass momentum distribution of the pairs, and this was done by projecting the pairs in
the many-body system onto tightly bound molecules by ramping across the Feshbach resonance.
Further conﬁrmation of superﬂuidity followed in 2005 when the group at MIT found that at low

54

Figure 4.5: BCS-BEC crossover phase diagram An experimental measurement of condensate
fraction, N0 /N , as a function of temperature and interaction strength allows one to map out thesuperﬂuid region in the BCS-BEC crossover. The right side of the plot is the BCS side and the left
side is the BEC side. The transition temperature is the highest near the center of the crossover( kF1 a = 0) and on the BEC side; Tc falls oﬀ quickly on the BCS side. On the vertical axis thetemperature reported, (T /TF )0 , is that of a weakly interacting gas before an adiabatic ramp that
increases the interaction strength. This ﬁgure is taken from [6] and was ﬁrst published in [7].

55
temperatures they could observe a vortex lattice by rotating the gas [62]. Since then a number ofexperiments have been consistent with the observation of a superﬂuid state including measurementsof thermodynamics [63, 13] and collective excitations [64, 65]. It turns out the method developed
at JILA to measure the pair condensate is quite sensitive to the transition and therefore is good formeasuring Tc and mapping out the phase diagram. This was done in Ref. [7] and the main result
is reproduced in Fig. 4.5. As can be seen in Fig. 4.5, the critical temperature decreases on the
BCS side of the diagram and levels out to about 0.15TF on the BEC side. It is important to notethat the temperature given is that of a weakly interacting Fermi gas before it was adiabaticallyramped to the strongly interacting regime. In this ramp, entropy is conserved but the temperature
is expected to increase. On the BCS side, and even at unitarity, this eﬀect is rather small (about
a 15% increase in TTF at Tc at unitarity [66]) but the temperature increase can be quite large for
adiabatic ramps into the BEC regime [66].
After this great experimental achievement, the goal of the next generation of experiments
became a bit more murky. Clearly, we had the ability to generate a very interesting system, where
correlations were so strong that theories based on approximations were uncontrolled and insight
into the system was more diﬃcult. Furthermore, it was a system that was relevant to other strongly
correlated systems like quark-gluon plasmas, neutron stars, and strongly correlated electron systems. So, the goal had to be to somehow make measurements that could give physical insight andguide theoretical frameworks in a way that would lead to a better overall understanding of thisstrongly correlated fermion system. A great number of experiments were done that helped contribute to this goal, including measurements of momentum distributions [67] and potential energy[30] in our group and other measurements of thermodynamic quantities, collective excitations, rf
spectroscopy (to name only a few) in other groups across the world [4]. All of these measurements
could be compared with theories and with quantum Monte-Carlo simulations to test their accuracy
and to try to gain physical insight.
In this thesis, I will present some experimental measurements that I think have been particularly helpful in getting to the heart of this strongly correlated system. These measurements are56
based on momentum-resolved rf spectroscopy and they directly measure the single-particle spectral function of the fermion system in a way that is analogous to angle-resolved photoemission
spectroscopy of strongly correlated electron materials. The spectral function tells us about thesingle-particle states and knowledge of it allows us to describe the system in the language of Fermi
liquid theory and standard quantum many-body theory. Measuring the spectral function will help
us answer some basic questions, such as can we describe this system in terms of quasi-particles
and does the normal gas behave as a Fermi liquid or does a pseudo-gap state exist? Furthermore,
the spectral function is an excellent quantity to directly compare with theories because it is afundamental quantity of the system.
In the next chapter, I will explain what the single-particle spectral function is and how it
behaves for the paradigms of Fermi liquids and BCS superﬂuids. I will also explain how we canmeasure it experimentally and what some initial experiments can tell us about the BCS-BEC
crossover.

Chapter 5

Fermi liquids, spectral functions, and photoemission spectroscopy

In this section I introduce the language of Fermi liquid theory and spectral functions and show
how these quantities help us understand the nature of interacting Fermi systems. I discuss how tomeasure the spectral function of an atomic Fermi gas using a momentum-resolved rf spectroscopy
technique. I present our ﬁrst measurements of the spectral function in the BCS-BEC crossover,
however I try to be brief as these experiments are discussed in detail in Jayson Stewart’s thesis[34]. I also describe how we improved the signal-to-noise ratio of these measurements by changing
our imaging scheme, and I present some recent higher-signal-to-noise data. My discussion of Fermi
liquids and spectral functions borrows often from Refs. [56, 68, 69] as well as from course notesfrom Jim Shepard and Victor Gurarie for many-body physics courses taught at CU. The original
photoemission data I present was published in Ref. [9].

5.1Landau’s Fermi liquid theory
As discussed in Chapter 2, the single-particle eigenstates of a non-interacting Fermi gas arekplane-wave states characterized by a wave-vector k and energy ~2m. In the many-body ground2 2

state, fermions ﬁll these states up to the Fermi surface at wave-vector kF . Excitations on topof this ground state can be classiﬁed as particle or hole excitations. A particle excitation is theoccupation of an extra state k > kF above the Fermi surface with an excitation energy of2|~2 |k2 −kF
2mcompared to the ground state of the N + 1 particle system. If k − kF ≪ kF the excitation energy
can be approximated as ~vF |k − kF | where vF = ~kmF is called the Fermi velocity. A hole excitation

58
is the absence of a particle in a state k < kF and has the same energy as the particle excitation,with k now being the wave-vector of the empty state.
For an interacting Fermi gas, the plane-wave states are no longer the eigenstates and it is notclear how to describe the system in terms of a simple picture as we did for the non-interacting gas.
Landau’s bold insight was to adapt the non-interacting picture of single-particle plane-wave states
for the interacting system with a few simple modiﬁcations. A thought experiment shows that thisshould be a valid approximation for excitations near the Fermi surface. In this thought experiment,
we ﬁrst postulate that there is still a well-deﬁned Fermi surface in the interacting system. Then,
we add a particle excitation at momentum ⃗k above the Fermi surface and ask what the probability
is for that particle to scatter into another state. The mechanism for scattering is to collide withanother particle with |⃗k ′ | < kF into two ﬁnal states above the Fermi surface, say ⃗k1 and ⃗k2 , seeFig. 5.1. If the matrix element for collisions around the Fermi surface is roughly constant, this isa question of phase space. In other words, how many possible collision partners and ﬁnal states
exist? We require conservation of momentum and energy, as well as |⃗k ′ | < kF and |⃗k1 |, |⃗k2 | > kF .
We can satisfy conservation of momentum by taking ⃗k ′ = ⃗k1 + ⃗k2 − ⃗k. Then we can write the total
number of possible collisions for a single-particle excited above a ﬁlled Fermi sea as∫
N (k) =

∫
3⃗

d k1

d3⃗k2 δ(|⃗k|2 +|⃗k1 +⃗k2 −⃗k|2 −|⃗k1 |2 −|⃗k2 |2 )θ(|k1 |−kF )θ(|k2 |−kF )θ(kF −|⃗k1 +⃗k2 −⃗k|)(5.1)

where θ(x) = 1 for x > 0 and θ(x) = 0 for x < 0; this is the zero temperature limit of the Fermi
Dirac function. To model a ﬁnite temperature gas, the θ functions can be replaced by Fermi Diracfunctions, which give the probability of occupation, (θ(kF − k) → f (E, T )). An analogous equation
can be written down to describe the phase space for a hole excitation below the Fermi surface todecay (see Fig. 5.2).
It turns out for k − kF ≪ kF , N (k) ∝ (k − kF )2 or N (k) ∝ E(k)2 [56, 68, 69] (see Fig. 5.2).
This is remarkable because it says the number of collision possibilities for a particle at k to scatterto another state goes to zero near the Fermi surface. Thus excitations near the Fermi surface are

59

k1
k
Fermi Seak’

k2

Figure 5.1: Quasi-particle decay process A particle excitation is made by placing a fermion in
the momentum state ⃗k above the Fermi surface. It can decay by colliding with another fermion
below the Fermi surface in state ⃗k ′ . The ﬁnal states are ⃗k1 and ⃗k2 , which must be above the Fermi
surface. The number of possible collisions that conserve energy and momentum is proportional to|k − kF |2 . As k → kF , the number of possible collisions decreases to zero because of Pauli blocking.

60

Hole Decay Phase Space

1

0.8

0.6

0.4

0.2

0
0

0.2

0.4

0.6

0.8

1

k/kF
Figure 5.2: Hole-decay phase space The total number of collisions, or phase space, for a hole
excitation to decay is plotted versus k/kF for diﬀerent temperatures. The y-axis is plotted inarbitrary units. From lowest to highest, the curves shown correspond to temperatures of TTF = 0,0.1, 0.2 and 0.4. The dotted line is a quadratic curve meant to show the low energy limit where
N (k) ∝ |kF −k|2 . As temperature increases, the phase space for collisions at k = kF is proportional
to T 2 at low TTF . A hole excitation decays when two fermions inside the Fermi sea collide, withthe ﬁnal state being one fermion ﬁlling the hole at k and the other excited out of the Fermi
sea. The equation describing this process is analogousand∫ 3 ∫to3Eq. 5.1, 2but for2 hole excitations
2
⃗
⃗
⃗
⃗
⃗
⃗
⃗
ﬁnite temperature. It can be written N (k) = d k1 d k2 δ(|k1 | + |k2 | − |k| − |k1 + k2 −⃗k|2 )f (k1 , T )f (k2 , T )(1 − f (|⃗k1 + ⃗k2 − ⃗k|, T )), where f (k, T ) is the Fermi Dirac function describing
2 k2
the probability for a state with wave-vector k, and energy ~2m, to be occupied at temperature T .

61long-lived. Because the inverse collision time goes to zero faster than the excitation energy, theexcitations near the Fermi surface can be treated as quasi-particles. An identical argument can bemade for hole excitations near the Fermi surface, as plotted in Fig. 5.2. This justiﬁes our postulatethat there will still be a well-deﬁned Fermi surface in the interacting system. However, just because
a particle or hole excitation near the Fermi surface has a well-deﬁned energy, the magnitude of theexcitation energy does not have to be identical to the non-interacting case. Even if there is no phasespace for momentum changing collisions, there are still interactions and virtual processes that canlead to a shift in the excitation energies. This leads us to call these excitations quasi-particles,
because they now consist of a particle excitation plus that particle’s interactions with all the otherfermions. For fermions interacting via a short-range potential, the 1/e lifetime of an excitation at
k is given byτ −1 =|k − kF |2
EF 2(kF a)2
~ π
kF2

(5.2)

in the limit k − kF ≪ kF and kF a ≪ 1 [56]. Note that the scattering rate is proportional to thetwo-body scattering cross-section 4πa2 .
Near the Fermi surface, we can write the quasi-particle excitation energy as ~mk∗F |k − kF |2

where m∗ is called the eﬀective mass of the quasi-particles. For fermions interacting via a short∗82
range potential, the eﬀective mass is given by mm = 1 + 15π
2 (7 ln 2 − 1)(kF a) in the limit kF a ≪ 1[56]. The properties of the quasi-particles can be used to determine numerous basic properties of
the interacting Fermi gas such as the heat capacity (proportional to m∗ ), compressibility, soundvelocity, and transport properties [56, 68, 69]. This treatment of interacting Fermi gases in terms ofquasi-particles is known as Landau’s Fermi liquid theory and has been very successful in describinga large class of electronic materials and interacting Fermi gases [56, 68, 69]. However, it is importantto remember that: 1) the quasi-particle theory is only strictly valid near the Fermi surface, where
the quasi-particles represent well-deﬁned excitations and that 2) the total energy of the systemdoes not equal the sum of the quasi-particle energies (since this would ignore interactions between
quasi-particles).

62
Landau’s Fermi liquid theory can be contrasted with the low energy theory for bosons (alsodeveloped in part by Landau), which predicts that the low-lying excitations are phonon-like andcannot simply be thought of as renormalized particles [69].

5.2

Green’s function and many-body quantum theory
One way to put Fermi liquid theory into a more formal framework is to examine the propertiesof the Green’s function of a many-body system. First, let’s examine the Green’s function for a single
particle. In this case, the Green’s function is the probability that a particle at an initial positionxi is found at xf after a time t and can be written [36]G(xf , xi , t) = −i∑e−iEn t ψn (xf )ψn∗ (xi )

(5.3)

n

For particles in a box, the Fourier transform of eq. 5.3 gives
G(k, E) =1
2 k2
E − ~2m
+ iϵ

,

(5.4)

where ϵ is needed for convergence but is to be taken to zero. This shows that the energy eigenstates
are actually poles of the Green’s function [36]. This turns out to be a general structure for any
one-particle Hamiltonian [36].
For a many-body problem, the Green’s function is deﬁned in an analogous way. Here we add
or subtract a particle at position xi and ti in the many-body system, let the system propagate, andthen check the overlap of that system with one where we add or subtract a particle at xf and tf[56]. Analyzing the properties of the Green’s function in Fourier space shows that, in analogy to thesingle-particle problem, the poles of the Green’s function represent the energy due to an excitationfrom adding or subtracting a particle at k [56] (either Ek (N + 1) − E(N ) for particle excitationsor E(N ) − Ek (N − 1) for hole excitations where Ek indicates the system has an excitation at k).
For a Fermi liquid, the Green’s function, G(k, E) can be writtenG(k, E) =θ(kF − k)θ(k − kF )+~2 k 2~2 k 2E − 2m − Σ(k) − iϵ E − 2m − Σ(k) + iϵ

(5.5)

63
where Σ(k) is called the self energy and contains all the physics due to the interactions [56].
The two terms correspond to quasi-hole and quasi-particle excitations. The poles will occur atk+ ReΣ(k) and this is then the energy of quasi-hole or quasi-particle excitation at energy
E = ~2m
2 2

E and momentum k. An important diﬀerence from the single-particle case is that the self energy canalso have an imaginary component that gives a ﬁnite width or lifetime to the quasi-hole excitation.

kF

Non-Interacting

kF

Fermi-Liquid

Figure 5.3: Fermi liquid spectral functions Left) A representation of the spectral function fora non-interacting gas. Hole excitations are shaded. In this case, the spectral function consists ofdelta-function peaks at the non-interacting plane-wave energies. Right) A representation of thespectral function for a Fermi liquid. Here the sharp peaks represent the quasi-particles, located at
the poles of G(k, E), and the broad peaks represent the incoherent contributions to G(k, E). Near
the Fermi surface, the quasi-particle peaks become sharper and contribute a greater fraction of thespectral weight. These ﬁgures are adapted from Ref. [8].

An important quantity is the spectral function A(k, E) = − π1 ImG(k, E). For a non-interactingkFermi gas, Σ = 0 and the spectral function is given by A(k, E) = δ(E − ~2m) (Fig. 5.3). For a
2 2

Fermi liquid, using eq. 5.5, we see that
A(k, E) =−1(πImΣ(k))2 ()2
E − 2m − ReΣ(k) + ImΣ(k)
~2 k2

(5.6)

which is the equation for a Lorentzian resonance centered at the quasi-particle excitation energy~2 k2
2m +ReΣ(k) and with a width given by ImΣ(k) (see Fig. 5.3). For a Fermi liquid with short-range

interactions, ImΣ(k) near the Fermi surface is given by ~τ −1 where τ is the collision time given in1
eq. 5.2. The eﬀective mass m∗ is found from m∗ = ( m+ ~21kF ∂ReΣ(k)|kF )−1 .
∂kIn the superﬂuid phase, the spectral function of a Fermi gas changes markedly from the Fermi

64
liquid phase. The spectral function splits into two branches and the energy of the quasi-particles
is given by eq. 4.8 and shown in Fig. 4.3. Importantly, at the Fermi surface (E = µ), the spectral
function is zero because the gap prevents any states from existing in this region.
Ultimately, the knowledge of the spectral function tells us what has happened to the singleparticle states in the interacting system. The nature of those single-particle states tells us how weshould think about the physical system we are studying and can help give us physical intuition.
Is it a Fermi liquid described by fermionic quasi-particles, a superﬂuid described by pairing, or is
it something completely diﬀerent? Furthermore, the Green’s function is a fundamental quantitythat many-body perturbation theory directly computes and so is a good quantity to measure andcompare to theory. If a theory does not at least qualitatively get properties of the spectral function
right, then it is not a good theory of the underlying physics in the system.

5.3

Photoemission spectroscopy
While several observables of a many-body quantum system can be related back to the Green’s

function, experiments that directly remove or add single fermions are the most direct probes of thisquantity. In materials, angle resolved photoemission spectroscopy (ARPES) directly measures
the spectral function of the electrons [8]. ARPES removes electrons via the photoelectric eﬀect.
The energy and momentum of the ejected electrons are measured with a hemispherical electron
spectrometer (Fig. 5.4). As long as the electron is able to leave the material without colliding withany other electrons, its momentum is preserved. Thus, by using kinematics and knowledge of thephoton energy and emission angle, one can back out the electron energy and momentum in thematerial [8]. This energy and momentum information allows one to reconstruct the electron spectral
function by making a two-dimensional plot of the intensity of measured electrons as function ofenergy and momentum (see Fig. 5.5). Of course, this technique can only measure occupied singleelectron states. To measure the unoccupied states, or excitations, there is inverse photo-emission
spectroscopy or (IPES) as well as tunneling experiments [8].
ARPES has proven to be an extremely important measurement tool for understanding the

65

Figure 5.4: Angle-resolved photoemission spectroscopy In ARPES, a high energy photon
kicks an electron out of the sample via the photoelectric eﬀect. The electron is measured in anelectron analyzer that determines the electron energy. This ﬁgure is adapted from Damascelli etal. [8].

66

Figure 5.5: ARPES data A surface plot is shown representing an ARPES data set obtained froma Bi2 Sr2 CaCu2 O8+δ for a cut in momentum space, which is shown in the upper left corner andwhich goes through the Fermi surface (ω = 0). The electron counts are indicated by the red andyellow colors. The quasi-particle dispersion, marked by the white dots, can be followed up to the
Fermi surface where the quasi-particles are the sharpest and best deﬁned. Cuts through the dataat constant energy (ω = 0) and constant momentum (k = kF ) are shown at the top and lower rightrespectively. These cuts are called momentum distribution curves (MDC) for ﬁxed energy andenergy distribution curves (EDC) for ﬁxed momentum. The EDC shows both the quasi-particle
feature and broad incoherent background and can be compared to the interacting Fermi gas pictureof Fig. 5.3. This ﬁgure is taken from Ref. [8].

67electronic properties of solids and, in particular, for learning about the nature of strongly correlated
electron systems such as high-Tc superconductors, colossal magneto-resistors, graphene, and manyothers. In particular, ARPES has been important in the discovery of the d-wave symmetry of the
gap and elucidation of the properties of the pseudogap state in high-Tc cuprates [8].
It is natural to seek a way to make the same kind of measurement in ultracold atomic Fermi
gases. With photoemission spectroscopy for a Fermi gas in the BCS-BEC crossover, one could
address questions about the basic nature of the gas. How do the single-particle states evolve fromthe BCS-like quasi-particles to deeply bound molecules on the BEC side? Can a Fermi liquid statepersist with resonant interactions? How large is the pairing gap? At what point do pairing andsuperﬂuidity become separate phenomena and does a pseudogap state of many-body pairs exist in
the crossover?

b

E

hn

k,

(E

k)

a

f
hn

0

kF

k

Figure 5.6: Momentum-resolved atom rf spectroscopy a) In momentum-resolved rf spectroscopy experiments, one outcouples atoms by ﬂipping them to a weakly interacting spin state via
an rf transition. This is analogous to ejecting an electron from a material with a high energy photon
Fig. 5.4. b) The rf photon drives a vertical (momentum conserving) transition. By measuring theenergy and momentum of the outcoupled atoms (upper curve), we can determine the quasi-particle
excitations and their dispersion relation (lower curve). Φ is the Zeeman energy diﬀerence between
the two spin states of the atom and is analogous to the work function energy of a material. Thisﬁgure is taken from [9].

Momentum-resolved rf spectroscopy gives us a tool to achieve an analogue of the photoemission spectroscopy measurement for ultracold atomic gases [9]. I will also refer to this measurement
as atom photoemission spectroscopy or simply PES. The spectroscopy takes advantage of the many

68
spin states of the atoms in these cold gases. In a magnetic ﬁeld, the degeneracy of these statesis split by the Zeeman interaction and at magnetic ﬁeld strengths around the Feshbach resonance(≈ 200 G), the splitting is on order of h × 50 MHz. This Zeeman splitting is much larger than otherenergy scales in the system, for example EF ≈ h × 10 kHz. Fortunately, all the spin-relaxation
mechanisms available to atoms in the two lowest energy spin-states are either forbidden or strongly
suppressed, so a gas mixture of atoms in those spin states happily stays that way without relaxing.
In 40 K, our strongly interacting Fermi gas consists of a mixture of atoms in the |9/2, −9/2⟩ and|9/2, −7/2⟩ states with all the other spin states being unoccupied. An rf magnetic-ﬁeld transverseto the quantization axis with frequency νrf can induce spin changing transitions when hνrf equals
the energy splitting of the spin states. In particular, an atom in the |9/2, −7/2⟩ state can be
ﬂipped by an rf photon to the |9/2, −5/2⟩ state with νrf ≈ 47 MHz at ﬁelds near 200 G. Near a
Feshbach resonance, the interactions between atoms is highly dependent on the spin-states. Whileinteractions between atoms in the |9/2, −9/2⟩ and |9/2, −7/2⟩ states are resonantly enhanced nearmagnetic ﬁeld strengths of 200 G, interactions between an atom in one of those states and an atomin the |9/2, −5/2⟩ state remain near their background values characterized by scattering lengths
on the order of 100a0 . This is the key to achieving photoemission spectroscopy. An atom in the|9/2, −7/2⟩ state that is strongly interacting can be “outcoupled” from the interacting system witha spin-changing rf transition. This is analogous to outcoupling an electron from a material byejecting it with a high energy photon (see Fig. 5.6). In both cases, the energy of the fermion in
the ﬁnal state, either a weakly interacting atom or a free electron, is trivially known. The spin
ﬂipped atoms in the |9/2, −5/2⟩ state can be counted because absorption imaging resolves spinstates selectively when the Zeeman splitting is greater than the width of the absorption lineshape(approximately 6 MHz).
In a traditional (momentum-integrated) rf spectroscopy experiment, one measures the number of atoms outcoupled, or transferred to the weakly interacting spin-state, as a function of thefrequency νrf (see Fig. 8.1). This sort of experiment has now been used to infer information about
the pair wave-function [40] and pairing gap [19] in the BCS-BEC crossover as well as to measure

69
the contact [10], a subject of Chapter 8. However, a momentum-integrated rf spectroscopy experiment cannot resolve important questions about the single-particle states such as the existence of apseudogap state or Fermi-liquid-like behavior [70, 71].
Momentum-resolved rf spectroscopy (Fig. 5.6), takes advantage of the fact that the rf photon
has a negligible momentum compared to the atom’s momentum and so the spin-ﬂip transition does
not change the atom’s momentum state (this is a vertical transition in the language of photoemission
spectroscopy). The momentum of the spin-ﬂipped atom, and thus the momentum of the atom inside
the interacting system, can be measured in a time-of-ﬂight experiment. With this information,
we can reconstruct the fermionic spectral function. Speciﬁcally, the rate of atoms transferred at
momentum k and rf frequency νrf , I(νrf , k) is related to the spectral function via
I(νrf , k) ∝ k 2 A(k, E)f (E)

(5.7)

k
with E = ~2m− h(νrf − ν0 ), where hν0 is the Zeeman splitting of the two spin-states [72]. f (E) is
2 2

the Fermi-Dirac distribution function, which appears because photoemission spectroscopy measures
only the occupied part of the spectral function. The energy E is oﬀset so that E = 0 corresponds tothe energy of a non-interacting atom at rest (as opposed to the usual condensed matter convention
where E is measured with respect to the Fermi energy or chemical potential). To understand where
this formula comes from, we can look at energy conservation for an rf transition. Initially, we havea photon of energy hνrf and the energy of the interacting system with N atoms. Then, we removek
one atom and put it into another spin-state where it will have energy ~2m+ hν0 . Thus, the energy
2 2kchange to the interacting system from removing one particle at momentum k is ~2m− h(νrf − ν0 ).
2 2

The factor of k 2 out front in eq. 5.7 is just the volume factor for the number of states at momentumk.
Momentum-resolved rf spectrsocopy compared to ARPES for materials removes some complications but also introduces some new ones. For example, ARPES suﬀers from ejected electrons
colliding with other electrons on their way out of the material [8]. This introduces a backgroundsignal and limits ARPES to probing near the material surface. For two-dimensional materials, this70
limitation is just technical but for other materials it can make measurements of bulk properties
diﬃcult [8]. With atoms, the interactions between the atoms in the outcoupled spin state and therest of the atoms is so weak that this is not a problem. In other words, the mean free path of theoutcoupled atom is much larger than the system size [9]. Furthermore, in ARPES there is a matrix
element for the process of removing an electron with a photon that depends on the angles andwave-vectors of the photon and electron [8]. This matrix element is not always known well enoughto divide out [8]. In the atom case, the emission process is simply a Zeeman spin-ﬂip transition andthe matrix element is constant [72]. Experimentally, the photon source in the atom case (an rf ﬁeld)
is far easier to produce than a synchrotron x-ray radiation source (somewhat of an understatement)
or even a laser based UV ARPES source. Also, the detection of atoms by time-of-ﬂight absorption
imaging is quite a bit simpler (requiring only a low power laser, CCD camera and imaging optics)
than the detection of elctrons with an electron spectrometer. A further general advantage is theability of ultracold atom gas experiments to reproduce identical samples repeatedly whereas thiscan be a real challenge in materials science.
However, it is important to note a number of new issues introduced in momentum-resolved rf
experiments for atoms. In the atom case, to acquire the full spectral function a range of frequencies
are needed, as opposed to ARPES where the entire spectral function for a 2-dimensional material
can be measured with a single photon frequency (due to the non-conservation of momentum andenergy in the direction perpendicular to the material surface [8]). In ARPES, the energy resolutionof the measurements is typically set by the detector resolution and can be made very small comparedto the Fermi energy (meV compared to eV) [8]. In the atom case, our energy resolution will be
Fourier limited by the duration of the rf pulse, which must be kept much shorter than the oscillation
period of the conﬁnement potential in order to keep the momentum of the outcoupled atoms fromchanging. In our experiments, this limits us to a resolution of approximately 0.2EF , although there
is some room for improvement here if the aspect ratio of the conﬁnement trap can be reduced.
Perhaps the most challenging complication of the ultracold atom gas experiments comes from thedensity inhomogeneity introduced by the conﬁnement trap. Since the trap is harmonic, the density

71
is greatest in the center and then decreases to zero with increasing distance from the center. This
inhomogenous system can be thought of in terms of a local density approximation. This means
locally, inside the atom gas, we can think of the properties of a given shell of the gas with constantdensity n as having the properties of a homogenous system at that density. The total gas iscomposed of many of these shells, each with a diﬀerent density. That means the spectral function
we measure will actually be the equivalent of averaging the spectral function of many homogenous
gases with diﬀerent densities and therefore diﬀerent values of quantities like EF , kF and µ, which
will range from their peak values at the center of the cloud all the way to zero near the edge.
However, this is not as bad as it sounds. Because we know the trap is harmonic, a theory thatcan predict the spectral function for a homogenous gas can easily be extended to predict the trapaveraged spectral function. The real disadvantage is that this trap averaging can wash out sharp
features and make it more diﬃcult to do a quantitative analysis of the data in the absence of theory.
I will discuss this issue more and devote a later chapter to some work we have done to decrease thedensity inhomogeneity and understand its eﬀect on the data.

