Probing a strongly interacting Fermi gas
byJohn Stewart
B.S. Physics and B.S. Mathematics, Texas A&M
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
2009This thesis entitled:
Probing a strongly interacting Fermi gas
written by John Stewart
has been approved for the Department of Physics

Dr. Deborah S. Jin

Dr. Eric A. Cornell

Date

The ﬁnal copy of this thesis has been examined by the signatories, and we ﬁndthat both the content and the form meet acceptable presentation standards ofscholarly work in the above mentioned discipline.

iii
Stewart, John (Ph.D., Physics)
Probing a strongly interacting Fermi gas
Thesis directed by Professor Dr. Deborah S. Jin

This thesis presents experimental work probing a strongly interacting Fermi
gas of atoms. The work presented here contributed to the demonstration thatstrongly interacting Fermi gas systems are accessing universal physics. The universality of these systems means that we can use an atomic Fermi gas to gain
an understanding of strongly interacting Fermions occurring elsewhere in nature,such as nuclear and neutron matter, high transition temperature superconductors, and the quark-gluon plasma. Once the universality of these systems was
veriﬁed, experiments were performed to extract the microscopic quantities of an
atomic Fermi gas. These experiments employ a technique akin to photoemission
spectroscopy for electrons found in condensed matter physics. The photoemission spectroscopy experiments presented here directly reveal the spectral function
which contains many valuable microscopic quantities such as the energy dispersion
and quasi-particle lifetime. These experiments provide stringent tests of manybody theories.

iv

Acknowledgements

It is hard to know where to begin when acknowledging everyone that has
led me to the path I’m on. So, I will begin at the beginning. Firstly, and mostimportantly, I would like to thank all of my wonderful family. My family has been
incredibly supportive and encouraging in my schooling ever since I was young.
Somehow, they had complete faith that I could achieve anything I set my mind
to and keeping their faith has been my driving force when the classes got hard
and the hours got long. Through my parents, I grew up believing I could literally
do anything that I set my mind to. Their acceptance of my choices and desire formy happiness has made my life such a joy. My siblings have always been there
for me and I trust they always will. My brothers have been a constant for me inthese last years of college and grad school. Even though we’ve been thousands ofmiles apart, hearing from them every week has made me still feel very close tothem and I consider them my very best friends.
My high school years were an important time for me. I think I was one ofthose nerds that loved learning. My many great teachers: Mrs. Comstock, Mrs.
Anderson, Mr. Ladley, and Mrs. Vice, to name a few, really opened my mind
to the world of math, science, and critical thinking. I would especially like tothank Mrs. Comstock for her wonderful teaching of this weird class of “physics”
which just blew me away. Physics, to me, was so bizarre and counter-intuitive,
and hard, that I just had to get my mind around it. I would also like to thank all

vof my friends from high school who all loved learning and really made school fun.
I would especially like to thank Justin McAllister who has been a true friend over
the years.
During my college years at Texas A&M I developed into the person I am
today. While there I learned that my future is in the laboratory. It really is like aplayground where we get to explore the secrets of the universe with expensive toys.
I have to thank Dr. Thomas Walther for that ﬁrst opportunity to work in the lab.
I knocked on his oﬃce door my ﬁrst week at A&M and he graciously accepted
my oﬀer and taught me the joys of experimental physics. I’d also like to thankhim for recommending me to come to Boulder. It was also during college that I
learned that the ﬁeld of physics was more expansive then I could ever imagined.
I would like to thank Dr. Agnolet, in particular, for being a fantastic teacher
and giving me a deep appreciation for quantum mechanics and thermodynamics.
Perhaps most importantly, in college I met my best two friends Marcus Teague
and Michael Hatridge. College certainly would not have been the experience it
was without them. We somehow meshed countless hours of homework sessions
with ample amounts of video games, queso, and margaritas. I’d like to thank Mikefor teaching me how to screen-look and Marcus for teaching me how to appreciatequality b-class sci-ﬁ.
My graduate studies have been so much better then I could ever have hopedfor. My advisor, Deborah Jin, is one of the great scientiﬁc minds of our time.
If I have picked up even a fraction of her problem solving skills, intuition, andexcitement for cool physics, I will consider myself fortunate. I have always appreciated that she gave us so much experimental freedom and the possibility tosteer the direction of the experiment. JILA as a whole has been just fantastic.
The infrastructure is designed to succeed: from the administrative staﬀ and theelectronics and machine shops, to the overall collaborative mind set of the Fel-vilows and students. In particular, I’d like to thank Eric Cornell and the tri-group.
Preparing some recent data for a tri-group meeting or talk has taught me how
to organize and motivate my research. If you can give your talk in front of thetri-group and ﬁeld their questions, you can give a talk with conﬁdence anywherein the world. I would also like to thank Krista Beck, she has been a huge help
throughout my graduate career and while I was a JILA graduate representative.
I have to thank John Bohn of teaching me the secrets of the tasty home brew.
During graduate school I have been blessed with some really great friends.
John Gaebler showed up ready to go on day one and everything in this thesis, andmuch more that is not, was done in equal participation with him. I have cometo value John’s opinion on anything from physics, economics, to politics. He
generally has a unique and compelling argument that contradicts the convenientor status-quo. Juan Pino and I have shared many a trip up to the mountains,
many a concussion trying some stupid trick, and many a conversation about life.
I will always value his opinion as real and challenging. There are many JILAns
that I have the pleasure to call friends in a variety of diﬀerent settings including:snowboarding, climbing, hiking, climbing, homework sessions, video games, trigroup challenges, and just talking physics. For this reason I’d like to thank: Russ
Stutz, Aaron Leanhardt, Giacomo Lamporesi, Laura Sinclair, Galen O’Neil, Amy
Bloom, Shihkuang Tung, Rob Wild, Huanqian Loh, Ty Cumby, John Perrault,
and Kang Kuen Ni, to name a few. I’d also like to thank Ian Schenck for all his
help, knowledge and friendship.
To my wife, Kara Renee Nelson-Stewart, I am forever indebted. She pickedup and left her family and friends so that we could be together here in Colorado.
She has been the most supportive and encouraging partner, way beyond anything
I could have hoped for. Many nights it was only at her insistence that I stayed late
to ﬁnish some experiment or analysis. I am continually amazed by her selﬂessness.

vii
She has been the glue that held us together for the last ﬁve years. It is to Karathat I dedicate this thesis.

Contents

Chapter
1 Introduction

1

1.1

Strongly interacting Fermi systems . . . . . . . . . . . . . . . . .

1

1.2

Ultra-cold atomic Fermi systems . . . . . . . . . . . . . . . . . . .

3

1.3

The work presented here . . . . . . . . . . . . . . . . . . . . . . .

6

1.4

Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92 Creating an ultra cold Fermi gas with strong interactions
2.1

2.2

11

Creating a cold sample . . . . . . . . . . . . . . . . . . . . . . . .

11

2.1.1

MOTs and magnetic trap evaporation . . . . . . . . . . . .

12

2.1.2

Optical trap evaporation . . . . . . . . . . . . . . . . . . .

12

2.1.3

Absorption imaging and thermometry . . . . . . . . . . . .

14

Controlling interactions . . . . . . . . . . . . . . . . . . . . . . . .

15

2.2.1

Fano-Feshbach resonances . . . . . . . . . . . . . . . . . .

16

2.2.2

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

22

3 Apparatus

29

3.1

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

30

3.2

Computer control . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

3.3

Vacuum pressure . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

ix
4 Universality in Fermi gases

38

4.1

The only scale: EF . . . . . . . . . . . . . . . . . . . . . . . . . .

38

4.2

Measuring β in an atomic Fermi gas . . . . . . . . . . . . . . . . .

41

4.2.1

β from the potential energy . . . . . . . . . . . . . . . . .

43

4.2.2

β from the kinetic energy

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

5 RF spectroscopy: A historical perspective

53

5.1

RF basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

5.2

RF spectroscopy and interactions . . . . . . . . . . . . . . . . . .

58

5.3

Using RF to measure energy gaps . . . . . . . . . . . . . . . . . .

62

5.4

Final-state eﬀects and trap inhomogeneity . . . . . . . . . . . . .

65

5.4.1

Final-state eﬀects . . . . . . . . . . . . . . . . . . . . . . .

67

5.4.2

Trap inhomogeneity . . . . . . . . . . . . . . . . . . . . . .

71

5.4.3

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

6 Photoemission spectroscopy

75

6.1

Photoemission spectroscopy and many-body theory . . . . . . . .

76

6.2

Photoemission spectroscopy for atoms . . . . . . . . . . . . . . . .

79

7 Conclusions and future directions

95

7.1

Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

7.2

The work not presented here . . . . . . . . . . . . . . . . . . . . .

96

Bibliography

98

Appendix
A Fermi gas thermometry110

x
A.1 Ideal Fermi gas thermometry . . . . . . . . . . . . . . . . . . . . . 111
A.2 Third spin-state thermometry . . . . . . . . . . . . . . . . . . . . 115
A.3 Three spin states near a Fano-Feshbach resonance . . . . . . . . . 116

Chapter 1
Introduction

1.1

Strongly interacting Fermi systems
Strongly interacting Fermi systems (sometimes called strongly correlated

Fermi systems) present some of the most intriguing and unsolved problems inmodern physics [1]. This is quite remarkable considering many of these stronglyinteracting systems have been studied for many decades. Perhaps the reason thesesystems have stayed at the forefront of physics for so long is that a complete theory
built from the ground up is computationally impractical in reasonable timescales.
In addition, experimental progress can be diﬃcult (experiments may require extremely low temperatures, high densities, very high magnetic ﬁelds, extremely
clean samples, etc.). Ultimately, I suspect it is the varied and counterintuitive
properties of strongly interacting systems that keeps the human mind so baﬄed
and intrigued. From a practical point of view, strongly correlated systems canhave useful properties for device applications, giving further motivation to understand these systems.
Of course, there are many systems that do not require a many-body understanding to explain observed phenomena. For example, the properties of simple
metals can be explained rather accurately by simply assuming the electron is a
non-interacting particle [2]. Non-interacting in this sense means assuming theelectron (fermion) does not interact with the other electrons in the system, it is2
a free particle. The atomic gas analog here would be the ideal Fermi gas. When
allowing for interactions between the fermions, amazingly, many phenomena canbe described by again assuming that the electrons are free particles but that theeﬀect of interactions is to simply renormalize the system, for example the fermion
mass as in Fermi liquid theory. Despite the success of these simple theories, there
exist interesting phenomena in interacting Fermi systems that cannot be describedwithin the free-particle framework.
The special types of interacting Fermi systems I refer to exhibit strong pairing interactions between pairs of spin-up and spin-down fermions. These stronglyinteracting Fermi systems can be found in superconductors, neutron and nuclear
matter [3, 4, 5, 6], the quark-gluon plasma [7], and now include the atomic Fermi
gases we study in this thesis. An important feature for these systems is that they
exhibit universality [5, 8, 9, 10]. Universality here means that for these stronglyinteracting Fermi systems, the details of the microscopic interaction are not important and at zero temperature the interparticle spacing becomes the only lengthscale. This is a most important feature because it means we can study any one of
the systems to learn about the others. This is quite remarkable when you consider
that the atomic Fermi gas system stands apart from its counterpart systems by
many orders of magnitude in density and temperature.
Intrinsic to this special class of strongly interacting Fermi systems is fermionicsuperﬂuidity. The existence a well deﬁned Fermi surface and attractive interactions between spin-up and spin-down fermions can lead to superconductivity inmetals or, more generally, fermionic superﬂuidity [11, 12, 13]. Superconductors
exhibit zero resistance, or persistent current, while superﬂuids exhibit persistent
ﬂow and vortices [14, 15]. Throughout this thesis I will mostly refer to superﬂuidity because we will be working with neutral atoms. Regardless, many of theproperties of the atomic Fermi gas we study in this thesis such as high transition

3
temperature, small pair size, and pseudogap, are relevant for charged superconductors [16, 17].
At this point there is a fairly natural question: why study strongly interacting Fermi systems using ultra-cold atoms if they are already so thoroughly
studied? Firstly, some of the other strongly interacting Fermi systems examples
are hard or even impossible to achieve in the laboratory (such as degenerate neutron matter) and therefore are hard to study thoroughly. Even condensed matter
experiments, which can be achieved in the laboratory setting, are subject to impurities, lattices, grain boundaries, etc., which can easily complicate what is alreadycomplicated physics. The atom gas, in contrast, is impeccably pure and simple.
Secondly, atomic systems introduce a degree of control not easily found in theother strongly interacting Fermi systems. For example, one cannot easily change
the density of a neutron star, or change its temperature. Or, in electron systems,
one cannot typically change the interaction strength between fermions, somethingeasily tuned in atomic systems. Using our simple, pure, tunable atomic Fermi gas,
we can help to understand unexplained phenomena found in these other systems
at a very basic level. Additionally, we can easily use our tunability to enter into
new and interesting regimes that may have not even been realized previously.
1.2

Ultra-cold atomic Fermi systems
Atomic systems provide an opportunity to study many-body problems ina system that is relatively free of complexity and very well understood at the
few-body level [18]. In particular, atomic Fermi gases are very valuable in testing
many-body theories because they are simple, pure, tunable samples with smoothdensity proﬁles. Then, as the underlying physics becomes known, one can thenintroduce lattices [19], impurities [20], disorder [21, 22], etc., to the atomic system
to simulate real material properties with a ground-up understanding.

4
Before addressing the interesting physics we can study with our atomic
Fermi gas, I think it is worth appreciating all the recent and rapid progress inatomic physics that allowed us to get to this point. Considering Bose-Einsteincondensation (BEC) was achieved experimentally only 14 years ago [23, 24], itis quite remarkable that we can already use our atomic Fermi gas to gain newunderstanding of such a wide variety of strongly interacting Fermi systems. The
major breakthroughs in atomic physics that led to BEC, namely laser cooling andevaporative cooling, were applied to cool fermions. Because the quantum statisticsof the particles does not manifest until very cold temperatures, a gas of fermionscan be initially cooled using the same techniques as bosons. However, below atemperature of order 20 µK, the atoms are cold enough that they collide only via
s-wave scattering, where the orbital angular momentum of the colliding particles
is zero [25]. However, since identical fermions cannot collide via s-wave scattering(the two-particle wave function must be anti-symmetric upon exchange of theparticles), the cooling of fermions requires at least two distinguishable particles.
The ﬁrst experiments to overcome this obstacle and create a quantum degenerate
gas of fermionic atoms occurred here at JILA, in 1999, on the same machine as
the experiments in this thesis [26].
Although the creation of a quantum degenerate gas of fermions was alreadya remarkable achievement, it was not the ultimate goal of the experiment. As itturns out, when one creates a degenerate non-interacting Fermi gas it does notundergo a phase transition as in the case of bosons. Instead the fermions simply
ﬁll every available energy level up to the Fermi energy, forming the so-called Fermi
sea [26, 2]. The next natural step was to tune the interactions in the system. In
fact, it had been proposed that if one could increase interactions to be suitably
strong, by using a scattering resonance known as a Fano-Feshbach resonance,then the transition temperature to the superﬂuid state could be within reach for5experimenters [27, 28, 29]. The requirements for achieving superﬂuidity: a colder
and more strongly interacting gas, were non-trivial and pursued here at JILA
with the culmination of fermionic superﬂuidity [30, 31]. Obviously, JILA was not
the only place where experimenters were creating these exciting new Fermi gases.
The creation of other degenerate interacting Fermi gases, including 40 K [32], 6 Li
[33, 34, 35, 36, 37] and now 176 Yb [38], as well as some I’ve certainly forgotten,
are being pursued all over the world.
The ability to tune interactions is one of the unique features of the atomicgas. The Fano-Feshbach resonance allows one to access attractive or repulsive
interactions at arbitrary strength. On one side of the Fano-Feshbach resonance,
there exists attractive interactions and one can in principle study BCS-like superﬂuidity. On the other side of the resonance, there exists a two-body bound stateand pairs of fermions, which are composite bosons, can Bose condense. In between these two limiting cases is the so-called BCS-BEC crossover. This crossoverfrom BCS physics to BEC physics was proposed [39, 40, 41] well before atomicgas experiments had achieved quantum degeneracy. However, the ability to tuneinteractions in atomic gases makes them well suited for realizing this crossover.
Indeed, the next step in the evolution of atomic Fermi gases was to control interactions for just this purpose [42, 43, 44, 45, 35].
A particularly exciting regime within the BCS-BEC crossover occurs forvery strong attractive interactions in which a pseudogap exists. The concept of apseudogap will be discussed later in more detail in this thesis but for now it serves
as another connection to work done in condensed matter physics. The pseudogap
has been discussed in relation to high-Tc superconductors [46] and is a featureof virtually every BCS-BEC crossover theory. Finally, for a review of the earlyworks on the BCS-BEC crossover I recommend the theses of C. A. Regal [47], M.
W. Zwierlein [48], and G. B. Partridge [49].

6
1.3

The work presented here
In the previous sections I attempted to set the stage for the work done inthis thesis. I was handed a new shiny toy and told to play with it. The hard
work of many groups throughout the world had eﬀectively opened a new ﬁeld ofphysics: ultracold strongly interacting atomic Fermi gas physics. To help set thestage for this thesis, I would recommend the thesis of my two predecessors who
were instrumental in opening this new area of physics: Brian DeMarco [50] andCindy Regal [47]. I began my thesis work in the Summer of 2004. At my disposal
was an ultracold Fermi gas with tunable interactions. Temperatures on order ofhalf the superﬂuid transition temperature were already achievable. The phase
diagram throughout the BCS-BEC crossover had been mapped out [30] (see Fig.
1.1), collective excitations [51, 52] and momentum distributions [53] had beenstudied, and vortices had been observed [54].
By the time I began my thesis work, superﬂuidity in atomic Fermi gas
systems was so thoroughly established that we were really in a position to ask:
what can this experiment teach us about strongly interacting systems? As wehave argued, atomic Fermi gases are simple, pure, tunable systems well poised totest many-body theories. Therefore, a natural way to proceed in answering theabove question is to design experiments that can provide stringent tests of (anddiﬀerentiate between) various many-body theories. In this thesis, I will describe
two experiments that we performed towards answering the question of just how
much atomic Fermi gases have to oﬀer.
The ﬁrst experiment I will describe brings us back to the topic of universality.
In order to use our Fermi gas experiment to shed light on the general class ofproblems dealing with strongly interacting Fermi systems, we ﬁrst had to provethat our system was universal. In a very clear way we were able to prove this by

7

0.15
0.10

N0 /N
0.05
0
-0.05
0
0.05
-0.4

-0.2

DB

0.10
0

0.2

(g a u

ss)

0.15
0.4

0.6

0.8

T/TF

0.20

Figure 1.1: Phase diagram of condensate fraction across the BCS-BEC crossover.
Figure taken from Ref. [30] showing condensate fraction as a function of temperature and magnetic ﬁeld. Positive magnetic ﬁeld tends toward the BCS side andnegative magnetic ﬁeld tends toward the BEC side. Measurements such as thisestablished superﬂuidity in the strongly interacting Fermi gases.

