Radio-Frequency Spectroscopy of Ultracold
Atomic Fermi Gases
byAndre Schirotzek
Submitted to the Department of Physics
in partial fulﬁllment of the requirements for the degree ofDoctor of Philosophy
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2010
c Massachusetts Institute of Technology 2010. All rights reserved.
⃝

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
February 8, 2010
Certiﬁed by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wolfgang Ketterle
John D. MacArthur Professor of Physics
Thesis Supervisor
Certiﬁed by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Martin W. Zwierlein
Assistant Professor of Physics
Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Krishna Rajagopal
Professor of Physics, Associate Department Head for Education

Radio-Frequency Spectroscopy of Ultracold Atomic Fermi
Gases
byAndre Schirotzek
Submitted to the Department of Physics
on February 8, 2010, in partial fulﬁllment of therequirements for the degree ofDoctor of Philosophy

Abstract
This thesis presents experiments investigating the phase diagram of ultracold atomic
Fermi gases using radio-frequency spectroscopy. The tunability of many experimental parameters including the temperature, the interparticle interaction strength andthe relative population of diﬀerent fermions allows to access very diﬀerent physicalregimes. Radio-frequency spectroscopy has been developed into an ideal tool to probecorrelations between particles in these diﬀerent phases.
In particular, radio-frequency spectroscopy of highly population imbalanced atomic
Fermi systems gives access to the impurity problem: A single fermion, or boson, immersed in a sea of fermions constitutes a polaron, which can be described by Landau’s
Fermi liquid theory. A critical interaction strength can be identiﬁed separating theregime of a fermionic polaron and a bosonic polaron. Radio-frequency spectroscopyof the polarized superﬂuid phase allows an accurate measure of the superﬂuid gap ∆and allows to identify the importance of Hartree Mean-ﬁeld energies. Furthermore,
it is shown how these diﬀerent physical regimes are connected.
Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacArthur Professor of Physics
Thesis Supervisor: Martin W. Zwierlein
Title: Assistant Professor of Physics

To Larry...

Acknowledgments
Five years is a long time but it certainly seems short when you are having a good time.
And I will miss the cordial, open and driven environment in our hallway. I would
like to thank everyone in building 26 for making my PhD a memorable experience I
will always look back to with great fondness. A few people I would like to thank inparticular:First and foremost, I want to thank Wolfgang for giving me the opportunity to
do research in his lab and for being an inspirational ﬁgure throughout the years.
There are not enough words to express my gratitude to Martin for being thebest friend, colleague, former room mate, and supervisor one could wish for. I willremember countless nights inside and outside the lab spent discussing physics andmore worldly things, appropriate and inappropriate, be it at Bertucci’s, in the Student
Center or in the bathroom. There are more than a few things I admire and appreciate
about Martin: His deep and intuitive understanding of physics, his boundless energy
and excitement, his body’s ability to trade sleep for coﬀee for a week or longer, his
endearing outbursts of deafening laughter followed by an equally deafening ”YEAH!”,
his ability to outrun frat-boy opponents on the soccer ﬁeld right after consumptionof a large Big-Mac value meal and the fact that you can always count on him when
you need him.
I have always had the luck to share the lab with great people - not only are they
outstanding physicists but I have always truly enjoyed working with them during every
day and many a long night. My recent labmates Ariel and Mark are no exception tothis rule and they are possibly the two nicest people on the face of the earth. It has
been a pleasure to see Ariel “growing up” to take over BECI in no time. His thoroughand critical way of thinking, his understanding and grasp of physics have amazed me
more than once. I am also indebted to him for enduring my proclivities to listening
to loud metal music as well as wrap- and unwrap music followed by “Wrong!”.
I am glad to have had a few months overlap with Mark. Besides being the gentle,
caring and thoughtful person that he is, he has already proven to be an integral andproductive member of the group. With these two at the helm, I have no doubt that
BECI will be a place of excellent spirit and research for the years to come.
The contest for the nicest person would be a close one with my former lab mate
Christian. Always smiling, always helpful, always positive and calm. Some my dearest
memories at MIT include Christian, Martin and me having the best of times in the
middle of the night, working in the lab.