5.4

Photoemission spectroscopy experiments with a strongly interacting
Fermi gasOk, enough discussion, it’s time to look at some data! In Fig. 5.7, momentum-resolved rf

spectroscopy data for a Fermi gas near a Feshbach resonance are shown. For these experiments,
the Fermi gas was cooled to ( TTF )0 = 0.16 at a magnetic ﬁeld strength of approximately 203.5 G
where the gas is weakly interacting ( kF1 a ≈ −3). The subscript 0 indicates the measurement wasmade in the weakly interacting regime. To change interactions, the gas was then adiabaticallyramped to a ﬁeld B around the resonance. In Fig. 5.7a, the measured spectral function of a
weakly interacting Fermi gas with EF = h × 9.3 kHz and kF =

√
2mEF
~

= 8.6 µm−1 is plotted. The

Fermi energy is determined by EF = ~ω(3N )1/3 , where ω is the geometric mean of the conﬁnement
trap frequencies and N is the total number of atoms. Interactions are decreased by ramping themagnetic ﬁeld to B = 208.43 G, near the zero-crossing of the resonance, resulting in an interaction

Single-particle energy (kHz)

72

20

20

a

c

10

10

0

0
5

0

10
k ( mm )

15

-1

-10

-20

b

20

Single-particle energy (kHz)

-30
10
-40
0
-50
-10
-60
-20
-70
-30
0

5

10
-1

k ( mm )

15

0

5

10

15

20

-1

k ( mm )Figure 5.7: Atom photoemission spectra a) An atom photoemission spectra, obtained with
momentum-resolved rf spectroscopy, for a weakly interacting atom gas. The quasi-particle dispersion (white dots) obtained from gaussian ﬁts to vertical cuts through the data (EDCs) is quadratic
and indistinguishable from the expected non-interacting dispersion (black line). b) The spectrum
for a strongly interacting Fermi gas near the center of the BCS-BEC crossover. The quasi-particle
dispersion is BCS-like, exhibiting backbending near k = kF . c) The spectrum for a Fermi gas in themolecular regime. The top feature is a result of unpaired atoms, which can be treated as a weaklyinteracting Fermi gas. The lower feature is due to the pairs, which are in the molecular regime andexhibit a negatively dispersing spectrum. EDC’s from this data set are shown in Ref. [10].

73
strength of kF1 a = −9.8. For this gas, the data reveal a spectral function that follows the expectedkparabolic curve ~2m(black line in Fig. a) of a non-interacting Fermi gas. Each white dot is the
2 2center position of a ﬁt to a vertical cut through the data at constant momentum called an energy
distribution curve (EDC). An EDC at k gives information about a single-particle excitation at
momentum k. Speciﬁcally, the location of the peak of the feature is the energy or real part of theself energy ReΣ(k). The width of the feature is related to ImΣ(k). We obtain values for the center
and width by ﬁtting a gaussian Ae−(E−E0 )2
2w2

to each EDC. For the data in Fig. 5.7, we ﬁnd the

EDC widths are constant and equal to 2.2 ± 0.1 kHz, which is consistent with our expected Fourierlimited energy resolution of 2.15 kHz (rms width). It is expected that the intrinsic quasi-particle
widths are much smaller than our energy resolution, with ImΣ(k) ≪ EF for a weakly interacting
gas, as can be seen in Eq. 5.2. Momentum states are occupied up to kF with a smearing in kconsistent with a harmonically trapped gas at a temperature of TTF = 0.16.
In Fig. 5.7 b., the momentum-resolved rf spectroscopy data for a gas at B = 202.1 and1
kF a= 0.17 [73] is plotted. The Fermi energy, as deﬁned above, is h × 10.4 kHz and kF = 9.1µm−1 . Here, we observe a downward shifted dispersion that back-bends near kF . This backbending indicates pairing and is well ﬁt by a BCS-like dispersion curve (white line in Fig. b.)√
2 k2. The best ﬁt gives µ = h(12.6 ± 0.7 kHz) andwith E = µ − (ϵk − µ)2 + ∆2 where ϵk = ~2m∆ = h(9.5±0.6 kHz). However, these ﬁt values cannot be immediately interpreted as measurementsof the chemical potential and the gap for two reasons: the spectral function is obtained froma trapped gas with inhomogenous density, and the gas is in the BCS-BEC crossover, where thedispersion from BCS theory may not be correct. I discuss BCS ﬁts to atom-photoemission data in
the BCS-BEC crossover in more detail in the next chapter.
We ﬁnd the widths of the EDCs are now broadened beyond our energy resolution indicatingthat the intrinsic energy widths of the quasi-particles may be large. Another possible source ofbroadening is averaging the signal over the trap. I will present a more detailed comparison of thewidths and amplitudes of the EDCs with theory in Chapter 6. It is worth noting that this gas
is very close to the transition temperature ( TTF )0 = 0.17 ± 0.01 measured in Chapter 6 (Fig 6.5).

74
The fact that even at this high temperature we observe a BCS-like dispersion with back-bendingthat is consistent with a large gap is already an indication of pseudogap behavior, since it cannot
be explained by BCS theory (Fig. 4.2). The temperature dependence of this behavior and its
relation to the existence of a pseudogap phase is investigated in Chapter 6. It is important to notethat due to the attractive interactions we expect the gas to contract as we adiabatically increaseinteractions; this leads to a greater atom density in the center of the cloud as compared to theweakly interacting gas. Based on previous measurements [30], we expect this to result in a local
Fermi energy of EF (r = 0) = 12.4 ± 0.7 kHz at the trap center.
Momentum-resolved rf spectroscopy data for a gas on the BEC side of the Feshbach resonance,at B = 201.51 G and kF1 a = 1.1 is shown in Fig. 5.7c. For this gas, EF = h × 9.3 and kF = 8.6µm−1 . Here, we observe that the dispersion is clearly split into two features. The high energy
feature follows the free particle dispersion (black line) and is similar to the data for a weaklyinteracting gas plotted in panel a. This feature is due to unpaired atoms, which far on the BEC
side can be considered to be weakly interacting with each other and with the pairs. The lower
energy feature is broad and negatively dispersing at all momenta. This feature is due to the pairs,
which can now be considered to be in the molecular regime. The fact that the spectrum disperses
downward starting at zero momentum instead of at kF indicates that Fermi statistics do not playa signiﬁcant role in the physics of the molecular gas. Furthermore, the large widths of the EDCs
suggests that the single-particle excitations are extremely poor quasi-particles and the gas should
not be thought of as consisting of fermionic degrees of freedom; this is consistent with the fact that
we are in the molecular regime of the BCS-BEC crossover. The downward energy shift at zero
momentum represents the amount of energy required to break apart a pair, and hence the pairbinding energy. We measure this oﬀset to be h × 28 kHz, which is close to the measured two-bodybinding energy at this ﬁeld of h×25±2 kHz. Because the pairing is a two-body eﬀect, we can model
the rf dissociation process and produce a Monte-Carlo simulation that shows remarkable agreementwith the experimental data, (see Fig. 5.8). In this model, we assume a thermal distribution forthe molecule center-of-mass motion and use the predicted distribution of relative kinetic energy

75of the atoms for rf dissociation of weakly bound molecules [74]. In this simulation, the widths of
the EDCs are a result of the center-of-mass motion of the molecules. See Appendix D for more
information on the simulation.

5.5

Improvements to the signal-to-noise ratio
The results presented above represent some of our ﬁrst attempts at atom photoemission

spectroscopy and an immediate subsequent goal was to improve the signal-to-noise ratio of thedata. In order to produce the intensity plots shown in Fig. 5.7, we acquire images of outcoupledatoms for a range of rf frequencies. Each image contains information relating to a parabolic cut
in the spectral function (eq. 5.7). To extract that information, the momentum distribution of
the outcoupled atoms must be derived from the image. The absorption image actually shows themomentum distribution projected onto a two-dimensional surface and an inverse Abel transform isrequired to convert this to a 3-d momentum distribution [9]. Because the momentum distribution
is isotropic, we can perform an azimuthal average of the images about the center of the cloud
before doing the inverse Abel transform. The signal-to-noise ratio of the images is directly related
to the ﬁnal signal to noise of the photoemission spectra. If we directly image the outcoupledatoms in the |9/2, −5/2⟩ state using the optical transition to the |11/2, −7/2⟩ excited state, thesignal-to-noise ratio is poor because this is not a cycling transition. The |11/2, −7/2⟩ state has abranching ratio of 0.9 to fall back into the |9/2, −5/2⟩ state, which means that if an atom absorbs
7 photons it has a 50% chance of being pumped into another spin state and “going dark” (i.e.
1 − 0.97 ≈ 0.5). This means that to avoid saturation eﬀects, we have to keep the average number ofphotons absorbed well under 7 per atom, which severely limits the signal-to-noise ratio. In contrast,
atoms in the |9/2, −9/2⟩ state can be imaged on a transition to the |11/2, −11/2⟩ state, which is a
closed transition. With this transition, we typically scatter 100 photons per atom. Therefore, it isobvious we would prefer to transfer the outcoupled atoms to the |9/2, −9/2⟩ state for imaging. Thiscan be done quite simply with two rf π-pulses that transfer the atoms in two steps, going throughthe |9/2, −7/2⟩ state. However, there is a catch. The |9/2, −9/2⟩ and |9/2, −7/2⟩ states are already76Single-particle energy (kHz)

0

a

-10
-20
-30
-40
-50
-60
-70
0

5

10
15
-1
k ( mm )20

Figure 5.8: Molecule spectrum simulation A Monte-Carlo simulation of momentum-resolved
rf spectroscopy of a molecular gas with the same conditions as Fig. 5.7 c. The white line is the ﬁt
to the data in Fig. 5.7 c. See Appendix D for more information.

77
populated by the atoms in the strongly interacting Fermi gas, and because we only outcouple asmall percentage of the atoms in each experiment, the atoms in those states far outnumber theatoms in the |9/2, −5/2⟩ state that we want to image. The rf pulses are coherent and, in principle,
a perfect π pulse would completely switch the populations of the states however, a more realistic
eﬃciency of 98% transfer would leave a large background signal. This is why we did not attempt
this scheme in the original experiments.

8

Optical Trap
7.85

RF Pulses
6

1.5

Blast Beams
Trap Off

Image
Time

Figure 5.9: Imaging scheme A typical timing sequence for imaging outcoupled atoms with an
8 ms time-of-ﬂight expansion. All times are in ms. The ﬁrst rf pulse outcouples atoms from thestrongly interacting system into the |9/2, −5/2⟩ state and is typically 0.3 ms long. After a 6 ms
wait, the ﬁrst blast pulse removes atoms in the |9/2, −7/2⟩ state and then a second one removes
atoms in the |9/2, −9/2⟩ state. Each blast pulse is typically 0.1 ms long. After the two blast pulses,
two rf π pulses transfer the outcoupled atoms to the |9/2, −9/2⟩ state for imaging.

The technique we later developed works by ﬁrst removing the atoms in the |9/2, −9/2⟩ and|9/2, −7/2⟩ states with resonant light pulses. If these pulses remove 98% of the atoms and then we
have an rf pulse that is 98% eﬃcient, we get a combined eﬃciency of 99.96%, which is adequate forour purposes (for 2 × 105 atoms, this would leave a background of only 20 atoms, compared to thethousands we typically outcouple). The resonant light pulses must remove atoms from the desired
states and not perturb the outcoupled atoms in the |9/2, −5/2⟩ state. In order to accomplish this,both resonant pulses drive the atoms to the |9/2, −9/2⟩ excited state. This is a σ − transition foratoms in the |9/2, −7/2⟩ state and a π transition for the |9/2, −9/2⟩ atoms. These transitions are

78
favorable because atoms in the |9/2, −9/2⟩ excited state have zero probability to fall back into the|9/2, −5/2⟩ ground state. Also, the branching ratio to fall into the F = 7/2 ground-state hyperﬁnemanifold is approximately 0.6, which means the atoms will only need to absorb a couple photons
before they are removed to a dark state. Finally, the frequency of these transitions is not close
to any transition for the |9/2, −5/2⟩ ground state (see Appendix A for transitions and branchingratios). Nevertheless, we ﬁnd that excessive power in these blast beams can oﬀ-resonantly excite
a small percentage of the atoms in the |9/2, −5/2⟩ and some of these excited atoms can fall backinto the |9/2, −5/2⟩ ground state with increased momentum due to the absorption and emission ofa photon. This then gives an artiﬁcial contribution to the photoemission spectra. To minimize thiseﬀect, we ﬁnd that the power in the removal beams should be kept only high enough to remove
95% − 98% of the atoms and that the removal should happen as close in time to the imaging as
possible. The timing of the sequence is shown in Fig. 5.9. Using this imaging method, we increased
the signal-to-noise rato in our images by a factor of 3 to 4. A data set taken using this imaging
technique is shown in Fig. 5.10 and can be compared with Fig. 5.7b. EDC’s from the data set areplotted in Fig. 5.11.

79

10

-3

10

-2

10

-1

1

E/EF

2
0

-2
-4
0

1
k/kF

2 0

1
k/kF

2

Figure 5.10: High signal-to-noise ratio PES Data taken with the similar conditions to the dataof Fig. 5.7b using the new imaging scheme to improve the signal-to-noise ratio. On the left, thedata is plotted on a linear color scale, as in Fig. 5.7b. On the right is the same data but with alog color scale to better show the data at high momentum.

Spectral Weight (arb.)

80

k = 0.10
0.25
0.40
0.55
0.70
0.85
1.00
1.15
1.30
1.45

-4

-2

0

2

Single Particle Energy (E/EF)
Figure 5.11: EDCs EDCs from the data set shown in Fig. 5.10 are plotted with values of kkF = 0.1(top) to kkF = 1.45 (bottom). The EDC at kkF = 1.0 is shown in bold. Black dots indicate thecenters of the gaussian ﬁts. Each plotted EDC is the average of EDCs over a range of approximately
0.15kF and has been normalized to have an area of unity.

Chapter 6

The Pseudogap state of a strongly interacting Fermi gas

In this chapter I will address the issue of the normal state of a strongly interacting gas in the
BCS-BEC crossover, and, in particular, the possible existence of a pseudogap phase that arises fromincoherent fermion pairs. The existence and meaning of a pseudogap phase has been a contentious
issue in this ﬁeld, but it is a question of utmost importance if we are to fully understand the
BCS-BEC crossover and make new contributions to the ﬁeld of strongly correlated systems. I will
present atom photoemission spectroscopy data, both as a function of temperature and interaction
strength, in order to explore the normal state in the BCS-BEC crossover and to compare to a theoryof the pseudogap phase. The majority of this work is published in two articles, Refs. [18, 16].

6.1

The pseudogap phase in the BCS-BEC crossover
The ground state of a strongly interacting Fermi gas across the BCS-BEC crossover is always

a paired superﬂuid, even if the nature of those pairs changes dramatically (consider the pair size andbinding energy, for example). In contrast, the normal state, the state that exists at temperatures
above the critical temperature for superﬂuidity, Tc , cannot be described by the same qualitative
framework in both limits. In the BCS limit, which is characterized by weak attractive interactions,the normal state is a Fermi liquid, which is a state similar to a non-interacting Fermi gas but withrenormalized parameters, as described in the previous chapter. In this limit, the pairs and the gap
completely vanish above Tc , leaving no trace of the superﬂuid ground state. However, in the BEC
limit, where the pair binding energy is much larger than kb Tc , the normal state is a molecular gas.

82
Here, the pairing of the fermions is not connected to the formation of a superﬂuid, but rather is a
result of the presence of a deeply-bound molecular state in the two-body potential. In this case,which I also refer to as the molecular regime, the pairs only dissociate at temperatures T ≈ Eb /kb ,where Eb is the pair binding energy. It should be noted that, because there is no phase transition in
the crossover, the Fermi liquid state must smoothly connect to the molecular gas in the BCS-BEC
crossover.
This raises an interesting question: can only deeply bound pairs exist outside a superﬂuid
state or might it also be possible for many-body Cooper pairs to form outside the superﬂuid statenear the center of the BCS-BEC crossover? A state where many-body pairing exists outside thesuperﬂuid state is termed a pseuodogap phase. A potential phase diagram is shown in Fig. 6.1. In
the BCS state, pairing leads to the formation of two excitation branches separated by an energygap (see Fig. 4.3) where no single-particle states exist. In the pseudogap phase, it is predicted thatpairing will lead to a qualitatively similar spectral function, except that the quasi-particles will
acquire energy widths and the density of states will not go all the way to zero in the gap (hence
the name pseudogap) [15, 11]. The dispersion of the spectral function in the pseudogap regime canbe written√(
E(k) = µ ±

)2
~2 2
2
2
(k − kL ) + ∆
2m

(6.1)

where the +(−) is for the upper (lower) branch, kL is the wave-vector where the two branches areat their point of closest approach (in the BCS limit kL = kF and in the BEC limit kL = 0), ∆ isthe magnitude of the pseudogap, and µ determines the midpoint in energy between the upper andlower branches [15, 16].
At high enough temperatures, pairs in the pseudogap and molecular regimes will dissociateleaving a thermal Fermi gas. The temperature marking the characteristic temperature above whichpairs dissociate is called T ∗ , and is shown as the upper dashed line in Fig. 6.1. Pairs that form
above Tc are also called pre-formed pairs because they form before entering the superﬂuid phase.
There are at least two possible deﬁnitions to mark the boundary of T ∗ and identify the pseudogap

83

Figure 6.1: Phase diagram of BCS-BEC crossover This plot shows a schematic phase diagramof the BCS-BEC crossover as a function of interaction strength, 1/kF a and temperature. The solidline shows the location of the critical temperature for the superﬂuid ﬂuid phase transition, Tc . The
thin dashed line marks the characteristic temperature for pairing T ∗ , which merges with Tc in the
BCS limit. The thick gray dashed line marks the crossover from the pseudogap phase where manybody Cooper pairs exist to the molecular phase where the pairing is expected from the two-body
physics. The phases denoted are NF for a normal Fermi liquid, PG for the pseudogap phase, SF
for superﬂuid, and MB for the molecular regime. All of the lines except the solid line marking
Tc represent crossovers and not actual phase transitions. This phase diagram is partially adopted
from Ref. [11]84
regime based on the spectral function [11, 75]. One deﬁnition is to look at the density of states,
obtained from the spectral function by integrating over the momentum axis. A dip in the densityof states near the Fermi surface would indicate the presence of pre-formed pairs and a pseudogap
phase. Another deﬁnition is to look at whether the EDC at k = kL has two distinct peaks. If
two distinct peaks are present this clearly indicates a pseudogap. These two deﬁnitions do notnecessarily coincide and, depending on the situation, either one may lead to a higher characteristic
pairing temperature T ∗ (see for example Ref. [11]).
Distinguishing whether the pseudogap state is mainly present in the density of states oralso present in a two peaked EDC near the Fermi surface is an interesting question and may help
diﬀerentiate between various theories of the pseudogap [75]. For example, theories that emphasize
pairs in the pseudogap state with predominantly low momentum (extended BCS-Leggett theories)
tend to predict well separated quasi-particle branches, while theories that consider pairs with abroad range of momenta (NSR type theories) tend to predict a pseudogap only in the density ofstates for intermediate temperatures because the large energy widths in the spectral function render
the branches indistinguishable near the Fermi surface [75].
The distinction between the pseudogap phase, where many-body pairs exist, and the molecular phase where thermal molecules exist is important. In the pseudogap phase, the fermionic natureof the gas should still be apparent and, in particular, Pauli blocking must play an important rolein pair formation, as in the classic Cooper pair case. Pairing in this regime is highly-non-trivial. In
the molecular phase, one expects pairs simply based on the existence of a deeply bound two-bodystate. In this phase, the underlying Fermi statistics would not play an important role and thepairing is a rather trivial phenomenon. The crossover from the pseudogap phase to the molecularphase will be an important topic in the second half of this chapter.
In the prediction of a pseudogap phase, we have a true question about the nature of thesingle-particle states of the strongly interacting Fermi gas in the BCS-BEC crossover. Are thosestates similar to a Fermi-liquid or are they paired states? The later case would represent a breakwith the highly successful Landau Fermi liquid theory that has been so successful at describing

85interacting Fermi gases. Either way, it is clear that these basic questions raised in the BCS-BEC
crossover are questions we must address in our quest to better understand strongly interacting
Fermi systems.