8
measuring the potential energy of the Fermi gas a function of interaction strength[55]. Measuring the potential energy allowed us to extract a universal parameter,β, that was actually ﬁrst proposed in nuclear theory and later applied to atomic
Fermi gases [5]. The value of β we measured agreed well with previous results
using a diﬀerent atom, 6 Li [43, 56, 45, 35, 57, 58, 59], as well as many diﬀerent
theoretical approaches [6, 60, 61, 62, 63, 64, 65, 66, 67]. This agreement showedthat although the microscopics of two atomic Fermi gases may be quite diﬀerent
they are indeed probing the same universal physics. More importantly, it meansthat we can invoke universality to apply our results to a wide class of problems.
The measurement of β, as well as every other previous experiment in atomic
Fermi gases, was a probe of the Fermi gas as a whole. Indeed, many of the earlyexperiments on Fermi gases were probing macroscopic quantities of the system.
On one hand, macroscopic quantities are already very interesting. For example, aprecise measurement of β or collective oscillations allows the delineation between
diﬀerent many-body theories. On the other hand, if one could measure microscopic
quantities of the system then one could have a much more stringent test of manybody theories. This goal was the driving motivation for the second experiment I
will describe.
Let us take for example measurements in condensed matter systems. In
condensed matter systems, if one can make a large variety of bulk measurements
on a material (thermal and electrical conductivity, transition temperature, heat
capacity, and many others) then there are enough constraints to say something
meaningful about the system. However, in condensed matter physics, there also
exists a measurement that has greatly improved the understanding of the microscopic quantities. Using angle resolved photoemission spectroscopy (ARPES) [68],experimenters have been able to map out the spectral function for many interesting strongly interacting systems and determine quantities such as the dispersion,

9quasi-particle lifetime, energy gaps, etc. This technique is particularly powerful
because once the spectral function is known these microscopic quantities can bedirectly compared to theory at a very fundamental level. Additionally, one can inturn determine many useful macroscopic quantities.
Using the condensed matter paradigm, it was natural to consider designingan ARPES type experiment for our atomic Fermi gas. In the second experiment
I will discuss, we have implemented an ARPES technique for our Fermi gas andindeed we have uncovered valuable microscopic information about our system[69]. At the time of the writing of this thesis, the ARPES result we published
is less than one year old. However, the microscopic information we obtained has
already generated substantial theoretical interest and our data is being used tovalidate theories and point out possible discrepancies [70]. Also, due to the broad
applicability of our new technique it will be exciting to see it implemented into
other atomic gas experiments, including interacting bosons, optical lattices, spinimbalanced systems, rotating systems, lower dimensional systems, etc. If we again
look to the condensed matter experiments, ARPES can be used to study dynamics
as well. Therefore, I suspect the ARPES technique described in this thesis will
be a very valuable tool for many years to come.
1.4

Thesis Outline
The two main topics that I will discuss in this thesis (universality in Fermi

gases and photoemission spectroscopy for Fermi gases) were described in Ref. [55]and Ref. [69], respectively. I will begin, in Chapter 2, by brieﬂy describing theexperiment techniques and the atomic physics used to create an ultra cold Fermi
gas with strong interactions. In Chapter 3, I will discuss signiﬁcant changes in
the apparatus that have occurred during my tenure. Chapter 4 will describe universality in Fermi gases and our measurement of the parameter β. The rest of the10thesis will cover radio frequency (RF) spectroscopy and ARPES for Fermi gases.
In Chapter 5, I will give a historical review of radio-frequency (RF) spectroscopy
and the lessons learned. Chapter 6 will describe ARPES and our experiments
on atomic Fermi gases. Finally, in Chapter 7, I will conclude and remark about
future prospects.

Chapter 2
Creating an ultra cold Fermi gas with strong interactions

In this chapter I will discuss both the machinery needed to create a cold gas
as well as the atomic physics used to create an interacting sample. For a more complete account of the cooling machinery I recommend the thesis of Brian DeMarco[50]. For a more complete account of the physics of Fano-Feshbach resonances and
BCS-BEC crossover physics I recommend the thesis of Cindy Regal [47]. My goal
here is simply to cover the basics needed for creating a cold interacting sample andto cover aspects of the experiment I think are particularly interesting or relevant
for the rest of this thesis. Signiﬁcant changes to the experimental apparatus are
discussed in Chapter 3.
2.1

Creating a cold sample
Brieﬂy, the procedure for creating a cold interacting Fermi gas in our exper-

iment begins by collecting atoms in magneto optical traps (MOTs). From here,
the atoms are loaded into a magnetic trap for a ﬁrst stage of evaporative cooling.
Finally, the atoms are transferred into an optical trap for a ﬁnal stage of evaporative cooling. Once we have conducted an experiment, we extract informationfrom the system by letting the gas expand from the optical trap and destructively
imaging the atoms. I will now go into more detail to give a brief account of thetrip an atom takes during one of our two minute experiment cycles.

12
2.1.1

MOTs and magnetic trap evaporation

As I mentioned earlier, many of the cooling techniques used in this experiment are re-creations of the techniques used to make the original BEC here
at JILA. The double-MOT setup for this experiment is no exception [71]. The
double-MOT takes advantage of two diﬀerent vacuum chambers that are connected by a transfer tube. One vacuum chamber, the collection chamber, has ahigher pressure due to the atomic sources that allow for high atom number capture. Then, the trapped atoms are transferred into a second chamber throughthe transfer tube. In the new chamber, the science chamber, the pressure is much
lower allowing for trapped atom lifetimes on the order of 100 seconds. It is in thisnew chamber that the rest of the experiment takes place.
Atoms from the science chamber MOT are then transferred into a magnetic
trap after using optical pumping to ensure that the atoms are in magneticallytrappable spin states. Evaporative cooling is performed on the mixture of two
spin states by removing the most energetic atoms via microwave spin-ﬂip transitions to magnetically untrapped states. As the cloud rethermalizes to a lower
temperature, the microwave frequency is reduced to continue removing the mostenergetic atoms. This process continues until we have a near quantum degenerategas. At the end of this stage of the experiment, which takes on order 60 seconds,
we have approximately ten million atoms at 5-7 µK. The degeneracy of the cloud,expressed as a ratio of the temperature to the Fermi temperature (TF = EF /kb ,where kb is Boltzmann’s constant) is around T /TF = 3.
2.1.2

Optical trap evaporation

Once we have achieved these conditions in the magnetic trap we transfer
the atoms into a far-oﬀ-resonance optical dipole trap. The motivation behind13this is three-fold. Firstly, the Fano-Feshbach resonance requires the atoms to bein particular Zeeman states. The exact states depend on the particular FanoFeshbach resonance and atoms in these states are not necessarily magneticallytrappable. Also, manipulating the atom-atom interactions near the Fano-Feshbachresonance requires one to tune the magnetic ﬁeld. This can be complicated if one istrying to simultaneously trap the atoms using spatially inhomogeneous magnetic
ﬁelds. More ideal is to optically trap the atoms and then apply a simple uniform
magnetic ﬁeld. Lastly, we have found that lower temperatures can be achieved if
we ﬁnish the evaporation using optical trap evaporation as compared to magnetic
trap evaporation. The most obvious reason is that the heating rate in the opticaltrap is very low, around 5 nK/s [47].
We achieve evaporation in the optical trap by reducing the power in the
optical trap beam. A nice explanation of optical traps can be found in Ref.
[72]. Reducing the power lowers the depth of the trapping potential the atoms
experience. In this way, a similar forced evaporation occurs as in the magnetic
evaporation stage because the most energetic atoms spill over the top of the potential and leave the trap. Evaporation in the optical trap is also much quicker
due to the higher trapping frequencies. We evaporate the atoms transferred into
the optical trap in about ten seconds and can typically produce a cloud of 105
atoms per spin state at a degeneracy of T /TF = 0.10(1).
We conduct evaporation in the optical trap with at mixture of atoms in two
Zeeman states, the |f, mf ⟩ = |9/2, −9/2⟩ state and the |9/2, −7/2⟩ state, where
f is the total atomic spin and mf is the magnetic quantum number. These states
are stable against spin-changing collisions and with them we reach our coldest
temperatures. We conduct the majority of the optical trap evaporation at highmagnetic ﬁeld. Typically, we set the magnetic ﬁeld at 205.7 G (on the atom side of
the |9/2, −9/2⟩ + |9/2, −7/2⟩ Fano-Feshbach resonance located at 202.1 G) where

14
the s-wave scattering length is approximately as = −800a0 , where a0 is the Bohrradius. The large magnitude scattering length provides a large scattering crosssection without a large inelastic loss rate (at our densities, which are typicallybetween 1012 cm−3 and 1013 cm−3 ). In addition, at this magnetic ﬁeld we canthen easily access the Fano-Feshbach resonance.

2.1.3

Absorption imaging and thermometry

Absorption imaging is the method by which we obtain nearly all the information we gather from the atoms. Once the ﬁnal stage of evaporation in the
optical trap has occurred and we have performed an experiment, we turn oﬀ theoptical trap and let the gas ballistically expand for approximately ten milliseconds. We then apply a short resonant laser pulse to the atoms. Atoms in thecloud scatter photons and we capture the shadow of the atoms on a CCD camera.
This type of imaging is referred to as time-of-ﬂight absorption imaging.
Our imaging is done almost exclusively at magnetic ﬁelds near the FanoFeshbach resonance, around 200 G, or “high-ﬁeld imaging” as we call it. At theseﬁeld strengths the transitions for imaging diﬀerent Zeeman states are far enoughseparated that we can image spin-selectively. Typically, we are able to nearly
simultaneously image atoms in the two spin states for each experiment. To achieve
the highest signal-to-noise, we try to always image atoms in the |9/2, −9/2⟩ state.
Imaging atoms in the |9/2, −9/2⟩ state gives the best signal-to-noise ratio because
there is a closed cycling transition to the |11/2, −11/2⟩ excited state.
Because we image the gas after ballistic expansion, we can obtain information about the atoms’ momenta. In the simplest case of non-interacting atoms,
an atom continues in the trajectory it had at the moment the trap was turned oﬀ(modiﬁed by gravity of course). Thus, for suﬃciently long expansion times, the absorption image yields the momentum distribution. We then apply a least-squares

15
surface ﬁt to the expected Fermi-Dirac momentum distribution to determine thenumber of atoms, their temperature and quantum degeneracy (T /TF ).
Ultracold Fermi gases have some intriguing behavior that make determining
the temperature challenging. For example, for temperatures higher than about
0.5 TF the Fermi-Dirac distribution is well approximated by a classical distribution, and deviations from this classical distribution are small and hard to extract.
Of course, in this limit, the temperature can be extracted from the width of thecloud. On the other hand, for temperatures lower than about 0.1 TF , the momentum distribution has approached the zero temperature limit where the width ofthe distribution reﬂects only the Fermi energy. In this limit, any deviations fromﬁnite temperature occur right around the Fermi energy and are hard to detect inthe 3D trapped distribution.
A substantial portion of my early lab work was set toward understanding
temperature in these ultra-cold Fermi gases and trying to design new thermometers. I composed a simulation to understand the fundamental limits of our imaging
and ﬁtting routines. I also attempted to use condensate fraction as a thermometerwith limited success. Finally, I attempted a clever technique mentioned in Cindy
Regal’s thesis, which was to use a third fermion as a thermometer. Again, thistechnique had limited success. Since I ﬁnd this topic interesting and the results of
my ﬁndings were not trivial (although not that applicable to the rest of this thesis)
I include additional information in Appendix A where I discuss thermometry ofultracold Fermi gases.
2.2

Controlling interactions
In this section I will discuss how we manipulate the interactions between

particles in our experiment. As I mentioned earlier, the use of Fano-Feshbachresonances was already a fairly standard technique by the time I began my thesis

16work. As such, it will not be my goal here to give a complete overview of FanoFeshbach resonances, but simply to outline their basic properties and how we usethem in the experiment. This section will also be useful for understanding someof the terminology for the rest of the thesis. In particular, the way we characterize
the interactions will be discussed. Finally, I will brieﬂy explain how Fano-Feshbachresonances are used in the study of the BCS-BEC crossover.

2.2.1

Fano-Feshbach resonances

Fano-Feshbach resonances [73, 74, 75] are used to vary the interactions between particles in our system. As we learned in the previous section, these gases
are in the ultracold regime and therefore are dominated by s-wave collisions. As
such, the interactions can be understood in terms of the s-wave collision cross section, σ = 4πa2 , where a is the s-wave scattering length. To change the interactions
we need to change a.
Away from a Fano-Feshbach resonance the background scattering length in40

K is abg = 174a0 [31], where a0 is the Bohr radius. A Fermi gas at this interaction

strength, for our typical temperatures and densities, is a very weakly interactinggas. Or, to put it another way, the mean-ﬁeld energy, 2~(na), for a gas at thisminteraction strength is on the order of 0.01 of the Fermi energy. Therefore, tocreate a strongly interacting Fermi gas we need a way to change the scattering
length by orders of magnitude.
To give a feel for how a Fano-Feshbach resonance allows us to vary theinteraction length in such a great way, we can solve a related problem we saw
back in our quantum mechanics class (assuming you took Tom DeGrand’s class).
First, however, let us recall some important parameters regarding low energy
scattering. A straight forward case is to imagine two atoms are interacting via asquare well potential of height V0 , see Fig. 2.1. Recall, the wave function rises

17linearly from zero and should be of the form u(r) = sin(kr + δ), where δ is thethe scattering phase shift. Then, in limit of low energy scattering (k → 0), thescattering length a is related to the scattering phase shift and the eﬀective range
r0 of the potential, via
1 1
k cot δ ≈ − + r0 k 2 .
a 2

(2.1)

Consequently, we can now calculate the cross section, using the deﬁnition σ =4π limk→0 |k cot δ − ik|−2 . To ﬁrst order we arrive at the expected cross section
σ = 4πa2 . If the potential V0 is positive, then a linear extrapolation of the wave
function at the barrier would reveal a positive scattering length a, Fig. 2.1a. If
the potential is negative, then a can be either positive, negative, or divergent,depending on the depth of the potential, see Fig. 2.1b-c.
Returning now to our quantum mechanics example, a nice intuition for
Fano-Feshbach resonances can be developed by solving the low energy scattering
problem of a spherical square well with a bound state just below threshold. I will
outline this problem here. Assume the potential has the form V (r) = −V0 for
r < R and V = 0 otherwise. The incoming state is has positive energy, Escatt , so
the wave function isu(r) = sin(Kr)

if r < R sin(kr + δ) if r > R

(2.2)

K
k
where ~2m
= Escatt + V0 and ~2m= Escatt . Now, if we go ahead and assume there
2

2

2 2

is a bound state with energy Eb then its solution would be sin(Gr) if r < R
u(r) =
 e−λr
if r > R

(2.3)

λ
G
where ~2m
= |Eb | and ~2m= V0 −|Eb |. Using the usual matching of wave functions
2 2

2

2

and their derivatives at the boundary conditions will allow the determination ofthese wave functions. However, the trick here is to recall that we are interested

18

b)

a)

U

U
V

0

a

a

r0

r

r

V

0

d)

c)

U

U

a
a

r

r
V

0

V

0

Figure 2.1: Schematic for understanding the scattering length, a. Consider thewave function of two atoms interacting via a simple square well potential (withrange r0 ) as a function of interparticle separation, r. a)Repulsive hard-core potential. V0 suppresses the wave function under it and a > 0. b) An attractive
potential can also give a > 0. c) A weaker attractive potential can give a < 0
as well. d) An intermediate attractive potential can give a divergent scattering
length.

19in the physics when a bound state is just coming through threshold. Therefore,
we want to consider |Eb | → 0 and since we are considering low energy scattering
we also want Escatt → 0. Solving this boundary condition problem (using all theappropriate substitutions and Taylor expansions) we ﬁnd k cot δ = −λ + 21 (λ2 +k 2 )R. We can now directly relate this result to Eqn. 2.1. In this way, we ﬁnd thescattering length to be11= λ − Rλ2 ,
a
2

(2.4)

and R is just the eﬀective range r0 . Now we see that as the bound state approaches threshold, Eb → 0, and correspondingly λ → 0, the scattering length
diverges. Additionally, we ﬁnd the scattering length is positive for a true boundstate. Conversely, a is negative when the bound state is just “above” threshold.
Therefore, assuming we can “tune” the height of the potential we have a technique
for tuning the scattering length.
Although the physics of the Fano-Feshbach resonance is considerably more
complicated, this simple example captures most of the essential features. It is
important to note the main diﬀerences. First, a Fano-Feshbach resonance is a
two-channel eﬀect [76]. This means that a bound state is not coming throughthreshold as we describe above, but rather that a bound state in an energeticallyclosed channel happens to coincide with the energy of two colliding atoms in the
open channel. The second major diﬀerence is that we do not tune the height of thepotential to move the bound state energy but rather change the energy diﬀerence
between the thresholds of the two channels. This is possible because the magnetic
moment of the closed channel bound state is diﬀerent than the magnetic momentof the open channel state. Therefore, we can simply tune the relative energy
diﬀerence by changing the magnetic ﬁeld.
Fig. 2.2 b) shows how we can tune the scattering length with the magnetic

a (104 a 0 )scatteringcross section

20

108

a)

106
104
102
1

b)

0
-1
200

204

208

212

Magnetic Field (G)Figure 2.2: Calculated scattering length and cross-section near a Fano-Feshbachresonance. These are the parameters for the Fano-Feshbach resonance usedthroughout this thesis. The width is 7.8 G, the location of the resonance is 202.1
G (dotted vertical line), and the background scattering length is 174 a0 . The
zero crossing of the scattering length occurs at 209.9 G (dashed vertical line). a)
The scattering cross section (4πa2 ) as a function of magnetic ﬁeld. b) Scattering
length a in units of the Bohr radius, a0 , as a function of magnetic ﬁeld. We cantune the scattering length to arbitrary values using the magnetic ﬁeld.

21
ﬁeld near the Fano-Feshbach resonance [76]. The resonance shown here aﬀects
the scattering between 40 K atoms in the |9/2, −9/2⟩ and |9/2, −7/2⟩ states. Thisis the resonance used throughout this thesis work. The scattering length as afunction of the magnetic ﬁeld, B, is given by the following:(a(B) = abg 1 −w )B − B0(2.5)

where abg = 174a0 is the background scattering length introduced earlier, w = 7.8
G is the width of the resonance, and B0 = 202.1 G is the location of the resonance[31]. It is important to note here that this is a broad Fano-Feshbach resonance[77, 47]. Broad in this sense means that the eﬀective energy width (the range ofcollision energies for which there is resonant scattering) is much bigger than the
Fermi energy of our sample. This is important because it means that when weapproach the Fano-Feshbach resonance the entire sample is interacting resonantly
and not just a portion of a cloud.
Fig. 2.2 a) shows the scattering cross section (4πa2 ) over the same span ofmagnetic ﬁeld. As expected, the cross section becomes very large near the FanoFeshbach resonance (vertical dotted line). However, one of the unique properties of40

K is that it has an easily accessible zero crossing of the Fano-Feshbach resonance(vertical dashed line), located at 209.9 G, see Eqn. 2.5. It is extremely useful tobe able to systematically compare properties of a strongly interacting gas to a
non-interacting gas. We will take advantage of this fact throughout this work andespecially when we study universality in Chapter 4. In contrast, in 6 Li, there isno nearby zero crossing of the Fano-Feshbach resonance.
Fano-Feshbach resonances really opened the way to study new physics in
Fermi gas systems. In addition to superﬂuidity, they allow the study of the BCSBEC crossover, which we will discuss in the next section. However, Fano-Feshbachresonances have simultaneously found their way into the study of bosons, inter-22
species mixtures, Bose-Fermi mixtures, atoms in optical lattices, etc. For now,
they are the way to control interactions in atomic systems. There are also interesting Fano-Feshbach resonance physics on the horizon. To name one example, it
may be possible to use external ﬁelds to modify the properties of some resonances,giving one the ability to tune the properties of the resonance [78, 79, 80].