Almost ﬁve of my total of six years at MIT, Yong was either my oﬃce mate or lab
mate, or both. He taught me the nuts and bolts in the lab and the basics of ultracold
atoms. Even though he is the role model of an ideal physicist, he would be the last
to know because he is one of the most modest people I have ever known. We becameclose friends over the years and I wish him and his family all the best in Korea and I
hope that our paths will cross again soon.
I would also like to thank the “new” New Lab aka Fermi1 consisting of the evercheerful Peyman, the tireless Cheng with his extremely witty and surprisingly dirty
humor, the strongest pole-vaulter on the hallway Caroline (“Arrrrgh!”), the widely
knowledgeable Ibon, the energetic Sara and Peter.
And then there is Aviv, my favorite urban achiever, whose “word generator”
has helped me keep my sanity through some seemingly endless runs. But even if it
wasn’t for his engineering ingenuity, his ever-positive and uplifting presence would besomething that is high on my list of things I will miss most. We have shared countlesslaughs, puns, pranks and we have quoted to each other more movie lines than anyone
should know by heart. Am I wrong? No, I’m not wrong...
Speaking of BECII, if you ﬁnd yourself in building 26 between midnight and 6am
you are likely to encounter Christian Sanner, the late-night philosopher. I would
highly recommend that, after he has generously given you the piece of equipment you
were in desperate need for and BECII had in stock, you spend the few extra minutes
to partake in his wisdom. He will invariably oﬀer a perspective and insight you havenot considered before, no matter what the topic might be.
Ye-ryoung, or in short Ye, and I have started grad school almost at the same time
and we have shared many ups and downs over the years, be it general exams, a severelack of motivation or complicated matters with the other sex. Having Ye around, theclosest thing I have to a little sister, certainly made things seem not as bleak as theysometimes appeared to be.
Before my gym-buddy Jae went domestic, we used to spend endless hours torturing
each other in the Z-center. I have many fond memories of crazy squad workouts andof him telling me to “check out his traps”. I would also like to thank all the otherpeople in the hallway for always being kind, helpful and fun to be around, including
Gyu-boong, Caleb, David, Patrick, Hiro, Ed, Tout, Tony, Monika, Ian, Thibault,
Sebastian, Haruka, Jon, Marko, Dave, Vladan and Joanna. Fond memories fromearlier times include Jamil, Dan, Aaron, Tom, Michele, Sebastian and Gretchen.
Outside the lab, I want to thank Andi for having been a truly amazing friend andfor many unforgettable memories.

My special thanks go out to my loved and loving girlfriend Paige. I am not sure
where she took the patience from to endure the unusual work hours and my passion
for working out when I was not in the lab, but I am grateful that she did.
Most of all, I would like to thank my parents who have always believed in me andwho have been supportive and patient with me all my life. Without them, I would
not be where I am today.

Contents
1 Introduction
1.1 A Brief History of Time from 2005 to 2010:
Ultracold Fermions in BEC1 . . . . . . . . . . . . . . . . . . . . . . .
1.2 Feshbach Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 BEC-BCS Crossover: A Mean-Field Description . . . . . . . . . . . .
1.4 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Radiofrequency Spectroscopy: An Overview . . . . . . . . . . . . . .
1.5.1 A Little Bit of History . . . . . . . . . . . . . . . . . . . . . .
1.5.2 A Little Bit of Theory . . . . . . . . . . . . . . . . . . . . . .
1.5.3 Radiofrequency Spectroscopy on the Microscopic Level . . . .

122 Imbalanced Fermi Systems and the Phase Diagram
2.1 BEC-BCS Crossover with Density Imbalance: A Mean-Field Description (and Beyond) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Theoretical Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . .
2.3 Discussion of the Theoretical Phase Diagram . . . . . . . . . . . . . .
2.4 Experimental Phase Diagram . . . . . . . . . . . . . . . . . . . . . .
2.5 Phase Diagram at Finite Temperatures . . . . . . . . . . . . . . . . .

30

3 The Fermi-Polaron
3.1 N+1 = The Impurity Problem . . . . . . . . . . . . . . . . . . . . . .
3.2 Another Angle on the Impurity Problem . . . . . . . . . . . . . . . .
3.3 Landau’s Fermi Liquid Theory . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Fermi Liquid Theory on a Microscopic Level . . . . . . . . . .
3.3.3 Calculating m∗ and Z . . . . . . . . . . . . . . . . . . . . . .
3.4 A Special Case: A |↓⟩ Swimming in a |↑⟩ Fermi sea . . . . . . . . . .
3.4.1 Radiofrequency Spectroscopy of a Fermi Liquid . . . . . . . .