6.2(Some) History of the pseudogap phase and relation to high temperature
superconductors
It is interesting and useful to examine some of the history of the pseudogap phase, both intheory and experiments, and in particular the connections to the high Tc cuprates. This section is
not a complete literature review of the subject, but rather a snapshot of a few interesting points.
The term pseudogap originally came from the high Tc cuprates where a number of experiments discovered anomalous properties of the normal state in the underdoped region of the holedoped cuprate phase diagram (Fig. 6.2). Both tunneling and ARPES experiments that probed
the single-particle electron structure found that a superconducting-like gap feature, the pseudogap,
persisted above Tc to a much larger temperature T ∗ . ARPES experiments showed that the pseudogap retained the same d-wave structure as the superﬂuid gap and transformed smoothly into thesuperconducting gap [8] as a function of hole-doping. This behavior is not observed in conventional
superconductors, where the gap is present only in the superconducting state, as predicted by BCS
theory [5].
While no complete microscopic theory of the high-Tc superconductors exists, various phemenological theories have been proposed to explain the pseudogap phase. These theories fall mainlyin either the pre-formed pairs category, which would be similar to the situation of the proposed
pseudogap phase in the BCS-BEC crossover discussed above, or in the non-pairing or competingorder category, where the pseudogap forms due to some other phenomenon.
In order to shed light on whether the pseudogap phase might originate from pairing, experimentalists have looked for evidence of whether the quasi-particle dispersion in the pseudogap phaseresembles the BCS-like dispersion of Eq. 6.1 that originates from pairing. Experiments were done
indicating that the dispersion was indeed BCS-like; with back-bending occuring near the Fermi

86

Figure 6.2: Typical hole-doped high-Tc phase diagram A typical phase diagram for a high-Tc
superconductor as a function of temperature and hole doping. SC stands for the superconducting
state and AFM stands for the anti-ferromagnetic insulator state. The superconducting phase exists
for a dome-shaped region of ﬁnite hole concentration. The hole concentration where Tc is maximal
is called optimally doped. The pseudogap phase exists in the temperature region of Tc < T < T ∗ inthe underdoped region of the phase diagram. Theories of the phase diagram based on the BCS-BEC
crossover predict that varying hole concentration plays the role of changing interaction strengthwith smaller hole concentration corresponding to stronger interacting strength (and hence a larger
pseudogap region) [12]. This ﬁgure is taken from Ref. [12].

87
surface [76], a symmetric upper branch indicating the particle-hole mixing of the quasi-particles
expected from Cooper pairing [77, 78], and a smooth connection to the superconducting gap [79].
However, some of these observations are contradicted in other experiments such as Ref. [80, 81].
Ultimately, this important issue remains unresolved and highly controversial.
For the preformed pairs theory of the pseudogap, there have been many attempts to understand the pseudogap in the BCS-BEC crossover and apply that to qualitatively understand thepseudogap of the high-Tc superconductors, for example Refs. [61, 15, 12]. However, it is clear thatto really model the cuprates one should take into account the two-dimensional nature as well as
the lattice structure and pairing symmetry [12]. The Fermi-Hubbard model has been proposed toaddress these issues, however, the doped Fermi-Hubbard model remains unsolved [82].
Nevertheless, if the pseudogap in high-Tc superconductors does arise, at least in part, frompre-formed pairs due to strong pairing correlations, it is reasonable to think that we may gain
some insight by studying the pseudogap regime in the strongly correlated BCS-BEC crossover. In
particular, experiments of Fermi gases in the BCS-BEC crossover can conﬁrm whether or not a
pseudogap state consisting of preformed pairs can arise from strong interactions and whether thatstate indeed leads to BCS-like features of the spectral function. Perhaps more importantly, experiments with atomic gases in the BCS-BEC crossover can test the validity of theoretical approaches
to understand the pseudogap phase in strongly interacting Fermi gases. This approach can lead toa better understanding of strongly interacting fermions and ultimately progress towards the evenmore diﬃcult problem of developing a microscopic theory of the high-Tc superconductors. It is
important to note that in the BCS-BEC crossover realized by atomic gases, the presence of a pseudogap state could be unambiguously attributed to pairing interactions as no other explanations
exist, in contrast to the competing explanations of the pseudogap for the high-Tc superconductors.
In atomic Fermi gases, despite a large body of theoretical work on the pseudogap phase[83, 84, 85, 15, 12, 86, 87] (to name a few), there has not been convincing experimental evidence
to conﬁrm the presence of the phase in the BCS-BEC crossover, other than the work presented
in this thesis. Previous rf spectroscopy experiments claimed to identify pairing above Tc , such as

88
Ref. [88], but these experiments turned out to be diﬃcult to interpret and ultimately unable todistinguish between a paired and unpaired state [34].
Furthermore, recent measurements at ENS of the equation of state for a Fermi gas at thecenter of the BCS-BEC crossover (1/kF a = 0) ﬁnd a linear dependence of the pressure with (T /µ)2 ,which they interpret as consistent with a Fermi liquid above the critical temperature [13, 89].
The density of states near the Fermi surface determines how a system will respond to changes intemperature and thus thermodynamic measurements such as the ENS data can indirectly probe
this quantity. A model of the pseudogap using a BCS-like dispersion (Eq. 6.1) with no energy
width, such that a real gap exists in between the upper and lower quasi-particle branches, can becompared to the ENS data. This comparison shows that the ENS data would only be consistent witha pseudogap smaller than 0.05EF [89]. However, a more realistic model of the spectral function withﬁnite width, such that the pseudogap was partially ﬁlled in, would allow the data to be consistent
with a larger value for the pseudogap [89]. Indeed, NSR theories of the pseudogap predict that the
gap should become signiﬁcantly ﬁlled in at temperatures above the critical temperature as shownin the inset of Fig. 6.3 (see also Refs. [11, 75]). Fig. 6.3 shows that the predicted behavior of thepressure from an NSR theory (black line) is indeed linear and consistent with the ENS data (bluecircles). This also explains how the quantum Monte Carlo data from Ref. [90], which predicts a
pseudogap phase, is in good agreement with the ENS data [91]. The ENS data, NSR theory, andQuantum Monte Carlo calculations indicate that thermodynamic measurements may be a poor
probe of the nature of the single-particle states near the Fermi surface in the strongly interacting
Fermi gas due to the large energy widths of those states [92].

6.3

Atom photoemission spectroscopy and the pseudogap phase in the BCSBEC crossover
The ideal measurement tool for identifying the pseudogap regime would be one that had directaccess to the single-particle fermion states, allowing one to directly check the density of states andproperties of the spectral function. The technique of atom-photoemission spectroscopy, based on

89

Figure 6.3: Pressure of a normal unitary Fermi gas Main Figure) Red circles are experimental
measurements of the pressure for a homogeneous Fermi gas at 1/kF a = 0, obtained in Ref. [13].
Blue squares show the result of a quantum Monte Carlo calculation obtained in Ref. [14]. The blackline is the prediction from a ﬁnite temperature NSR theory of the pseudogap [15, 16]. The linear
behavior of the pressure in the theory, despite the existence of a pseudogap in the density of states
up to temperatures as high at T /Tc = 1.6, demonstrates that thermodynamic measurements ofRef. [13] cannot exclude a pseudogap phase. Inset) The inset shows the density of states obtained
from the same theory versus the temperature T /Tc . The black dashed curve shows the density ofstates for a non-interacting Fermi gas. This ﬁgure is taken from Ref. [16].

90
momentum-resolved rf spectroscopy, has the potential to achieve exactly that by directly measuring
the single-particle spectral function. There is however one catch. Photoemission spectroscopy
measures only the occupied single-particle spectral function as can be seen from the presence of the
Fermi function in Eq. 5.7. To measure a gap in the density of states or to resolve the upper branchof the quasi-particle dispersion, one needs to examine the full spectral function. In principal, thefull spectral function can be obtained by dividing out the Fermi function from the photoemission
data if the temperature and chemical potential are known. However, this would require atomphotoemission data for a homogenous sample. In a trapped sample, as studied in the previous
chapter and this chapter, the Fermi-function has a radial dependence and so cannot be divided out
from trap-averaged data [16]. I will present progress towards obtaining photoemission data from ahomogenous sample in the next chapter.
However, this is not to say that the trap-averaged data cannot help resolve the issue of thepresence of the pseudogap in the BCS-BEC crossover. From the occupied spectral function, we will
be able to examine the dispersion of the lower quasi-particle branch and look for an indication of
a BCS-like dispersion with back-bending near the Fermi surface. We will also be able to compare
spectral functions at diﬀerent temperatures and look for evidence (or lack thereof) of a change
from a paired state to a Fermi-liquid state at the critical temperature, as would be predicted byconventional BCS theory. We can also compare the trap averaged photoemission data to theoriesof the pseudogap state to determine if they are consistent.
In this series of experiments, we prepare an ultracold gas of fermionic 40 K in the |9/2, −9/2⟩
and |9/2, −7/2⟩ states near 203.5G as in Ref. [9]. At the end of the evaporation, we increase theinteractions adiabatically with a slow magnetic-ﬁeld ramp to the Feshbach scattering resonance.
To vary the temperature of the cloud, we either truncate the evaporation or parametrically heat
the cloud by modulating the optical dipole trap strength at twice the trapping frequency. To
determine the temperature of the Fermi gas, we expand the weakly interacting gas and ﬁt the
momentum distribution to the expected distribution and extract (T /TF )0 , where the subscript
indicates a measurement made in the weakly interacting regime, before ramping the magnetic ﬁeld

91
to the Feshbach resonance. For the data presented here, we obtain clouds with ﬁnal temperatures
ranging from (T /TF )0 = 0.12 to 0.43 with N = 1.0 × 105 to 1.8 × 105 atoms per spin state. The
trap frequencies vary depending on the ﬁnal intensity of the optical trap and range from 180 to
320 Hz in the radial direction and 18 to 27 Hz in the axial direction. Correspondingly, the Fermi
energy, EF ranges from h × 8 kHz to h × 13 kHz, where h is Planck’s constant. The Fermi energy
is obtained from N and the geometric mean trap frequency, ν, as EF = hν(6N )1/3 . We deﬁne the√Fermi wave-vector as kF = 2mEF /~ and the Fermi temperature as TF = EF /kB . The energy
resolution of the experiments is given by the inverse duration of the rf pulse and is equal to 2.15
kHz (rms width) or a range of 0.16 − 0.27EF .

Figure 6.4: Data in the pseudogap regime We take data along a vertical cut (arrow) in thephase diagram at 1/kF a = 0.15 ± 0.03 by varying the temperature of the gas.

The dimensionless parameter that characterizes the interaction strength for this data is
1/kF a = 0.15 ± 0.03. At this interaction strength we measure the critical temperature to be( TTFc )0 = (0.17 ± 0.02) based on the condensate fraction measurement technique [7], see Fig. 6.5.
By varying the temperature, we take photoemission data along a vertical line in the phase diagram through the pseudogap region as shown in Fig. 6.4. The photomession data is shown in Fig.

Condensate Fraction

92

0.2

0.1

0
0

0.1

0.2

0.3

(T/TF)0
Figure 6.5: Measuring Tc Using time-of-ﬂight expansion at B = 202.1 G, where our photoemission
experiments are performed, we map out the condensate fraction. Temperature is measured in the
weakly interacting regime before the adiabatic ramp to strong interactions. We ﬁnd ( TTFc )0 =0.17 ± 0.02. Note that the density of the trapped cloud decreases with increasing distance fromthe trap center, and therefore, in a local density picture, even at TTF = 0.17 only the part of thegas at the very center of the trap is below Tc . The horizontal error bars represent one standard
deviation of systematic uncertainty in the temperature and the vertical error bars represent a
standard deviation of statistical error in the measurements.

93
6.6 and Fig. 6.7. In Fig. 6.6, we plot the fraction of out-coupled atoms as a function of theirsingle-particle energy and momentum for temperatures encompassing the pseudogap regime. In
the intensity plots, white dots indicate the centers derived from unweighted gaussian ﬁts to eachof the energy distribution curves, or EDCs, (vertical trace at a given wave-vector). The energy
dispersion mapped out with these ﬁts (white dots) can be contrasted to the expected free particle
dispersion for an ideal Fermi gas (black curve) and are also plotted in Fig. 6.8 along with theory
curves (solid lines) that are discussed below. In Fig. 6.7 we show the same data plotted as EDCs forwave-vectors ranging from k/kF = 0.1 to k/kF = 1.4. In order to show the evolution of the spectral
function from below Tc through the pseudogap regime the data are shown for four temperatures,(T /TF )0 = 0.13, 0.21, 0.25 and 0.35, corresponding to (T /Tc ) = 0.74, 1.24, 1.47 and 2.06.
For the data below Tc (Fig. 1a), we see a smooth back-bending that occurs near k = kF ,similar to what was found in the data of Fig. 5.7b and Fig. 5.10, which were taken at a slightly
higher temperature but the same interaction strength. The white curve in Fig. 1a shows a BCSlike dispersion curve, Eqn. 6.1, obtained by ﬁtting to the white dots for momenta in the range
0 < k < 1.4 kF . While we cannot use this ﬁt to extract the gap and chemical potential in a modelindependent way due to the harmonic trapping conﬁnement, the BCS-like ﬁt is consistent with alarge pairing gap, on order of EF , consistent with the data of Fig. 5.10, and as expected for a
Fermi gas near the center of the BCS-BEC crossover [6, 4].
In the case of a pairing gap, we expect the spectral function to exhibit a BCS-like dispersion
with back-bending at k near kF . For the three lowest temperatures, we observe this behavior. In
fact, we observe no qualitative change from the data at T /Tc = 0.74 to the data at T /Tc = 1.24.
We interpret this as strong evidence in favor of the existence of a pseudogap regime above Tc
comprised of uncondensed pairs in the strongly interacting Fermi gas. The data at the highest
temperature, T /Tc = 2.06, also displays back-bending but in a qualitatively diﬀerent way. In thisdata, we observe a sharp kink in the dispersion (obtained from the Gaussian ﬁts to the EDCs) at a
momentum of k = 1.5kF . This may seem strange because at high temperature, above T ∗ where nopairs exist, one naively expects to observe a purely upwardly dispersing feature. However, it turns

94Single-particle energy (ES/EF)

2 a

c

b

100

d

0
10

-1

-2
10-2

-4

0

T/Tc = 0.74

= 1.24= 1.47= 2.06(T/TF)0 = 0.13

= 0.21= 0.25= 0.35

1

k/kF

2 0

1

k/kF

2 0

1

k/kF

2

0

1

2

k/kF

Figure 6.6: Photoemission spectra across the pseudogap regime. Spectra are shown for
Fermi gases at four diﬀerent temperatures, each with an interaction strength characterized by(kF a)−1 ≈ 0.15. The intensity plots show the fraction of out-coupled atoms as a function of
their single-particle energy (normalized to EF ) and momentum (normalized to kF ), where E = 0
corresponds to a non-interacting particle at rest. The spectra are normalized so that integratingthem over momentum and energy gives unity. White dots indicate the centers extracted fromgaussian ﬁts to individual energy distribution curves (traces through the data at ﬁxed momentum).
The black curve is the quadratic dispersion expected for a free particle. a At T = 0.74 Tc , we observea BCS-like dispersion with back-bending, consistent with previous measurements [9]. The whitecurve is a ﬁt to a BCS-like dispersion, Eq. 6.1 b,c At T = 1.24 Tc and T = 1.47 Tc , respectively,
the dispersion with back-bending persists even though there is no longer any superﬂuidity. d At
T = 2.06 Tc , the dispersion does not display back-bending in the range of 0 < k < 1.5 kF . In all
the plots there is a negative dispersion for k/kF > 1.5. We attribute this weak feature (note thelog scale) to a universal property of the Fermi gas related to short-range correlations.

10-3

Spectral Weight (arb.)

95

T/Tc = 0.74

T/Tc = 1.24

T/Tc = 1.47

T/Tc = 2.06

k = 0.10
0.25
0.40
0.55
0.70
0.85
1.00
1.15
1.30
1.45

-4

-2

0

2

-4

-2

0

2

Single-Particle Energy (ES/EF)
Figure 6.7: Energy distribution curves (EDCs) EDCs are obtained by taking vertical traces atﬁxed k through the photoemission spectra shown in Fig. 1. We show EDCs between kkF = 0.1 (top)and kkF = 1.4 (bottom) for the four data sets with T /Tc labeled above each ﬁgure. Each plotted
EDC is an average of EDCs over a range of approximately 0.15kF . The EDC at kkF = 1.0 is shownin bold. Black dots indicate the centers of the gaussian ﬁts to the EDCs. Each EDC is normalized
to have an area of unity. Vertical dotted lines are placed at the local EF that corresponds to theestimated average density of the gas.

Single-particle energy (ES/EF)

96

2
1
0
-1
-2
-3
0

1

2

k/kF
Figure 6.8: Single-particle dispersion curves The ﬁts to the EDC centers are shown for thefour temperatures in Fig. 6.6 and Fig. 6.7 a,b,c and d, represented by black, blue, red, and green,
respectively. We observe a BCS-like dispersion, smooth back-bending near kkF = 1, for temperatures
below and moderately above Tc . For the highest temperature, we observe a quadratic dispersion
near kkF = 1 and a sharp discontinuity near kkF = 1.5. The lines are theory curves that include
eﬀects of the harmonic trap, as described in the text. The error bars on the points represent one
standard deviation of uncertainty from our ﬁts.

97
out that the short-range correlations between the fermions, due to the contact interaction, lead tothe existence of a broad, downwardly dispersing feature at high momenta regardless of whether
the system is a superﬂuid, Fermi-liquid, or other state, see Fig. 6.9. It should be noted that thestrength of the broad, downwardly dispersing feature is dependent on the state of the system, and,moreover, the strength of the feature can be related to various other quantities in the system, suchas the total energy, in a universal way. This is the subject of Chapter 8 and Ref. [10].
If back-bending is possible even in a Fermi liquid system, how can we tell whether the spectral
functions in the intermediate temperature regime actually indicate pairing? First of all, as notedin the caption of Fig. 6.9, the back-bending present in a Fermi-liquid is actually the result of two
separate features in the spectral function: one upwardly dispersing feature that is eventually cutoﬀ
by the Fermi function, and one downwardly dispersing feature. In this situation, one would expect
a discontinuity in the ﬁts to the EDC’s as the ﬁts jump from the upward feature to the downward
feature. This should be evident both in a sharp cusp in the dispersion occurring at k > kF anda change in the amplitudes of the gaussian ﬁts. A sharp cusp in the dispersion at T /Tc = 2.06
is evident in Fig. 6.6 d and Fig. 6.8 (green circles). A sharp drop in the amplitude is evident in
Fig. 6.10 (green circles). For the data sets in the intermediate temperature regime, T /Tc = 1.24
and 1.47, we do not observe this behavior. In these data sets, the smooth back-bending of thedispersion near kF is strong evidence for BCS-like pairing.
Furthermore, if the back-bending in the intermediate temperature regime were indeed duesolely to the universal feature and not pairing one would expect the maximum energy of the occupied
spectral function, Emax , to increase approximately proportional to the temperature of the gas as
the upwardly dispersing quasi-particle branch became occupied up to higher and higher energy. We
do not observe this smooth dependence as shown in Fig. 6.11. Instead, we observe a sharp increasein Emax for temperatures above (T /TF )0 = 0.25, where the spectral function appears to revert toFermi-liquid like behavior. This is further evidence for a suppression in the density of states in theintermediate temperature range due to a pseudogap phase.
In Fig. 6.8, we compare our atom-photoemission spectra to a BCS-BEC crossover theory

98

Figure 6.9: Universal structure of an interacting Fermi liquid A logarithmic intensity plot ofA(k, w)/(kF a)2 for a Fermi liquid with repulsive interactions (kF a = 0.1) taken from Ref. [17]. Thisﬁgure shows both a quasi-particle branch dispersing upwards (red line) as well as a comparatively
broader and weaker feature that is downwardly dispersing (black dashed line). The downwardly
dispersing feature can be associated with short-range correlations due to the contact interactionthat are not related to pairing [17]. In photoemission spectroscopy we measure the occupied spectral
function, A(k, w)f (w) where f (w) is the Fermi function. The Fermi function serves as a cutoﬀ forsignal at high energy which results in the upper quasi-particle feature terminating at some ﬁnite
momentum. Beyond this momentum only the lower feature will persist and the EDC’s will reveal
a dispersion that jumps to the downwardly dispersing feature, as seen in Fig. 6.6 d or the green
curve in Fig. 6.8.

99

Amplitude

0.3
0.2
0.1
0
1.0

1.5
k/kF

2.0

Figure 6.10: EDC amplitudes We show the amplitudes from the gaussian ﬁts to the EDCs forthe same experimental data as shown in Figs. 6.6,6.7, and 6.8. The ﬁt amplitudes evolve smoothly
for the lower temperatures (black, blue, and red circles in order of increasing temperature) but
jump discontinuously for the highest temperature gas (green circles).

100

Emax/EF

1.5
1.0
0.5
0
0.1

0.2

0.3

(T/TF)0
Figure 6.11: Maximum energy of the dispersion We show the maximum energy of the dispersion, Emax , for temperatures ranging from (T /TF )0 = 0.13 to (T /TF )0 = 0.35. Emax is thehighest energy of the dispersion obtained from the gaussian ﬁts to the EDCs. The vertical dashedline marks the critical temperature, Tc at (T /TF )0 = 0.17. There is a clear jump that occurs fortemperatures above (T /TF )0 = 0.25 indicating that for temperatures below that value Emax is
suppressed by an energy gap. In future experiments, it would be interesting to take more points
in the region of (T /TF )0 = 0.25 to 0.35 to map out the sharp increase in Emax . From left to right:
the ﬁrst, third, fourth, and sixth points are from the data in Fig. 6.6 published in Ref. [18], thesecond point is the from the second panel on the top of Fig. 6.13 published in Ref. [16], and theﬁfth point, at (T /TF )0 = 0.3 is from an unpublished data set.

101
that is based on a diagrammatic t-matrix approximation that includes pairing ﬂuctuations and is
described in Ref. [15] (also referred to as NSR theory [75]). To compare to the experimental data,
we ﬁt theoretical EDCs to single Gaussians to extract the centers; the results are shown as lines
in Fig. 6.8. The theory, which predicts the existence of a pseudogap in the density of states forthe intermediate temperatures of (T /Tc )0 = 1.24 and 1.47, is in excellent qualitative agreementwith the experimental data. In particular, the theory lines show smooth BCS-like back-bending
for the lower temperatures and a kinked dispersion for the highest temperature, (T /Tc )0 = 2.06,
where a pseudogap state is no longer present. We will compare predictions of the spectral function
from theory to experimental data in more detail in the next section of this Chapter. For now, I
will simply note that the good agreement of our data above Tc with a theory of the pseudogap isfurther evidence in support of the existence of a pseudogap state in the strongly interacting Fermi
gas. In a similar manner, the theoretical work of Ref. [93] ﬁnds that back-bending of the trap
averaged photoemission data corresponds to the existence of a pseudogap state in the center of
the trap. I should also remark that other theories that have predicted a pseudogap state for thestrongly interacting Fermi gas appear to at least qualitatively agree with our data, see for example
Refs. [94, 72].
For temperatures below Tc , the theory [15] predicts a dispersion that is shifted to lower energy
than the data. This disagreement can be attributed to a sharp variation of the order parameterwith temperature close to Tc . This may suggest the theory predicts too high of an order parameterjust below Tc .

6.4

Pseudogap to molecular gas crossover
To learn more about the pseudogap phase, I will present, in this section, atom photoemission

data for diﬀerent interaction strengths in the BCS-BEC crossover, ranging from the center of thecrossover, at 1/kF a = 0, to well into the molecular regime, at 1/kF a = 1.1, at temperatures near orjust above the critical temperature Tc (see Fig. 6.12). It is important to recall that at T = Tc only
the very center of a trapped atom gas is actually at the critical temperature, and the vast majority

102
of atoms should be considered to have a temperature above the local Tc . Thus, the photoemission
data presented here probe the pseudogap regime. Furthermore, by investigating this region ofinteraction strengths, we will probe the crossover from the pseudogap state, where the underlying
fermionic properties of the system remain important, to the molecular regime where the pairing
can be considered primarily a two-body eﬀect. We will see that the value of the wave-vector, kL ,(see the pseudogap dispersion 6.1), which determines where the two quasi-particle branches are attheir point of closest approach, can characterize whether a gas is in the pseudogap or molecularregime. A value of kL near kF will indicate the presence of a remnant Fermi surface, indicating
the importance of the fermionic nature of the system. In the molecular regime, kL approaches 0. I
will also show detailed comparisons of the data to the NSR theory of Ref. [15], which predicts thisbehavior of kL . The experimental data and theory comparisons are in Ref. [16].

Figure 6.12: Data taken near Tc in the pseudogap and molecular regime We take data near
Tc for diﬀerent interaction strengths ranging from kF1 a = 0.0 to 1.1. This region is the heart of thepseudogap phase and the crossover to a molecular phase.