2.2.2

BCS-BEC crossover

The goal of the work presented in this thesis is to use atomic Fermi gases tostudy the general problem of strongly interacting Fermi systems. That being said,
these experiments inherently probe the physics of the BCS-BEC crossover and we
often compare to BCS-BEC crossover theories. Further, many atomic Fermi gas
experiments of the last few years studied BCS-BEC crossover physics, for example
see the theses of C. A. Regal [47], M. W. Zwierlein [48], and G. B. Partridge [49].
As such, most of our terminology and intuition for the happenings of atoms andpairs near the Fano-Feshbach resonance is built upon the studies of BCS-BEC
crossover physics. Therefore, in this section I will discuss the BCS-BEC crossoverand its connection to Fano-Feshbach resonance physics. Without repeating too
much that is already discussed in the theses listed above I will use this section to
introduce how we have come to understand interactions and pairing in a stronglyinteracting Fermi gas.
We learned in the previous section that we have control of interactions
through the use of a Fano-Feshbach resonance. To develop an intuition for BCSBEC crossover physics we will follow a thought exercise similar to A. J. Leggett’s
1980 paper [40]. Imagine we begin with a zero temperature Fermi gas at the zerocrossing of the Fano-Feshbach resonance. This is the textbook Fermi gas withevery state ﬁlled to form a perfect Fermi surface. Now, as we move to lower magnetic ﬁeld we begin to turn on arbitrarily small attractive interactions (see Fig.

23
2.2 b).
Notice that these are the exact requirements for BCS theory. Recall BCS
theory is the theory developed by Bardeen, Cooper and Schrieﬀer [11, 12, 13]that successfully explained superconductivity in electron systems. BCS theory
is based on an observation made by Cooper that a spin-up electron and a spindown electron in the presence of a well deﬁned Fermi surface, and an attractive
interaction, can pair up and lower their energy. These pairs of fermions (called
Cooper pairs) behave as a composite boson that can Bose condense. One shouldnote here that the pairing between fermions in BCS theory is in momentum space.
Or, in other words, the pair size is much bigger than the interparticle spacing.
Also, for completeness, the presence of the Fermi surface is required and this istruly a many-body eﬀect (two fermions in vacuum will not form a bound state on
their own).
Continuing on with our gedanken experiment, if we adiabatically decrease
our magnetic ﬁeld to well below the location of the Fano-Feshbach resonancethen there exists a closed channel bound state. Here, our spin-up and spin-down
fermions are paired as a bound molecule. Note that this eﬀect is relevant for atoms
and would not be possible for electrons. This composite boson will undergo Bose
condensation. In contrast to BCS physics, this pairing mechanism is no longer amany-body eﬀect (or rather, it is not density dependent) and this pair would exist
even in vacuum. Also, in contrast to the BCS state, the pairing is in real spacesince the pair size is much smaller than interparticle spacing. Another useful way
to understand the pairing is in terms of interaction strength. We know that as
we decrease the magnetic ﬁeld we increase the attractive interaction between thefermions. In essence, we have increased the attractive interaction between the two
fermions so much that they have become paired.
Now, for our gedanken experiment, we want to know what happens in be-Diatomic
Molecules

Strongly
Interacting
Pairs

24

Cooper
Pairs

kF

BCS

BEC
Crossover region

Figure 2.3: Cartoon drawing of pairing throughout the crossover region. In the
BEC limit, fermions are tightly bound in molecules. In the BCS limit, pairing isa many-body eﬀect occurring in momentum space. At the cusp of the crossover
region, pairs have characteristics of both Cooper pairs and molecules, and pairing
is a many-body eﬀect.

tween these two limits. We know that at T = 0 on one side of the resonance wehave a BCS superﬂuid, and on the other side of the resonance we have a BEC ofbound pairs. The essence of the BSC-BEC crossover is that these two limits arevery much linked and one does not pass through a phase transition to get from
one to the other [39, 40, 41]. By increasing interactions one can smoothly andcontinuously move from a many-body pairing with a large pair size into a boundpair with a small pair size. The cusp of the BCS-BEC crossover occurs near the
Fano-Feshbach resonance, where the binding energy of the two-body bound stategoes to zero. This is a very interesting region. The pair size is very small ascompared to Cooper pairs but very large as compared to molecules. The pair size
is on the order of the interparticle spacing and the pairing would not be possible
without the presence of the other fermions (it is a many-body eﬀect). Using ourunderstanding from the previous section, the cusp of the crossover occurs near the
point where the scattering length diverges (i.e., the location of the Fano-Feshbachresonance). Our standard cartoon drawing of pairing in these three regions, theBCS, BEC, and cusp of the crossover, are shown in Fig. 2.3.

25
The crossover region occurs when interactions between two fermions are toostrong to be described by BCS theory and yet too small to form tightly boundmolecules that can be described by BEC theory. A convenient way to characterize
the interaction strength in a Fermi gas is to compare the scattering length a to√
the Fermi wave vector kF = 2mEF /~. Recall in a homogeneous Fermi gas that~
the Fermi energy is related to the density via EF = 2m
(6π 2 n)2/3 . Therefore, kF is
2

related to the inverse interparticle distance and (kF a)−1 is essentially a comparisonof the scattering length to the interparticle distance. In addition, (kF a)−1 is thedimensionless quantity that characterizes interaction strength used in most BCSBEC crossover theories [40, 41]. The BCS limit, where the interaction is weak andattractive, occurs for (kF a)−1 ≪ −1. Conversely, the BEC limit of tightly boundmolecules occurs for (kF a)−1 ≫ +1. The crossover region occurs for interaction
strengths not covered by either theory, and one ﬁnds signiﬁcant deviations fromeither theory in the vicinity of −1 < (kF a)−1 < +1 [16, 17]. To give a feel forthe span of the BCS-BEC crossover in our experimental parameters, the crossoveris approximately a 1 G region surrounding the Fano-Feshbach resonance position(see Fig. 2.2 b).
In this thesis I have not tried to give an exhaustive review of superﬂuidity.
However, there is one important aspect of superﬂuidity that I will cover brieﬂy,
which is the gap parameter. The gap will become important later in the thesis when we discuss angle resolved photoemission spectroscopy (ARPES). BCS
superﬂuids exhibit their “superness” because they require a minimum amount ofenergy to produce an excitation. This minimum energy required to produce an excitation, referred to as the gap energy, 2∆, eﬀectively protects the system against
arbitrarily small excitations. For easy reference I repeat a ﬁgure from the thesisof Cindy Regal that qualitatively shows the gap ∆ (and the chemical potentialµ) as a function of (kF a)−1 [47], see Fig. 2.4. In the BCS limit, the gap falls

26
oﬀ exponentially with interaction strength, ∆ = EF e−π/kF a . In the BEC limit,
the energy needed to break apart a pair is the binding energy and ∆ is half thebinding energy. Again, the interesting regime is near the cusp of the crossover
where many-body eﬀects continue to play a role in the pairing and so we expect∆ ∝ EF . As can be seen from Fig. 2.4, the gap in this regime is a large fraction of
the Fermi energy. Recent Quantum Monte Carlo results put the zero temperature
gap at (0.45 ± 0.05)EF [81], and recent experiments also show the gap to be alarge portion of the Fermi energy [69, 82]. This is a very large gap as compared
to condensed matter superconductors where the gap is ∼ 10−4 EF .
There is one ﬁnal topic I would like to cover regarding the BCS-BEC
crossover physics, which is temperature. It was instructive in our gedanken experiment to assume a T = 0 gas. In reality, our Fermi gas is not at zero temperature,
which means we cannot access the superﬂuid state for all interaction strengths.
This can especially be seen in the BCS limit where the transition temperature, Tc ,depends exponentially on the interaction strength, Tc ∝ TF e−π/2kF a [83]. In fact,
we cannot experimentally achieve temperatures much below 0.1 TF and therefore
cannot observe a phase transition for values of (kF a)−1 less than about −1. In
the BEC limit, we are simply condensing bosons and the transition temperature
is density independent and can be determined from straight forward BEC theory.
Again, throughout the crossover there is a smooth connection of the transition
temperature from the BCS to BEC limits. For reference, in Fig. 2.5, I adapt a
ﬁgure from Ref. [84] showing a sketch of the expected transition temperature.
While discussing temperature in the presence of strong interactions, there is
yet another temperature scale which is referred to as the pairing temperature, T ∗ .
In conventional superconductivity, pairing and superconductivity occur at thesame temperature. In these strongly interacting systems, however, pairing andsuperﬂuidity can occur at diﬀerent temperatures. This is because the very strong

27

2.0

BEC limit
BCS limit

m/EF

D/EF

1.5
1.0
0.5
0

2

1

0

-1

-2

1
0
-1
-2
-3
-4
-5
-6

2

1

0

-1

-2

1/kFa

1/kFa

Figure 2.4: The gap ∆ and chemical potential µ as predicted by NSR theory.
On the BCS side the gap falls oﬀ exponentially with interaction strength. Fromcomparison to the red and blue curves (BCS theory and BEC theory) we see thatthe crossover region indeed occurs from −1 < (kF a)−1 < 1. Figure from Cindy
Regal’s thesis [47].

Tc /TF

TBEC

1

0

-1

(k F a )-1

Figure 2.5: A sketch of the expected superﬂuid transition temperature as a function of (kF a)−1 . On the BCS side the transition temperature falls of exponentially.
Approaching the BEC limit the transition temperature approaches normal BEC
transition temperature. The two limits are connected smoothly although the transition temperature is highest at the location of the Fano-Feshbach resonance.

28
interactions allow for incoherent pairing correlations at temperatures above thesuperﬂuid transition temperature [85, 86, 87, 88, 89, 90, 91, 92, 93]. The region
between the pairing temperature and the superﬂuid transition temperature isreferred to as the pseudogap phase. In the pseudogap phase the Fermi gas retains
some of the characteristic superﬂuid signatures (such as a gap in the spectral
function) and yet it is not a superﬂuid. We will study the pseudogap phase in
Chapter 6.
Throughout this chapter we have noted that the region of (kF a)−1 near the
Fano-Feshbach resonance holds some of the richest physics. A careful observer
might question why the atomic cloud does not collapse or explode near the FanoFeshbach resonance. After all, according to Fig. 2.4 b), the two-body scattering
length is diverging near the resonance. Indeed, the Fermi gas does not implode or
explode near this region but rather its density evolves smoothly from one side of
the resonance to the other. In chapter 4 we explore why the Fermi gas is so well
behaved in this regime, which will in turn help us to understand and introduce
universality in strongly interacting Fermi systems.

Chapter 3
Apparatus

The apparatus needed to cool our sample of 40 K atoms to ultracold temperatures is a collection of lasers, hundreds of optics, vacuum equipment, atomic
sources, magnetic ﬁelds, control computers, and more. Many of the core pieces,such as the vacuum equipment and magnetic coils, have remained untouched forover ten years. I bring this up because although servos have been modiﬁed andoptical beam paths have changed many times, the core of this experiment was
built extremely well and provides an awesome foundation to build upon. Beginning with a solid foundation (along with 10+ years of tweaking laser locks andevaporation trajectories) has led to an extremely robust experiment. Assuming
the gremlins did not steal into the lab during the night and move some mirrors,
we can regularly produce a degenerate Fermi gas within thirty minutes of turningon the experiment. It is not uncommon for the experiment to continue to operate
well into the night, if needed, with minimal adjustment during the day.
The core components of this experiment are covered in detail in the thesesof Brian DeMarco [50] and Cindy Regal [47]. In some ways, almost all of thechanges to the experiment during my tenure were to simplify and modernize theexperiment. These changes include, but are not limited to: upgrading to a newcomputer control system giving us better timing and voltage control, new RF
delivery schemes, new and more powerful lasers, fancier and more robust locking

30
schemes, higher resolution imaging, etc. Here, I will highlight a few of the mostsigniﬁcant changes including our implementation of a crossed dipole trap and newcomputer control. I will also discuss the vacuum pressure of our system, which has
become a bit of an issue. My feeling leaving this experiment is that my changes
have resulted in an experiment that is considerably simpler and less intimidating.
3.1

Crossed optical dipole trap
As discussed in Chapter 2, the ﬁnal stage of evaporation in our systemoccurs in a far oﬀ resonant optical dipole trap. All of the work in this thesis wasdone in an optical trap potential slightly diﬀerent from that described in Cindy
Regal’s thesis. In an eﬀort to improve axial thermalization times we introduced
another optical dipole trap beam in an orthogonal direction to the original trap
beam. This is often called a crossed dipole trap. Our trap now consists of two
focused beams with waists of approximately 30µm × 30µm along the horizontal
direction and 200µm × 200µm along the vertical direction. This new verticalbeam works to eﬀectively cap-oﬀ the large Raleigh range of the main conﬁning
potential and reduce the aspect ratio. For typical conditions, the trap frequencies
are ωr /2π ≈ 200 Hz along the radial direction and ωz /2π ≈ 20 Hz along the axial
direction. This is approximately an order of magnitude reduction in the trap
aspect ratio as compared to the previous setup.
Imperative to the implementation of our new optical trap setup was thecareful designing of the new geometries and powers. In an atomic Fermi gas,
there are many considerations for cooling such as density, trap frequencies, trap
depth, single-photon scattering, etc. Of particular help has been a Mathematica
notebook that was started by Cindy Regal and brought to its ﬁnal form by me. In
this notebook, we address many of the concerns for cooling a Fermi gas in orderto design a great optical trap the ﬁrst time. A 3D plot from our Mathematica

31
notebook for our new crossed dipole trap is shown in Fig. 3.1. As I understandit, this notebook has become a standard for optical trap design in JILA.
The laser light is provided by a 20W IPG ﬁber laser (which really appears tojust be a ﬁber-ampliﬁed diode laser) which operates single-mode at 1064nm withan 100 kHz linewidth. (As a side note, while our laser has worked ﬁne for years
now, other IPG single-frequency lasers around JILA and other labs around theworld have been notoriously unreliable and I would probably not recommend thislaser.) The light is split into two beams and each beam is individually intensity
controlled using acoustic optical modulators (AOMs). The AOMs are then used
to ramp down the laser power for evaporation. Only when a signiﬁcant portion of
the optical trap evaporation has occurred is the vertical optical trap applied. Thisis because the vertical trap has a relatively weak intensity and only contributes
signiﬁcantly to the trap strength near the end of evaporation.
3.2

Computer control
Ultra cold atomic gas experiments are suﬃciently complicated that theyrequire computer control. Additionally, since every experiment eﬀectively destroys
the sample, we require the sort of reproducibility that only a computer can provide.
The experiment, as it was handed to me, ran on an old computer that did noteven boot into Windows. Instead, the control code was written in QBasic and ran
from DOS. The two major faults with this setup were that because the computer
was so old, replacement parts could only be found on eBay. And, because theequipment was so old, we were at capacity for its capabilities. Newer systems
have better timing, more voltage control, and more channels. The downside ofmost newer systems is that they tend to be programmed in LabView, which canbecome very tedious and slow with such complicated experiments. The question
was then: can we keep the low-level programming we like (Basic) while upgrading

32

Y
z
x
Figure 3.1: A 3D plot of our crossed optical dipole trap. The potential is in µK
and the axes are in meters. The second optical trap (vertical) acts to cap oﬀ thelarge Raleigh range of the main conﬁning trap (horizontal).

33
the hardware?
We decided to go with Visual Basic because the old QBasic syntax could becopy and pasted into the newer version. The main challenge was to modify theQBasic code to talk to the new digital (NI PXI DIO64) and analog boards (NIPXI 6733) in Visual Basic. This hard work was initiated by Travis Nicholson andﬁnished by Tara Drake. We also decided to go with a PXI chassis (NI PXI-10420)
from National Instruments, into which the analog and digital boards are inserted.
In this way, the chassis is connected to the computer via a single board and it will
be easy to replace a computer if it breaks or parts become outdated.
Fig. 3.2 is a picture of the control setup during the switch from old to new.
The process was quite tedious but it paid oﬀ. With the new control computer
setup we have approximately 50% more digital and analog lines. We have also
gained about a factor of thirty in timing and voltage resolution. The new control
computer has such better capabilities that we removed about six diﬀerent commercial pulse generators (and the sub-routines to talk to them) and numerous
JILA-made pulse-summers, inverters, and pulse generators. I was able to remove
hundreds of meters of BNC cables, power cables and GPIB cables and correspondingly able to remove hundreds of lines of control code. All in all, the upgrade to anewer control system has been a great success and resulted in an experiment thatis a lot less intimidating and much more transparent. Our experimental control
computer and code is so reliable that we typically go months without restartingthe code or the computer. I suspect that this system will also become a new
standard for computer control in JILA.
3.3

Vacuum pressure
Over the last couple years we have been having problems with the vacuum

pressure in the science cell. Because we do not have a vacuum gauge in our system,

34

Figure 3.2: A picture of our computer control setup while we were in the middleof switching to our new setup. Switching to a newer setup was a daunting task
but thorough preparation resulted in less than one week of downtime.

35
we use the atoms to determine to the vacuum pressure. This is typically done bymeasuring the “lifetime” of atoms held in the magnetic trap, using either MOT
recapture or the magnetic trap after evaporation. In atomic systems, the lifetimeof the atoms is either long enough or not long enough. In other words, only when
the lifetime of trapped atoms becomes comparable to evaporation timescales doesit pose a problem.
In Fig. 3.3, I plot the atom lifetime as measured with MOT recapture (ormagnetic trap) since the ﬁrst measurement in the early months of 1998. In theearly days, atom lifetimes on the order of a few hundred seconds were regularly
measured. In fact, there are spans of many months in which a lifetime was nevermeasured because it was always good enough. As time went on, there were downturns in the lifetime but these were usually promptly ﬁxed by either switching
to a new getter or ﬁring the ti-sub pump and even turning oﬀ one of the ion
pumps. However, there has been a steady decrease in the atom lifetime in thelast few years, especially during my tenure. The horizontal dashed line indicates atypical evaporation timescale. As all previous measurements of the lifetime wereabove this line there was never much concern for the vacuum. However, as themost recent lifetime measurements are approaching this timescale we have trouble
evaporating to quantum degeneracy and it becomes a problem we have address.
We have three pumps in our vacuum system: two ion pumps (a 40 L/s
and a 20 L/s) and a titanium-sublimation (ti-sub) pump. As of a few years
ago (before I came to the lab), the 20 L/s ion pump was turned oﬀ because
it appeared to be hurting the atom lifetime. So, we are left with a single ionpump (which is continually on) and the ti-sub pump. Whereas in the theses ofBrian DeMarco and Cindy Regal the ti-sub pump was ﬁred less than once a year,
we ﬁnd ourselves ﬁring every few months now, usually with mixed results. The
options I see for improving the vacuum lifetime are either turning on the ion

36

Science cell vacuum
lifetime (s)

600
500
400
300
200
100
0

0 2 4 6 8 10 12time (years)

Figure 3.3: The atom lifetime in the science cell. Determined from a combinationof measurements including MOT recapture and holding at the end of the magnetic
trap. The vacuum was ﬁrst pumped down and baked in the early months of 1998[50]. Since then, the lifetime in the science cell has slowly drifted down. The
dashed line indicates a typical evaporation timescale. If the lifetime dips below
the dashed line we will have diﬃculty evaporating to quantum degeneracy.