39
39
40
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42
44
46
49
51

7

13
18
20
23
24
24
27
28

30
31
33
35
35

3.5

Generalization of the Impurity Problem . . . . . . . . . . . . . . . . .
3.5.1 N + M = M (any) Impurities . . . . . . . . . . . . . . . . . .
3.5.2 Radiofrequency Spectroscopy for Finite Impurity Concentration
3.5.3 Polaron vs Molaron or Fermi Liquid vs Bose Liquid . . . . . .
3.5.4 Finite Size Eﬀects of the Impurity Problem . . . . . . . . . . .
3.6 Applicability of Fermi Liquid Theory to the N+1 Body System . . . .
3.7 Experimental Results:
Radiofrequency Spectroscopy of the Polaron . . . . . . . . . . . . . .
3.7.1 Preparation and Parameters . . . . . . . . . . . . . . . . . . .
3.7.2 Results I:
Chemical Potential µ↓ of the N + 1-body System . . . . . . .
3.7.3 Results II: Quasiparticle Residue Z of the Polaron . . . . . . .
3.7.4 Results III: The Contact Coeﬃcient C . . . . . . . . . . . . .
3.7.5 Results IV: Eﬀective Mass m∗ and Quasiparticle Interactions .
3.8 Experimental Results: Collective Oscillations . . . . . . . . . . . . . .
3.8.1 Optical Potential, I . . . . . . . . . . . . . . . . . . . . . . . .
3.8.2 Optical Potential, II . . . . . . . . . . . . . . . . . . . . . . .
3.8.3 Magnetic Gradient, I . . . . . . . . . . . . . . . . . . . . . . .
3.8.4 Magnetic Gradient, II . . . . . . . . . . . . . . . . . . . . . .
3.8.5 Magnetic Gradient, III . . . . . . . . . . . . . . . . . . . . . .
3.8.6 Size Measurement . . . . . . . . . . . . . . . . . . . . . . . . .
4 Quasiparticle Spectroscopy
4.1 Quantitative Studies of the BEC-BCS Crossover . . . . . . . . . . . .
4.2 Extracting the Superﬂuid Gap from Radiofrequency Spectra . . . . .
4.3 BCS Theory: Quasiparticles . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Radiofrequency Spectroscopy of Quasiparticles . . . . . . . . .
4.3.2 “Cold Infusion” of Quasiparticles or:
The Importance of Being Imbalanced . . . . . . . . . . . . . .
4.3.3 Experimental Results and a Surprise in the Lab . . . . . . . .
4.3.4 Hartree: Cause and Eﬀect . . . . . . . . . . . . . . . . . . . .
4.3.5 Why Should That Work?! . . . . . . . . . . . . . . . . . . . .
4.4 Connecting the Polaron Phase to the Superﬂuid Phase . . . . . . . .

52
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55
56
58
58
59
62
64
68
74
75
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76
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79
81
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82
83
84
86
88
91
93
96

5 The Equation of State of a Strongly Interacting Fermi Gas at Finite
Temperature
98
5.1 Introduction: Equation of State of a Balanced Fermi Mixture . . . . . 98
8

5.2
5.3

Inhomogeneous Potential . . . . . . . . . . . . . . . . . . . . . . . . .
Temperature Measurement at Unitarity . . . . . . . . . . . . . . . . .
5.3.1 Virial Expansion . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Eﬀects of the Trap: Harmonic vs Gauss . . . . . . . . . . . . .
5.3.3 Determination of the Temperature . . . . . . . . . . . . . . .
5.4 Preparation: Equation of State of the Non-Interacting Fermi gas . . .
5.5 The Equation of State of the Unitary Fermi Gas I . . . . . . . . . . .
5.6 The Equation of State of the Unitary Fermi Gas II (preliminary) . . .
6 Conclusion and Outlook

99
100
101
101
105
107
109
110
112

A Realization of a Strongly Interacting Bose-Fermi Mixture from a
Two-Component Fermi Gas
113
B Phase diagram of a two-component Fermi gas with resonant interactions
118
C Observation of Fermi Polarons in a Tunable Fermi Liquid of Ultracold Atoms
128
D Determination of the Superﬂuid Gap in Atomic Fermi Gases by
Quasiparticle Spectroscopy
137
Bibliography

145

9

List of Figures
1-1
1-2
1-3
1-4
1-5
1-6
1-7
1-8

Evidence for superﬂuidity in the BEC-BCS crossover . . . . . . . . .
Evidence for superﬂuidity in imbalanced Fermi mixtures . . . . . . .
Direct observation of resonance superﬂuidity . . . . . . . . . . . . . .
Phase separation in a strongly interacting Fermi gas . . . . . . . . . .
In-situ three-dimensional density proﬁles . . . . . . . . . . . . . . . .
Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A toy model for the BEC-BCS crossover . . . . . . . . . . . . . . . .
The chemical potential and the superﬂuid gap as a function of interaction strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-9 Quasiparticle excitation spectrum in the BEC-BCS crossover . . . . .
1-10 Energy level structure of 6 Li in the ground state . . . . . . . . . . . .