The photoemission data is shown in Fig. 6.13. Some general observations regarding the data
can be made immediately. In the data sets at 1/kF a = 0.45 and higher, two distinct spectral

103

2

Single Particle Energy (Es/EF)

0
-2

-4

1/kFa = 0.0

0.15

0.45

0.6

0.8

1.1

2
0

-2

-4
0

1

20

1
k/kF

20

1

2

Figure 6.13: Photoemission spectra near Tc in the pseudogap and molecular regime
The six panels show data taken at diﬀerent interaction strengths, parameterized by 1/kF a, and
T /Tc = 1.0 ± 0.1, (see Fig. 6.12). The interaction strengths span the center of the pseudogap
regime to the molecular regime. The black line is the free particle dispersion and the white dots
are obtained from Gaussian ﬁts to the EDCs. In the ﬁrst two data sets (1/kF a = 0.0 and 0.15)
only a single spectral feature is resolved and so the ﬁts are single Gaussians. In the other four
data sets (1/kF a = 0.45, 0.6, 0.8 and 1.1) two spectral features are resolved and the ﬁts are double
Gaussians. The white lines are ﬁts to the back-bending spectral features with Eq. 6.1.

104features become distinguishable with the higher feature being upwardly dispersing and close to thefree-particle dispersion. Furthermore, as the value of 1/kF a is increased, the lower feature shifts
downwards and changes shape to become ﬂatter and eventually monotonically decreasing in themolecular limit.
These general features can be interpreted as follows. In the molecular limit, unpaired fermions
are weakly interacting and so should have a spectral function identical to a weakly interacting
Fermi gas. It has been previously found that in this temperature range, the conversion of atoms tomolecules after a magnetic-ﬁeld sweep from the BCS to BEC limits is only 75% eﬃcient, see Ref.
[95]; and so we expect a ﬁnite number of unpaired atoms to contribute to the photoemission signal.
Because the three-body recombination rate is relatively slow in our gas, these unpaired atoms can beout of chemical equilibrium with the molecules. This explains the upwardly dispersing feature that
follows the free-particle dispersion in the data sets closest to the molecular limit (1/kF a = 0.8 and
1.1). As for the data sets closer to the pseudogap regime, such as the spectrum at 1/kF a = 0.45 and
1/kF a = 0.6 , it might seem surprising to observe a spectral feature near the free-particle dispersion,as we expect the atoms in this regime to be strongly interacting. However, it is important to recall
that this photoemission data is for a trapped gas and that atoms at the edge of the cloud will be at
low density, and hence weakly interacting, regardless of the scattering length. Thus, the spectralfeature near the free particle line observed in data sets in the strongly interacting regime are likely
due to atoms at the edge of the cloud, see for example Ref. [93].
In the spectra nearest to the center of the crossover, 1/kF a = 0.0 and 0.15, only one spectralfeature is observed. However, in this situation the lower branch of the spectral function is closeenough to the free particle line that the contribution from weakly interacting atoms at the edge ofthe cloud cannot be clearly resolved. For data sets in the strongly interacting regime, in addition
to weakly interacting atoms at the edge of the cloud, one might expect some signal due to unpaired
atoms near the center of the cloud, where they would be strongly interacting. These excitations
would be similar to the quasi-particles we derived in the BCS theory and would have a dispersion
diﬀerent from the free-particle dispersion, see Fig. 4.3. However, we expect the presence of the

105
Fermi function in the photoemission signal, Eq. 5.7, to suppress any contributions from quasiparticle excitations. Furthermore, near the Fermi surface, where quasi-particles are most easily
excited, the two branches of the spectral function may not be resolved due to the large energy
widths of the quasi-particles [75, 16, 11]. The inhomogenous density introduced by the trapping
potential and signal from atoms at the edge of the cloud would further serve to obscure any signof the upper quasi-particle branch. These reasons may also explain why the upper branch of thequasi-particle dispersion is not observed in data taken at higher temperature, (see Fig. 6.6).
The changes in the dispersion of the lower spectral feature that we observe as a function of
1/kF a are important. The shift of the dispersions to lower energy for increasing 1/kF a is due tothe increasing binding energy of the pairs as we cross into the molecular regime. Perhaps moreinterestingly, the shape of the dispersion also changes in the data. Near the center of the crossover,in the data at 1/kF a = 0.0 and 0.15, the dispersion is qualitatively similar to the BCS quasi-particle
dispersion, Eq. 4.8, with an increasing dispersion at low momenta and back-bending occurring near
the Fermi surface k/kF = 1. Back-bending near k/kF = 1 signiﬁes that Fermi statistics still play
an important role in the gas and indicates that the back-bending dispersion is caused by manybody Cooper pairs. As we approach the molecular regime with larger values of 1/kF a, we observethat the dispersions ﬂatten and eventually become purely downwardly dispersing, as expected formolecules, see Fig. 5.8. Here, Fermi statistics do not play an important role and k/kF = 1 is nolonger a special point.
The point where back-bending occurs is determined by the parameter kL in Eq. 6.1. The
value of kL extracted from BCS ﬁts to the data (white lines in Fig. 6.13) is shown in Fig 6.16.
The molecular regime should be marked by a two-body molecular binding energy greater than thetemperature, leading to the formation of pairs in the absence of many-body eﬀects. The two-bodypair binding energy compared to the Fermi energy is given by Eb /EF = 2(kF a)2 (see Eq. 3.9) andthis can be compared to the approximate critical temperature of (T /TF )0 = 0.17, which is given interms of the temperature of the gas before the adiabatic ramp to strong interactions. We ﬁnd that
EB /kB TF = 0.0, 0.05, 0.41, 0.72, 1.28 and 2.42 for the data sets shown in Fig. 6.13.

106

6.5

Comparisons to a theory of the pseudogap
In this section, we compare the data of Fig. 6.13 to a theory of the pseudogap introduced inRef. [15]. This theory predicts a pseudogap in the density of states over a wide range of temperatureand interaction strength for a gas in the BCS-BEC crossover [75, 15, 11] (see inset of Fig. 6.3).
For comparison with the trap-averaged data of Fig. 6.13, EDCs are produced by taking theoretical
spectra for homogenous gases and averaging them over predicted density proﬁles of the gas, which
are obtained from the same theory. For details of the theory, I refer the reader to Refs. [15, 16].

Figure 6.14: Comparison of experimental and theoretical EDCs Here we plot experimental
EDCs (red circles), obtained from vertical cuts through the ﬁrst 5 data sets of Fig. 6.13, and thecorresponding theoretical EDCs (blue lines) from an NSR theory based on Ref. [15]. This ﬁgure istaken from Ref. [16]. See text for discussion.

The comparisons of the EDCs can be seen in Fig. 6.14. The dispersions and energy widths
obtained from the EDCs are shown in Fig 6.15. A ﬁrst remark that can be made regarding
Figs. 6.14 and 6.15 is that the theory reproduces the amplitude, center, and energy widths of theexperimentally measured EDCs quite well. This suggests that the spectral functions produced by
NSR-type theories [75, 15, 11] capture the essential physics of the pseudogap regime in the BCS-

107

(a)

(b)

2

6
-1

(kFaF) =0.0

1

4

0
-1

2

-2
-3

0
-1

(kFaF) =0.15

1

4

0

2

-2
-3

0
(kFaF)-1=0.45

1

4

0
-1

2

-2
-3

0
(kFaF)-1=0.57

1

FWHM (EDC left peak) / EF

EDC left peak position / EF

-1

4

0
-1

2

-2
-3

0
-1

(kFaF) =0.78

1

4

0
-1

2

-2
-3

0
0

0.5

1
k/kF

1.5

0

0.5

1
k/kF

1.5

2

Figure 6.15: Comparison of experimental and theoretical dispersions and widths (a) We
plot dispersions (red circles) obtained from unweighted Gaussian ﬁts to the EDCs of the backbending feature (left peak) of the data in Fig. 6.13. Solid green lines mark the free-particle dispersion.
(b) We plot the widths (full width at half maximum), shown as red circles, that are obtained fromunweighted Gaussian ﬁts to the EDCs of the backbending feature (left peak) of the data in Fig.
6.13. Blue lines are obtained from the theoretical spectra, shown in Fig. 6.14, that include theeﬀects of trap averaging. Dashed lines are obtained from theoretical spectra for a homogeneous gas
with density equal to the predicted average density of the trapped gas. This ﬁgure is taken from
Ref. [16]. See text for discussion.

108
BEC crossover. The theory also conﬁrms that the parameter kL decreases rapidly in the area ofthe crossover from the pseudogap regime to the molecular regime and marks the disappearance ofa remnant Fermi surface in the strongly interacting gas, as can be seen in Fig. 6.16.
In some of the data sets in Fig. 6.14, it is clear that the theory and experiment disagree on
relative weight of the upper and lower peaks. However, this relative weight is very sensitive to thetemperature and chemical potential. As such, the discrepancy is not surprising since the theory
overestimates Tc by approximately 30% [16].
Given that the theory reproduces the essential features of the data, we can ask the question,
what does the theory tell us about the eﬀect of trap averaging? To investigate this, dispersions andenergy widths obtained from theoretical EDCs for a homogenous gas with a density equal to theaverage density of the trapped gas are shown as dashed lines in Fig 6.15. From this we can make a
few remarks. Both a homogeneous gas and the the trapped gas show similar qualitative features.
In particular, the BCS-like shape of the dispersion with back-bending at the wave-vector kL is not
aﬀected by the trap. Furthermore, Fig. 6.15 shows that the large single-particle excitation energy
widths, which are on the order of the Fermi energy, near the Fermi surface, do not appear to becaused by the trapping potential but, according to the NSR theory are actually intrinsic energy
widths. This would imply the gas is far from the regime of a Fermi liquid where single-particle
excitations at the Fermi surface correspond to narrow energy peaks in the spectral function andcan be treated as weakly interacting quasi-particles [68].
Fig. 6.17 shows that, according to this NSR theory, the fact that we can map out a single
back-bending dispersion for the entire range of momenta k/kF = 0 to 2.0 is actually due to the
suppression of high energy states from the Fermi function in the photoemission signal, Eq. 5.7. In
the full spectral function predicted by the theory, the upper and lower branches are only distinguishable at low (k/kF < 0.9) and high (k/kF < 1.5) momenta and the two branches merge into asingle broad feature near the Fermi surface (see also Refs. [75, 11]). It is important to note that,along with the existence of two distinguishable branches (with similar spectral weight) at low andhigh momenta, the spectral functions obtained from NSR theory contain a dip in the density of

109

1
0.8

kL 0.6
0.4
0.2
0

0

0.5

1.0

1.5

1/kFa
Figure 6.16: Comparison of experimental and theoretical kL The value of kL (Eq. 6.1)
obtained from theoretical dispersions (black line) and from the data (red squares). This ﬁgure istaken from Ref. [16]. See text for discussion.

110

Figure 6.17: Dispersion with and without Fermi function at 1/kF a = 0.15 The plot showsthe experimental dispersion obtained from trap averaged photoemission data at Tc at 1/kF a = 0.15
(red circles). The dashed line is obtained from a theoretical prediction for a homogeneous gas
with density equal to the predicted average density of the atom cloud. The black line is obtained
from the theoretical prediction but without the inclusion of the Fermi function. The black line
thus traces out the dispersion of the lower branch in the full spectral function. In the region of(0.9 < k/kF < 1.5) the two branches of the dispersion cannot be distinguished due to the large
energy widths. This ﬁgure is taken from Ref. [16].

111
states at the Fermi surface due to a pseudogap, see Fig. 6.3 inset.

6.6

Extracting information from BCS-ﬁts to the spectral functions
Fits to the experimentally obtained dispersions using the expected pseudogap dispersion, Eq.

6.1, are shown as white lines in Fig. 6.13. The functional form of Eq. 6.1 obviously matches the data
very well. However, it is not clear if the best ﬁt values of the parameters are particularly meaningful.
After all, the photoemission spectra are obtained for trap-averaged data where the gas does nothave a well-deﬁned value for the gap and chemical potential due to the density inhomogeneity.
Furthermore, we obtain the dispersions from gaussian ﬁts to the EDCs, but the intrinsic EDCs
may have a diﬀerent functional form and could be asymmetric. Another complication is that any
excitations due to ﬁnite temperature will tend to pull the gaussian ﬁts of the EDCs to higher
energies. For all these reasons, it is diﬃcult to directly extract a value for gap (or pseudogap) andchemical potential from these ﬁts.
However, the ﬁts qualitatively capture three important features in the dispersions that are
predicted by theory. As a function of increasing 1/kF a, we observe that the wave-vector where
back-bending occurs shifts to lower momentum. This is clearly captured by the parameter kL andis discussed in the previous section. We also observe that the dispersions shift to lower energy withincreasing 1/kF a. This reﬂects a decrease in the chemical potential, which gives a constant energy
oﬀset in the dispersion given by Eq. 6.1. Also, with increasing 1/kF a, the dispersions become ﬂatover a larger range of k near kL , which gives an increase in ∆. The values obtained in the ﬁts areshown as red circles in Fig. 6.18 and both of these features can be observed.
However, the values obtained for µ and ∆ are larger than the expected values for a zerotemperature homogeneous gas, which are shown as stars in Fig. 6.18 and are obtained from Refs.
[20, 19, 21]. This may seem counterintuitive since the density inhomogeneity due to the trap meansthat most of the cloud will have a lower density and presumably smaller values of the gap andchemical potential. This can be understood by considering the eﬀect that atoms at low density will
have on the data. Atoms at low density are weakly interacting and thus contribute spectral signal

112

Gap

2

1

m

0
1
0
-1
0

0.3

0.6

0.9

1/kFa
Figure 6.18: BCS ﬁts Top) Red circles show the values of ∆ obtained from ﬁts to the data in Fig
6.13 using Eq. 6.1. The error bars are due to uncertainty in the ﬁts. Black circles show the valueof ∆ obtained by taking half the distance between Emax and the free-particle line as described in
the text and in Fig. 6.19. For these points I have included an additional systematic error due
to the uncertainty in Emax from error in the calibration of the magnetic ﬁeld. Both the red andblack circles are values of ∆ in units of the Fermi energy for the trapped gas, which is obtained
from the total number and trap frequency (Eq. 2.4). The stars show the value of the gap for ahomogeneous gas at zero temperature, inferred from momentum-integrated rf spectra, in units of
the Fermi energy of the homogenous gas (Eq. 2.2) [19]. The line is a guide to the eye. Bottom) Red
circles show the values of µ obtained from ﬁts using Eq. 6.1. Black circles show the values given by
the midpoint of Emax and the free-particle line. The stars show the values of µ for a homogeneousgas at zero temperature obtained from Refs. [20, 21].

E/EF

113

m

2D

k/kF
Figure 6.19: Extracting the gap and chemical potential In this example, the minimum of2 k2. This occurs when
the upper quasi-particle branch intersects the free-particle dispersion line ~2mµ + ∆ ≈ kL2 . If this case, we can estimate the gap and chemical potential from the maximum energyof the lower quasi-particle dispersion. The gap will be half the distance between this maximum
energy and the free-particle line. The chemical potential will be the energy halfway in between
the maximum of the lower√ dispersion and the free-particle line. In this example, values used areµ = 0.4, ∆ = 0.4 and kL 0.8, which are very close to the values predicted for a zero temperature
gas at 1/kF a = 0 [19].

114
close to the free-particle line. Furthermore, because the local value of kF is also smaller, atoms
at low density contribute signal at low momenta. This has the eﬀect of pulling the trap-averaged
dispersions to higher energy for small k/kF . This shift to higher energy leads to larger values of µand a ﬂattening of the dispersion that gives larger values of the gap ∆. This at least partly explains
why the values obtained from the ﬁts to Eq. 6.1 for µ and ∆ are larger than expected.
An alternative approach for extracting µ and ∆ from the data is to use information about
the position of the upper branch for quasi-particle excitations. The upper and lower branches ofthe dispersion are symmetric about µ and separated by a minimum energy of 2∆ (Fig. 6.19).
In the BCS limit, where the chemical potential is equal to the Fermi energy, the minimum ofkthe upper branch is above the free particle line ( ~2m) by an energy equal to the gap. In the
2 2molecular limit, the upper branch is simply the free-particle line, since unpaired atoms are weaklyinteracting. Interestingly, theoretical predictions and measurements based on rf spectra [20, 96, 19]
put the minimum energy of the upper quasi-particle branch at 1/kF a = 0 within 0.01EF of thefree-particle dispersion. Since both at resonance and in the molecular limit, the minimum of theupper quasi-particle branch touches the free-particle line, it is a reasonable assumption that theminimum of the upper branch is near the free-particle line in the range of interactions strengths
examined in this chapter, 0 < 1/kF a < 1.1. The gap can then be estimated from half the distance
between the measured maximum of the lower branch and the free-particle line, and the chemical
potential can be estimated by taking the energy halfway in between that maximum and the freeparticle line (Fig. 6.19). The values obtained for µ and ∆ using this analysis are shown in Fig.
6.18. While this analysis technique does not resolve the problem of density inhomogeneity, it does
appear to lead to a better estimation for µ and ∆ than the parameters extracted directly from ﬁts
to the data with Eq. 6.1.
The value of the maximum energy of the lower dispersion√is best obtained by reparametrizing
(
)2
2
~2
2
2
Eq. 6.1 using Emax = µ − ∆ such that E(k) = Emax + ∆ −+ ∆ . While ﬁts
2m (k − kL )to the data using Eq. 6.1 suﬀer from a strong correlation between the gap and chemical potential
parameters, the values obtained for Emax are robust and not strongly correlated with the other ﬁt

115
parameters; this may reduce systematic errors.

6.7

Conclusions
At this point, we can summarize a few conclusions that can be made from the photoemission

experiments presented thus far. The data reveal a BCS-like back-bending dispersion in the normal
state over a large range of temperatures (T /Tc = 1.0 − 1.5) and interaction strengths (1/kF a =0.0 − 1.1). At higher temperatures (T /Tc > 1.5), we observe a spectral function consistent withtheories of a Fermi liquid with short-range interactions (Fig. 6.9). As a function of interaction
strength, we observe a crossover from a pseudogap regime where the dispersion back-bends at
the Fermi wave-vector (kL ≈ kF ), indicating the existence of a remnant Fermi surface, to themolecular regime where kL = 0. The crossover occurs in the range of 1/kF a ≈ 0.45 − 0.8. A
comparison to a ﬁnite temperature NSR theory of the BCS-BEC crossover reveals good agreement
and provides further evidence for the existence of pseudogap phase. In addition, the theory gives
an indication that the presence of a trapping potential does not qualitatively alter the features ofthe photoemission spectra. In particular, this serves as evidence that the large energy widths ofthe single-particle excitations that we observe may be intrinsic widths, indicating an absence ofquasi-particles in the strongly interacting gas. Finally, while the NSR theory of the pseudogap isconsistent with linear behavior observed in recent thermodyanmic measurements [13] (Fig. 6.3),
theories based on an extended BCS-Leggett theory, for example Refs. [12, 75], that predict a strong
dip in the density of states and relatively narrow energy widths are probably not consistent with the
ENS data [89]. The excellent agreement of our data with the NSR theory of the pseudogap serves
as further evidence that the spectral function in the pseudogap regime, in the range of interaction
strengths investigated here, is likely dominated by single-particle excitations with large energy
widths, possibly caused by pairs with large center-of-mass momentum [75], and thus a density ofstates with a gap that becomes signiﬁcantly ﬁlled in above Tc .
So far our experimental evidence for the pseudogap state is : BCS-like back-bending of the
trap-averaged occupied spectral function (Figs. 6.6 and 6.13), the non-linear behavior of Emax (Fig.

116
6.11) and the strong agreement with a theory of the pseudogap phase in the BCS-BEC crossover(Fig. 6.14). The inhomogenous density of the atomic gas, which is a consequence of the trapping
potential, prevents us from directly measuring the pseudogap by dividing out the Fermi functionin the photoemission data. In the next chapter, I will present progress towards photoemission
spectroscopy of a homogenous sample. This data provides further evidence for the existence of apseudogap regime and single-particle exctitaions with anomalously large energy widths.

Chapter 7

Towards atom-photoemission spectroscopy of a homogeneous gas

In the previous chapters, I have presented atom-photoemission data for a trapped gas, which
necessarily has a spatially inhomogeneous density. This inhomogeneity can lead to broadening ofspectral features and makes it impossible to divide out the Fermi function from the photoemission
data. Spatial inhomogeneity can also be an issue for ARPES experiments on electronic materials
containing impurities, non-uniform doping, or a non-uniform chemical potential. However, there isone major advantage of the trapped gas compared to an ARPES experiment with a non-uniform
material, which is that the density proﬁle of the gas is smooth and can be predicted and measured.
This means we can still compare to theories, via a local density approximation. We can also predict
some of the general features we expect to arise due to the inhomogeneity, such as an increased
spectral weight near the free-particle dispersion due to weakly interacting atoms near the edge ofthe gas. Also, importantly, every trap-averaged sample we study is identical, as opposed to diﬀerent
material samples.
In the last chapter, I showed that we can obtain a signiﬁcant amount of information about
the spectral function in the BCS-BEC crossover from trap-averaged photoemission data. Here,
we examined qualitative features of the data and also compared to a theory of the crossover.
However, it is clear that photoemission spectra of a homogeneous atomic gas would be very useful.
In particular, one would like to measure the energy widths for the homogeneous gas, and thedispersion of the quasi-particles. In addition, for data on a homogeneous gas, one could divide
out the Fermi function and measure the gap or pseudogap in the density of states. Ultimately,

118
photoemission data obtained from a homogeneous sample would allow for more sensitive tests oftheories of the crossover and advance our goals of developing a better understanding of stronglyinteracting Fermi gases.
In this chapter, I will present progress towards photoemission spectroscopy of a homogeneouscloud. Unfortunately, the obvious solution of creating a uniform trapping potential, or box trap,
instead of a harmonic trap, turns out to be technically very diﬃcult. In particular, it would be
diﬃcult to make a box trap small enough to achieve the same atom densities obtained in theharmonic trap. Instead, we tried a diﬀerent approach, which would allow us to keep the harmonic
trap, but remove photoemission signal arising from atoms at the edges of the cloud. We refer tothis approach as the hollow light beam method (or, around the lab, as the donut beam method).
The data provide further evidence that the energy widths of quasi-particles near the Fermi surface
are a signiﬁcant fraction of the Fermi energy as well as show what the eﬀect of the inhomogeneousdensity is on the quasi-particle dispersions. The work presented in this chapter will be the subjectof forthcoming publications.
It should be noted that other experiments have shown it is possible to obtain momentumintegrated rf-spectra for a homogeneous Fermi gas by imaging the atoms transferred by the rf ﬁeld
in-situ, without a time-of-ﬂight expansion [97]. However, it is not possible to use this technique forphotoemission spectroscopy, which requires momentum information that can only be obtained bytime-of-ﬂight imaging. Rather, we needed a technique that would give both spatial selectivity andmomentum information for the atomic gas.

7.1

Focusing on the center of the cloud
The method of using hollow light beams to investigate eﬀects of density inhomogeneity isbased on the idea that removing atoms at low density leaves behind a sample that is more homogeneous. This can be seen from Fig. 7.1, which shows the atom density as a function of scaled radiusfrom the cloud center (a), and the number of atoms in the cloud as a function of scaled atom density(b) for a zero temperature Fermi gas in a harmonic trap. If we can remove atoms at low density,

119

1.5

1.0

N(n)

n(r)

b

a

0.8

1.0

0.6
0.4

0.5

0.2
0

0

0.2

0.4

0.6

r/RTF

0.8

1.0

0

0

0.2

0.4

0.6

n/n0

0.8

1.0

Figure 7.1: Atom density distribution of a trapped Fermi gas a) The radial density proﬁleof a T = 0, harmonically trapped Fermi gas. r/RT F is the scaled radius, where each component(x,y,z) is divided by the Thomas Fermi radius in that direction. The total number of atoms at agiven radius is 4πn(r)r2 dr. b) The number of atoms in the cloud at a given density n/n0 is plotted,
where n0 is the density at the center of the cloud. There are, of course, 0 atoms at n/n0 = 0 andalso 0 atoms at density n/n0 = 1, because this corresponds to a singular point at the center of thecloud. The average density of atoms is approximately navg /n0 = 0.51 and the median density is
0.54.