37
pump which has been decommissioned, or doing a bake of the vacuum system. A
bake is particularly challenging in our science chamber where the magnetic trap
is basically a permanent ﬁxture of the vacuum. Thus, a bake would need to be
mild and probably of a long duration to be eﬀective at removing contaminatesfrom the cell walls. If the contaminates are primarily 40 K atoms then this type ofbake could have some nice improvement to the atom lifetime since alkali atoms
have weak bonding with the cell walls.
In conclusion, I cannot say enough good things about the experimental apparatus I have had the pleasure to work with. It is a robust experiment that has
produced some amazing results over the years. I suspect this is due to the strong
foundation and relative simplicity of the experiment (only one atomic species).
Assuming that the getters continue to produce 40 K and the vacuum lifetime problems can be kept at bay, this experiment should continue to produce stronglyinteracting degenerate Fermi gases for some time to come.

Chapter 4
Universality in Fermi gases

Universal behavior in physics is often very powerful. In principle, by studying just one system, one can learn about every other system for which the universal
behavior applies. Universal, as deﬁned here, will refer to behavior that does notdepend on the microscopic details of the interactions, and instead only depends on
density and temperature. In this chapter, we will study universality that occurs in
the limit of very strong interactions. In this limit, in which the length which characterizes the interaction between particles is much larger than the interparticle
spacing, fermion systems display universal behavior and universal thermodynamics. This universal regime is related to nuclear and neutron matter and is also
expected to be found in the quark-gluon plasma. Demonstrating universal behavior means showing that the physics we observe is not speciﬁc to our atomic
species. The experiment shown here serves as a good example of how studying
universal properties an atomic Fermi gas can inform us about an entire class ofproblems.
4.1

The only scale: EF
Generically, as a gas of fermions is cooled from the classical regime to quan-tum degeneracy, the Pauli exclusion principle becomes manifest in the propertiesof the ultracold gas [26, 94]. This is evident in a conﬁned zero-temperature ideal

39
Fermi gas as a ﬁnite energy and a ﬁnite size. The Pauli exclusion principle acts as“pressure,” called the Fermi pressure, and is responsible for the stability of whitedwarf and neutron stars. In this chapter, we investigate what happens if we begin
with a quantum degenerate non-interacting Fermi gas and then adiabatically turnon attractive interactions.
We can characterize the interactions between two fermions using the twobody scattering length, a. It is instructive to imagine we can then tune a however
we like, for example by using a Fano-Feshbach resonance. If we tune the scattering
length to be negative, corresponding to attractive interactions, one would expectthat the gas should be compressed due to attractive interactions and eventuallypairing eﬀects [95]. As the magnitude of the negative scattering length is increased,
and the interactions become arbitrarily strong and attractive, one might expect
the trapped gas to collapse to high density, or implode. In fact, the Pauli exclusion
principle prevents the gas from imploding and the Fermi gas exhibits universal
behavior [5, 3, 8].
We can gain some insight into the eﬀect of diverging interactions by considering a simple mean-ﬁeld approach. While this approach neglects pairing andtherefore is not suﬃcient to fully describe the behavior, it provides a ﬂavor of how
the Fermi gas is aﬀected by interactions. Following the argument outlined in Ref.
[96], the equation of state for a conﬁned zero-temperature Fermi gas isµ = ϵF (x) + UM F (x) + Utrap (x),(4.1)

where µ is the chemical potential, ϵF (x) is the local Fermi energy, UM F (x) is
the mean-ﬁeld contribution, and Utrap (x) is the trapping potential. We can relate~~
ϵF (x) = 2m
kF2 (x) to the density n(x) = 6π1 2 kF3 (x) via ϵF (x) = 2m
[6π 2 n(x)]2/3 . The
22interactions appear in the density dependent mean-ﬁeld contribution, UM F (x) =4π~2 an(x).
m40
For the case in which a diverges, the equation given above for UM F (x)becomes unphysical. As a → ±∞, the only ﬁnite energy scale in the problem isthe Fermi energy. To incorporate this we can approximate the scattering length by
an eﬀective scattering length set by the Fermi energy, aef f = −1/kF . Using thissubstitution into UM F (x) we see that the local mean-ﬁeld energy is proportional
to the Fermi energy, and one can deﬁne a constant of proportionality β given by
UM F (x) = βϵF (x) [97, 43]. Now, we arrive at the new equation of state for astrongly interacting Fermi gas:µ = (1 + β)ϵF (x) + Utrap (x).

(4.2)

From this new equation of state it is particularly clear that the interactions have
dropped out of the problem and we are left with a universal result that depends
solely on the Fermi energy [5, 3, 8]. Notice that the new equation of state issimply a re-scaled version of the non-interacting equation of state (β → 0 in the
non-interacting case). Also note that only a zero temperature gas will depend
solely on the Fermi energy. More generally, a strongly interacting Fermi gas will
also depend on the relative temperature T /TF .
In our simple mean-ﬁeld estimate, we can make the substitutions to ﬁnd4ϵF (x), or βM F = −0.42. The negative sign is not obvious fromUM F (x) = − 3π

this approach, but a more sophisticated many-body approach shows the meanﬁeld interaction should be attractive [5]. Theories and Quantum Monte Carlo(QMC) simulations that include the eﬀects of pairing ﬁnd β to be more negative,
or rather, the ground-state energy is even lower than one would predict from thesimple mean-ﬁeld approach. Theoretical and QMC values for β all come in around
β = −0.6 [6, 60, 61, 62, 63, 64, 65, 66, 67].
If one can measure a property of the gas that is related to the equationof state and compare measurements for a strongly interacting gas and a non-

41
interacting gas, then the universal parameter β can be determined. As we learnedin Chapter 2, we have the ability to control the scattering length with a FanoFeshbach resonance and we can study both the strongly interacting and noninteracting Fermi gases. In the next section, I discuss how we measure β, and forthe ﬁrst time its temperature dependence, using our atomic gas.
4.2

Measuring β in an atomic Fermi gas
We learned in the previous section that by studying the equation of state fora Fermi gas with and without strong interactions we can determine the universal
parameter β. To see this more explicitly for the trapped atomic Fermi gas, wewill ﬁrst consider the ideal gas case. We can use the equation of state to express∫ x0
the number of atoms N = 0 F n0 (x)d3 x in terms of the chemical potential µ0 .
Here, x0F is the Fermi radius and the superscript (0 ) refers to a quantity of a
non-interacting Fermi gas. Using (4.1) with UM F (x) = 0 and expressing ϵF (x) interms of n(x), we can solve for the number1
N = 2 4π
6π
0

∫ x0F
0

[
]3/2
1
0
2 2 2mx (µ − mω x ) 2
dx
2
~
2

(4.3)

where Utrap (x) = 12 mω 2 x2 is the trapping potential and m is the mass of 40 K andω is the angular trap frequency for a spherically symmetric trap. At the edge ofthe trap the density goes to zero (n(x0F ) = 0 and µ0 = 21 mω 2 (x0F )2 ) so evaluating(4.3) we ﬁnd

( )3
1 µ0
.
N =
6 ~ω
0

(4.4)

This analysis (determining the number of atoms in terms of the chemical
potential) works similarly in the limit of strong interactions. Using the modiﬁed
equation of state (4.2), we see that the only diﬀerence from the non-interacting
case is that the Fermi energy is multiplied by (1 + β). Thus, we ﬁnd the atom

42
number at unitarity to be
(
N=

1
1+β

)3/2 ( )3
1 µ
.
6 ~ω

(4.5)

Equating the number in both cases,(4.4) and (4.5), we ﬁnd√µ=
1 + β.
µ0(4.6)

Now we have arrived at the relation between the chemical potential for the stronglyinteracting trapped Fermi gas and the universal parameter β.
The question is then, how can we relate the quantity µ/µ0 to a measurable
quantity in Fermi gases. In the non-interacting limit, we can relate the chemical
potential µ0 to the potential energy through the virial theorem. Recall that thevirial theorem states that in a conﬁning potential of the form V (r) = αr2 , then the
average total potential energy Epot is related to the average total kinetic energy
via Ekin = Epot . Further, we know the average total energy is 34 EF for a trapped
Fermi gas in a harmonic trap [50]. Thus, in the non-interacting limit, we can relate0
the potential energy to the chemical potential via Epot
= 83 µ0 . As it turns out, thevirial theorem also holds on resonance as ﬁrst proposed by J. E. Thomas in Ref.
[98]. More recently, a generalized virial theorem for all interaction strengths has
been proposed [99], and on resonance the ideal gas virial theorem is recovered.
Thus, by using the virial theorem for both a weakly and strongly interacting Fermi
gas, we can directly relate the chemical potential to a measurable quantity: thepotential energy.
By the time we began our experiments to determine β, there were several
experiments reporting or inferring β in 6 Li [43, 97, 45, 35, 57, 58, 59]. For the6

Li experiments, the values for β were all coming in around β = −0.6. Whilethe 6 Li values were in good agreement with theory, we felt that a measurement
using a diﬀerent atomic species was essential to demonstrate the universality of43strongly interacting atomic Fermi gases. Since 40 K and 6 Li are diﬀerent atomic
species with diﬀerent atomic interatomic potentials and diﬀerent Fano-Feshbachresonance properties, agreement on β using these two diﬀerence species would bea clear demonstration of universality.
As a bit of a side note, a Fano-Feshbach resonance is not a suﬃcient requirement for realizing the universal regime. In fact, the resonance must be abroad Fano-Feshbach resonance in order for the gas properties to not depend on
the details of the interatomic potential [100, 101, 77, 102]. In other words, good
agreement between the two species would also put to rest some theoretical claims
that the Fano-Feshbach resonance in 40 K is not a wide resonance [103, 104].

4.2.1

β from the potential energy

At the time we set out to measure β in 40 K, measuring the potential energy
seemed the most obvious route since we could make a direct comparison with βobtained in similar measurements done in 6 Li. In addition, the potential energy
can be obtained from a measurement of the Fermi gas in the trap and can be
related directly to the chemical potential via the virial theorem. As it turns out,
Cindy Regal had measured the momentum distribution throughout the BCS-BEC
crossover and we were able to use her measurement to determine β as well. I will
discuss how to extract β from the momentum distribution in 4.2.2. Regardless,our initial experiment was to measure β using the potential energy and I will coverthat approach ﬁrst.
Recall that we can extract β from the ratio of interacting to non-interacting
potential energies. Therefore, to extract β, we must measure the potential energy
after an adiabatic ﬁeld sweep to the Fano-Feshbach resonance and the potential
energy after an adiabatic ﬁeld sweep to the zero crossing of the Fano-Feshbach resonance. Adiabaticity here requires that the time over which we make a signiﬁcant

44change to the scattering length is long compared to the axial trapping period. Thismeans that ideally we want ȧ/a ≪ 2π/ωz , where ωz is the angular trap frequency
in the axial direction. However, the magnetic ﬁeld sweeps cannot be too slow, because we observe signiﬁcant heating rates near the Fano-Feshbach resonance due
to inelastic collisions. For this reason we had to add additional axial conﬁnement
to our optical trap setup, as discussed in Chapter 3. By increasing ωz /2π fromapproximately 2 Hz to 20 Hz, the ramp duration required for adiabaticity was
reduced to tens of milliseconds from hundreds of milliseconds. After each ramp
the Fermi cloud was allowed to expand for a short time, 1.876 ms. During thisshort expansion time there is signiﬁcant expansion in the radial direction, whichhelps to reduce the optical depth of the gas, but negligible expansion in the axial
direction. Therefore, by imaging along one of the radial directions we can probe
the density distribution of trapped gas in the axial direction.
For each absorption image we perform a 2D surface ﬁt to a ﬁnite temperature
Fermi-Dirac function(

OD(y, z) = pk g2

2

2

2σy

2σz

− y2− z 2−ζe)/g2 (−ζ)where ζ, σy , σz , and pk are independent ﬁtting parameters and gn (x) =

(4.7)
∞ k
∑
x
k=1

kn

[50]. This is the expected optical depth (OD) distribution for a non-interacting
cloud both in trap and after expansion. Empirically we ﬁnd that this function also
ﬁts well in the strongly interacting regime. The potential energy of the trapped
gas is obtained from the cloud proﬁle in the axial direction. The potential energy
per particle in the axial direction is given by1
g4 (−ζ)Epot = m ωz2 σz22g3 (−ζ)(4.8)

where m is the mass of 40 K and ωz is the axial trapping frequency.
In previous experiments using 6 Li atoms, the measured cloud sizes and energies were normalized to a calculated value for the non-interacting gas. This can

45
introduce systematic errors because the calculation relies on the atom number andtrap frequencies, which can have systematic errors. The reason that measuring
the non-interacting size or energy is impractical in 6 Li is that the Fano-Feshbachresonance is extremely wide and one would have to move the magnetic ﬁeld hundreds of Gauss to access the zero crossing. In 40 K, the zero crossing is roughly 10
0
G away and we can measure both Epot and Epot
directly. Therefore, systematics indetermining β are substantially reduced. Experiments accessing the zero crossingof the 6 Li have now been accomplished [105].
In Fig. 4.1, I show the measured potential energy ratio as a function of
1/kF0 a, where again a superscript naught indicates a measurement made in the
non-interacting regime. As expected, the data show that as attractive interactions
are increased there is a strong reduction in the potential energy due to a compression of the trapped gas. Strictly speaking, β is only deﬁned at the location of
the resonance, but the potential energy should behave universally as a functionof 1/kF0 a as well. The solid line is a mean-ﬁeld calculation valid for the BCS-sideof the resonance [53] and has good agreement for 1/kF0 a < −1 . Notice, however,that it breaks down as interactions approach the BCS-BEC crossover regime. On
the BEC side of the resonance one would expect that Epot will depend on condensate fraction. For temperatures similar to these experiments we ﬁnd a maximum
condensate fraction of approximately 15% on resonance; this fraction decreases as
detuning from the resonance is increased [30].
The error bars in Fig. 4.1 include statistical uncertainty in repeated measurements as well as an uncertainty due to heating during the magnetic-ﬁeld
ramps. The magnetic-ﬁeld ramps must be suﬃciently slow to be adiabatic; however, heating during the ramp can be a problem for slower ramps. For ﬁnal
magnetic ﬁelds around the resonance and especially on the BEC side of the resonance we would observe some heating which can alter the determination of Epot .

46

0Epot /Epot

1.0
0.8

1.0
0.8

0.6
0.6

0.4

0.4
0.5

0

-5

0

-0.5

-10

-15

-1.0

-20

0

1/kF aFigure 4.1: Measured potential energy, Epot , normalized to the value measured
0in the non-interacting regime, Epot
, versus 1/kF0 a. Here (0 ) denotes a quantity
measured in the non-interacting regime, i.e., at the zero crossing of the FanoFeshbach resonance. The resonance is located at 202.10 ± 0.07 G [30]; the dashedlines show the uncertainty in the resonance location. Data points toward the BCS
limit show good agreement with a zero temperature mean-ﬁeld calculation (solidline). The larger error bars on the BEC side of the resonance reﬂect uncertainties
due to heating of the gas due to inelastic loss. In the strongly interacting region
there exists ±0.1 uncertainty in 1/kF0 a due to uncertainty in the resonance position. (inset) Subset of the data focusing on the strongly interacting region nearresonance.

47
We investigate the dependence of the measured potential energy by varying theduration of a linear ramp. An example of this is shown in Fig. 4.2 where thegas was ramped from the magnetic ﬁeld used for evaporation to the resonanceposition.
To determine the optimum ramp rate, as well as the eﬀect of heating on thepotential energy measurement, we ﬁt data such as that shown in Fig. 4.2 to an
exponential decay (due to adiabaticity) plus linear (due to heating). From the ﬁt
we determine the ﬁnal potential energy of the cloud if heating were not present.
This introduces a correction that is applied to the data and error bars shown in
Fig. 4.1. On the BCS side of the resonance we see little or no heating due tomagnetic-ﬁeld ramps, and the error bars are dominated by shot-to-shot statisticaluncertainty.
0
From the data in Fig. 4.1, we ﬁnd Epot /Epot= 0.70 ± 0.02 on resonance,from which we can use Eq. (4.6) giving β ∗ = −0.51 ± 0.03. Whereas β is normally only deﬁned for T = 0, I introduce (∗ ) to denote that the system is at aﬁnite temperature (T /TF )0 = 0.08 ± 0.01. Including uncertainty in the resonanceposition we ﬁnd β ∗ = −0.51+0.04−0.12 . Error bars include statistical error as well as
the heating eﬀects mentioned above.
At the time we made this measurement it was not clear to us that β ∗ at atemperature of (T /TF )0 = 0.08 ± 0.01 should be the same as zero temperature
β. In fact, this was a point seemingly ignored by all previous measurements. To
determine the temperature dependence of β ∗ we measured the potential energy
ratio at resonance for diﬀerent starting temperatures. In Fig. 4.3, I show the0
as a function of initial temperature (T /TF )0 . Clearly the universal
ratio Epot /Epot

many-body parameter β depends strongly on temperature.
To determine the initial temperature we prepare a gas as described previously and then ramp to the zero crossing of the Fano-Feshbach resonance and ﬁt

48

Epot (arb.)

1.4

1.2

1.0
0

500

1000

Ramp duration (ms)Figure 4.2: Measured potential energy at the Fano-Feshbach resonance versus
magnetic-ﬁeld ramp duration. For very fast ramps, we measure a higher energy
because of nonadiabaticity. For very slow ramps, heating due to inelastic collisions increases the measured energy. We use a simple empirical ﬁt to determine
optimum ramp rate and Epot .

49

1.1

0

Epot /Epot

1.0
0.9
0.8
0.7
0.6
0

0.5

1.0

(T/ TF )0
Figure 4.3: Potential energy Epot normalized to the measured energy in the non0
interacting regime Epotverses the noninteracting gas temperature (T /TF )0 . The
cloud is heated by parametrically modulating the trapping potential. For thesedata the trapping frequencies in the radial direction vary from ∼ 180 to 450 Hz
and in the axial direction from ∼ 18 to 21 Hz.