14
14
15
16
17
18
20

2-1
2-2
2-3
2-4

31
32
36

21
22
27

A mean-ﬁeld phase diagram of strongly interacting fermions . . . . .
A numerical phase diagram of strongly interacting fermions . . . . . .
Density proﬁles of strongly interacting fermions . . . . . . . . . . . .
Experimental phase diagram including results from density proﬁles andradiofrequency spectroscopy . . . . . . . . . . . . . . . . . . . . . . .
2-5 Finite temperature phase diagram in the unitary limit . . . . . . . . .

37
38

3-1 Two-body energy vs many-body energy . . . . . . . . . . . . . . . . .
3-2 Momentum distribution of a Fermi liquid . . . . . . . . . . . . . . . .
3-3 Momentum distribution of the polaron system . . . . . . . . . . . . .
3-4 The eﬀective mass in radiofrequency spectra . . . . . . . . . . . . . .
3-5 Polaron vs Molaron . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-6 Finite size eﬀects . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-7 Radiofrequency spectroscopy of the polaron . . . . . . . . . . . . . .
3-8 FWHM of polaron spectra . . . . . . . . . . . . . . . . . . . . . . . .
3-9 Stability of polaron spectra against polaron concentration . . . . . . .
3-10 Averaged polaron spectrum . . . . . . . . . . . . . . . . . . . . . . .

41
43
48
54
56
57
60
61
62
63

10

3-11 Polaron binding energy as a function of interaction strength . . . . .
3-12 Determination of the quasiparticle residue from spectra . . . . . . . .
3-13 Quasiparticle residue vs interaction strength and polaron concentration
3-14 Quasiparticle residue as a function of interaction strength for low polaron concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-15 A plateau in the spectra multiplied by ω 3/2 . . . . . . . . . . . . . . .
3-16 Dimensionless contact coeﬃcient as a function of interaction strength3-17 Minority spectrum from variational Chevy-Ansatz . . . . . . . . . . .
3-18 Polaron energy as a function of polaron concentration . . . . . . . . .
3-19 Spin dipole oscillation in the weakly interacting limit . . . . . . . . .
3-20 Overdamped spin dipole oscillations in the unitary limit . . . . . . .
3-21 Quadrupole oscillation of a minority cloud . . . . . . . . . . . . . . .
4-1
4-2
4-3
4-4

4-8
4-9

Principle of quasiparticle spectroscopy . . . . . . . . . . . . . . . . .
Radiofrequency spectra for balanced systems for various temperatures
Creation of cold quasiparticles . . . . . . . . . . . . . . . . . . . . . .
Radiofrequency spectra for various density imbalances in the unitary
limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Two-dimensional representation of radiofrequency spectra in the unitary limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Peak positions in the polarized superﬂuid . . . . . . . . . . . . . . . .
Stability of the extracted superﬂuid gap and the Hartree energy against
variation of assumed chemical potential . . . . . . . . . . . . . . . . .
Comparison to mean-ﬁeld spectra . . . . . . . . . . . . . . . . . . . .
Generalized quasiparticle residue as a function of density imbalance .

5-1
5-2
5-3
5-4
5-5
5-6
5-7

Gaussian potential . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Equipotential lines for a hybrid potential . . . . . . . . . . . . . . . .
Pressure vs Potential . . . . . . . . . . . . . . . . . . . . . . . . . . .
Determination of βµ from the virial expansion . . . . . . . . . . . . .
Equation of state for a non-interacting Fermi gas . . . . . . . . . . .
Equation of state of a unitary Fermi gas I . . . . . . . . . . . . . . .
Equation of state of a unitary Fermi gas II . . . . . . . . . . . . . . .