120
we will be left with a distribution of atoms that has a higher average density and less variation indensity. This can be done by removing atoms at large radius. In the limit where one removes all
the atoms except just those at r = 0, the remaining gas would be completely homogeneous with adensity equal to the peak density of the original atom cloud. Unfortunately, there would also be
no atoms left. Instead, we use the hollow light beams to remove 30 − 70% of the total atoms, andthereby signiﬁcantly decrease the number of atoms at low density.
The hollow light beams remove atoms by optically pumping them to another spin state thatinteracts only weakly with the atoms in the spin states we are interested in. When we take a pictureof the gas, only the atoms that have not been optically pumped will show up. The hollow lightbeam is dark in the center, and focused onto the cloud, and so will have no eﬀect on atoms at thecenter of the cloud. We require two (cylindrical) intersecting hollow light beams in order to remove
atoms at the edges of the three-dimensional cloud. Laguerre-Gaussian beams, with a proﬁle in the
far-ﬁeld given by−r 2

I(r) = I0 r2l e 2w2(7.1)

where l is the order of the beam and w is the waist, are obtained using absorption masks patternedwith a forked diﬀraction grating with l dislocations, see for example Ref. [98].

1ω√The atom cloud is cylindrically symmetric with a Thomas Fermi radii (given by RT F =2EFm ) of approximately 10 µm and 125 µm in the radial and axial directions, respectively. For

these experiments, we use a hollow light beam with l = 1 and w = 11 µm propagating along theaxial direction to remove atoms in the radial directions and a beam with l = 2 and w = 90 µm
propagating perpendicular to the axial direction to remove atoms in the axial direction. Fig. 7.2
shows the atom removal probability as a function of scaled radius.

7.1.1

Optical pumping

Because we are interested in measuring the momenta of the remaining atoms, it is necessary
for the optical pumping to be highly eﬃcient. If an atom absorbs a photon from a hollow lightbeam and then returns to the initial state, instead of being optically pumped out, it will have anAxial direction

121

Radial direction
Figure 7.2: Atom removal probability The surface plot shows the probability for atoms to remain after optical pumping with the hollow light beams as a function of scaled radius in the radial
direction and in the axial direction. The radii are scaled to the Thomas Fermi radii in those directions. The color map shows the probability to remain, with red corresponding to high probability
to remain and blue corresponding to high probability to be optically pumped and removed fromthe imaging. Integrating over all radii, approximately 65% of the atoms in a degenerate Fermi gas
would be optically pumped.

122
increased momentum due to the photon recoil. To make matters worse, the photon recoil, ~k wherek = 2πλ , is approximately equal to a typical Fermi momentum ~kF in our experiments. In order tominimize this, we choose the hollow light beams to be resonant with transitions to excited statesthat have small branching ratios to decay back into the initial state.
In this work, we have tried two diﬀerent transitions, both resonant with the |9/2, −5/2⟩
ground state that we use for the atom photoemission experiments. One transition uses the |9/2, −7/2⟩
electronically excited state. This state has a 0.02 probability to decay back to the |9/2, −5/2⟩ groundstate, or in other words, only a 1 in 50 chance. The downside for this transition is that 3 MHz
away there is a π transition (∆mF = 0) to the |9/2, −5/2⟩ excited state, which has a 0.6 branchingratio for decay back to the |9/2, −5/2⟩ ground state. The higher branching ratio also means theπ transition is stronger and easier to excite. As a result, a small error in the polarization of thehollow light beam can cause signiﬁcant excitations with the π transition, and this results in poor
optical pumping. The second transition we tried is to the |5/2, −3/2⟩ excited state. This state
has a slightly larger branching ratio back to the original state (0.095), however, there are no othernearby transitions that can be oﬀ-resonantly excited.
With either transition, we still have to worry about a small number of atoms that absorb a
photon from one of the hollow light beams and then fall back into the |9/2, −5/2⟩ ground state withan increased momentum. Increasing the power of the hollow light beam does not help with thisissue because the center of the beam is dark, and so there will always be a region where the intensityof the beam is such that atoms will only absorb a single photon on average (the probability that an
atom will absorb two photons and still remain in the same state is negligible for both transitions).
The size of this region can be made smaller by using beams parameterized by larger values of l.
However, the limited resolution of our optics delivery system prevents us from eﬀectively using abeam with l > 1 for the beam with the 11 µm waist.
The number of atoms that will absorb a photon and fall back into the same state with achanged momentum can be modeled using the radial proﬁles of the atom cloud and light beams.
We ﬁnd that these atoms typically correspond to 5 − 10% of the total number of atoms that remain

123(see Fig. 7.3, or Appendix B for details). These atoms are an undesirable background signal. To
subtract out the background, we model the momentum distribution of these atoms by convoluting
the momentum distribution of the atoms that absorbed light from the hollow light beams witha spherical shell of momentum of momentum ~k, where k is the amplitude of the photon wavevector. We ﬁnd the measured size of the background signal is approximately equal for both possible
transitions, despite the diﬀerence in branching ratios.

7.2

Application to a weakly interacting Fermi gas
As a ﬁrst application of the hollow light beams, we can measure the momentum distributionof atoms at the center of a quantum degenerate, weakly interacting trapped Fermi gas. The
momentum distribution of weakly interacting fermions is described by the Fermi-Dirac distribution(remember all the way back in Chapter 2 Eq. 2.3). The hallmark of the Fermi-Dirac distribution is a
sharp Fermi surface at wave-vector kF . At zero temperature, the distribution is characterized by thefull occupation of momentum states below kF and zero occupation above. At ﬁnite temperature,
the distribution near the Fermi surface becomes smeared over a region of k-states spanning anenergy width on order of kb T . However, in a trapped Fermi gas, the Fermi surface is not apparent
in the momentum distribution. This can be understood from a local density approximation. The
wave-vector marking the Fermi surface of a homogeneous Fermi gas (Eq. 2.1) depends on thefermion density, which in a trapped gas varies as a function of distance from the trap center. A
trapped gas can be thought of as a collection of Fermi gases, each described by a diﬀerent value ofkF . When we measure the momentum distribution of all the atoms in the trap, we average together
Fermi-Dirac distributions parameterized by values of kF ranging from 0 (corresponding to atoms
at the edge of the trap) to the maximum value of kF (corresponding to atoms at the center of thetrap). In fact, the shape of the momentum distribution is identical to the density distribution [33](this is a special feature of harmonically trapped gases) and looks like Fig. 7.1 a) with the x-axis
as atom momentum.
If we could measure the momentum distribution of a select group of atoms in the trap, ones

124

25

N(k)

20
15
10
5
0

0

0.5

1.0

1.5

k/kF
Figure 7.3: Background signal The measured number of atoms, as a function of k/kF , whereN ∝ n(k)k 2 , is shown with (black line) and without (red line) hollow light beams. The experimental
data with the hollow light beams contain a contribution from the background signal (green line),
which is calculated from a model. It is clear a background signal is present in the measured density
distribution because the data with atoms removed by hollow light beams actually has higher signal
for k > 1.15k/kF than the data where no optical pumping occurs. The data here have beensmoothed to better show the diﬀerence between the curves at large k/kF .

125with nearly constant density, then we could actually observe a sharp Fermi surface. The hollow
light beams allow us to accomplish exactly this. In this experiment, we cool an equal mixture of
atoms in the |9/2, −7/2⟩ and |9/2, −9/2⟩ ground states to a temperature of T /TF = 0.10 ± 0.02 at
a magnetic ﬁeld of approximately 203.5 G. Then, we ramp the magnetic-ﬁeld strength to a value
in the range of 205 − 209 G, corresponding to scattering lengths of −285 to −15 Bohr radii, where
the Fermi gas can be considered to be weakly interacting. We then use an rf π-pulse to transfer
all of the atoms in the |9/2, −7/2⟩ state to the |9/2, −5/2⟩ state where they will be resonant withthe hollow light beams. After the rf pulse, the trapping potential is turned oﬀ and the hollow lightbeams are turned on for 40 µs (this is a short enough duration that the expansion and movementof the cloud due to gravity are negligible). The intensities of the hollow light beams are adjusted tocontrol the total number of atoms optically pumped away. The momentum distributions for clouds
where 0% and 60% of the atoms are removed is shown in Fig. 7.4. The momentum distributions
for the cloud with 60% of the atoms removed clearly shows a sharp Fermi surface. To parameterize
the sharpness of the Fermi surface we can ﬁt the clouds to Fermi-Dirac distributions. In these ﬁts,
we let the temperature and value of kF vary, as well as an overall scaling factor. For the cloud with
60% of the atoms removed, the ﬁt gives T /TF = 0.19 ± 0.04, which is much lower than the trap
averaged cloud, which ﬁts to T /TF = 0.6 ± 0.1.
In Fig. 7.5, we show the value of T /TF obtained from a ﬁt to the momentum distribution
versus the fraction of atoms remaining after optical pumping with the hollow light beams. The
fraction is varied by varying the intensity of the beams. The line is an expected value obtained
from a Monte Carlo simulation using the Fermi Dirac distribution and calculated atom removal
probability (Fig. 7.2). In the limit of blasting away all of the atoms, we would expect T /TF = 0.1,
which corresponds to the degeneracy of atoms at the center of the trap.
The fact that the hollow light beams allow us to observe a Fermi surface conﬁrms that thetechnique works as we expect and demonstrates the potential to uncover properties of a homogeneous Fermi gas from a trapped gas. The technique is ideally suited for observing a Fermi surface,in part because of the favorable dependence of kF on the density (kF ∝ n1/3 ). As an example,

126

2.0

n(k)

1.5
1.0
0.5
0

0

0.5

1.0

1.5

k/kF
Figure 7.4: Observing a Fermi surface The fermion momentum distribution, n(k) with (black
circles) and without (red circles) optical pumping from the hollow light beams. The distributions
are separately normalized to have an area of 1. Fits to the data with a Fermi-Dirac distribution
for a homogeneous gas are shown (black and red lines). The ﬁt to the data with optical pumping
gives T /TF = 0.19 ± 0.04 and the ﬁt to the data with no optical pumping gives T /TF = 0.58 ± 0.14.
The optical pumping intensity was set to remove approximately 60% of the atoms. The error bars
shown are statistical error; there may be residual systematic errors due to imperfections in theimaging system.

127

Fitted T/TF

0.8
0.6
0.4
0.2
0

0

0.25 0.50 0.75
Fraction Remaining

1

Figure 7.5: Varying optical pumping intensity The value of T /TF obtained from ﬁts to themomentum distribution of atoms after optical pumping with hollow light beams versus the fractionof atoms remaining after the optical pumping. The black line is a prediction from a model, which
is described in the text. The temperature of the trapped gas before blasting is 0.11 ± 0.02TF . The
power ratio of the two beams was set by measuring the number of atoms optically pumpbed byeach beam on its own. The number of atoms optically pumped by the beam propagating along theaxial direction was approximately twice the number of atoms optically pumped by the other beam.

128
this means that if we eliminate all the atoms from regions with a density less than half the peakdensity, the total variation in kF would only be 20%.

7.3

Studying the eﬀect of the trapping potential on atom-photoemission
spectroscopy
We can now apply the hollow light beam technique to atom photoemission spectroscopy andbegin to answer some questions regarding the eﬀect of the trapping potential on the photoemission
data. In particular, I will address the eﬀect of the trapping potential on the shape of the singleparticle dispersion and the energy widths of the quasi-particles. The ﬁrst issue is important because
BCS-like ﬁts to the trap-averaged dispersions produce signiﬁcantly larger values for the gap andchemical potential than what has been predicted by theory (see Fig. 6.18). While it is obvious thatthe trap conﬁnement can aﬀect the shape of the dispersion, and thus the values of ﬁt parameters, wewould like to investigate this eﬀect experimentally and see if the data is consistent with theoreticalpredictions of the quasi-particle dispersion of a homogeneous gas.
The second issue is also very interesting. In the trap-averaged photoemission data, we measured energy widths near the Fermi surface that were a signiﬁcant fraction of the Fermi energy (seeFig. 6.15). Answering the question of whether or not the widths are due to averaging over thedensity inhomogeneity, or are reﬂective of large intrinsic energy widths is a critical issue. The quasiparticle energy widths are an important component of understanding the nature of the stronglyinteracting gas in the pseudogap region and large widths would indicate a departure from typicalFermi-liquid behavior.
The comparisons to theoretical predictions presented in the previous chapter suggest answers
to both of these questions. In the case of the shape of the dispersions, the theory, which predicts
reasonable values for the gap and chemical potential of a homogeneous Fermi gas [99, 100], agrees
with our trap averaged data after using a local density approximation. This suggests the trap
averaged data are consistent with theoretically predicted dispersions for a homogenous gas. The
theory also predicts that, while the energy widths of trap-averaged data are somewhat broadened

129
by the density inhomogeneity, they are also reﬂective of large intrinsic quasi-particle energy widths.
It is the goal of this chapter to provide further experimental evidence for these claims by using themethod of hollow light beams to obtain photoemission signal from the center of the Fermi gas.
In these experiments, the trap is parameterized by a radial trap frequency of ωr = 2π · 210
Hz and an axial trap frequency of ωz = 2π · 16 Hz. We obtain a 50/50 mixture of atoms in the the|9/2, −9/2⟩ and |9/2, −7/2⟩ spin states. Our ﬁnal stage of evaporation occurs at a magnetic ﬁeldof 203.5 G. At the end of the evaporation, we have 1.2 × 105 atoms per spin state at a normalized
temperature TTF = 0.13 ± 0.02.
After the evaporation, we increase the interaction strength adiabatically with a slow magneticﬁeld ramp to either 202.17G or 202.10G near the center of the Feshbach scattering resonance.
Here, the interaction strengths can be characterized by the parameter kF1 a = 0.05 ± 0.05 and√
0.20±0.05 where kF is the Fermi wavevector given by 2mEF /~. Using our previous measurementsof the the size of an interacting Fermi gas [30], we estimate the peak and average density of theinteracting cloud for the interaction strengths of (0.05, 0.20) to be given by (1.44, 1.51) ± 0.1n0 and(0.70, 0.73) ± 0.05n0 where n0 is the density of a homogenous gas with a Fermi energy of EF andis given by Eq. 2.2.
For the photoemission experiments, the strongly interacting atomic gas is probed with an rf
pulse to transfer a small fraction of the atoms in the |9/2, −7/2⟩ state to the |9/2, −5/2⟩ state.
Immediately after the transfer, the trapping potential is switched oﬀ, so the atoms in the |9/2, −5/2⟩
state can ballistically expand and be imaged to obtain their momentum distribution. To perform
photoemission spectroscopy on the core of the gas, we use the technique of crossed hollow lightbeams to optically pump away |9/2, −5/2⟩ atoms in the low density regions at the edges of theatom gas. The hollow light beams are pulsed on for 40 µs immediately after the trap is switched oﬀ(and thus, immediately after the rf pulse). The intensity of the beams is such that the integrated
photoemission signal is reduced by approximately 60%. As in the case of a weakly interacting gas(Fig. 7.3), the optical pumping from the hollow light beams leaves a small number of atoms in
the |9/2, −5/2⟩ with increased momentum, creating a small high-momentum background in the

130photoemission signal. We model this background and subtract it out as described in Fig. 7.3 andthe Appendix B. The rf pulse amplitude envelope is gaussian, with a full width, 1/e4 amplitudeduration ranging from 300 µs to 100 µs. A duration of 300 µs is used for rf frequencies near thefree-particle dispersion, and shorter durations are used for larger detunings of the rf frequency,
because the atoms outcoupled at those frequencies have large momenta and the hollow light beams
must be pulsed on before the positions of the atoms can signiﬁcantly change.
The results of photoemission experiments at two magnetic-ﬁeld strengths with (b,d) andwithout (a,c) the hollow light beam are shown in Fig. 7.6. The white circles in Fig. 7.6 are thecenters obtained from Gaussian ﬁts to vertical cuts at constant k in the spectra and map out thedispersion for the occupied single-particle states. From Fig. 7.6, it is clear that removing atoms
from the edge of the trap does not qualitatively change the spectra. For both kF1 a = 0.05 and 0.20,
we observe a BCS-like dispersion with back-bending near the Fermi wavevector kF for the trap
center data.
To quantitatively compare the spectra taken with and without the hollow light beam, weshow in Fig. 7.7 the results of Gaussian ﬁts to the EDCs for both the trap-averaged spectra(black circles), and the trap center spectra obtained with the hollow light beams (red squares). In
the top panel, we plot the centers obtained from the ﬁts. For kkF < 0.75 the trap center data isclearly shifted to lower energy compared to the trap averaged data. This is expected because atoms
near the edge of the trap are at low density where the local 1/kF a is large and the gas is weaklyinteracting. Therefore, these atoms contribute signal near the free particle line at low kkF . In fact,in comparing the two sets of spectra in Fig. 7.6, it is clear that the main eﬀect of the hollow lightbeams is to decrease intensity near the free-particle line at k < kF . This can also be seen in the
momentum distributions, which are plotted in Fig. 7.8. Here, the trap center data has lower signal
at low momenta.
To compare the trap center data obtained with the hollow light beams to theory for the limitof a homogeneous cloud, we show in Fig. 7.7 a dashed line corresponding to the expected dispersion
for atoms at the center of the trap, Eq. 6.1. We use theoretical values for a homogeneous T = 0

131

Trap-Averaged2aTrap Center

b

10

0

10

-1

10

-2

10

-3

0
-2
-4
1/kFa = 0.05
2

c

d

0
-2
-4
1/kFa = 0.20
0

1
k/kF

2 0

1
k/kF

2

Figure 7.6: Photoemission data with and without hollow light beams (a) Trap averaged
photoemission data for a gas at kF1 a = 0.05 at TTF = 0.13. (b) Trap center photoemission data forthe same gas obtained by removing 60% of the atoms using the intersecting hollow light beams. c)
Trap-averaged data at kF1 a = 0.20. d) Trap center data at kF1 a = 0.20. White dots show the centersof gaussian ﬁts to the EDCs of each data set. While the spectra are qualitatively similar, and bothshow BCS-like backbending of the dispersion, the intensity at low momenta near the free-particle
dispersion (black line) is decreased in both of the trap center data sets. This shifts the centers of
the EDCs to lower energy.

132

E/EF

0
-1
-2

1/kFa = 0.05

Width (EF)-3

1/kFa = 0.20

1.5
1.0
0.5
0

0

0.5

1.0

k/kF

1.5

2.0

0

0.5

1.0

1.5

2.0

k/kF

Figure 7.7: EDC ﬁt results We show the centers (top) and rms widths (bottom) from gaussianﬁts to the EDCs of the spectra shown in Fig. 7.6. The left side is data for a gas at kF1 a = 0.05
and the right side is for a gas at kF1 a = 0.20. Trap averaged data is shown as black circles and trap
center data is shown as red squares. The dashed lines in the top panels are the expected T = 0
dispersion for atoms at the center of the cloud obtained using our estimate of the peak density andthe values reported in Refs. [19, 20]. In the lower panels, the horizontal dashed lines are placedat the experimental energy resolution due to the ﬁnite duration of the rf pulse. The resolution isapproximately 0.25EF .

133

2.0

1/kFa = 0.05

1/kFa = 0.20

n(k)

1.5
1.0
0.5
0.0

0

0.5

1.0

0

0.5

1.0

1.5

k/kF
Figure 7.8: Momentum distributions Single-particle momentum distributions with (red squares)
and without (black circles) hollow light beams for the data at kF1 a = 0.05 (left) and the data at
1
kF a = 0.20 (right) at T /TF = 0.13±0.02. The eﬀect of the hollow light beams is to remove fermions
with low momenta. The distributions are normalized to have an area of 1. The distributions are
obtained by integrating over the energy axis of the photoemission data of Fig. 7.6.

134
gas for µ of 0.38EF and 0.22EF for 1/kF a = 0.05 and 0.20 respectively, from Ref. [20]. For thevalues of ∆, we use 0.47EF and 0.58EF and for kL we use 0.91 and 0.85 for 1/kF a = 0.05 and 0.20
respectively, measured for a low temperature gas in Ref. [19]. When values for the exact couplingstrength were not available, we used a linear extrapolation between the closest points. These values
where then scaled using our estimate for the peak density in the trapped gas given above. For low
momenta (k < 0.75kF ), the dispersions obtained from the data with the hollow light beams can beseen to be approaching this limit. For larger momenta (k > 0.75kF ), the data with and without
hollow light beams appear to be similar. This indicates that atoms near the edge of the cloud
may not contribute much signal at large momenta. In order to see the data approach the expected
dispersion for the trap center at momenta near k/kF = 1, it is likely that a more homogenous
sample is necessary. It should be noted that, because the data is at ﬁnite temperature, even if
we could only measure atoms at the exact trap center, the dispersion would not fully approach
the theoretical dispersion, which is obtained in the zero temperature limit. Rather, we expect the
maximum energy of the dispersion curve to move to higher energies at ﬁnite temperature, due tothe closing of the gap and presence of signal from excitations.
Fitting the modiﬁed BCS dispersion to the trap averaged and trap center data we obtainvalues of (µ,∆,kL ) in terms of EF and kF shown in table 7.1.
Table 7.1: BCS Fit Results
Parameterµ∆kL

1
kF a = 0.05

Trap average
0.85 ± 0.14
0.59 ± 0.15
1.01 ± 0.03

Trap center
0.54 ± 0.12
0.16 ± 0.16
1.03 ± 0.03

1
kF a = 0.20

Trap average
0.93 ± 0.18
0.98 ± 0.19
0.87 ± 0.04

Trap center
0.47 ± 0.12
0.54 ± 0.11
1.00 ± 0.04

The trap averaged values are on the left and the trap center values are on the right. From
this data, we can conclude that when ﬁtting the dispersion obtained from trap-averaged data one
obtains larger values of µ and ∆ than for trap center data. This is because signal from atoms at
low density near the trap edge shift the centers of the EDCs at low momenta to higher energy.

135
As expected, the size of the gap is larger for the data at 1/kF a = 0.20 compared to the data
at 1/kF a = 0.05 and the value of kL is smaller (the expected trends are shown in Fig. 6.18). One
would also expect the chemical potential to be slightly smaller at 1/kF a = 0.20, however, the error
in µ is too large to determine the trend. Interestingly, the value of the maximum energy of thedispersion appears robust against removing atoms from the edge of the cloud. This may explain
why our estimates of the gap and chemical potential based on this value for the trap-averaged
data in Chapter 6 appear more reasonable than the values obtained directly from the BCS ﬁt (Fig.
6.18). However, it is not clear if this value would change if the gas were made more homogeneous.
Furthermore, it should be mentioned that the EDC centers in the region kF < k < 1.4kF for the
trap-center data can be aﬀected by the background subtraction (discussed in Fig. 7.3 and Appendix
B).
The lower panels in Fig. 7.7 shows the widths of the gaussian ﬁts to the EDCs of the trapaveraged and trap-center data. Surprisingly, there is little change in the EDC widths, indicatingthat the widths may come from large intrinsic energy widths. This is particularly clear in thedata taken at kF1 a = 0.20. This suggests that the gas is far from the limit of a Fermi-liquid where
the EDC energy widths should become small near the Fermi surface. In fact, it is not surprisingthat sharp quasi-particles are absent if one looks at the momentum distribution Fig. 7.8. Heren(k) is obtained by summing over the energy axis in the data of Fig. 7.6 and dividing out the( kkF )2 phase-space factor. Unlike the data for a weakly interacting gas shown in Fig. 7.4, the trap
center momentum distribution for the strongly interacting gas does not show a sharp Fermi surface.
However, a sharp Fermi surface is required for application of Fermi liquid theory [68].
In summary, using two intersecting hollow light beams, we can measure the momentum
distribution of atoms at the center of a trapped Fermi gas by optically pumping atoms at large
radius to another spin state. Using the hollow light beams, we can observe a sharp Fermi surface in a
weakly interacting Fermi gas. We can also use the hollow light beams to obtain atom-photoemission
spectra for strongly interacting atoms at the center of the gas. These spectra allow us to study theeﬀect of the trapping potential on atom-photoemission spectroscopy. We ﬁnd that the trap center

136
data is qualitatively similar to the trap-averaged data, but with less signal at low k/kF near thefree-particle dispersion. The widths of the EDCs near the Fermi surface remain large, and no sharp
Fermi surface is observed. These observations indicate the gas is far from the regime of a normal
Fermi liquid. One consequence of the large energy widths is that it may be diﬃcult to understandtransport phenomena in terms of standard Fermi-liquid theory, which assumes quasi-particles, andstrongly interacting Fermi gases are a novel system for research of dynamics in strongly interacting
systems [101].