50
the expanded cloud to Eq. (4.7). We extract (T /TF )0 from the fugacity ζ using
g2 (−ζ) = −(T /TF )−3 /6. We only measure the temperature for a non-interacting
gas because interactions distort the expansion of a strongly interacting gas. However, because we know the temperature for our non-interacting gas and our ﬁeld
ramps are adiabatic, theory can be used to extract the new temperature [63, 106].
To vary the temperature we parametrically modulate the optical trap strength.
To ensure harmonic conﬁnement the trap strength is chosen to be substantiallyhigher than the Fermi energy.
For the purpose of extracting a value for the zero temperature β we ﬁt aquadratic function to the data points below (T /TF )0 = 0.25, from which we ﬁnd
β = −0.54+0.05−0.12 . The error bars reﬂect the uncertainty in the extrapolation toT = 0 and the uncertainty in the resonance position. This value of the universal
many-body parameter β, as well as the value at (T /TF )0 = 0.08, is in good
agreement with the rough value of β = −0.6 found in theory and Monte Carlocalculations. These values are also in good agreement with multiple experimental
reports in 6 Li: β = −0.73+0.12−0.09 , −0.61 ± 0.15, −0.49, and −0.54 ± 0.05 in Refs.
[107], [57], [59], and [58] respectively. In particular, I would like to note the mostaccurate measurement to date from the group of J. E. Thomas at Duke University,where they ﬁnd β = −0.60(2) and −0.62(2) using two diﬀerent techniques [105].
Our measurement of the universal many-body parameter β was the ﬁrst of
its kind in 40 K, and when added to the 6 Li measurements our result clearly demonstrated universality in atomic Fermi gases. We also made the ﬁrst temperaturedependent measurement of β and found that it can depend strongly on temperature, even at low temperatures. Also, the technique used here was expanded upon
to determine the entropy and critical temperature in Ref. [108].

51
4.2.2

β from the kinetic energy

It was not until after we had made our measurement of β using the potential
energy that we realized β could also be determined from a previous result by
Cindy Regal et al [53]. In the previous measurement, the momentum distribution
was mapped out throughout the BCS-BEC crossover and the kinetic energy on
0= 2.3(0.3) [53]. To see how β can beresonance was determined to be Ekin /Ekinderived from the kinetic energy I will brieﬂy sketch out the relation here.
The energy per particle of an ultracold Fermi gas in the universal regime
scales as [109]3
E(x) = Ekin (x) + Eint (x) = (1 + β)EF (x)
5

(4.9)

where Eint (x) refers to the interaction energy. Recall from the discussion in Section
4.1 that Eint (x) = UM F (x) = βϵF (x). Thus we have(3

2 )Ekin (x) =− β EF (x).
5 5(4.10)

If we use the same procedure to determine the atom number as we did for Eqs.
(4.4) and (4.5) and integrate Eq. (4.10) over the trapped gas, we ﬁnd
Ekin =

1 3 − 2βµ.
8 1+β(4.11)

0
Recall that in a non-interacting Fermi gas the kinetic energy is Ekin= 38 µ0 . As-

suming we keep the atom number ﬁxed for each case we have from Eq. (4.6) that√µ/µ0 = 1 + β and after simpliﬁcation we can relate the kinetic energies
1 − 23 β
Ekin
√
=
.
0
Ekin
1+β

(4.12)

Now we can clearly see that β can be determined by measuring the ratio of
kinetic energies. If we use the mean-ﬁeld estimate for β = −0.42 one ﬁnds that0
Ekin /Ekin= 1.69, which is substantially smaller than the experimental result.

Again, this shows that more sophisticated theories that include the eﬀects of520
pairing are needed. The measured value of Ekin /Ekin= 2.3(0.3) gives a value forβ of −0.62(7), which is in good agreement with theory and previous measurements.
As we have seen with the measurement of β, atomic Fermi gases can alreadyprovide constraints on theoretical models. In particular, we have seen a large
discrepancy between a simple mean-ﬁeld estimate for β = −0.42 and the more
complicated theoretical and experimental works that put the value closer to β =−0.6. This example serves as a reminder that pairing in these systems cannot
be ignored but in fact makes up a substantial portion of the physics. The rest of
this thesis will study pairing using radio frequency spectroscopy, providing much
more detailed microscopic information as well as stringent tests for many-bodytheories.

Chapter 5
RF spectroscopy: A historical perspective

Probing an atomic gas can require a fair amount of ingenuity. This is because
classical probes such as thermometers or electrical leads are not applicable. All
measurements are eﬀectively “hands oﬀ.” Basically, the only tools one has are E &M ﬁelds: light, electric ﬁelds and magnetic ﬁelds. One of the experimental toolsthat has had continued success probing atomic Fermi gases is radio-frequency(RF) spectroscopy, for example see Refs. [42, 110, 111, 44, 112, 113, 114, 115, 69,82]. In Chapter 6, we discuss a new probing technique where an RF photon isused to “photo-emit” atoms into another Zeeman state in an experiment similar
to photoemission spectroscopy (PES) experiments in condensed matter physics.
Therefore, it is my goal in this chapter to introduce the role of RF in atomic
systems and its role as a spectroscopic probe, which was already a powerful tool
before we applied it to PES for atoms. I will present RF spectroscopy from achronological point of view to better illustrate its evolution into a spectroscopic
probe.
As we will review, RF spectroscopy has proven to be a very powerful probeof interactions in atomic gases. It was instrumental in determining the physics
around Fano-Feshbach resonances. It can also be used to associate and disassociate molecules and is a microscopic probe of strongly interacting atomic gases.
I will begin by outlining the simple theory regarding the coupling of states using

54
RF and then move on to its use as a spectroscopic probe.
Our chronological tour of RF spectroscopy as powerful probe will eventually
come to a “bump in the road.” This is because RF spectroscopy is only valid whena particular set of conditions are met. If these conditions are not met, such as caneasily be the case in 6 Li, then they can lead to results that are misleading andcan be misinterpreted. I will try to highlight the pitfalls of RF as a spectroscopic
probe, the lessons learned, and how the ﬁeld eventually got back on the righttrack.
5.1

RF basics
Typically, we use RF in the lab for internal spin-state manipulation. Ex-

amples include removing high energy atoms during magnetic trap evaporation,achieving correct spin states for optical trap evaporation, or for spectroscopy. For
RF spectroscopy, atoms are transferred between diﬀerent Zeeman states, see Fig.
5.1 a). The RF frequency is typically in the 10-100 MHz range. Hence, the wavelength of the RF radiation is very long, on order of 3-30 m, and the momentum
transferred to the atom is completely negligible. Therefore, we can think of RF
spectroscopy as only changing the internal state of the atom and leaving its kinetic
energy and momentum unchanged.
It is useful to remind ourselves how one can drive a transition between two
internal states of an atom. Due to the long lifetime of the internal states weare dealing with we can assume a two-level system: the state we are startingout in |g⟩ and the state we wish to couple to |e⟩, where the energy diﬀerence is
Eg − Ee = ~ω0 . For the coupling ﬁeld we will assume the RF is monochromaticand “strong” so we can treat it classically. Then, the coupling ﬁeld can be written
as E = E0 cos(ωL t), where E0 is the electric ﬁeld strength and ωL is the frequencyof the RF. Then, we can write down our Hamiltonian as H = Hatom + Vatom−light ,

55

a)

b)
1.0

Pe

|e
0.5

RF

|g

0.0
0.0

0.5
1.0
t ( 2p/Weff)

Figure 5.1: RF basics. a) We apply radio frequency to transfer atoms between two
internal Zeeman states. b) Population in state |e⟩ as a function of RF duration
assuming resonant RF.

56
where in matrix form:









 0 0  0 dE 
Hatom =  , Vatom−light = 
,
0 ~ω0
dE 0

(5.1)

where d is the dipole matrix element d = ⟨e|e π̂ · ⃗r|g⟩. We want to know what
happens to the states of the atom as a function of the strength, duration andfrequency of applied RF. To do so we need to solve the time dependent Schrodinger
equation i~ ∂ψ= Hψ with our wave function ψ = c1 (t)|g⟩ + c2 (t)|e⟩eiω0 t , where∂tc1 and c2 are the probability amplitudes. Using the rotating wave approximation(i.e., assume the RF frequency is near resonance) we end up with the set of
equations








−iδt

i

Ωe
d  c1   0

=
dt
c2
Ωe−iδt0  c1 

,
c2

(5.2)

where Ω = dE0 /~ is the Rabi frequency and δ = ωL − ω0 is the detuning of theapplied RF. Solving these coupled equations with the boundary condition that all
atoms begin in state |g⟩ at t = 0 we ﬁnd the probability to ﬁnd an atom in state|e⟩ is( √δ 2 + Ω2 )Ω2Pe (t) = 2
sint .
δ + Ω22(5.3)

Pe (t) is plotted in Fig. 5.1 b). To fully transfer all of the atoms into the |e⟩state one should apply resonant RF (δ = 0) for t = π/Ω; this is referred to as aπ−pulse.
The energies for our atomic Zeeman states in a magnetic ﬁeld can be calculated exactly using the Breit-Rabi formula. Therefore, we can choose ωL to
be exactly resonant and transfer atoms between Zeeman states by applying RF
through an RF antenna [47]. An example of such a spin-state transfer is shown in
Fig. 5.2. We deliver the RF in a gaussian envelope pulse to minimize unwanted
frequencies. The RF lineshape, for a non-interacting gas, has a linewidth limited

Fraction transferred

57

1.0
0.8
0.6
0.4
0.2
0.0
47.18

47.20
47.22
47.24
RF frequency (MHz)Figure 5.2: RF can be used to spectroscopically determine the magnetic ﬁeld. On
the vertical axis is the fraction of atoms transferred to a new internal Zeeman
state. The horizontal axis shows the RF frequency. The red and black lineshapes
are taken at two diﬀerent magnetic ﬁelds approximately 100 mG apart.

58
by the inverse pulse duration. Our linewidths are ultimately limited by noise in themagnetic ﬁeld. Typically, we can measure linewidths to 1 kHz or less, consistent
with magnetic ﬁeld stability on the order of a few mG. Good frequency sensitivity
and magnetic ﬁeld stability are imperative for using RF as a spectroscopic probe.
Using RF to determine the magnetic ﬁeld and its stability is already a great
example of RF as a spectroscopic probe. By varying ωL one can take a lineshape
to determine ω0 , see Fig. 5.2. Then, the Breit-Rabi formula can be used todetermine the magnetic ﬁeld. We use this technique quite often in the laboratory
to calibrate the magnetic ﬁeld as a function of current through the magnetic-ﬁeld
coils.
5.2

RF spectroscopy and interactions
One of the most illustrative examples of how RF can be used as a spec-

troscopic probe is also one of the earliest experiments that used RF to study
interactions [111, 44]. In particular, RF spectroscopy can be used to determine
the mean-ﬁeld energy, UM F ∝ na, where n is the density and a is the s-wave
scattering length. Near a Fano-Feshbach resonance a can be either positive or
negative and therefore so can the mean-ﬁeld energy. By taking RF lineshapes as
the scattering length a is varied, one can determine how the resonant frequency
is shifted by the mean-ﬁeld energy.
When discussing RF spectroscopy in context of measuring interactions I will
generally be discussing an experiment that involves three spin states. I will try
to be consistent and refer to the two interacting states as | ↑⟩ and | ↓⟩, and athird state |3⟩, which I will call the probe state. Typically, RF spectroscopy willrefer to the transfer of atoms in the | ↓⟩ state into the |3⟩ state. For example, theFano-Feshbach resonance in 40 K at 202.1 G aﬀects the interactions between atoms
in the |9/2, −9/2⟩ and |9/2, −7/2⟩ states (| ↑⟩ and | ↓⟩) and the probe state would

59

a)

b)
|3

RF

|3
RF

|

|

|

|

Figure 5.3: RF spectroscopy (when interactions are involved) will typically involve
three internal states. RF spectroscopy can be used to probe the interactions
between | ↑⟩ and | ↓⟩ in two complimentary ways. a) Sending some atoms fromthe non-interacting state |3⟩ to the interacting state | ↓⟩, will be sensitive to meanﬁeld energy shifts. b) Alternatively, energy shifts can be probed by sending some
atoms from the interacting state | ↓⟩ into the non-interacting state |3⟩.

b)

a)
+ |3

RF

U MF

|

+|

1.0

N remaining in | 3

|

0.5

U MF
0.0

-20 -10 0 10 20
RF frequency (kHz)

Figure 5.4: Measuring the mean-ﬁeld energy using RF spectroscopy. a) Atoms
from state |3⟩ are transferred into state | ↓⟩. The mean-ﬁeld energy shifts therequired RF frequency with respect to the single-atom resonance. b) Numberof atoms remaining in state |3⟩ as a function of the applied RF frequency. Theinteracting lineshape (red solid line) is shifted and broadened as compared to
the non-interacting lineshape (black dashed line). UM F ∝ na and therefore theinteracting lineshape can be at higher or lower frequency than the non-interacting
lineshape, depending on the sign of a.

60then be the |9/2, −5/2⟩ state (|3⟩). One can determine the mean-ﬁeld energy via
RF spectroscopy by preparing a sample of | ↑⟩ and |3⟩ states and then applying
an RF pulse to send some of the atoms from |3⟩ state into state | ↓⟩, see Fig. 5.3a). Conversely, one can prepare a mixture of | ↑⟩ and | ↓⟩ states and then apply
an RF pulse to send | ↓⟩ to |3⟩, see 5.3 b).
To measure the mean-ﬁeld energy, experimenters measured the frequency
shift needed to transfer atoms from the |3⟩ state to the | ↓⟩ state [111], see Fig.
5.4 a). The frequency is shifted because of the mean-ﬁeld energy, given by∆ν =2~n↑ (a↑,↓ − a↓,3 ),
m

(5.4)

where n↑ is the density of atoms in state | ↑⟩, a↑,↓ is the scattering length between
states | ↑⟩ and | ↓⟩, and a3,↑ is the scattering length between states | ↑⟩ and |3⟩.
In fact, this technique was not only used to measure the mean-ﬁeld shift ∆ν but
it was also used to determine a↓,↑ [111]. Notice the frequency shift, Eq. (5.4),
can be both positive or negative. In the experiment, this is determined from araising or lowering of the resonant frequency required for transfer, see Fig. 5.4 b).
This mean-ﬁeld shift in energy is referred to as the “clock shift” in atomic clock
experiments, as it is one of the dominant sources of systematic error [116].
I will now review for which cases we would expect to see a mean-ﬁeld energy
shift. This material is covered thoroughly in Ref. [36] and only outlined here; itwill illuminate why fermions require the use of three spin states to measure energy
shifts. As we established earlier, the mean-ﬁeld energy between two states | ↑⟩and | ↓⟩ is ∝ na↑,↓ . Suppose we begin with a spin-polarized sample in state | ↑⟩and apply RF to transfer to state | ↓⟩. One might suspect that the RF transition
should already begin to be shifted by UM F as atoms are transferred into the newstate; however, this is not the case because we are dealing with fermions and thetransfer is coherent. The RF is coherent in that it creates a superposition of | ↑⟩

61
and | ↓⟩ simultaneously for all the atoms. Or rather, they are still polarized intheir new superposition state and since they are fermions, s-wave collisions are
forbidden and there is no mean-ﬁeld shift. Note, if there was a frequency shift wewould not be able to use one-state RF spectroscopy to determine the magnetic
ﬁeld strength as described above.
Imagine now that we have a mixture of two spin states, | ↑⟩ and | ↓⟩.
While the atoms’ energies are shifted by the mean-ﬁeld energy, an RF pulse toﬂip atoms between | ↑⟩ and | ↓⟩ is not shifted. The argument follows similarly tothe spin polarized case. Atoms originally in | ↑⟩ are transferred coherently intoa superposition of | ↑⟩ and | ↓⟩, while atoms originally in | ↓⟩ are transferredcoherently into the orthogonal superposition. The RF transition is not modiﬁedby the mean-ﬁeld energy and RF transfer between two interacting spin statescannot be used to probe their interactions.
To probe interactions, one needs to use three spin states. If we now createa mixture of | ↑⟩ and | ↓⟩ states and apply RF to couple the transition from | ↓⟩to |3⟩ we are sensitive to the interactions between | ↑⟩ and | ↓⟩. This is because
while the transition between | ↓⟩ to |3⟩ does not see a mean-ﬁeld shift, due to thearguments above, the energy of state | ↓⟩ is indeed modiﬁed by state | ↑⟩. Ideally,
for RF spectroscopy to be an eﬀective probe, state |3⟩ should be unoccupied andweakly-interacting with either | ↑⟩ or | ↓⟩. Of course, mean-ﬁeld energy is not the
only way the atoms’ energy can be shifted. For example, far on the BEC side of
the Fano-Feshbach resonance, atoms can pair and acquire a binding energy which
is not density dependent. At JILA, RF spectroscopy was again employed to study
this new state [110].

62
5.3

Using RF to measure energy gaps
RF spectroscopy similar to that used to measure mean-ﬁeld energies can also

be used to probe molecular binding energy. This was demonstrated in an atomic
Fermi gas when RF spectroscopy was used to measure the binding energy of FanoFeshbach molecules [110]. In this experiment, atoms were prepared in a mixture of| ↑⟩ and | ↓⟩ states and adiabatically ramped through the Fano-Feshbach resonanceto create two-body bound pairs. Then, RF was applied to transfer atoms fromthe | ↓⟩ state into the |3⟩ state. In order to dissociate the molecule, the RF
frequency must be large enough to incorporate both the Zeeman energy as well as
the binding energy of the molecule. An example of such a spectra is shown in Fig.
5.5 a). The RF spectra has the characteristic “double-peak” structure expected
when probing an energy gap. There exists a narrow peak at the location of thefree atom transition due to unpaired atoms in the sample and a separated second
asymmetric broad peak that arises from dissociating bound molecules. The free
atom feature is narrow because it is a two level transition between diﬀerent longlived internal states of the atom. The molecule feature is wide and asymmetricbecause the RF photon can drive a transition from the molecule to a continuum of
scattering states of two atoms. These scattering states can have diﬀerent relative
momentum, depending on the RF photon energy, see Fig 5.5 b). The lineshapethen depends on the overlap of the molecule wave function and free atom wave
function [110, 117].
The separation between the peaks results from the energy gap in the system,
which in this case comes from the binding energy of the molecules [110, 117]. RF
spectroscopy is useful for probing energy gaps because the probe (a spin-ﬂip of oneof the | ↓⟩ atoms) is inherently a single-particle measurement [70]. For example, inthe dissociation of a molecule, the RF photon results in a single-particle excitation,

63

kinetic energy (MHz)transfer (arb)

1.0

a

0.8
0.6
0.4
0.2
0.0
0.4

b

0.3
0.2
0.1
0.0
49.8

50.1 50.4 50.7
rf frequency (MHz)Figure 5.5: RF spectra of weakly bound Fano-Feshbach molecules. a) Fraction of
atoms transferred into the probe state as a function of RF frequency. The narrow
feature comes from unpaired atoms. The broad asymmetric feature results fromthe dissociation of molecules. b) Atoms from the dissociated molecules can carryaway excess RF energy in the form of kinetic energy. Energy and momentum
are conserved because the constituent atoms ﬂy apart with equal and opposite
momentum. Figure taken from Ref. [47].