Chapter 8

Universal relations and the contact

8.1

The contact
A typical treatment of a weakly interacting Fermi gas with short-range interactions involves

perturbation theory in the small parameter 1/kF a. However, it is obvious that the short-range
potential will have certain non-perturbative eﬀects on the wave-functions of particles at shortdistances. For example, a strongly repulsive short-range potential of range r0 will preclude the wavefunctions of any two particles from overlapping at distances smaller than r0 . This implies a strong
correlation between the particles that is clearly not captured at a mean-ﬁeld level approximation,
which nevertheless generally predicts the correct behavior of the system in other respects. For
example, in the Hartree-Fock approximation for a homogeneous Fermi gas, one takes the manybody wave-function to be a Slater determinant of plane-wave states [56], yet the approximation
scheme is accurate for calculating the total energy of the system in the limit of small 1/kF a. It
is clear that this short-range behavior is not important for most properties of the system (except
in that it controls the value of the scattering length) as long as the physical interaction occurs at
length scales much smaller than other length scales in the problem.
However, a new set of relations, referred to as the Tan relations, demonstrate that the shortrange behavior of the wave-functions can actually be linked to several macroscopic properties of thegas, such as the total energy, in a universal way, valid for all scattering lengths and phasesof the gas [102, 103, 104]. This amazing revelation is at ﬁrst quite diﬃcult to believe.
For particles interacting via a contact interaction, the wave-function describing two particles

138
at small inter-particle separation, r, must behave aslim Φ(r) = (1/r − 1/a)A

r→0

(8.1)

where r is the internuclear distance, a is the scattering length [104, 36], and A is a constant that
depends on the problem. A quantity called the contact C, can be deﬁned by⃗ = lim (4/s4 )Npair (R,
⃗ s)
C(R)
s→0

(8.2)

⃗ is the contact density at position R⃗ and Npair (R,⃗ s) is the probability of ﬁnding two
where C(R)⃗ The contact, C isparticles of opposite spin inside a sphere of radius s centered at position R.
∫⃗⃗ s) can beobtained from the contact density by C = d3 RC(R).
The s4 behavior of Npair (R,obtained from Eq. 8.1 by noting that the probability to ﬁnd the ﬁrst atom simply scales as the∫svolume, 43 πs3 , and the probability of ﬁnding the second atom can be obtained from 4π 0 Φ(r)2 r2 dr,
which is proportional to s in the limit s ≪ a. Thus, Npair (R, s) ∝ s4 . If the positions of the particles⃗ s) would go as s6 , and then we would ﬁnd C(R)⃗ = 0. If the short-range
were not correlated, Npair (R,potential has an eﬀective range r0 , the relation holds for s smaller than other length scales in thesystem but larger than r0 .
In a gas with only short-range interactions, the interaction energy is determined by theprobability for particles to be close to each other and this is parameterized by C. If one tunes thescattering length, the change in the energy of the system that this produces should be proportional
to the number of fermions that are close enough together to be interacting. A formal proof, ﬁrst
given by Shina Tan, showed that this relation, called the adiabatic sweep theorem can be written
2π

dE
=C
d(−1/(kF a))(8.3)

where E is the total energy in units of the EF and C is the contact in units of kF [102].
This amazing relation links the short distance properties of the particle wave-functions to thetotal energy of the system. Tan’s derivation does not make any assumptions about the phase of thegas (superﬂuid, normal, ect.), temperature, size, interaction strength, or even quantum statisticsof the particles and so is universally valid for all Fermi and Bose gases that can be characterized

139
by short-range 2-body interactions [102, 103, 104, 105]. Although the contact may seem like aquantity purely governed by two-body physics, it is linked to the many-body properties of thesystem through the overall normalization of the wave-function that determines the constant A in
Eq. 8.1, and actually encapsulates all of the many-body physics of the system, see for example Ref.
[106].
While we do not have a good experimental method for measuring C based on Eq. 8.2,
an alternative deﬁnition involving the momentum distribution is easily derived. The momentum
distribution n(k) can be obtained by squaring the Fourier transform of Eq. 8.1, and the 1/r term
leads to a 1/k 4 tail for large momenta. The amplitude of this 1/k 4 is equal to C, with
C = lim k 4 n(k)

(8.4)

k→∞

where the limit is understood to mean in the limit where k is larger than other momenta in thesystem, but less than 1/r0 .
The momentum distribution is a natural quantity to experimentally measure in atomic gaseswith time- of-ﬂight expansion. It can also be directly obtained from the occupied spectral function
measured by photoemission experiments. Indeed, a high momentum feature in the spectral function
is predicted to be a universal feature in the spectral function of Fermi gases (see Ref. [17] and Fig.
6.9).
Besides Eq. 8.3, a number of other relations (the set of relations involving the contact are
termed the Tan relations) have been derived by Tan and others [103, 105, 22, 107, 101]. These
relations link the contact to a large number of experimentally measurable quantities that are listed in
the recent review of the subject in Ref. [108] and include: the pressure, rf spectra, photoassociation
rates, closed channel molecule fraction, static structure factor, and viscosity. In particular, ageneralized viral theorem for a harmonically trapped gas has also been derived [103], and I will
present an experimental veriﬁcation of the relation in the next section. The generalized virial
theorem is given by
E − 2V = T + I − V = −C
4πkF a(8.5)

140
where E is the total energy and can be broken up into three terms T, I, and V , [103] where T is
the kinetic energy, I is the interaction energy, and V is the potential energy due to a harmonic
trapping potential.
Before moving on to the experimental measurement of the contact and veriﬁcation of theTan relations, I would like to discus one more important point. The reader may have noted that a
momentum distribution that falls oﬀ as 1/k 4 produces a divergent kinetic energy. While, the issueof an obviously unphysical inﬁnite energy is resolved by the ﬁnite range cutoﬀ, this still leaves thequestion of what happens to the extremely large but ﬁnite kinetic energy of the gas (large comparedto say the Fermi energy). The answer to this question, resolved by Tan, is that there is an equallylarge (or divergent, in the case of r0 = 0) but negative interaction energy that comes from the 1/r
behavior of the wave-function at short distance [104]. Thus, while the individual terms in the energy,
T and I, diverge for a zero-range potential, the sum T + I is ﬁnite. This again demonstrates how
a basic perturbation theory can miss the short-range and large-momentum physics of the contact
interaction but nevertheless get the total energy correct.

8.2

Measuring the contact
A couple of recent works have measured the contact for a Fermi gas and compared results totheoretical predictions, which are obtained by inserting calculations of the energy from Monte Carlosimulations into Eq. 8.3. Speciﬁcally, photoassociation measurements to determine the closedchannel fraction [109, 22], and inelastic Bragg scattering [107] were used to extract the contact
and compare to theory. The recent measurements of the pressure and energy of a Fermi gas in the
BCS-BEC crossover can also be used to extract the contact [13, 110, 89]. The work I present inthis chapter was published in Ref. [10] and was undertaken after the photoassociation experiments,
and concurrently with the Bragg and equation-of-state measurements. Our work is unique inthat we measured the contact using several techniques, including rf spectroscopy, the momentum
distribution, the release energy, and the potential energy and applied these measurements to directly
verify the adiabatic theorem and the generalized virial theorem in a model-independent way. In

141particular, to my knowledge, no other work has extracted the contact from rf spectroscopy ormeasurements of the momentum distribution and no other work has directly tested the generalized
virial theorem.
Our measurements are done in an ultra cold gas of fermionic 40 K atoms conﬁned in a harmonic
trapping potential with a 50/50 mixture of atoms in two spin states the |9/2, −9/2⟩ and |9/2, −7/2⟩
spin states. The trap is parameterized by a radial trap frequency, which varies for these data fromωr = 2π · 230 to 2π · 260 Hz and an axial trap frequency, which varies from ωz = 2π · 17 to 2π · 21 Hz.
Our ﬁnal stage of evaporation occurs at a magnetic ﬁeld of 203.5 G, where the s-wave scattering
length, a, that characterizes the interactions between atoms in the |9/2, −9/2⟩ and |9/2, −7/2⟩
states is approximately 800 a0 , where a0 is the Bohr radius. At the end of the evaporation, we
have 105 atoms per spin state at a normalized temperature TTF = 0.11 ± 0.02, where the Fermi
temperature TF corresponds to the Fermi energy, EF = kb TF = ~ω(6N )1/3 . Here, N is the atom
number in one spin state, ω is the geometric mean trap frequency given by ω = (ωr2 ωz )1/3 , and kb is
the Boltzmann constant. After the evaporation, we increase the interaction strength adiabaticallywith a slow magnetic-ﬁeld sweep to a Feshbach scattering resonance.
To measure the contact, we obtain n(k) using ballistic expansion of the trapped gas, where weturn oﬀ the interactions for the expansion. We accomplish this by rapidly sweeping the magnetic
ﬁeld to 209.2 G where a vanishes, and then immediately turning oﬀ the external trapping potential[67] and taking an absorption image of the cloud after 6 ms of expansion. The probe light for theimaging propagates along the axial direction of the trap and thus we measure the radial momentum
distribution. Assuming the momentum distribution is spherically symmetric, we obtain n(k) withan inverse Abel transform. We normalize n(k) for a 50/50 spin mixture such that∫n(k) 3d k = 0.5.
(2π)3

(8.6)

Fig. 8.1a shows an example n(k) measured for a strongly interacting gas with a dimensionlessinteraction strength (kF a)−1 of −0.08 ± 0.04. The measured n(k) exhibits a 1/k 4 tail at large k andwe extract the contact C from the average value of k 4 n(k) for k > kC , where we use kC = 1.85 for

142

0
0

4

0.5 1.0 1.5

Rate (G/ms)

2
00

0.5 1.0 1.5 2.0 2.5

k23/2p2n3/2 G(n)2k4 n(k)

6

6

a

C

8

b

4
2
0
0

2

4

6

8

10 12

n

Figure 8.1: Extracting the contact from the momentum distribution and rf lineshape(a) Measured momentum distribution for a Fermi gas at kF1 a = −0.08 ± 0.04. Here, the wave
number k is given in units of kF , and we plot the normalized n(k) multiplied by k 4 . The dashedline corresponds to 2.2, which is obtained by averaging k 4 n(k) for k > 1.85. (Inset) The measured
value for C depends on the rate of the magnetic-ﬁeld sweep that turns oﬀ the interactions before
time-of-ﬂight expansion. (b) RF lineshape measured for a Fermi gas at kF1 a = −0.03 ± 0.04. Here,ν is the rf detuning from the single-particle Zeeman resonance, given in units of EF /h. We plot thenormalized rf lineshape multiplied by 23/2 π 2 ν 3/2 , which is predicted to asymptote to C for large ν.
Here, the dashed line corresponds to 2.1, from an average of the data for ν > 5.

143
(kF a)−1 > −0.5 and kC = 1.55 for (kF a)−1 < −0.5. These values for kC are chosen empirically suchthat for k ≥ kC the momentum distributions are in the asymptotic limit to within our statisticalmeasurement uncertainties. One issue for this measurement is whether or not the interactions are
switched oﬀ suﬃciently quickly to accurately measure the high-k part of n(k). The data in Fig. 8.1aG
were taken using a magnetic-ﬁeld sweep rate of 1.2 µs
to turn oﬀ the interactions for the expansion.

In the inset to Fig. 8.1a, we show the dependence of the measured C on the magnetic-ﬁeld sweep
rate. Using an empirical exponential ﬁt (line in Fig. 8.1a inset), we estimate that for our typicalGsweep rates of 1.2 to 1.4 µs, C is systematically low by about 10%. We have therefore scaled C

measured with this method by 1.1.
The contact is also manifest in rf spectroscopy, where one applies a pulsed rf ﬁeld andcounts the number of atoms that are transferred from one of the two original spin states intoa third, previously unoccupied, spin state [39]. We transfer atoms from the |9/2, −7/2⟩ state to
the |9/2, −5/2⟩ state. It is predicted that the number of atoms transferred as a function of the rf
frequency, ν, scales as ν −3/2 for large ν, and that the amplitude of this high frequency tail is givenby [111, 112, 113]lim Γ(ν) =ν→∞C
23/2 π 2ν −3/2 .

(8.7)

Here, ν = 0 is the single-particle spin-ﬂip resonance, and ν is given in units of EF /h. This prediction
requires that atoms transferred to the third spin-state have only weak interactions with the otheratoms, so that “ﬁnal-state eﬀects” are small [113, 74, 114, 115, 116, 117, 118, 119], as is the casefor 40 K atoms. In Fig. 8.1b, we plot a measured rf spectrum multiplied by 23/2 π 2 ν 3/2 . The rf
spectrum, Γ(ν), is normalized so that the integral over the rf lineshape equals 0.5. We observe
the predicted 1/ν 3/2 behavior for large ν, and obtain the contact by averaging 23/2 π 2 ν 3/2 Γ(ν) forν > νC , where we use νC = 5 for (kF a)−1 > −0.5 and νC = 3 for (kF a)−1 < −0.5. These valuesfor νC are chosen empirically such that for ν ≥ νC the rf spectrum is in its asymptotic limit.
The connection between the rf spectrum and the high-k tail of the momentum distribution
can be seen in the Fermi spectral function, which can be probed using photoemission spectroscopy

144
for ultra cold atoms [9]. Recent photoemission spectroscopy results on a strongly interacting Fermi
gas [18] revealed a weak, negatively dispersing feature at high k that persists to temperatures well
above TF . This feature was attributed to the eﬀect of interactions, or the contact, consistent with arecent prediction [17]. Atom photoemission spectroscopy, which is based upon momentum-resolved
rf spectroscopy, also provides a method for measuring n(k). By integrating over the energy axis,
or equivalently, summing data taken for diﬀerent rf frequencies, we obtain n(k). This alternative
method for measuring n(k) yields results similar to the ballistic expansion technique, but avoids
the issue of magnetic-ﬁeld sweep rates.
In Fig. 8.2 we show the measured contact for diﬀerent values of 1/kF a. The contact isshown in units of the Fermi momentum kF . Our data do not extend to larger values of 1/kF awhere we ﬁnd that our magnetic-ﬁeld sweeps are no longer fully adiabatic [44]. Fig. 8.2 shows
the contact extracted using the three diﬀerent techniques described above to probe two distinct
microscopic quantities, namely n(k) and Γ(ν). We ﬁnd that the amplitude of the 1/k 4 tail of n(k)and the coeﬃcient of the 1/ν 3/2 tail of Γ(ν) yield consistent values for C. The error bars shownin Fig. 8.2 include both statistical and estimated systematic uncertainties, which are roughly
equal in magnitude. In extracting the contact from the rf measurements, the largest source ofsystematic error comes from residual interactions with atoms in third spin state [113]. For theballistic expansion measurements, the systematic uncertainty is dominated by the eﬀect of the
ﬁnite magnetic-ﬁeld sweep rate. For comparison with the data, the solid line in Fig. 8.2 shows aprediction for the contact that was reported in Fig. 1 of Ref. [22]. This zero temperature prediction
consists of the BCS limit, interpolation of Monte Carlo data near unitarity, and the BEC limit fora trapped gas and uses a local density approximation.

8.3

Testing the Tan relations
In the preceding section, I presented three techniques for measuring the contact that probe

the small-length scale physics of the gas through measurements of the momentum distribution andrf lineshape. Now, we would like to test whether those measurements can indeed be used to make

145

5

C

4
3

MomentumRF lineshape
PES

2
1
0
-3

-2

-1
-1

0

(kFa)

Figure 8.2: The contact We measure the contact, C, as a function of (kF a)−1 using three diﬀerent
methods. Filled circles correspond to direct measurements of the fermion momentum distributionn(k) using a ballistic expansion, in which a fast magnetic-ﬁeld sweep projects the many-body state
onto a non-interacting state. Open circles correspond to n(k) obtained using atom photoemission
spectroscopy measurements. Stars correspond to the contact obtained from rf spectroscopy. The
values obtained with these diﬀerent methods show good agreement. The contact is nearly zerofor a weakly interacting Fermi gas with attractive interactions (left hand side of plot) and thenincreases as the interaction strength increases to the unitarity regime where (kF a)−1 = 0. The line
is a theory curve obtained from Ref. [22].

146
predictions about macroscopic properties of the gas, such as the total energy. As mentioned above,
the total energy can be split into two well-behaved parts, the potential energy to the harmonic
trap, V , and the release energy T + I, which is the sum of the kinetic and interaction energy. We
use measurements of both of these quantities to test the adiabatic sweep theorem, Eq. 8.3, and thegeneralized virial theorem, Eq. 8.5.
We measure V by imaging the spatial distribution of the atom cloud [30]. We allow thecloud to expand for 1.6 ms to lower the optical density and then image along one of the radial
directions in order to see the density distribution in the axial direction. Because the expansion
time is 40 times shorter than the axial trap period, the density distribution in the axial direction
reﬂects the in-trap density distribution. The potential energy per particle, in units of EF , is thenV = E3F 12 mωz2 ⟨z 2 ⟩, where ⟨z 2 ⟩ is the mean squared width of the cloud in the axial direction and wehave assumed that the potential energy is distributed equally in x, y, and z.
To measure T + I we turn oﬀ the trap suddenly and let the cloud expand for t = 16 ms (withinteractions) before imaging along one of the radial directions; this is similar to measurements
reported in Ref. [120]. The total release energy is the sum of the release energy in the tworadial directions and the release energy in the axial direction. For the radial direction, the releaseenergy per particle, in units of EF , is simply Tr + Ir = E2F 12 m ⟨yt2 ⟩ where t is the expansion time2and ⟨y 2 ⟩ is the mean squared width of the expanded cloud in the radial direction. For the axial
direction, the expansion is slower and the expanded cloud may not be much larger than the in-trap
density distribution. This is especially true near the Feshbach resonance where the cloud expands
hydrodynamically [121]. Accounting for this, the axial release energy is Tz + Iz = E1F 12 m

⟨z 2 ⟩−z02,
t2

where z02 is mean squared axial width of the in-trap density distribution.
We extract the mean squared cloud widths from surface ﬁts to the images, where we ﬁt
to a ﬁnite temperature Fermi-Dirac distribution. Rather than being theoretically motivated, we
simply ﬁnd empirically that this functional form ﬁts well to our images. In order to get the best
estimate of the energy in the clouds, we perform a weighted ﬁt where each point in the image
is weighted by the square of the distance from the center of the cloud. To eliminate systematic

147
error due to uncertainty in the trap frequencies and imaging magniﬁcation, we measure the releaseenergy and potential energy of a very weakly interacting Fermi gas at TTF = 0.11, where we expect
T + I = V = 0.40EF . We then use the ratio of 0.40EF to our measured values as a multiplicative
correction factor that we apply to the data. This correction is within 5% of unity. For the pointwith kF1 a > 0 we add the binding energy of the molecules, −1/(kF a)2 , to the release energy T + I.
We show our data for the V and T + I versus (kF a)−1 in the inset of Fig. 8.3. The error bars
shown in Fig. 8.3 include both statistical and systematic sources of uncertainty, which are roughly
equal in magnitude.
We can now test the predicted universal relations connecting the 1/k 4 tail of the momentum
distribution with the thermodynamics of the trapped Fermi gas. We ﬁrst consider the adiabatic
sweep theorem. The inset to Fig. 8.3 shows E obtained by summing the measured values fordE, simply by
T + I and V . To test the adiabatic sweep theorem, we ﬁnd the derivative, d(−1/(kF a))calculating the slope for pairs of neighboring points in the inset of Fig. 8.3. In the main partof Fig. 8.4, we compare this point-by-point derivative, multiplied by 2π, to C obtained from theweighted averages of the the data shown in Fig. 8.2(◦). Comparing these measurements of the left
and right sides of Eqn. 8.3, we ﬁnd good agreement and thus verify the adiabatic sweep theorem
for our strongly interacting Fermi gas.
To test Eq. 8.5, in Fig. 8.4 we plot the measured diﬀerence T + I − V versus (kF a)−1along with 4πkCF a , where we use our direct measurements of C. We ﬁnd that these independentmeasurements of the left and the right sides of Eqn. 8.5 agree to within the error bars, which
include both statistical and systematic sources of uncertainty. It is interesting to note that themeasured energy diﬀerence T + I − V is small (in units of EF ), so that even a Fermi gas with astrongly attractive contact interaction nearly obeys the non-interacting virial equation.
In conclusion, we have measured the integrated contact for a strongly interacting Fermi gas
and demonstrated the connection between the 1/k 4 tail of the momentum distribution and the highfrequency tail of rf spectra. Combining a measurement of C vs (kF a)−1 with measurements of thepotential energy and the release energy of the trapped gas, we verify two universal relationships

148

0.40

E

-2pdE/d(1/kFa)

4

0.35
0.30

T+I
V

0.25

2

-3

-2

-1

0

0
-2

-1 -1

0

(kFa)

Figure 8.3: Testing the adiabatic sweep theorem (Inset) The measured potential energy, V ,and release energy, T + I, per particle in units of EF are shown as a function of 1/kF a. (Main)dE
Taking a discrete derivative of the data shown in the inset, we ﬁnd that 2π d(−1/(k(•) agrees
F a))well with the average value of the contact C obtained from the measurements shown in Fig. 8.2
(◦).

149

0.06
T+I-V

0.03
0
-0.03
-0.06
-2

-1 -1

(kFa)

0

Figure 8.4: Testing the generalized virial theorem The diﬀerence between the measured releaseenergy and potential energy per particle T + I − V is shown as ﬁlled circles. This corresponds tothe left hand side of Eq. 8.5. Open circles show the right hand side of Eq. 8.5 obtained fromthe average values of the contact shown in Fig. 8.2. The two quantities are equal to within themeasurement uncertainty.

150[103, 104], namely the adiabatic sweep theorem and the generalized virial theorem. These universal
relations represent a signiﬁcant advance in the understanding of many-body quantum systems withstrong short-range interactions. Furthermore, these relations could be exploited to develop novel
experimental probes of the many-body physics of strongly interacting quantum gases.

Chapter 9

Conclusion and Outlook

9.1

Summary
In this thesis, I presented ultracold atomic Fermi gases as an ideal system for studying theeﬀects of strong interactions in quantum systems. Feshbach resonances are the key tool that allow
us to vary the interaction strength between atoms in the gas and enter into the regime of the
BCS-BEC crossover. In the crossover, strong interactions cause the fermions to form pairs with atypical pair size on order of the interparticle spacing. This pair size puts the system well beyond
the reach of the standard BCS theory of pairing but not yet to the limit where the pairs can beunderstood as two-body molecules. Instead, the gas is a strongly correlated manybody system.
I introduced an atom photoemission spectroscopy technique, which is based on momentumresolved rf spectroscopy, to probe the strongly interacting Fermi gas in the BCS-BEC crossover.
This technique allows one to directly measure the energy and momentum of the occupied singleparticle states in the strongly interacting system. This information can be used to construct theoccupied spectral function and compare directly to many-body theories.
Using this method, we obtained atom photoemission data as a function of temperature andinteraction strength in the crossover. This data shows many of the unusual features of the gas,including a pairing gap on order of the Fermi energy, large energy widths of the single-particle states,
and the disappearance of the Fermi surface on the BEC side of the crossover. Perhaps the mostunusual behavior of the gas revealed by the data is the persistence of a BCS-like dispersion above
the superﬂuid transition temperature. This is strong evidence for the existence of a pseudogap

152
phase, consisting of incoherent pairs of fermions.
In order to understand the eﬀect of the conﬁnement trap on the photoemission data, I introduced a measurement technique to obtain spatially resolved momentum information in a trapped
atomic gas. We demonstrated that this technique allows for the direct observation of the Fermi
surface in a weakly interacting Fermi gas. We then applied this technique to study the eﬀects of
the harmonic trapping potential for the Fermi gas on the photoemission data.
Finally, I presented measurements of the amplitude of the large-momentum tail of the fermion
momentum distribution, termed the contact. I showed how this quantity could be linked to anumber of other measurable quantities in the system, such as the rf spectrum, total energy, andvirial theorem. The relations involving the contact are known as the Tan relations and are valid
for all interaction strengths and phases of the system (normal, superﬂuid, ect.). As such, they
represent an important new development in the study of strongly interacting systems with shortrange interactions.