64
namely removing an atom in the | ↓⟩ state. An RF pulse with a frequency lessthan the binding energy will not be absorbed by the molecule. Only when the RF
photon has suﬃcient energy to dissociate the molecule can it be absorbed. Or, inother words, RF spectroscopy as we have described it is not a probe of two-particle
excitations. We will review this concept in our discussion of momentum-resolved
RF spectroscopy next chapter.
Following the success of RF spectroscopy in probing energy gaps for weakly
bound molecules, it was extended to probe the rest of the BCS-BEC crossover
region by the group of R. Grimm in Innsbruck in 2004 [118]. The RF spectra
observed by the Innsbruck group using 6 Li atoms exhibited “double-peak” structure throughout the crossover region and was attributed to a paring gap. Recall
from our discussion of the BCS-BEC crossover that in the regime of (kF a)−1 ≤ 0,
superﬂuidity is a many-body eﬀect and the gap (or superﬂuid order parameter)
is an important parameter for describing the superﬂuid. Therefore, the work of
the Grimm group as the ﬁrst measurement of the superﬂuid gap throughout thecrossover region was initially very well received. Giving the experimental work additional scientiﬁc weight was a “back-to-back” theoretical article published by thegroup of P. Torma, which conﬁrmed the gapped spectra observed by the Innsbruckgroup as evidence for a superﬂuid gap [119].
In the months and years that followed, however, a number of theoreticalworks were published calling into question the simple interpretation of doublepeak spectra as evidence for an energy gap. While the interpretation worked ﬁne
for density-independent energy gaps, such as the weakly bound molecules studied
at JILA, several theoretical papers questioned the eﬀect of interactions on atoms
transferred into state |3⟩. These eﬀects are often referred to as “ﬁnal-state eﬀects.”
In addition, it is important to appreciate the role that the density inhomogeneity
for atoms in a trap plays in RF spectra. In the 6 Li data, these eﬀects are not at

65
all negligible. We will discuss the importance of these eﬀects in the next section.
5.4

Final-state eﬀects and trap inhomogeneity
As I began my thesis work it was understood that the superﬂuid gap had

already been measured throughout the BCS-BEC crossover. The determinationof the superﬂuid gap came from the interpretation of double-peak structure as
an energy gap [118], seen in an RF experiment by Grimm and co-workers. I
schematically recreate their ﬁndings in Fig. 5.6 as inferred from Figs. 1 and 3 inRef. [118]. Here, I have re-scaled the RF frequency in terms of the Fermi energy.
They observed a single peak at the free atom Zeeman resonance for temperatures
well above the pairing temperature, Fig. 5.6 a). As the temperature was reduced,
a second peak was observed to appear at higher frequencies. This was attributed
to paired atoms Fig. 5.6 b). As the temperature was reduced further, the free
atom peak ﬁnally disappeared completely and only the peak attributed to paired
atoms remained Fig. 5.6 c).
At the time, the interpretation of double-peak structure as evidence of apairing gap seemed like a logical conclusion because of the success of the previous experiment here at JILA on weakly-bound two-body molecules [110]. The
JILA experiment, as described in the previous section, correctly determined thebinding energy of density − independent weakly-bound molecules from the observation of two well-separated peaks in the RF spectra. Extending this simple experiment into the strongly interacting region near a Fano-Feshbach resonance, however, must be done carefully, especially when pairing is a manybody eﬀect. Since the Innsbruck experiment and initial theoretical description[118, 119], numerous theoretical works have made it clear that the interpretationof RF spectroscopy near a Fano-Feshbach resonance must be handled with care
[117, 121, 122, 123, 124, 125, 126, 89, 127, 128, 129, 15, 130, 131, 132, 133]. In par-

66

1

a)

T> T *

Intensity (arb.)

0.5

0
1

b)

T< T *

0.5

0
1

c)

T< Tc

0.5

0
-0.5

0

0.5

1

hn /EF
Figure 5.6: A schematic diagram showing RF spectra with double-peak structuresuch as observed for 6 Li by the Innsbruck group and later by the MIT group (seeFigs. 3 and 4 in Ref. [118] and Fig. 2 in Ref. [120]). Here, the x-axis is the RF
frequency in units of the Fermi energy. For reference, compare these spectra to
the expected pair dissociation spectra, solid line in Fig. 5.7. It is clear that the
RF spectra are strongly modiﬁed by ﬁnal-state eﬀects.

67
ticular, it became evident that double-peak structure could arise very genericallyand was not necessarily an indication of an energetically gapped state. Theories
for double-peak structures tend to fall into one of two categories. The ﬁrst considers the eﬀect that interactions with the third spin state, or the probe spin state,
can have on the RF spectra (as it turns out this is quite negligible for 40 K but
is a huge eﬀect for 6 Li). The second considers the density dependent mean-ﬁeld
energy shift and how one can observe double-peak structure in a harmonic trap.
Although a complete theory should contain both eﬀects, either one can give adouble-peak structure even when the system does not have an energy gap.

5.4.1

Final-state eﬀects

Recall from section 5.2 that one can probe interactions between the | ↑⟩and | ↓⟩ states by using an RF pulse to transfer atoms to or from the |3⟩ state.
There, we assumed that state |3⟩ was an ideal probe state, meaning that it hada small interaction strength with atoms in the | ↑⟩ or | ↓⟩ states. In 40 K, this isalmost always the case, and the scattering length between the probe state andthe strongly interacting states is less than a couple hundred Bohr radii. In 6 Li,
however, there exist extremely wide Fano-Feshbach resonances (on the order ofhundreds of Gauss wide) between all three spin states in close proximity to eachother [82]. Thus, in 6 Li, when studying the Fano-Feshbach resonance between | ↑⟩and | ↓⟩, the close proximity of wide resonances between | ↑⟩ - |3⟩ and | ↓⟩ - |3⟩means state |3⟩ is almost never an ideal probe [132].
A complete review of how interactions between the ﬁnal state and the initialstate aﬀect the RF spectra is beyond the scope of this thesis. Indeed, there are
now many theoretical papers discussing ﬁnal-state eﬀects on RF spectra for avariety of situations including weakly bound molecules, BCS-type pairing, BCSBEC crossover, spin polarized, ﬁnite temperature, etc. As an illustrative example

68of how ﬁnal-state eﬀects can distort the RF spectra, I will follow the argumentof C. Chin and P. S. Julienne for weakly bound molecules found in Ref. [117].
In this paper, the authors calculate bound-to-free transitions as well as bound-tobound transitions from weakly bound molecules. Normally, one would not needto consider bound-to-bound transitions if the ﬁnal state was an ideal probe state.
However, as we established, in the case of 6 Li the ﬁnal state may also be neara Fano-Feshbach resonance so that the scattering length can be of arbitrary sizeand sign. To calculate the bound-free RF spectra one needs to use Fermi’s goldenrule:Γbf (E) =2π|⟨E|V |i⟩|2 ,
~

(5.5)

where i is the initial state, E is the ﬁnal state, V is the RF coupling that we
can write as ~Ω/2 where Ω is the Rabi frequency. Note that energy conservation
requires E = ~2 k 2 /2µ = ERF − E0 − Eb where ERF is the photon energy, E0 is thecontinuum threshold, and Eb is the binding energy. Because we are considering
weakly bound molecules we can write down the initial wave function:√
2 −r/ae|i⟩ = ϕm (r) =
a

(5.6)

where a is the s-wave scattering length. We can write the outgoing ﬁnal state
wave function as√|E⟩ = ψE (r) =2µsin(kr + δ ′ )
π~2 k

(5.7)

where (′ ) indicates a parameter in the outgoing channel. Calculating the expectation value, squaring, and using the low energy expansion of the phase shift one
eventually ﬁnds√a′ )2 1~2 Ω2 (
E
Γbf (E) = 2 1 −3/2
2π
a Eb (1 + E/Eb ) (1 + E/Eb′ )(5.8)

where k 2 a2 = E/Eb and k 2 a′2 = E/Eb′ [117]. It can be useful here to make atranslation E → E ′ − Eb , so that the binding energy can be read directly from69
the spectra as the diﬀerence from zero, such as is the case for the experiments. In
the limit of no ﬁnal-state interactions, this formula simpliﬁes to the well known
pair-dissociation formula which falls oﬀ as E −3/2 for large E, see solid line in
Fig. 5.7. However, as interactions in the ﬁnal state are increased, the lineshape
becomes substantially modiﬁed, see dashed line in Fig. 5.7. In this example, I have√set Eb = EF and a′ = a/ 10, for which one ﬁnds the bound-to-free integrated
spectra for the interacting case only contains 25% of the spectral weight of the
non-interacting case. Where has the rest of the spectral weight gone? In thisexample, the remaining 75% of the spectral weight has gone into the bound-tobound transition. This concept is covered explicitly for the case of 6 Li in Ref.
[132]. Essentially, a bound pair made up of | ↑⟩ and | ↓⟩ states makes a two-bodytransition to a diﬀerent bound pair made up of | ↑⟩ and |3⟩ states. Additionally, thehigh energy tail falls oﬀ with the wrong power, E −5/2 . Even from our relatively
straightforward example it is already clear that ﬁnal-state eﬀects can severely
aﬀect RF spectroscopy.
The technique outlined here to determine RF spectra can be applied throughout the BCS-BEC crossover and into the BCS limit as long as one knows the appropriate wave function. In the absence of ﬁnal-state interactions, the onset of the
RF spectra is determined by the binding energy Eb in the BEC limit [117] and thesuperﬂuid gap ∆2 /2EF in the BCS limit [122]. In the BCS-BEC crossover, withthe presence ﬁnal-state interactions, calculating the RF spectra is theoreticallychallenging and work is still ongoing [124]. However, more recent data for 6 Lifrom an experiment that included an eﬀort to reduce ﬁnal-state eﬀects shows that
the RF spectra evolve smoothly from the BEC-side to the BCS-side of the FanoFeshbach resonance [115], and in all cases the spectra retain the pair-dissociationlineshape shown in Fig. 5.7. Unfortunately, even in this best case scenario for 6 Li,
it is calculated that the bound-to-bound transition may still contain up to 59%

70

0.25

Intensity

0.2

0.15

0.1

0.05

0
0

1

2

3

4

5

E/EF
Figure 5.7: Theoretical RF spectra for weakly bound molecules (bound-to-freetransition). The solid line is the RF spectrum for the case of no ﬁnal-state eﬀects
for the parameters described√ in the text. The dashed line is for the case of ﬁnalstate interactions that are 10 weaker than the interactions between the original
two spin states. For the parameters given in the text, the interacting lineshape
only contains 25% of the total spectral weight. The rest is in the bound-to-boundtransition.

71of the total spectral weight [132], suggesting the lineshape may still suﬀer fromﬁnal-state eﬀects.
Perhaps one of the most striking features of the RF spectra of Grimm andco-workers is that the feature does not have the pair-dissociation look to it, seeFig. 5.6. Rather, as the authors mention, they ﬁt the their data to a Lorentzianplus a gaussian, which does not have the characteristic E −3/2 tail. Also striking
is that the entire lineshape occurs within a frequency span of about EF /2. We
can understand this in the case of 6 Li due to the close proximity of nearby FanoFeshbach resonances. As a′ approaches a, the bound-to-free transition vanishes(see Eq. 5.8) and only a bound-to-bound transition is left. Also, as one approaches
a′ → ∞ the RF spectra again approaches a bound-to-bound delta function eventhough a bound state does not exist in the outgoing channel [117]. Presumably, theclose proximity of the Fano-Feshbach resonance in the outgoing channel resulted
in the entire RF spectra looking much more bound-to-bound than than one might
have naively expected.
Here, I have only covered one theoretical explanation for double peak spectra
due to the presence of ﬁnal-state eﬀects. However, there are numerous alternativeand complementary explanations, see for example Refs. [122, 123, 124, 125, 126,
89, 127, 128, 129, 15, 131, 132, 133]. The universal conclusion is that double-peakstructure in the RF spectra of a strongly interacting Fermi gas subject to ﬁnalstate eﬀects is complicated and is not convincing evidence for observing a pairinggap. In the next section, we see how density inhomogeneity due to the trapping
potential can also produce double-peak structure.

5.4.2

Trap inhomogeneity

Typically, most atomic Fermi gases are conﬁned using a far oﬀ resonant
optical dipole trap. One or more focused laser beams intersect at the location

72
of the atoms to provide a trapping potential. The potential, while gaussian,can be well approximated as harmonic near the bottom of the trap. Because
the trapping potential is spatially inhomogeneous, the cloud density varies as afunction of distance from the trap center. The gas density is highest at the trap
center and falls oﬀ as one approaches the trap edge. Consequently, shifts in the
RF spectra due to density (mean-ﬁeld shift, for example) will eﬀectively depend
on which part of the cloud is being probed. Naively, one might expect that since
the density varies monotonically that any RF spectra would simply be broadened.
As it turns out, this is not always the case. In fact, the density inhomogeneity can
produce double-peak spectra even in the absence of ﬁnal-state eﬀects [121, 89, 130].
Indeed, as we will see, one does not even require pairing, much less superﬂuidity,
to observe double-peak structure.
The most general theories explaining double-peak structure arising fromdensity inhomogeneity were actually in response to a paper by the MIT group[120]. In short, the MIT group observed double-peaked spectra almost identicalto the Innsbruck spectra (Fig. 5.6) in a system known to not be superﬂuid. Regardless, the theoretical results are general because they show that even when asystem has no pairing, double-peak RF structure can arise from density inhomogeneity. If one were to take an RF lineshape at the center of the trap, we would
expect it to be shifted with respect to the free atom transition due to densitydependent eﬀects such as pairing and mean-ﬁeld energy. Then, as one moves
towards the edge of the trap, where the density goes to zero, we expect to recover
the free atom RF spectra. At issue, however, is that the total RF spectra is thesum of these individual spectra, where signal coming from atoms at a distance r
from the center of the trap is weighted by r2 n(r), where n(r) is the density atr. The resulting total RF can show two peaks. One peak, due to a signiﬁcantportion of the atoms near the center of the trap, is shifted with respect to the73

Figure 5.8: Figure from Stoof et al. in Ref. [89]. The ﬁgure is an intensity
plot showing transition intensity as a function of radius in the cloud and RF
frequency. This ﬁgure shows how double-peak structure can arise from density
inhomogeneity. Note, at any given radius the RF spectrum only shows one peak.
However, integrating over all radii give a total RF spectrum that has two separated
peaks.

74
free atom transition. The other peak, due to a signiﬁcant number of atoms at low
density (because of the r2 weighting) is centered at the free atom transition. I
include a ﬁgure from Ref. [89] which shows this eﬀect quite nicely, see Fig. 5.8.
Notice, at any given radius there is only one peak in the RF spectra; however,integrating over all radii it is clear that double-peak spectra would be present.
In 2007, in-situ tomographic techniques were used by the MIT group to observe
single-peaked RF spectra as a function of position in the cloud [114].

5.4.3

Conclusion

In this chapter, we have seen how RF spectroscopy can be a very powerful probe, with applications ranging from measuring magnetic ﬁelds to probinginteractions and pairing. However, we have also learned some important lessons
regarding the interpretation and application of RF spectroscopy. Namely, we
have to carefully consider ﬁnal-state eﬀects and density inhomogeneity in RF
spectroscopy, especially when the signatures we are looking for are density dependent. Previous experiments claiming to measure a superﬂuid gap using RF
spectroscopy [118, 120] were grossly aﬀected by these eﬀects. In the next chapter,
we will once again turn to RF spectroscopy for studying energy gaps, but thistime in an entirely new way.

Chapter 6
Photoemission spectroscopy

Throughout this thesis I have argued that the simplicity, purity, tunability,
and universality of atomic Fermi gases make these gases model systems in whichto test condensed matter theories. A challenge for experimenters is to ﬁnd ways
to probe these atom gases that relate directly to condensed matter ideas and enable sensitive searches for new phenomena that can advance our understandingof strongly correlated systems. For the most part, up until now, studies havemeasured macroscopic properties of the Fermi gas, such as: condensate fraction
and its phase diagram, collective mode oscillations, vortices, RF spectra, potential
energy, kinetic energy, etc. However, one would ideally like to probe the microscopic behavior directly and determine important microscopic quantities such as
the dispersion, energy gaps and quasi-particle lifetimes.
In condensed matter physics, photoemission spectroscopy (PES) has provento be a very powerful probe of the single-particle spectral function [134, 68]. I
will begin this chapter by reviewing PES for electronic systems and its connection
to the spectral function and many-body theory. Then, I will move on to our
new technique, photoemission spectroscopy for atoms using momentum-resolved
RF spectroscopy (in contrast to the momentum-integrated RF spectroscopy ofthe last chapter). With PES, we study new microscopic quantities such as theenergy dispersion, energy gaps, and quasi-particle lifetimes as well as macroscopic

76
quantities such as the momentum distribution and density of states. Many ofthese quantities were probed for the ﬁrst time in atomic Fermi gases with our
new technique. This new ability to probe the microscopic quantities of a stronglyinteracting Fermi gas will allow detailed comparison to many-body theories. Also,
we can study the pseudogap, which can have implications in both AMO andcondensed matter communities.
6.1

Photoemission spectroscopy and many-body theory
At a very basic level, the eﬀect of interactions is a modiﬁcation of the single-

particle states. As the strength of interaction is increased, the single-particle
eigenstates of the non-interacting case become quasi-particles and phase transitions manifest themselves as qualitative changes in the excitation spectrum, suchas the appearance of energy gaps. The single-particle excitation spectrum can bepredicted by many-body theory and is a fundamental property of any interacting
system.
For electronic systems, photoemission spectroscopy provides a powerful technique to probe the occupied single-particle states. In a typical photoemission spectroscopy experiment, see Fig. 6.1, electrons are ejected from a material throughthe photoelectric eﬀect. The photoelectrons are collected, energy- and momentumresolved, and counted to give a spectrum of intensity as a function of the measured
kinetic energy, ϵk = ~k 2 /2m, where ~ = h/2π, where h is Plancks constant, andm is the particle mass. By conservation of energy, the energy of the original
single-particle state, Es , isEs (k) = ϵk + ϕ + hν.

(6.1)

where hν is the photon energy, ϕ is the work function of the surface and thequantity (EF − ES ) is often referred to as the binding energy [68]. As we will see,

77

b

E

hn

k,

(E

k)

a

f
hn

0

kF

k

Figure 6.1: Photoemission spectroscopy for ultracold atom gases. a) In electron
photoemission spectroscopy, the energy of electrons emitted from solids, liquids
or gases is measured using the photoelectric eﬀect. Using energy conservation,
the original energy of the electrons in the substance can be determined. Similarly, in photoemission spectroscopy for atoms, a radio frequency photon withenergy hν transfers atoms into a weakly interacting spin state. b) The radiofrequency photon drives a vertical transition where the momentum k is essentiallyunchanged. By measuring the energy and momentum of the outcoupled atoms(upper curve) we can determine the quasi-particle excitations and their dispersion
relation (lower curve). Here ϕ is the Zeeman energy diﬀerence between the two
diﬀerent spin states of the atom.

78
Es (k) directly gives the dispersion of the system, and its width in energy gives
the quasi-particle lifetime.
Photoemission spectroscopy probes the single-particle spectral function which
is directly related to the single-particle Greens function predicted by many-bodytheories. I will also frequently refer to ARPES, which is angle resolved photoemission spectroscopy. ARPES allows one to measure the spectral function withrespect to energy and momentum. The intensity of photoelectrons emitted isgiven (assuming the sudden approximation) by
IARP ES (k, ω) = M0 (k, ω)f (ω)A(k, ω)(6.2)

where M0 (k, ω) is a matrix element which contains relevant selection rules [68, 70].
For our atom gas application, this matrix element will be unity. Now, we seethat the ARPES intensity is determined solely by the Fermi function f (ω) =(e~w/kB T + 1)−1 and the single-particle spectral function A(k, ω). In this way,
ARPES directly measures the occupied part of the spectral function.
I will now very brieﬂy highlight how the spectral function is connected to the“interesting” physics that we care to probe. For a more complete consideration
there exists much literature [1, 134, 68, 70]. The occupied single-particle spectral
function is related to the Green’s function via
1
A(k, ω) = − Im[G(k, ω)].
π

(6.3)

For the case of interacting fermions, the Green’s function is given by
G(k, ω) =1,
ω − ϵk − Σ(k, ω)(6.4)

where Σ(k, ω) is called the self energy. The self energy contains all the interesting
particle-particle correlations for the many-body system. The self energy can be
written in terms of its real and imaginary parts as Σ(k, ω) = Σ′ (k, ω) + iΣ′′ (k, ω),

79
where the real term contains all the information on the energy renormalization(in Fermi liquid theory this would be the eﬀective mass [135]) and the imaginary
term contains all the information on the quasi-particle lifetime [136, 137]. Finally,
we can write the spectral function in terms of the self energy to more clearly see
how ARPES, by being directly proportional the spectral function, directly probes
the “interesting” physics
A(k, ω) =1
Σ′′ (k, ω).
π [ω − ϵk − Σ′ (k, ω)]2 + [Σ′′ (k, ω)]2

(6.5)

Useful insight can be gained by working this out for the Fermi liquid case in whichΣ(k, ω) = αω + iβω 2 , where α gives the eﬀective mass and β the quasi-particle
lifetime.
6.2

Photoemission spectroscopy for atoms
As we learned in the previous chapter, radio-frequency spectroscopy has

been used to probe a strongly interacting atomic Fermi gas. In a typical experiment, a radio-frequency pulse drives atoms into an unoccupied Zeeman spin state,
where they are counted to yield a spectrum of counts versus frequency. Previously, the radio-frequency outcoupled atoms had not been energy- or momentumresolved. However, in analogy to electron photoemission spectroscopy, the momentum of the radio-frequency photon is negligible in comparison with the typical momentum of the atoms, and the momenta of the outcoupled atoms are
therefore characteristic of the original atom states. Eq. 6.1 applies to photoemission spectroscopy of atom gases, by means of momentum-resolved radio-frequency
spectroscopy, if we simply replace the work function ϕ with the Zeeman energy
splitting, see Fig. 6.1. The extension of photoemission spectroscopy from condensed matter to cold Fermi gases was discussed in [138].
There are two essential requirements for using momentum-resolved RF spec-

80
troscopy to determine the excitation spectrum. The ﬁrst is that the interactionenergy be small enough that ϵk = ~2 k 2 /2m holds and the data are not subject
to the complicated ﬁnal-state eﬀects we discussed in the last chapter. The second requirement is that collisions do not scramble the energy and momentum
information carried by the outcoupled atoms. In previous momentum-integrated
RF spectroscopy measurements in 6 Li, these requirements were not both satisﬁed[44, 118, 120, 115, 82]. In our 40 K gas, however, both of these requirements canbe satisﬁed. The interaction energy of the outcoupled atoms is approximately
h · 640 Hz, which is much smaller than EF . Furthermore, the mean free pathof the outcoupled atoms is much greater than the size of the gas: 1/σn ≈ 6RF ,where σ is the collision cross-section, n is the average density and RF is the Fermi
radius of the non-interacting gas.
Resolving the kinetic energy ϵk of the radio-frequency outcoupled atoms also
requires that we apply a radio-frequency pulse that is short in comparison withthe trap period. This ensures that the momentum of the outcoupled atoms is
not signiﬁcantly changed. We then immediately turn oﬀ the trap, let the gas
ballistically expand and measure the velocity distribution using state-selective
time-of-ﬂight absorption imaging, Fig. 6.2 a). We take the data by varying
the RF frequency and observing the velocity distribution of the atoms that areoutcoupled. For reference, in Fig. 6.3, I show an example of how the momentum
varies as a function of the applied RF. In Fig. 6.3 a), we see the fraction of
atoms transferred to the third spin state as a function of RF frequency. Shownin Fig. 6.3 b) is the gaussian size returned from a ﬁt of the absorption image
for the outcoupled atoms. From data such as this it is clear that RF at diﬀerent
frequencies couples to atoms at diﬀerent momenta.
To extract the kinetic energy ϵk of the outcoupled atoms it is not suﬃcient
to use the gaussian size as shown in Fig. 6.3 b). This is because we need the 3D

81

5

100
5
90
10
80

a

10
15

15
70

20

20
60

25

50
25

30

40
30

b

30
35
20
40
10
45

35
40
45
10

20

30

40

10

20

30

40

Figure 6.2: Extracting the three-dimensional momentum distribution. a) A timeof-ﬂight absorption image (145 µ × 145 µm) of atoms that have been transferred
into a third spin state is taken after applying a radio-frequency pulse to a Fermi
gas on the BEC side of the Feshbach resonance. b) After performing quadrant
averaging we use an inverse Abel transform to reconstruct the three-dimensional
momentum distribution. In this example, a two-dimensional slice at the center
reveals a shell-like structure for the momentum distribution of the outcoupledatoms.

gaussian size (arb.) Frac. trans. (%)

82

0.3
a)

0.2
0.1
0.0
1.5

b)

1.0
0.5
0.0

0.00
0.04
RF frequency (MHz)0.08

Figure 6.3: RF spectra for a strongly interacting degenerate Fermi gas. The
Fermi energy is h · 8.2 kHz, (kF a)−1 = 0.15, and T /TF = 0.16(2). a) Integrated
signal of outcoupled atoms as a function of RF frequency. The high-asymmetry
and non-zero tail results from the strong interactions. (Note the similarity to apair-dissociation lineshape as found in Fig. 5.7). b) Gaussian size (both radial
directions) from a 2D ﬁt to the outcoupled atoms’ absorption image. At higher
RF frequencies, we observe atoms with higher velocities. These high velocities
occur when the two atoms are at high densities or close distances as within a pair.

83
momentum distribution to determine ϵk whereas the image we retrieve from the
CCD camera is a 2D projection of the 3D momentum distribution. For example, ashell in 3D will look fairly gaussian when projected on to a 2D surface. Projecting
a 3D object onto 2D space is referred to as an Abel transform. We extract the 3D
distribution by performing an inverse Abel transform on the image which takes
advantage of our cylindrical symmetry. An example of this is shown in Fig. 6.2b)
where we have taken a slice through the center of the image after the inverse Abeltransform revealing a shell-like structure. Clearly, even though the 2D projection
can look fairly standard (Fig. 6.1a) the 3D momentum distribution may not be
trivial.
Once we have determined the kinematics of the outcoupled atoms we canapply Eq. 6.1 to determine the single-particle energy Es as a function of the
momentum. To check that the photoemission spectroscopy is working as intended,
we ﬁrst apply it to a weakly interacting cold Fermi gas. To create a very weaklyinteracting gas, we adiabatically ramp the magnetic ﬁeld to the zero crossing of
the Feshbach resonance. In Fig. 6.4a, we plot the intensity, which is proportional
to the number of atoms transferred into the third spin state, as a function of theoriginal single-particle energy Es and wavenumber k. Again, the data are obtainedby varying the radio frequency and counting the outcoupled atoms as a functionof their momenta. For the data here we deﬁne zero energy to be the energy of anoninteracting atom at rest in the initial spin state. This is in contrast to ARPES
in electronic systems where zero is typically deﬁned to be the Fermi energy. Fora non-interacting Fermi gas we would expect the photoemission spectra to revealdelta function peaks at Es = ϵk , since ϵk is the expected quadratic dispersion
for a non-interacting Fermi gas. To extract the dispersion we ﬁt each energy
distribution curve (EDC) to a single gaussian and mark the center as a white dot.
EDCs are slices through the spectra at a given k. The dispersion (white dots)

Single-particle energy (kHz)

84
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 6.4: Single-particle excitation spectra obtained using photoemission spectroscopy of ultracold atoms. Plotted are intensity maps (independently scaled
for each plot) of the number of atoms outcoupled to a weakly interacting spinstate as a function of the single-particle energy Es (expressed as frequency) andwavenumber k. Black lines show the expected dispersion curves for an ideal Fermi
gas. White symbols mark the center of each ﬁxed-k energy distribution curve. a)
Data for a very weakly interacting Fermi gas. The Fermi wavevector kF0 is 8.6±0.3µm−1 . b) Data for a strongly interacting Fermi gas where 1/kF0 a = 0 and T ≈ Tc .
The white line is a ﬁt of the centers to a BCS-like dispersion. c) Data for a gas
on the BEC side of the resonance where 1/kF0 a = 1 and the measured two-bodybinding energy is h · (25 ± 2 kHz). We attribute the upper feature to unpaired
atoms and the lower feature to molecules. The white line is a ﬁt to the centers
using a quadratic dispersion.

85
show good agreement with the expected free-particle dispersion (black curve).
The root-mean-square width in Es of the EDCs is h · 2.1 kHz and is due to anenergy resolution that derives from the radio-frequency pulse duration. In otherwords, at this level, our spectra is limited by measurement energy resolution andwe observe no other deviation from the expected delta function peaks.
Now, we apply our photoemission spectroscopy technique to a strongly interacting Fermi gas. To create a strongly interacting Fermi gas, we adiabaticallyramp the magnetic ﬁeld to the location of the Feshbach resonance, where thes-wave scattering length a diverges and the dimensionless interaction parameter 1/kF0 a equals zero and kF0 corresponds to the peak density of the original,
weakly interacting gas. Previous measurements have shown that after the ramp
to 1/kF0 a = 0, our Fermi gas initially with T /TF = 0.16 will be very near but
slightly below the superﬂuid transition temperature with T = (0.9 ± 0.1) × Tc
[30]. We extract the intensity map shown in Fig. 6.4b. The interactions lower theoverall energy and ﬂatten the dispersion curve. In addition, the energy width isbroadened well beyond our energy resolution.
As we established in the previous chapter, the interpretation of previous
radio-frequency spectroscopy measurements [118, 120] in terms of a pairing gap
is a diﬃcult problem that is still unsolved theoretically [15]. In contrast, the photoemission spectroscopy technique presented here directly measures the occupied
single-particle states, and is therefore well suited to measuring pairing gaps [68].
The energy gap in BCS theory results directly from a gap in the spectral function
as k approaches kµ . Mapping out the location of the peak of the spectral function
results in a BCS-like “back-bending” of the dispersion
Es = µ −√(ϵ − µ)2 + ∆2

(6.6)

where µ is the chemical potential and ∆ is the superﬂuid gap. I show a repre-

86

Es

2

1

0
0

0.5

1

1.5

k/k m
Figure 6.5: A sketch of BCS-like particle and hole dispersions. The occupied
lower branch (green curve) increases similar to a free-particle dispersion (black
dashed curve). However, near kµ , the dispersion turns around and an energy gap
forms in the excitation spectrum. The width of the lines loosely represents thequasi-particle lifetime.

87
sentative BCS-like dispersion curve in Fig. 6.5. Note that the dispersion loosely
follows the free-particle dispersion until it eventually turns around near kµ andthen begins to decrease. The minimum energy required for an excitation is twice
the superﬂuid gap, 2∆, and occurs at k = kµ . The energy gap and back-bending
dispersion are characteristic signatures of superconductivity in BCS-theory.
Recall that our data is taken at temperatures very near Tc . In BCS theory,
the gap vanishes at Tc and so one might naively expect our spectrum to reveal
a near-zero pairing gap. However, in the BCS−BEC crossover, a pseudogap due
to preformed pairs is predicted to exist above Tc [85, 16, 86, 87, 88, 139, 90]. In
the case of a pseudogap, the spectral function for a strongly interacting Fermi gas
near Tc is predicted to retain its gap-like double-peak structure. In this sense,
even though the system will not be superconducting (or superﬂuid), the spectral
function peaks continue to follow a BCS-like back-bending dispersion curve where
the BCS gap is replaced by the pseudogap [85, 16, 86, 87, 88, 139, 90]. As a ﬁrst
step in analyzing our data, we ﬁt the centers of the intensity at each value of k(white dots) to Eqn. 6.6, where we replace the superﬂuid gap with the pseudogap.
The best ﬁt, shown as the white curve in Fig. 6.4b, gives µ = h · (12.3 ± 0.7 kHz)and ∆ = h · (9.5 ± 0.6) kHz. Note that we have also performed photoemission
spectroscopy on a gas well below the superﬂuid transition temperature and ﬁnd
the data to be qualitatively similar.
A fairly natural question is to what extent does density inhomogeneity playa role in this spectra? Recall from the last chapter that in all trapped gas experiments the density is inhomogeneous and that the pairing gap will depend on thedensity, or equivalently on the local Fermi energy. We can come to a qualitative
understanding of how our spectra may look for a homogeneous gas by comparing
our strongly-interacting spectra to our weakly-interacting spectra. Namely, we
expect that atoms at the edge of the cloud are at low density and will therefore

88
follow the free-particle dispersion similar to Fig. 6.4a. While this may changeslightly the best ﬁt parameters, µ and ∆, it cannot cause the back-bending seenin the dispersion. In this way, our photoemission spectroscopy for ultracold atoms
provides a robust signature for pairing.
Because the photoemission spectroscopy intensity plots contain vast amounts
of information, one is left with the question of how to compare these new microscopic quantities to theory. In condensed matter ARPES experiments, the PES
data is often presented in the form of energy distribution curves (EDCs). An EDC
is a vertical slice through the photoemission spectra at a ﬁxed k. Conversely, a
momentum distribution curve (MDC) is a slice at ﬁxed Es . EDCs are especially
useful because they can be directly related to the spectral function at a givenk, such as we found in Eq. 6.5. Then, the peak position of the EDC indicatesthe dispersion at that k and the width of the feature indicates the lifetime of asingle-particle excitation at that k. As expected, the very strong interactions in
Fig. 6.4b result in a short lifetime for a particle in any given k and thus a verylarge width. In Fig. 6.6 we show energy distribution curves for Fig. 6.4b for
select values of k. As k increases, the peak position of the EDCs approach the
Fermi energy. However, as k approaches kF , the peak positions turn around andthen decrease for larger k. In this way, we clearly see the back-bending of thedispersion and we can directly compare to theoretical EDCs.
Continuing with our photoemission spectroscopy data, we now move to the
BEC side of resonance. Far on the BEC side of the resonance, for 1/kF0 a ≫ 1,
the pairing gap eventually becomes a two-body binding rather than a many-bodyeﬀect that depends on the local Fermi energy. We measure the excitation spectrumof the Fermi gas at 1/kF0 a ≈ 1, where the molecule binding energy measured fora low density gas is h · (25 ± 2 kHz). We observe two prominent features (Fig.
6.4c). The ﬁrst feature is narrow in energy, starts at zero energy and follows

89

-1

k ( mm )7.6

Intensity (arb.)

9.8
12.0
14.2
16.4

-30

-20

-10

0

10

20

Single-particle energy (kHz)
Figure 6.6: Energy distribution curves (EDCs) for a strongly interacting Fermi
gas. We plot the intensity for selected values of k. Each curve is the averagefor seven neighboring values of k in Fig. 6.4b. The data have been smoothedwith a 1.5-kHz-wide ﬁlter, and the energy is expressed in terms of frequency. The
peak position of the EDCs approaches the Fermi energy for increasing k; however,the peak positions are eventually turned around by the pairing gap and tend to
decrease for larger k.

90
the quadratic dispersion expected for free atoms (black line). We attribute thisfeature to unpaired atoms, which may be out of chemical equilibrium with thepairs. The second feature is very broad in energy, is shifted to lower energy andtrends towards lower energy for increasing k. This feature we attribute to atoms
in the paired state. An excitation gap separating the two features is evident in thedata. We ﬁt the centers of the molecule feature to a quadratic dispersion (whiteline) with the free parameters being the energy oﬀset and an eﬀective mass m∗ ,Es = ~2 k 2 /2m∗ + EB . In the BEC limit, where the molecules are tightly bound,
we would expect the energy oﬀset to be the molecule binding energy, which equals
2∆, and the eﬀective mass to be −m. This negative eﬀective mass reﬂects thefact that outcoupling an atom at high k leaves behind an excitation in the formof an unpaired atom. Or, in other words, as we discussed in the previous chapter,
RF spectroscopy is a single-particle probe and therefore we observe the dispersion
for only one of the constituent atoms. The best ﬁt to the data gives an energy
oﬀset of h · (28 kHz) and an eﬀective mass of m∗ = −1.25m.
The large energy width of the molecule feature seen in Fig. 6.4c is probably due to center-of-mass motion of the pairs. For comparison with the data, we
performed a simple Monte Carlo simulation, assuming a thermal distribution forthe center-of-mass motion and using the predicted distribution of relative kinetic
energy for RF dissociation of weakly bound molecules [117]. We assume that thepairs are in thermal equilibrium with the unpaired atoms and use the measured
temperature of the unpaired atoms in our simulation. Assuming a molecule binding energy of h · (25 kHz), we calculate the intensity map shown in Fig. 6.7a. To
easily compare the experimental data, we reproduce the white curve from Fig.
6.4c. The agreement is quite satisfactory.
We argued earlier that part of the appeal of studying the microscopics ofa Fermi gas was that one can in turn determine its macroscopic quantities. For

91Single-particle energy (kHz)

0

b

a
-10
-20
-30
-40
-50

c

-60
-70
5

Occupied densityof states (arb.)

0

10
15
-1
k ( mm )

20

-60

-40

d

0
-80

-20

0

20

Single-particle energy (kHz)
Figure 6.7: The occupied density of single-particle states. a) A calculated intensity
map for a T = 0.17 µK thermal distribution of weakly bound molecules. The whiteline is the ﬁt to the data shown in Fig. 6.4c. The agreement between theory anddata is quite satisfactory. b) The density of states for a weakly interacting Fermi
gas (black line) agrees well with a ﬁt (red curve) for a Fermi gas in a harmonic trap.
The ﬁt, whose only free parameter is the amplitude, includes our measurement
resolution. The dashed black vertical line shows EF . c) The density of statestaken at the peak of the Feshbach resonance is shifted to much lower energy andcontains a high (negative) energy tail. d) The density of states on the BEC sideof the resonance has two features: a peak due to unpaired atoms and a broader
feature due to molecules. This time, the red curve is the expected density of statesfrom the simulation shown in a). Energy is expressed in terms of frequency.