9.2

OutlookIn the short-to-medium term, I believe there are some interesting research opportunities

related to the issues presented in the last two chapters of this thesis. As far as photoemission
spectroscopy in the BCS-BEC crossover goes, I think obtaining a spectrum for a homogeneousgas is a worthy goal. In Chapter 7, we were able to see some of the eﬀects of the trapping
potential on the data. However, the hollow light beam technique by itself is inadequate to obtaina spectrum for a gas with density proﬁle homogeneous enough to divide out the Fermi function
from the photoemission data. The spectral function of a homogeneous gas would be a great help
in comparing directly to theories of the pseudogap and developing a better understanding of theproperties of this strongly correlated system. For example, it would allow one to directly measure
the gap in the density of states. In order to improve upon the current results, it may be necessary
to consider modifying the trapping potential to be ﬂatter in the center. The most diﬃcult partof making a box trap is obtaining sharp edges at the boundaries. However, the hollow light beam

153
technique is excellent for removing atoms at the edge of the cloud. Thus, a box trap is not necessary.
For example, a quartic trapping potential (V (r) ∝ r4 ) might prove adequate. Other experiments in
our ﬁeld have demonstrated the kind of beam shaping control necessary to achieve this. Without
data for a homogeneous gas, I fear that some questions regarding the pseudogap phase could gounresolved, such as the energy widths of the single-particle states near the Fermi surface and themagnitude and temperature dependence of the pseudogap in the density of states. Of course, othermeasurement techniques could also shed light on the pseudogap phase, such as measurements ofthe spin susceptibility [92].
Our work in Chapter 8 on the contact shows that measurements of this quantity can be auseful tool for probing a strongly interacting gas. As with photoemission spectroscopy, measurements of the contact for a homogeneous gas would be desirable. For example, the same theory
we have compared to our photoemission data also predicts a sharp increase in the contact near
the Tc [122]. However, it is predicted that this increase cannot be seen in a trapped gas withinhomogeneous density [122]. Luckily, measuring the contact for a homogeneous gas is signiﬁcantly
easier than photoemission spectroscopy for a homogeneous gas. This is because the contact can beobtained directly from momentum integrated rf spectra. In fact, spatially resolved rf spectroscopy
has already been demonstrated in Ref. [97]. However, in Ref. [97] the fermion studied was 6 Li,
where rf spectroscopy is complicated by the absence of a weakly interacting third spin state [40].
Thus our experiment with 40 K would have a distinct advantage. Measurements of the contact fora homogeneous gas as a function of temperature and interaction strength would be an importantbenchmark for theories and could help us develop a better understanding of the crossover. Furthermore, the behavior of the contact near Tc could give information about the pseudogap state
and the nature of the superﬂuid transition.
The medium-to-long term outlook is, obviously, more diﬃcult to predict. It seems alreadythat the projects we deem most promising, even in the short term, can completely change on a dime
when interesting results are presented by other experimental and theoretical groups. As such, themost likely thing to happen is there will be new predictions and observations regarding strongly154
interacting fermions that our experiment will be well suited to study. One recent development thatappears to remain an open question is the potential observation of a ferromagnetic state for strongly
repulsive fermions [57]. It would also be interesting to study Fermi liquid behavior in a low temperature repulsive Fermi gas using photoemission spectroscopy or other means. Non-equilibriumand dynamic behavior of strongly interacting quantum gases is a rich ﬁeld with many open questions. The ability to quickly change the magnetic ﬁeld and thus the interaction strength means we
can excite dynamic behavior. For example, measuring the time-dependence of the contact after amagnetic ﬁeld jump could be interesting, possibly using time-resolved photoemission spectroscopy.
Also, it has been pointed out that the value of the contact can be related to the shear viscosity ofa strongly interacting Fermi gas, which may itself may be approaching a lower limit predicted bystring theory [101].
Eventually, one can consider adding a degree of complexity to our system. Reducing thedimensionality via an optical lattice is an obvious possibility. The lower dimension could suppress
superﬂuidity and allow one to study the normal state down to lower temperature [12], perhaps also
sheading light on the physics of the pseudogap phase. The two-dimensional Fermi-Hubbard model
still remains the holy grail of atomic Fermi gas experiments and there are promising proposals for
reducing the entropy enough to observe magnetic behavior and map out the low-temperature phase
diagram [123].
Regardless, as experimenters develop new techniques and more sophisticated measurements totest theoretical predictions and study many-body quantum phenomena there is sure to be dramaticand exciting progress in this ﬁeld over the next several years. I look forward to following thosedevelopments.

Bibliography[1] C. A. Regal and D. S. Jin, Measurement of positive and negative scattering lengths in a Fermi
gas of atoms, Phys. Rev. Lett. 90, 230404 (2003).
[2] J. P. Gaebler, J. T. Stewart, J. L. Bohn, and D. S. Jin, p-wave Feshbach molecules, Phys.
Rev. Lett. 98, 200403 (2007).
[3] D. S. Jin, J. P. Gaebler, and J. T. Stewart, in Proceedings of the International Conferenceon Laser Spectroscopy, Telluride 2007, edited by L. Hollberg, J. Berquist, and M. Kasevich(World Scientiﬁc, 2008).
[4] W. Ketterle and M. W. Zwierlein, in Proceedings of the International School of Physics“Enrico Fermi”, Course CLXIV, edited by M. Inguscio, W. Ketterle, and C. Salomon (OSPressAmsterdam, 2008).
[5] J. R. Schrieﬀer, in Theory of Superconductivity, edited by D. Pines (Addison-Wesley Publishing Company, Inc., 1964).
[6] C. A. Regal, Experimental realization of BCS-BEC crossover physics with a Fermi gas of
atoms, Ph.D. thesis, University of Colorado, 2006.
[7] C. A. Regal, M. Greiner, and D. S. Jin, Observation of resonance condensation of fermionic
atom pairs, Phys. Rev. Lett. 92, 040403 (2004).
[8] A. Damascelli, Z. Hussain, and Z. Shen, Angle-resolved photoemission studies of the cuprate
superconductors, Reivews of Modern Physics 75, 473 (2003).
[9] J. T. Stewart, J. P. Gaebler, and D. S. Jin, Using photoemission spectroscopy to probe astrongly interacting Fermi gas, Nature 454, 744 (2008).
[10] J. T. Stewart, J. P. Gaebler, T. E. Drake, and D. S. Jin, Veriﬁcation of universal relations ina strongly interacting Fermi gas, Phys. Rev. Lett. 104, 235301 (2010).
[11] S. Tsuchiya, R. Watanabe, and Y. Ohashi, Single-particle properties and pseudogap eﬀects in
the BCS-BEC crossover regime of an ultracold Fermi gas above Tc , Phys. Rev. A 80, 033613(2009).
[12] Q. Chen, J. Stajic, and K. Levin, Applying BCS-BEC crossover theory to high temperature
superconductors and ultracoldatomic Fermi gases, Low Temp. Phys. 32, 406 (2006).

156[13] S. Nascimbene, N. Navon, K. Jiang, F. Chevy, and C. Salomon, Exploring the thermodynamics of a universal Fermi gas, Nature 463, 1057 (2010).
[14] A. Bulgac, J. E. Drut, and P. Magierski, Spin 1/2 fermions in the unitary regime: a superﬂuid
of a new type, Phys. Rev. Lett. 96, 090404 (2006).
[15] A. Perali, P. Pieri, G. C. Strinati, and C. Castellani, Pseudogap and spectral function fromsuperconducting ﬂuctuations to the bosonic limit, Phys. Rev. B 66, 024510 (2002).
[16] A. Perali, P. Pieri, P. F., G. C. Strinati, J. T. Stewart, J. P. Gaebler, T. E. Drake, and
D. S. Jin, Pseudogap phase and remnant Fermi surface in ultracold Fermi Gases, , arXiv:
1006.3406.
[17] W. Schneider and M. Randeria, Universal short-distance structure of the single-particle spectral function of dilute Fermi gases, Phys. Rev. A 81, 021601 (2010).
[18] J. P. Gaebler, J. T. Stewart, T. E. Drake, D. S. Jin, A. Perali, P. Pieri, and G. C. Strinati,
Observation of pseudogap behavior in a strongly interacting Fermi gas, Nature Physics 6, 569(2010).
[19] A. Schirotzek, Y. il Shin, C. H. Schunck, and W. Ketterle, Determination of the superﬂuid
gap in atomic Fermi gases by quasiparticle spectroscopy, Phys. Rev. Lett. 101, 140403 (2008).
[20] A. Bulgac, J. E. Drut, and P. Magierski, Quantum Monte Carlo simulations of the BCS-BEC
crossover at ﬁnite temperature, Phys. Rev. A 78, 023625 (2008).
[21] G. E. Astrakharchik, J. Boronat, J. Casulleras, and S. Giorgini, Equation of state of a Fermi
gas in the BEC-BCS crossover: A quantum Monte Carlo study, Phys. Rev. Lett. 93, 200404(2004).
[22] F. Werner, L. Tarruel, and Y. Castin, Number of closed-channel molecules in the BEC-BCS
crossover, Eur. Phys. J. B 68, 401 (2009).
[23] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wiemand, and E. A. Cornell, Observation of Bose-Einstein condensate in a dilute atomic vapor, Science 269, 198 (1995).
[24] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Evidence of Bose-Einsteincondensation in an atomic gas with attractive interactions, Phys. Rev. Lett. 75, 1687 (1995).
[25] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn,and W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett. 75,
3696 (1995).
[26] W. Ketterle, in Nobel Prizes 2001 (Nobel Foundation, 2001).
[27] P. W. Anderson, More Is Diﬀerent, Science 177, 4047 (1972).
[28] in Open Problems in Strongly Correlated Electron Systems, Proceedings of the NATO Advanced Research Workshop in Bled, Slovenia, 26-30 April 2000, edited by J. Bonca, P.
Prelovsek, R. A., and S. Sarkar (NATO Sicence Series II: Mathematics, Physics and Chemistry, Vol. 15, 2001).
[29] H. Heiselberg, Fermi systems with long scattering lengths, Phys. Rev. A 63, 043606 (2001).

157[30] J. T. Stewart, J. P. Gaebler, C. A. Regal, and D. S. Jin, Potential energy of a 40 K Fermi gas
in the BCS-BEC crossover, Phys. Rev. Lett. 97, 220406 (2006).
[31] E. Shuryak, Strongly coupled quark-gluon plasma: the status report, Prog. Part. Nucl. Phys.
53, 273 (2004).
[32] G. A. Baker, Neutron matter model, Phys. Rev. C 60, 054311 (1999).
[33] B. DeMarco, Quantum behavior of an atomic Fermi gas, Ph.D. thesis, University of Colorado,2001.
[34] J. T. Stewart, Probing a strongly interacting Fermi gas, Ph.D. thesis, University of Colorado,2009.
[35] V. Gurarie and L. Radzihovsky, Resonantly-paired fermionic superﬂuids, Annals of Physics
322, 2 (2007).
[36] J. J. Sakurai, in Modern Quantum Mechanics, edited by S. F. Tuan (Addison-Wesley Publishing Company, Inc., 1985).
[37] R. B. Diener and T.-L. Ho, The condition for universality at resonance and direct measurement of pair wavefunctions using rf spectroscopy, (2004), arXiv:cond-mat/0405174v2.
[38] C. Ticknor, C. A. Regal, D. S. Jin, and J. L. Bohn, Multiplet structure of Feshbach resonancesin non-zero partial waves, Phys. Rev. A 69, 042712 (2004).
[39] C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Creation of ultracold molecules from a
Fermi gas of atoms, Nature 424, 47 (2003).
[40] C. H. Schunck, Y. i. Shin, A. Schirotzek, and W. Ketterle, Determination of the fermion pairsize in a resonantly interacting superﬂuid, Nature 454, 739 (2008).
[41] T. Loftus, C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Resonant control of elastic
collisions in an optically trapped Fermi gas of atoms, Phys. Rev. Lett. 88, 173201 (2002).
[42] M. Greiner, C. A. Regal, and D. S. Jin, Probing the excitation spectrum of a Fermi gas in
the BCS-BEC crossover regime, Phys. Rev. Lett. 94, 070403 (2005).
[43] C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Tuning p-wave interactions in an ultracold
Fermi gas of atoms, Phys. Rev. Lett. 90, 053201 (2003).
[44] C. A. Regal, M. Greiner, and D. S. Jin, Lifetime of molecule-atom mixtures near a Feshbachresonance in 40K, Phys. Rev. Lett. 92, 083201 (2004).
[45] S. Durr, V. T., and G. Rempe, Dissociation of ultracold molecules with Feshbach resonances,
Phys. Rev. A 70, 031601(R) (2004).
[46] Y. Inada, M. Horikoshi, S. Nakajima, M. K. Kuwata-Gonokami, M. Ueda, and T. Mukaiyama,
Collisional properties of p-wave Feshbach molecules, Phys. Rev. Lett. 101, 100401 (2008).
[47] J. Fuchs, C. Ticknor, P. Dyke, G. Veeravalli, E. Kuhnle, W. Rowlands, P. Hannaford, andC. J. Vale, Binding energies of 6 Li p-wave Feshbach molecules, Phys. Rev. A 77, 053616(2008).

158[48] J. Levinson, N. R. Cooper, and V. Gurarie, Stability of fermionic gases close to a p-wave
Feshbach resonance, Phys. Rev. A 78, 063616 (2008).
[49] V. Gurarie, L. Radzihovsky, and A. V. Andreev, Quantum phase transitions across p-wave
Feshbach resonance, Phys. Rev. Lett. 94, 230403 (2005).
[50] C.-H. Cheng and S.-K. Yip, Anisotropic Fermi superﬂuid via p-wave Feshbach resonance,
Phys. Rev. Lett. 95, 070404 (2005).
[51] Y. Ohashi, BCS-BEC crossover in a gas of Fermi atoms with a p-wave Feshbach resonance,
Phys. Rev. Lett. 94, 050403 (2005).
[52] S. S. Botelho and C. A. R. S. de Melo, Quantum phase transition in the BCS-to-BEC evolutionof p-wave Fermi gases, J. Low Temp. Phys. 140, 409 (2005).
[53] N. Read and D. Green, Paired states of fermions in two dimensions with breaking of parity
and time-reversal symmetries, and the fractional quantum Hall eﬀect, Phys. Rev. B 61, 10267(2000).
[54] R. A. W. Maier, C. Marzok, and C. Zimmermann, Radio-frequency spectroscopy of 6 Li pwave molecules: Towards photoemission spectroscopy of a p-wave superﬂuid, Phys. Rev. A
81, 064701 (2010).
[55] Y. J. Han, C. Y. H., A. J. Daley, S. Diehl, P. Zoller, and L. M. Duan, Stabilitzation of thep-wave superﬂuid state in an optical lattice, Phys. Rev. Lett. 103, 070404 (2009).
[56] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (Dover PublicationsMineola, New York, 1971).
[57] G. B. Jo, Y. Lee, J. H. Choi, C. A. Christensen, T. H. Kim, J. H. Thywissen, D. E. Pritchard,
and W. Ketterle, Itinerant gerromagnetism in a Fermi gas of ultracold atoms, Science 325,
1521 (2009).
[58] L. N. Cooper, Bound electron pairs in a degenerate Fermi gases, Phys. Rev. 104, 1159 (1956).
[59] A. J. Leggett, Cooper pairing in spin-polarized Fermi systems., J. Phys. C (Paris) 41, C7(1980).
[60] P. Nozieres and S. Schmitt-Rink, Bose condensation in an attractive fermion gas: from weak
to strong coupling superconductivity, J. of Low Temp. Phys. 59, 195211 (1985).
[61] M. Randeria, J. M. Duan, and L. Y. Shieh, Superconductivity in two-dimensional Fermi gas:
Evolution form Cooper pairing to Bose condensation, Phys. Rev. B 41, 327 (1990).
[62] M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H. Schunck, and W. Ketterle, Vortices
and superﬂuidity in a strongly interacting Fermi gas, Nature 435, 1047 (2005).
[63] J. Kinast, A. Turlapov, J. E. Thomas, Q. Chen, J. Stajic, and K. Levin, Heat capacity ofastronglyinteracting Fermi gas, Science 307, 1296 (2005).
[64] J. Kinast, S. L. Hemmer, M. E. Gehm, A. Turlapov, and J. E. Thomas, Evidence for superﬂuidity in a resonantly interacting Fermi gas, Phys. Rev. Lett. 92, 150402 (2004).

159[65] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Denschlag, and R. Grimm,
Collective excitations of a degenerate gas at the BEC-BCS crossover, Phys. Rev. Lett 92,
203201 (2004).
[66] Q. Chen, J. Stajic, and K. Levin, Thermodynamics of Interacting Fermions in Atomic Traps,
Phys. Rev. Lett. 95, 260405 (2005).
[67] C. A. Regal, M. Greiner, S. Giorgini, M. Holland, and D. S. Jin, Momentum distribution ofa Fermi gas of atoms in the BCS-BEC crossover, Phys. Rev. Lett. 95, 250404 (2005).
[68] P. Nozieres, in Theory of Interacting Fermi Systems, edited by D. Pines (Addison-Wesley
Publishing Company, Inc., 1964).
[69] P. Noziéres and D. Pines, The Theory of Quantum Liquids, Vol. I Normal Fermi Liquids(Addison-WesleyCambridge, Massachusetts, 1966).
[70] R. Haussmann, M. Punk, and W. Zwerger, Spectral functions and rf response of ultracold
fermionic atoms, Phys. Rev. A 80, 063612 (2009).
[71] P. Massignan, G. M. Bruun, and H. T. C. Stoof, Twin peaks in rf spectra of Fermi gases atunitarity, Phys. Rev. A 77, 031601(R) (2008).
[72] Q. Chen and K. Levin, Probing the spectral function using momentum resolved radio frequency spectroscopy in trapped Fermi gases, Phys. Rev. Lett. 102, 190402 (2009).
[73] This interaction strength is calculated using the Feshbach resonance calibration of Chap. 3
and so is diﬀerent that what is reported in [9].
[74] C. Chin and P. S. Julienne, Radio-frequency transitions on weakly bound ultracold molecules.,Phys. Rev. A 71, 012713 (2005).
[75] C. Chien, H. Guo, Y. He, and K. Levin, Comparative study of BCS-BEC crossover theories
above Tc : the nature of the pseudogap in ultra-cold atomic Fermi gases, Phys. Rev. A 81,
023622 (2010).
[76] A. Kanigel, C. U., M. Randeria, M. R. Norman, G. Koren, K. Kadowaki, and J. C. Campuzano, Evidence for pairing above Tc from the dispersion in the pseudogap phase of cuprates,
Phys. Rev. Lett 101, 137002 (2008).
[77] M. Shi. et al., Spectroscopic evidence for preformed Cooper pairs in the pseudogap phase ofcuprates, EPL 88, 27008 (2009).
[78] H. B. Yang, J. D. Rameau, P. D. Johnson, T. Valla, A. Tsvelik, and G. D. Gu, Emergence ofpreformed Cooper pairs from the doped Mott insulating state in Bi2 Sr2 CaCu2 O8+δ , Nature
456, 6 (2008).
[79] K. Nakayama, T. Sato, Y. Sekiba, K. Terashima, P. Richard, T. Takahashi, K. Kudo, N.
Okumura, T. Sasaki, and N. Kobayashi, Evolution of a pairing-induced pseudogap from thesuperconducting gap of (Bi,Pb)2 Sr2 CuO6 , Phys. Rev. Lett. 102, 227006 (2009).
[80] M. Hashimoto et al., Particle-hole symmetry breaking in the pseudogap state of Bi2201,
Nature Physics 6, 414 (2010).

160[81] T. Kondo, R. Khasanov, T. Takeuchi, J. Schmalian, and A. Kaminski, Competition between
the pseudogap and superconductivity in the high-Tc copper oxides, Nature 457, 296 (2009).
[82] Handbook of High-Temperature Superconductivity, edited by J. R. Schrieﬀer and J. S. Brooks(Springer, 2007).
[83] M. Randeria, N. Trivedi, A. Moreo, and R. T. Scalettar, Pairing and spin gap in the normal
state of short coherence length superconductors, Phys. Rev. Lett. 69, 2001 (1992).
[84] C. A. R. Sá de Melo, M. Randeria, and J. R. Engelbrecht, Crossover from BCS to Bose superconductivity: Transition temperature and time-dependent Ginzburg-Landau theory, Phys.
Rev. Lett. 71, 3202 (1993).
[85] B. Janko, J. Maly, and K. Levin, Pseudogap eﬀects induced by resonant pair scattering, Phys.
Rev. B 56, R11407 (1997).
[86] G. M. Bruun and G. Baym, Bragg spectroscopy of cold atomic Fermi gases, Phys. Rev. A
74, 033623 (2006).
[87] P. Magierski, G. Wlazlowski, A. Bulgac, and J. E. Drut, The ﬁnite temperature pairing gapof a unitary Fermi gas by quantum Monte Carlo, Phys. Rev. Lett. 103, 210403 (2009).
[88] C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. H. Denschlag, and R. Grimm,
Observation of the pairing gap in a strongly interacting Fermi gas, Science 305, 1128 (2004).
[89] S. Nascimbene, Thermodynamics of ultracold Fermi gases, Ph.D. thesis, ENS, 2010.
[90] A. Bulgac, J. E. Drut, and P. Magierski, Thermodynamics of a trapped unitary Fermi gas,
Phys. Rev. Lett. 99, 120401 (2007).
[91] A. Bulgac, M. M. Forbes, and P. Magierski, in BCS-BEC crossover and the unitary Fermi
gas, edited by W. Zwerger (Springer, 2011), Chap. The unitary Fermi gas: From Monte Carloto density functionals.
[92] C. Chien and K. Levin, Fermi liquid theory of ultra-cold trapped Fermi gases: Implications
for pseudogap physics and other strongly correlated phases, Phys. Rev. A 82, 013603 (2010).
[93] S. Tsuchiya, R. Watanabe, and Y. Ohashi, Photoemission spectrum and eﬀect of inhomogeneous pairing ﬂuctuations in the BCS-BEC crossover regime of an ultracold Fermi gas:, Phys.
Rev. A 82, 033629 (2010).
[94] H. H., X.-J. Liu, P. D. Drummond, and H. Dong, Pseudo-gap pairing in ultracold Fermi
atoms, Phys. Rev. Lett. 104, 240407 (2010).
[95] E. Hodby, S. T. Thompson, C. A. Regal, M. Greiner, A. C. Wilson, D. S. Jin, E. A. Cornell,and C. E. Wieman, Production eﬃciency of ultra-cold Feshbach molecules in bosonic andfermionic systems, Phys. Rev. Lett. 94, 120402 (2005).
[96] J. Carlson and S. Reddy, Superﬂuid pairing gap in strong coupling, Phys. Rev. Lett. 100,
150403 (2008).
[97] Y. Shin, C. H. Schunck, A. Schirotzek, and W. Ketterle, Tomographic rf spectroscopy of atrapped Fermi gas at unitarity., Phys. Rev. Lett. 99, 090403 (2007).

161[98] N. R. Heckenberg, R. McDuﬀ, C. P. Smith, H. Rubinsztein-Dunlop, and J. Wegener, Laserbeams with phase singularities, Optical and Quantum Electronics 24, S951 (1992).
[99] A. Perali, P. Pieri, and G. C. Strinati, Quantitaive comparison between theoretical predictions
and experimental results for the BCS-BEC crossover, Phys. Rev. Lett. 93, 100404 (2004).
[100] P. Pieri, L. Pisani, and G. C. Strinati, Comparison between a diagrammatic theory for the
BCS-BEC crossover and quantum Monte Carlo results, Phys. Rev. B 72, 012506 (2005).
[101] E. Taylor and M. Randeria, Viscosity of strongly interacting quantum ﬂuids: spectral functions and sum rules, Phys. Rev. A 81, 053610 (2010).
[102] S. Tan, Large momentum part of a strongly correlated Fermi gas, Ann. Phys. 323, 2971
(2008).
[103] S. Tan, Generalized Virial Theorem and Pressure Relation for a strongly correlated Fermi
gas, Ann. Phys. 323, 2987 (2008).
[104] S. Tan, Energetics of a strongly correlated Fermi gas, Ann. Phys. 323, 2952 (2008).
[105] E. Braaten, D. Kang, and L. Platter, Exact relations for a strongly-interacting Fermi gas
from the operator product expansion, Phys. Rev. Lett. 100, 205301 (2008).
[106] S. Zhang and A. J. Leggett, Universal properties of the ultracold Fermi gas, Phys. Rev. A
79, 023601 (2009).
[107] E. D. Kuhnle, H. Hu, X.-J. Liu, P. Dyke, M. Mark, P. D. Drummond, P. Hannaford, andC. J. Vale, Universal behavior of pair correlations in a strongly interacting Fermi gas, Phys.
Rev. Lett. 105, 070402 (2010).
[108] E. Braaten, in Universal relations for Fermions with large scattering length, edited by W.
Zwerger (Springer, 2011).
[109] G. B. Partridge, K. E. Strecker, R. I. Kamar, M. W. Jack, and R. G. Hulet, Molecular probeof pairing in the BEC-BCS crossover, Phys. Rev. Lett. 95, 020404 (2005).
[110] N. Navon, S. Nascimbene, F. Chevy, and C. Salomon, The equation of state of a lowtemperature Fermi gas with tunable interactions, Science 328, 5979 (2010).
[111] P. Pieri, A. Perali, and G. C. Strinati, Enhanced paraconductivity-like ﬂuctuations in theradio frequency spectra of ultracold Fermi atoms, Nat. Phys. 5, 736 (2009).
[112] W. Schneider, V. B. Shenoy, and M. Randeria, Theory of Radio Frequency Spectroscopy of
Polarized Fermi Gases, arXiv:0903.3006v1 (unpublished).
[113] E. Braaten, D. Kang, and L. Platter, Short-time operator product expansion for rf spectroscopy of a strongly interacting Fermi gas, Phys. Rev. Lett. 104, 223004 (2010).
[114] Z. Yu and G. Baym, Spin-correlation functions in ultracold paired atomic-fermion systems:
Sum rules, self-consistent approximations, and mean ﬁelds., Phys. Rev. A 73, 063601 (2006).
[115] M. Punk and W. Zwerger, Theory of rf-spectrscopy of strongly interacting fermions., Phys.
Rev. Lett. 99, 170404 (2007).