92
example, if we sum the photoemission spectra over all k, then we recover thenumber of atoms at each Es , also referred to as the occupied density of states(DOS). We plot the DOS for our three spectra in Fig. 6.7b-d. For the nearly ideal
Fermi gas 6.7b, we ﬁnd good agreement with the expected DOS for a trapped
Fermi gas at T = 0.18 TF (red curve). The theory curve includes our measurement
resolution and the only free parameter in the ﬁt is the amplitude. For the stronglyinteracting gas (Fig. 6.7c), the occupied density of states becomes wider in energy
and the position shifts towards lower energies. The shift we attribute to the strong
attractive interactions in the system. We also observe a high (negative) energy tail,which indicates that strong interactions promote atoms into high energy states (aswhen two atoms are close together in pairing). On the BEC side of resonance (Fig.
6.7d), a pairing gap is readily apparent. This is because broken pairs are alreadyavailable in the form of free atoms. The red curve is the expected density ofstates determined from the simulation of a thermal distribution of weakly boundmolecules (Fig. 6.7a). The only free parameter in the simulation is an overallscaling factor for the amplitude.
Another macroscopic quantity easily obtained is the momentum distribution. This is obtained by summing the photoemission spectra over all Es , giving
the number of atoms as a function of k. The momentum distribution for an interacting Fermi gas was previously measured in Ref. [53] where an interacting
Fermi gas was quickly jumped to the weakly interacting regime. In contrast, themomentum distribution as obtained from photoemission spectroscopy does not require this magnetic-ﬁeld jump because the outcoupled atoms are already weaklyinteracting.
The momentum-integrated RF spectra we discussed last chapter, can also
be recovered, generating a spectrum similar to that in Fig. 6.3a. At this point I
would like to highlight, as we discovered last chapter, that while RF spectroscopy

93
is inherently a microscopic probe, in its momentum-integrated form it is inadequate for revealing the single-particle dispersion [89]. It is only when we resolve
the momentum that we can observe the back-bending dispersion of the paired
state. The MIT group has shown that if one is willing to make an assumptionof the quasi-particle dispersion, then momentum-integrated RF spectroscopy canbe used in a spin-imbalanced Fermi gas to determine the superﬂuid gap [82]. Unfortunately, this approach is not straightforward because it takes place at theinterface between the superﬂuid and normal phase. Additionally, the work in Ref.
[132] demonstrates that even in the best-case scenario for 6 Li, ﬁnal-state eﬀects
may still be non-negligible.
From an ultracold atoms point of view, we are very excited about our new
photoemission spectroscopy technique and its ability to directly probe the spectral
function and determine many microscopic quantities such as the energy dispersion,energy gaps, and quasi-particle lifetimes. Much of this information was a ﬁrst for
the atomic physics community. In light of the importance of these importance of
these concepts I would like to again acknowledge the work of Dao et al. [138] which
we found to be the ﬁrst proposal of ARPES for atomic systems. Additionally, I
want to acknowledge previous experiments which probed microscopic qualities ofa Fermi gas such as the Bragg spectroscopy experiments of Vale et al. [37] andthe photoassociation experiments of Hulet et al. [140].
Thus far, photoemission spectroscopy has already proved to be very valuable
for studying strongly interacting atomic Fermi gases. Currently we are alreadyextending this technique to include spatially resolved momentum-resolved RF
spectroscopy to study the eﬀects of density inhomogeneity. We are also beginning to study these strongly interacting gases as a function of temperature andin particular probe the controversial pseudogap regime for temperatures above
the superﬂuid transition temperature. In the future, it will be exciting to see

94photoemission spectroscopy applied to other atomic systems as it is a general andconceptually simple probe of strongly correlated atom gases. As such, it could
be applied to lattice systems to study band dispersions and the eﬀect of interactions therein, as well as low dimensional systems. For example, photoemission
spectroscopy has now been proposed as a possible tool for observing the FuldeFerrelLarkinOvchinnikov (FFLO) state in 1D Fermi gases [141]. Additionally,much like ARPES of electronic systems, our photoemission spectroscopy technique could be used for an atom system where the pairing is not isotropic as inhigher partial-wave pairing [142, 143, 144].

Chapter 7
Conclusions and future directions

The work in this thesis aimed to answer the question of what can these new
Fermi gas systems teach us about the general class of strongly interacting Fermi
gases? We began by contributing to the broad demonstration that atomic Fermi
gases are addressing universal physics. This demonstration of universality was
imperative to be able to say that we are studying the relevant underlying physics
found in nuclear and neutron matter, the quark-gluon plasma, and possibly evenhigh temperature superconductors. Secure in the fact that we were addressing
universal physics, we decided to answer the question head on by directly probing
the microscopic physics.
Intimately connected with answering the above question is ﬁnding ways tomake measurements that connect directly to many-body theories and ideas. In
this way, with detailed enough measurements, we can use our relatively pure andsimple system to constrain many-body theories and thereby modify our understanding of all strongly interacting Fermi gas systems. To do this, we created a
new measurement technique (for the atomic physics community) using photoemission spectroscopy. Photoemission spectroscopy allowed us to directly probe thesingle-particle spectral function, which is in turn directly predicted by many-bodytheories. For the ﬁrst time, we were able to probe the energy dispersion, the pairing gap, and the quasi-particle lifetimes of a strongly interacting atomic Fermi

96gas. These measurements are now allowing for the most detailed microscopic
comparisons between experiment and theory to date.
7.1

Future directions
In general, the future for atomic Fermi gases seems boundless. As thesesystems continue to address condensed matter and many-body ideas with greater
accuracy, I think they will continue to make contributions relevant to the greater
physics community. I think the photoemission spectroscopy technique developed
here will continue to have an important role to play as well. In particular, as
these gases are being introduced to lattices, higher partial-wave pairing, lower
dimensions, impurities, etc., this measurement technique will become even more
relevant for relating to condensed matter systems. Particular to these types ofexperiments, I would still like to see PES or IPES (inverse photoemission spectroscopy) used to observe the upper branch of the BCS-like dispersion (we tried
for many months with little success).
7.2

The work not presented here
In addition to the work I have presented here, I also worked on a few otherresults. Before the work in this thesis, I took part in a study using atom shot noise
to probe fermionic pairing [112]. All of the work I have discussed in this thesis wasdone in equal collaboration with John Gaebler and as such we divided up topics
for our theses. Another topic we addressed, which John will likely cover in histhesis, was the ﬁrst clear creation and observation of p-wave Feshbach molecules,
found in Ref. [142]. In the near future, we hope to publish our recent work inwhich we have used photoemission spectroscopy to address the center of the cloud(a more homogeneous sample) by using holographic techniques to remove atoms
at the edge of the sample. Also in the near future, we hope to publish our work in97
which we have studied the dispersion of a strongly interacting Fermi gas above thesuperﬂuid temperature, relevant for pseudogap physics. Lastly, we have recently
returned to this idea of universality in Fermi gases and hope to address this topic
more broadly by studying the Tan relations [99, 145, 146]

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Appendix A
Fermi gas thermometry

In this Appendix I will describe an experiment we pursued regarding Fermi
gas thermometry. The goal of these experiments was to use a novel technique todetermine the temperature of a strongly interacting Fermi gas. As we will shortly
see, the temperature of a very cold trapped Fermi gas is a tricky thing to measure,and strong interactions make the problem worse. In this appendix, I will motivate
why measuring temperature is important in these systems, why it is hard, andwhy, unfortunately, our new technique was of limited value. I am including thismaterial because I ﬁnd it interesting and with the hope that it may be of value
to anyone else who decides to develop a new thermometer for these gases.
Temperature is an important parameter when describing a Fermi gas. The
temperature relative to the Fermi temperature (T /TF ) separates a classical gas
from a quantum gas. When interactions are present, the temperature relative tothe transition temperature separates a superﬂuid from the normal state. Throughout the BCS-BEC crossover, the temperature and the transition temperature areagain important parameters, and of current theoretical interest. Unfortunately,
this regime of cold temperatures and strong interactions is particularly diﬃcult
for two reasons. First, we determine temperature from time-of-ﬂight expansion.
For a trapped ideal Fermi gas, as we will see, it is already diﬃcult to measure
the temperature of a gas much below 0.1 TF [47]. Second, as interactions are

111
turned on, the interaction energy becomes convoluted with the release energy intime-of-ﬂight expansion. To extract the temperature in this way would require amodel of the release energy. Thus, it would be nice to have a model independent
thermometer that would work for low temperatures and strong interactions.
A.1

Ideal Fermi gas thermometry
To measure the temperature of a weakly interacting Fermi gas, we release

the gas from its conﬁning potential and use time-of-ﬂight imaging. For suﬃciently
long expansion times, the momentum distribution is captured on the CCD. Recall
from Chapter 4, we ﬁt the 2D image using an appropriate Fermi-Dirac function(
OD(y, z) = pk g2

− ξe

2

2

2σy

2σz

− y2− z 2)/g2 (−ξ)where ξ, σy , σz , and pk are independent ﬁtting parameters and gn (x) =

(A.1)
∞ k
∑
x
k=1

kn

.

Recall the fugacity, ξ, appears in the Fermi-Dirac distribution function
FF D (ϵ) = (ξ −1 eϵ/kB T + 1)−1

(A.2)

as ξ = eµ/kb T , where µ is the chemical potential. In this way, the fugacity only
depends on the relative temperature
(
)1/3
6
T
=−
,
TF
g3 (−ξ)(A.3)

and therefore we can extract T /TF directly from our ﬁt.
This procedure for determining the temperature works very well for a rangeof temperatures, namely from 0.1 TF < T < 0.5 TF [47]. For temperatures above
0.5 TF , subtle diﬀerences between the Fermi-Dirac distribution and a classicaldistribution cannot be picked out by the ﬁt (but of course in this limit we can relyon the classical distribution to determine temperature). For temperatures much
below 0.1 TF , the ﬁnite temperature distribution approaches the zero temperature

112
distribution and it is dominated by the Fermi energy. In this limit, our ﬁtting
routine cannot diﬀerentiate between the two, at least within our signal-to-noise,
and we are in need of a new thermometer.
To better understand the challenge in ﬁtting a low temperature gas, let us
ﬁrst consider the simplest case (Eqn. A.2), the distribution function for an ideal
Fermi gas. In Fig. A.1 a), I show the distribution function as a function of k fora zero temperature gas (black line) and a ﬁnite but small temperature (red line).
The temperature of the red line distribution can be determined by counting theexcitations above kF (and correspondingly the holes below kF ). Generally, one
would say the width of the broadening of the distribution is proportional to kB T .
In Fig. A.1 b), I show a similar calculation assuming this time a spherical squarewell potential, and integrate through one direction similar to imaging an atom gas
in time-of-ﬂight imaging. The red curve represents the same ﬁnite temperature
as in Fig. A.1 a), but now the diﬀerences in the two distributions are much more
subtle. As this example shows, integrating through one of the directions of an
atom cloud (projecting a 3D object onto 2D) already masks much of the signal.
As it turns out, real potentials are not even spherical square wells and thisfurther complicates matters. Because, in a real trap, the density decreases asa function of the gas radius and the local Fermi temperature TF decreases, therelative temperature T /TF increases. In Fig. A.2, I reproduce the expected momentum distributions after integrating through one dimension of a gas expandedfrom a harmonic trap as found in Ref. [47]. The distributions are shown forvarious temperatures. The distribution at T /TF = 0.1 is very similar to the zerotemperature distribution. With our current signal-to-noise, the coldest relative
temperature we can accurately measure is T /TF = 0.1 . However, the distribution at T /TF = 0.2 is still clearly diﬀerentiated from the zero temperature
distribution and we can ﬁt it quite accurately. Mapping a temperature at or be-

113

a)

nk

1

b)
Zero T
Finite T

0.5

0

0

1
k/kF

2 0

1

2

k/kF

Figure A.1: Momentum distributions at zero temperature (black line) and a ﬁnite
but small temperature (red line). a) Fermi distribution function for a homogeneous
system. The temperature can be determined from the diﬀerence in the ﬁnite andzero temperature distributions. b) Here, I have integrated through one dimension(similar to imaging an atom cloud) of a 3D spherical square well potential. For thesame ﬁnite temperature, the distributions are much more similar. Determining
the temperature from such a distribution is much more diﬃcult.

114

Figure A.2: Momentum distributions after integrating through one dimension ofa gas expanded from a harmonic trap as found in Ref. [47]. There is very little
diﬀerence between the distributions at T /TF = 0 and 0.1.

115
low T /TF = 0.1 on to a T /TF = 0.2 distribution was at the heart of our attempt
at a new thermometer.
A.2

Third spin-state thermometry
The ability to measure temperatures accurately below 0.1 TF would be use-

ful for a couple reasons. Firstly, to create a gas below 0.1 TF it helps to be able
to measure those low temperatures. For example, how does one modify an evaporation trajectory to make a colder gas if the thermometer is stuck at 0.1 TF ?
Secondly, a colder gas would allow us to access a larger portion of the superﬂuid
phase diagram, particularly on the BCS side. A thermometer that could determine low temperatures and also worked for a strongly interacting gas would be
doubly beneﬁcial for BCS-BEC crossover studies.
The idea behind using a third spin state as a thermometer is as follows:
place a small population of atoms in a third spin state in thermal contact with
the Fermi gas. Because the atoms in the third spin state are in thermal contact
with the other atoms, they have the same temperature T . However, the number ofatoms in the third spin state is chosen to be substantially smaller than the states
we wish to study, so that the third spin state has a smaller Fermi temperature TF .
In this way, we can map the absolute temperature of the states we are interestedin (but have diﬃculty measuring) onto a state with a higher relative temperature
T /TF , that we can accurately measure.
For example, let us assume we begin with our typical conditions of 105
atoms per spin state (|9/2, −9/2⟩ and |9/2, −7/2⟩ states) with a degeneracy of
T /TF = 0.1. Notice this temperature is at the limit we can accurately measure.
Imagine now that we place this cloud into thermal contact with 2 ∗ 104 atoms
in a third spin state (|9/2, −5/2⟩). Recall that the Fermi temperature goes as
TF = ~ω(6N )1/3 /kB , so that our 5% population in the third spin state now has

116
a relative temperature of T /TF ≈ 0.27, a value we can again accurately measure.
This technique was introduced as an impurity thermometer in Cindy Regal’s thesis
and was shown, at low magnetic ﬁeld where the spin states are stable, to be able
to duplicate temperature as well as the 2D surface ﬁt explained above.
Practically, impurity thermometry has a couple of challenges. First, because we tend to study the Fano-Feshbach resonance between the |9/2, −9/2⟩ and|9/2, −7/2⟩ states, our impurity spin state is the |9/2, −5/2⟩ state. Because thisstate has does not have a closed cycling transition for imaging and the numberof atoms is chosen to be small, signal-to-noise tends to be quite low. Additionally, for this probe to be useful it should be applicable for magnetic ﬁelds near
the Fano-Feshbach resonance (around 200 G). However, we ﬁnd that these spinstate combinations are not stable at high ﬁelds. In particular, we ﬁnd rather
counter-intuitively that while the |9/2, −5/2⟩ and |9/2, −7/2⟩ states are stable,the |9/2, −5/2⟩ and |9/2, −9/2⟩ states are not (they are stable on the order of one
second). Presumably, spin-exchanging collisions are due to the instability, butrather puzzling is that likely exchange culprit for the |9/2, −5/2⟩ + |9/2, −9/2⟩
collision would be that both states leave as |9/2, −7/2⟩, which would mean entering in a s-wave channel and leaving on a p-wave channel, which should be
suppressed for our ultra cold temperatures. Regardless, the spin mixture is suﬃciently stable for the third spin state to come into thermal equilibrium the sample
and we measure good agreement between impurity thermometry and the 2D surface ﬁts, even at high magnetic ﬁelds.
A.3

Three spin states near a Fano-Feshbach resonanceAlthough we would like to have a good thermometer for a weakly-interacting

gas, the real potential with impurity thermometry is the possibility to probe thetemperature of a strongly interacting Fermi gas. As we mentioned earlier, it is117
diﬃcult to extract the temperature of a strongly interacting Fermi gas from typical
time-of-ﬂight expansion experiments. This is because the interaction energy can
become convolved with the expansion energy, and extracting the temperature
becomes model dependent. Our current method is to measure the temperatureof a weakly interacting Fermi gas and then adiabatically turn on the interactions.
In this way, various theoretical models can predict the temperature of the gas in
the strongly interacting regime.
Impurity thermometry, however, has the possibility to bypass the problemof the release energy eﬀecting the time-of-ﬂight expansion. This is because while
there may exist strong interactions between the |9/2, −9/2⟩ and |9/2, −7/2⟩ statesnear their Fano-Feshbach resonance, they each still interact fairly weakly with the
impurity state |9/2, −5/2⟩. Thus, if we use the |9/2, −5/2⟩ state as our impurity
state, we can measure its temperature in the traditional way. This could be very
helpful for, say, measuring the temperature (and transition temperature) of thestrongly interacting gas throughout the BCS-BEC crossover.
This idea of impurity thermometry on a strongly interacting gas seemedvery promising. In particular, the time required for thermalization between the
impurity spin state was almost an order of magnitude smaller then the lifetimeof the |9/2, −9/2⟩ and |9/2, −7/2⟩ states near their Fano-Feshbach resonance (astringent requirement if the impurity state was to be used as a thermometer).
Unfortunately, as we tried these experiments for the very ﬁrst time on an ultracold
sample, we found out that the entire sample became extremely unstable as thespin mixture approached the Fano-Feshbach resonance. We understand this now
as the third (distinguishable) spin state de-stabilizing the fragile fermion pairs.
To understand why impurity spin state thermometry fails to work near the
Fano-Feshbach resonance we return to the case of the strongly interacting gas
without a third spin state. A key property of these strongly interacting Fermi

118
gases is that they have a relatively long lifetime (with respect to collision timescales, trap periods, etc.) [147]. In 40 K, the lifetime of fermionic pairs at the FanoFeshbach resonance is 100s of milliseconds, and can be even larger in 6 Li. Whatis surprising, however, is that the sample is so stable (consider our discussion in
Chapter 2 in which we found these are very large, fragile pairs). Similar conditions
for a Bose gas will result in orders of magnitude smaller lifetimes [148]. So, where
does the increased lifetime of fermionic pairs originate? The answer lies in thefermionic nature of the constituent atoms. Namely, because the entire gas is madeup of spin-up and spin-down fermions, and a fermionic pair is made up of the same,a collision to de-stabilize the pair is suppressed because no two identical fermionsapproach each other (due to the Pauli exclusion principle). This is explained indetail in Ref. [47].
We can now see why adding a third (distinguishable) spin state causes a
problem. While the fragile fermionic pairs are still Pauli protected from atoms in
the identical spin states, they are not Pauli protected from the newly introduced
third spin state. Atoms in this third distinguishable spin state are able to collidewith the fermionic pairs and carry away their excess energy as the pairs decay into
deeper bound states. This process is suﬃciently detrimental to the Fermi gas that
we could not even observe a fermionic condensate in the presence of the impurity
spin state. One might argue that impurity thermometry may still be applicablefor temperatures above the pairing temperature, but our typical thermometry
techniques are already suﬃciently accurate in that regime. Alternatively, it may
also be useful for spin-imbalanced studies as found in Ref. [149].