162[116] S. Basu and E. J. Mueller, Final-state eﬀects in the radio frequency spectrum of stronglyinteracting fermions, Phys. Rev. Lett. 101, 060405 (2008).
[117] A. Perali, P. Pieri, and G. C. Strinati, Competition between ﬁnal-state and pairing gap eﬀects
in the radio-frequency spectra of ultracold Fermi atoms., Phys. Rev. Lett. 100, 010402 (2008).
[118] M. Veillette, E. G. Moon, A. Lamacraft, L. Radzihovsky, S. Sachdev, and D. E. Sheehy, Radio
frequency spectroscopy of a strongly imbalanced Feshbach-resonant Fermi gas, Phys. Rev. A
78, 033614 (2008).
[119] Y. He, C. C. Chien, Q. Chen, and K. Levin, Radio frequency spectroscopy of trapped Fermi
gases with population imbalance, Phys. Rev. A 77, 011602 (2008).
[120] T. Bourdel, J. Cubizolles, L. Khaykovich, K. M. F. Magalhães, S. J. J. M. F. Kokkelmans,
G. V. Shlyapnikov, and C. Salomon, Measurement of the interaction energy near a Feshbachresonance in a 6 Li Fermi gas, Phys. Rev. Lett. 91, 020402 (2003).
[121] K. M. O’Hara, S. L. Hemmer, M. E. Gehm, S. R. Granade, and J. E. Thomas, Observationof a strongly interacting degenerate Fermi gas of atoms, Science 298, 2179 (2002).
[122] F. Palestini, A. Perali, P. Pieri, and G. C. Strinati, Temperature and coupling dependenceof the universal contact intensity for an ultracold Fermi gas, Phys. Rev. A 82, 021605(R)(2010).
[123] T.-L. Ho and Q. Zhou, Universal colling scheme for quantum simulation, arXiv:0911.5506v1(unpublished).
[124] A. Corney, Atomic Laser Spectroscopy (Clarendon Press, Oxford, 1977).
[125] E. Arimondo, M. Inguscio, and P. Violino, Experimental determinations of the hyperﬁne
strucutre in the alkali atoms, Rev. Mod. Phys. 49, 31 (1977).
[126] R. Jördens et al., Quantitative determination of temperature in the approach to magnetic
order of ultracold fermions in an optical lattice, Phys. Rev. Lett. 104, 180401 (2010).

Appendix A
40

K Transitions

In a number of places in this thesis I discuss various optical transitions 40 K atoms in amagnetic ﬁeld. In this section, I explain how the energy and branching ratios for these transitions
are calculated and list all the transitions for 40 K atoms that are relevant for this thesis. The
references for this section are the book on laser spectroscopy, Ref. [124], and the review article
containing measurements of the hyperﬁne structure in alkali atoms, Ref. [125]. Also, when I wrote
the MATLAB program for this, I borrowed from a Mathematic code written by Jayson Stewart
that was a ﬁnal project for his graduate atomic physics class. A 40 K atom has nuclear spin I = 4
and electronic spin S = 1/2. In the ground state (4S1/2 ), where the orbital angular momentum
L is zero, we have J = L + S = 1/2. The total angular momentum F = J + I can be either 9/2
or 7/2. The F = 9/2 hyperﬁne state is the lower energy state that we use for experiments. The
relevant electronic transitions are to states with L = 1 and J = 3/2 (4P3/2 ), which are called D2
transitions. The electronically excited state can have F ′ = 11/2, 9/2, 7/2, 5/2, where the prime
signiﬁes we are talking about the excited state.
To compute the Zeeman shifts of the hyperﬁne states, one needs to diagonalize the Hamiltonian HHF S (B) = HHF S + HZ . The hyperﬁne Hamiltonian HHF S can be written
3
K(K + 1) − 2I(I + 1)J(J + 1)
1
HHF S = hAJ K + hBJ 2
2
2I(2I − 1)2J(2J − 1)

(A.1)

where K = 2I · J = F (F + 1) − I(I + 1) − J(J + 1), AJ is the magnetic hyperﬁne structure constant,and BJ is the electric quadrupole interaction constant. For a magnetic ﬁeld of strength B along

164
the z axis, the interaction with the magnetic ﬁeld is described by HZ and can be written
HZ = µB B(gj mj + gi

µN
mi )
µB

(A.2)

where mj and mi are the projections of J and I along the z-axis, µB is the bohr magneton, µN
is the nuclear magneton, and gj and gi are the g-factors for the electron and nuclear magnetic
moments. The Lande g-factor, gj depends on the g-factor for the electron, g, and can be written
gj = g2 (1 + J(J+1)+S(S+1)−L(L+1)).
2J(J+1)
J,I,L,S and the total spin projection, mF , are all good quantum numbers at any ﬁeld, while
F is also a good quantum number at B = 0. We label all the states by F and mF , where F thenrefers to the value for that state at B = 0.
The physical constants necessary are g = 2.002319, gI = −0.3245, µB = 1.3996 MHz/G,µN /µB = 0.000544617. For the ground state, AJ = −285.7308 and BJ = 0. For the excited state,
AJ = −7.59 and BJ = −3.5.
Figures A.1,A.2, and A.3, show all the D2 transitions from the three lowest energy Zeeman
levels in the F = 9/2 ground state at 202.2 G. Next to each excited state, the detuning (in MHz),and transition strength are shown for the transition indicated. The values are calculated for amagnetic-ﬁeld strength of B = 202.2 G (the location of the s-wave Feshbach resonance). The
detuning is calculated from the diﬀerence in energies obtained for the states after diagonalizing the
Hamiltonian H = HHF S + HZ . The transition strength is obtained by calculating the transition
matrix element between the new eigen-states. The probability that an atom in one of the excitedstates will spontaneously decay into a particular ground state is equal to the transition strength(i.e. a transition strength of 0.02 implies a 1 in 50 chance to decay to that state).

165968 / 9x10608 / 8x10

242 / 1.0

208 / 0.96

-3-5F’ = 7/2

563/ 0.018

9/2

172 / 0.90

11/2

-5/2
-7/2
mF = -11/2-9/2
Figure A.1: σ− Transitions.

F = 9/2

166

1375 / 1x10

653 / 0.67

253 / 2x10

-31016 / 0.02

964 / 0.05

610 / 0.64

566 / 0.6

219 / 3x10

-3183 / 3x10

mF = -9/2
Figure A.2: π Transitions

F’ = 5/2

7/2

9/2

-3

-5/2
-7/2

-4

11/2

F = 9/2

1671061 / 0.33

655 / 9x10

264 / 2x101422 / 0.04

1365 / 0.095 F’ = 5/21012 / 0.32

960 / 0.31

-3569 / 2x10-4-6613 / 2x10

230 / 5x10

-6194 / 7x10

-3/2
-5/2
-7/2
mF = -9/2
Figure A.3: σ+ Transitions

7/2

-3

-6

9/2

11/2

F = 9/2

Appendix B

Subtracting the high-momentum background signal from optical pumping

In chapter 7, I presented a technique for measuring the momentum distribution of atoms
from the center of a trapped gas using intersecting hollow light beams to optically pump atoms atlarge radius from the trap center. It was noted that this optical pumping is not perfectly eﬃcient,and leaves some atoms in the same state with an extra momentum kick due to absorbing andspontaneously emitting a photon. We subtract this background using a simple model. Fig. 7.3
shows the measured atom momentum distribution with, and without the hollow light beams, aswell as the predicted background contribution. In this appendix, I will sketch out the details of themodel.
The ﬁrst question to address is: how large do we expect the background signal to be basedon the branching ratio of the optical pump transition? This calculation is easily done for a cloud2

2

at zero temperature. The density proﬁle of the atoms is given by n = n0 (1 − RrT F )3/2 , wherer2 = x2 + y 2 + ( λz )2 and λ is the ratio of axial (along ẑ) to the radial (the (x-y) plane) trap√
2EF
frequency and RT F is the Fermi radius in the radial direction, given by RT F =. In ourmω 2experiment λ ≈ 10. A hollow light beam of order l has a proﬁle given by Eq. 7.1. For a closed
transition, the probability for an atom to absorb a certain number of photons, P (N ) is given
by a Poissonian distribution P (N ) = e−I IN ! , where I corresponds to the mean number of photonsN

absorbed per atom. For a non-cycling transition, with a probability T for the excited state to decay
back into the original state, the probability that an atom absorbs a single photon and falls backinto the original state is T P (1). The probability to absorb two photons and still be in the original

169
state is T 2 P (2), which we will consider to be negligible as T ≪ 1 for the transitions considered.
We consider a single, cylindrical hollow light beam with either l = 1 and w = 1.1RT F
propagating along the axial direction (ẑ) or an l = 2 and w = λ0.75RT F propagating perpendicular
to the axial direction (along ŷ). To compute the number of atoms that fall back into the samestate, we calculate the intensity of the light beam at each position in the cloud using Eq. 7.1, andthe corresponding probability to absorb a single photon and fall back into the original state. For
the l = 1 beam, we ﬁnd the number of atoms absorbing a single photon and falling back into thesame state as a fraction of the total initial number of atoms is approximately 1.12T N (0)(1 − N (0))
where N (0) is the faction of atoms that absorbed 0 photons (the fraction of atoms remaining after
the optical pumping excluding those that spontaneously decayed back to the original state). For√
the l = 2 beam we ﬁnd the number to be approximately 0.37T N (0)(1 − N (0)).
The atoms that absorb a photon and fall back into the same state will receive two momentum, where λph ≈ 767 nm. The ﬁrst momentum kick willkicks equal to the photon recoil k0 = h λ2π
phbe in the direction of propagation of the cylindrical beam due to the absorption of a photon. The
second will be in a random direction due to the spontaneous emission of a photon. For the l = 1
hollow light beam propagating along the axial direction of the cloud, we can ignore the eﬀect of theﬁrst photon because it is along the imaging axis (ẑ). The momentum distribution of atoms after
the spontaneous emission, nbg (k), is a convolution of the original momentum distribution of thoseatoms that absorb a single photon, n1 (k), and a spherical shell with a radius of k0 . This can be
written∫nbg (k) =1
δ(p − k0 ) 3
d p=
n1 (|⃗k − p⃗)|)
4πp2
2

∫ π

n1 (k 2 + k02 − 2kk0 cosθ)sinθdθ

(B.1)

0

In an experiment, we cannot directly measure the momentum distribution of just the atoms
that absorbed a single photon. However, a good approximation for n1 (k) is to take the diﬀerence
between the momentum distribution obtained with and without the hollow light beams, which is
the momentum distribution of the atoms that absorbed at 1 or more photons.
The other hollow light beam propagates in the imaging plane (along ŷ) and so there will

170
be two photon recoils. This can be modeled by taking nbg from Eq. B.1 and plugging it backinto the same equation as n1 to perform a second convolution. This neglects that the ﬁrst photon
kick will always be in a particular direction, but is a reasonable approximation since we take anazimuthal average of the images anyway. Typically, the background signal due to the l = 2 beams
is approximately half the signal due to the other beam (based on blasting 50% of the atoms withl = 1 beam and 20% of the atoms with the l = 2 beam).
The procedure for analyzing images begins by taking images of atom clouds after time-ofﬂight expansion with and without the hollow light beams. We perform inverse Abel transforms
on each image to obtain a 3D momentum distribution. We take the diﬀerence of the two momentum distributions to obtain n0 (k) and then use Eq. B.1 to obtain the momentum distributionof the background signal. We scale the ﬁrst background signal to have an amplitude given by
1.12T N (0)(1 − N (0)) where we use the fraction of atoms remaining with the hollow light beams
for N (0). We scale the second background signal to be half of that. Finally, we subtract thebackground signal from the momentum distribution measured with the hollow light beam. A momentum distribution with and without the hollow light beams, as well as a calculated backgroundsignal are shown in Fig. 7.3.

Appendix C

Accounting for interactions with the third spin-state in measurements of thecontact

In photoemission experiments, and measurements of the contact, we use an rf ﬁeld to transfer
atoms out of the strongly interacting system and into a weakly interacting state. In this thesis, thestrongly interacting system consists of atoms in the |9/2, −9/2⟩ and the |9/2, −7/2⟩ spin states,
which we will call states 1 and 2 respectively. The weakly interacting state is the |9/2, −5/2⟩ state,
which we will call state 3. It was pointed out in Ref. [111] that, due to residual interactions withthe third spin state, rf spectra should go as ω −5/2 at large detunning, as opposed to the ω −3/2 tailpredicted in the ideal case (Eq. 8.7). In Ref. [113], the exact form of the tail is derived and theauthors ﬁndlim Γ(ν) =ν→∞C
23/2 π 2ν −3/2()2(kF a12 )−1 − (kF a13 )−1(kF a13 )−2 + ν/2(C.1)

where a12 is the scattering length between the strongly interacting atoms, a13 is the scattering
length between states 1 and 3, and ν is the rf frequency in units of the EF /h with ν = 0 deﬁned as
the single atom resonance. The rf ﬁeld transfers atoms from state 2 to state 3. In the limit kF a13
small, this formula reduces to Eq. 8.7. In our experiments, the scattering length a13 is controlled
by a Feshbach resonance at 224.2G [6], which has a width of 7.5 G [126]. At 202.2 G, which is thecenter of the Fesbach resonance for the 1 and 2 states, this gives a scattering length a13 of 233a0 ,and for a typical value of kF , we ﬁnd (kF a13 )−1 = 9, leading to a few percent correction to thevalue of the contact (Fig. C.1). The scattering length a13 varies by less than 10a0 over the rangeof magnetic-ﬁeld strengths for the data presented here.

172
Final state corrections will also apply to the momentum distribution obtained from photoemission spectroscopy. To derive the high k tail from Eq. C.1, we use ν = 2k 2 , where k is thefermion wave-vector in units of kF . This relation can be checked in the limit of rf dissociation of adimer molecule. In this case, the initial state is a molecule with binding energy EB . The ﬁnal stateis two atoms, each with momentum of magnitude k. Conservation of energy gives ν = EB + 2k 2 .
At large k, we can neglect EB . Plugging this into Eq. C.1, we ﬁnd,

N (k)dk = Γ(ν)dν =

C −2
k
2π 2

(
)2
(kF a12 )−1 − (kF a13 )−1(kF a13 )−2 + k 2

dk

(C.2)

where N (k) is the number of atoms with wave-vector k. The momentum distribution, n(k), is
4π
2deﬁned as (2π)
3 n(k)k dk = N (k)dk, such thatn(k) = Ck −4()2(kF a12 )−1 − (kF a13 )−1(kF a13 )−2 + k 2

.

(C.3)

From this we see that the n(k) obtained from photoemission spectroscopy will have a k −6 tail atvery large k.
Fig. C.1 shows the contact extracted from rf spectra and photoemission data using theabove equations, which take into account interactions with the third spin-state (red points). For
comparison we also show the contact we reported in Ref. [10], where we neglected interactions
with the third spin state (black points). While the corrections due to interactions with the thirdspin-state generally shift the data closer to the expected theory line from Ref. [22], the magnitudeof the corrections is quite small. This justiﬁes the approximation used in Ref. [10], which neglects
the eﬀects of interactions with the third spin-state.

173

5
4
3

C/kF

2
1
0
4
3
2
1
0
-2.5

-2.0

-1.5

-1.0

-0.5

0

1/kFa
Figure C.1: Interactions with third spin-state Top) We plot the value of the contact extracted
from photoemission data using Eq. 8.4 (black circles) and using Eq. C.3 (red circles), which
accounts for the interactions with the third spin-state. On the BCS side (1/kF a < 0), the correction
leads to smaller values of the contact, while on resonance and on the BEC side (1/kF a < 0), thecorrection leads to slightly smaller values. The solid line is the prediction from Ref. [22]. Bottom)
We plot the value of the contact extracted from rf spectra using Eq. 8.7 (black stars) and using
Eq. C.1 (red stars).

Appendix D

Simulating photoemission spectroscopy of molecules

RF dissociation of weakly interacting Feshbach molecules is a well understood process because
the exact two-body wave-function of the molecule is known, see for example Refs. [74, 40]. Thismeans we can simulate an atom photoemission experiment in the BEC limit, where interactions
between the molecules can be ignored. In this appendix, I will explain in more detail how thesimulation of Fig. 5.8 was obtained. I will also use the model to explore the eﬀects of condensationand to look at the BCS-BEC crossover.
For a Feshbach molecule, the number of atoms outcoupled over a range of frequncies dν, asa function of rf frequency ν can be written [74]√√
2 EB ν − EB
Γ(ν)dν =
dν
π
ν2

(D.1)

where ν is in units of EF /h and EB is the molecule binding energy in units of EF . ν = 0 corresponds
to the transition energy for free atoms. The initial state is the bound molecule, and the ﬁnal stateis two free atoms with equal and opposite momentum kν , in the molecule center-of-mass frame.
The energy of each atom is kν2 , where kν is given in units of kF . From conservation of energy wehave ν = EB + 2kν2 . We plug this into Eq. D.1 to get the distribution of relative momenta fordissociated molecules√
27/2 EBkν2
N (kν )dkν =dkν .
π(EB + 2kν2 )2(D.2)

For non-Bose-condensed molecules, there will also be a center-of-mass momentum. We as2 ⟩. In asume this momentum distribution will be a gaussian distribution with a variance ⟨kcm

175
trapped T = 0 Fermi gas ⟨( kkF )2 ⟩ = 38 , which is the ratio corresponding to the average kinetic
energy per atom compared to the Fermi energy. Thus, an estimate for the variance in the pair2 ⟩ = 3 (twice the single-atom variance). In the case of Fig. 5.8, wemomentum distribution is ⟨kcm
4
2 ⟩ by assuming the molecules are in thermal equilibrium with unpaired atoms, whose
calculated ⟨kcm
2 ⟩ = 1.3. To simulate a gas with amomentum distribution we measure. For that case, we ﬁnd ⟨kcm

condensate of molecules, we can set a fraction of the molecules to have a center-of-mass momenta
equal to zero.
The total momentum the atom will be k = kν ± kcm /2. The single-particle energy of theatom, ES , is calculated from Eq. 5.7. Speciﬁcally, we have ES = k 2 − ν = k 2 − (EB + 2kν2 ).
To create simulated photoemission data: I ﬁx the binding energy EB , assume kcm and kν
are independent, generate a list of atoms obeying the probability distributions for knu and kcm ,calculate k and Es for each atom, and then plot the number of atoms versus Es and k. In these2 ⟩ = 1.1.
simulations I set ⟨kcm

The results of three simulated experiments with values of EB /EF = 2.5, 1.0, and 0.5 areshown in Fig. D.1. Fits to the EDCs are shown in Fig. D.2. As noted previously, in the limit of
weakly bound molecules this simulation will be exact, and for EB /EF = 2.5 the simulation agrees
well with the data in the BEC limit (as discussed in Chap 5 and Fig. 5.8). The center-of-mass
momenta of the pairs leads to the observed energy widths, as can be seen in Fig. D.3, where acondensate of pairs leads to a sharp spectral feature. It is interesting that the energy widths arelarge, even for the long-lived pairs in the deeply bound-molecule regime. This shows explicitly thatlarge energy widths of the single-particle excitations does not necessarily correspond to short-lived
pairs, strong interactions between the pairs, or large pair binding energy. Rather, in this case, thelarge energy widths arise only from the center-of-mass momentum distribution of the pairs.
It is interesting to look at simulations for a gas of molecules with small binding energy
compared to the Fermi energy. This is shown on the right hand side of Fig. D.1 and with alog-scale color map in Fig. D.4. Here the simulation produces spectral functions with a BCS-like
dispersion with back-bending. In fact, this simulation is qualitatively similar to photoemission

176
data taken near the center of the BCS-BEC crossover (Fig. 6.13). The qualitative agreement
is surprising because interactions between pairs must play an important role in the BCS-BEC
crossover. Nevertheless, it is suggestive that the pseudogap phase could consist of long-lived pairsthat have a large center-of-mass momenta.
In Fig. D.3, a simulation for a gas with EB = 2.5 and a 20% condensate fraction is shown.
The condensate leads to a sharp feature with a dispersion ES = −EB −k 2 . While this feature should
be observable in experiments, where we can obtain condensate fractions as high as 15%, it has notbeen seen. The feature could be washed out due to ﬁnite experimental resolution, or a condensate
that still has signiﬁcant center-of-mass motion of the pairs due to strong interactions (we have only
obtained condensates for 1/kF a < 1). This is an interesting subject for future experiments.

177

EB = 1.0

EB = 0.5

ES/EF

EB = 2.5

k/kF

k/kF

k/kF

Figure D.1: Simulated photoemission spectra Three simulated photoemission spetra are shownfor diﬀerent values of the molecule binding energy EB /EF , labeled in the upper right corner.
The upper white line is the free-particle dispersion. The lower white line is the dispersion thatwould be obtained in the limit of zero center-of-mass momentum of the molecules. The whitecircles are obtained from gaussian ﬁts to the EDCs. As EB decreases, the eﬀect of the centerof-mass motion of the molecules becomes important at low k/kF , and leads to a ﬂattening, andeventually a BCS-like back-bending in the dispersion. For the spectrum with EB /EF = 0.5, adashed line shows a BCS ﬁt to the centers of the gaussians. The best ﬁt parameters are µ, kL , ∆ =0.52 ± 0.14, 0.64 ± 0.07, 1.07 ± 0.15.

178

2.0
EB/EF = 2.5
1.0
0.5

Width (EF)

1.5
1.0
0.5

ES/EF

0
-2
-4
0

0.5

1.0

1.5

2.0

k/kF
Figure D.2: Dispersions and energy widths The results of gaussian ﬁts to the EDCs of simulations with the same parameters as those in Fig. D.1 are plotted versus k/kF . Black circles
correspond to the data with EB /EF = 2.5, red circles with EB /EF = 1.0, and blue circles with
EB /EF = 0.5. Top) The rms widths, in units of the Fermi energy, are plotted. They are similar
for all three data sets and increase as a function of k/kF . The large energy widths are due to thecenter of mass momenta of the molecules. These widths can be compared to those in Fig. 6.15.
Bottom) The centers of the gaussian ﬁts.

ES/EF

179

k/kF
Figure D.3: Simulated spectrum with a condensate A simulated photoemission spectrum fora gas with EB /EF = 2.5 and a 20% condensate fraction. The condensate creates a sharp signalthat exactly follows the lower white line, which is the dispersion in the limit of zero center-of-mass
momentum. In the simulation, the condensate was given a small center-of-mass momentum spread⟨kc m2 ⟩ = 0.003kF , so that it has a ﬁnite width in the simulated spectrum.

180

0

10

-1

ES/EF

10

-2

10

-3

10

k/kF
Figure D.4: Log plot of photoemission simulation This plot of the simulation with EB /EF =0.5 (right hand side of Fig. D.1) uses a log intensity colormap and can be compared to thephotoemission data in the pseudogap regime plotted in Figs. 6.6 and 6.13. The upper white line is
the free particle dispersion. The lower white line is a BCS-ﬁt to the data; the best ﬁt parameters
are given in the caption of Fig. D.1.