4-5
4-6
4-7

11

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103
104
106
107
108
109
110

Chapter 1
Introduction
All particles in nature are fermions. Fermions have half-integer spin and according
to the spin-statistics theorem [1, 2] the wavefunction of indistinguishable fermions is
antisymmetric with respect to exchange of two particles. The Pauli exclusion principle[3] is a direct result, which states that two identical fermions can not occupy the same
quantum state. Therefore, the ground state of fermions consists of particles “stackedup” in momentum space to the Fermi momentum. We will call this ground state a
Fermi sea in the following. For all temperatures, the distribution of fermions overenergy states is described by Fermi-Dirac statistics.
Bosons are either exchange particles mediating interactions between fermions orcomposite particles comprised of an even number of fermions and have integer spin.
Bosons obey Bose-Einstein statistics and can all occupy the same quantum state.
Below a critical temperature, bosons occupy the ground state of the system in large
numbers, forming a Bose-Einstein Condensate (BEC) [4].
Fermions and bosons thus exhibit a very diﬀerent behavior at low temperatures.
This immediately raises the question what makes a “composite particle”? To compose
a particle, interactions between fermions are required. Physically, it is clear that theenergy scale characterizing the interactions must be compared to the kinetic energyof the Fermi sea. If there is a bound state between two fermions whose binding
energy is much larger than the kinetic energy, we arrive at the simplest case of stablecomposite bosons which can subsequently form a BEC. For bound state energies
comparable to or lower than the Fermi energy, one expects no such stable composite
particles as surrounding fermions can trade places with one of the constituents ofthe pair. This is a highly correlated state of interacting fermions. The limit of weak
attractive interactions compared to the Fermi energy is again well understood andis characterized by the formation of Cooper pairs of fermions [5]. The ground state12

is a BCS superﬂuid, named after the authors of the seminal paper [6] providing atheoretical framework for the understanding of conventional superconductors.
For arbitrary interaction strength, temperature and particle numbers of a twocomponent fermion system, the phase diagram is still poorly understood on a quantitative and even qualitative level. Here, experimental data is of crucial importanceto gain a deeper insight into interacting Fermi systems. Since the experimental realization of Bose-Einstein Condensation in 1995 [7, 8] and degenerate Fermi gases in1998 [9] the ﬁeld of ultracold atoms has been developed into a powerful toy model
for many-body theories. Experiments with ultracold atoms oﬀer the unique possibility to control almost any parameter of the system at will, including the interaction
strength, temperature, density imbalance, dimensionality, disorder, lattice parametersand more. In particular, controlling the interaction strength between fermions allows
the experimenter to access the BEC regime of strongly bound diatomic molecules aswell as the regime of BCS pairing. The crossover region connecting these two regimes
is called the BEC-BCS crossover.
In this thesis, we will summarize experiments with ultracold atomic Fermi gases
performed at MIT in the past few years. In particular, the emphasis will be onquantitative studies of the BEC-BCS crossover using radiofrequency spectroscopy. In
the remainder of this chapter we will give a brief overview of the experimental eﬀorts
in BEC1 at MIT over the past ﬁve years. This overview will be followed by a few
basics, concepts and tools relevant to experiments with ultracold atomic Fermi gases.
Chapter 2 deals with the theoretical and experimental investigation of the phase
diagram of interacting fermions. In chapter 3 and 4 we present a quantitative studyof two limits of particular interest: The highly imbalanced normal phase allowing
access to Fermi liquid properties of the system in its normal state and the moderately
imbalanced superﬂuid region allowing access to the superﬂuid gap and the Hartree
energy. In chapter 5 an ongoing eﬀort will be presented on the extraction of theequation of state of a unitary Fermi gas.

1.1

A Brief History of Time from 2005 to 2010:
Ultracold Fermions in BEC1

The ﬁeld of ultracold atomic Fermi gases has matured considerably over the past
ﬁve years. When I joined the BEC1 lab in 2005, condensates of bosonic Feshbach
molecules comprised of two fermionic atoms had been created [10, 11] a little over a

13

Figure 1-1: The presence of vortex lattices is evidence for superﬂuidity throughout
the BEC-BCS crossover.

year earlier. However, in the unitary limit, where there is no two-body bound stateand pairing is a many-body aﬀair, evidence for superﬂuidity of ultracold fermionic
atoms remained elusive. There were several observations consistent with superﬂuidity[12, 13, 14] but conclusive evidence was only provided by the observation of vortex
lattices, see ﬁgure 1-1, in our lab in the spring of 2005 [15].

Figure 1-2: The presence of vortex lattices is evidence for superﬂuidity in imbalanced
Fermi mixtures. Superﬂuidity is stable against number imbalance up to a criticalvalue of 70(5)%. For higher number imbalance, the Fermi mixture is in the normal
phase.

Using the presence of vortex lattices as a marker for superﬂuidity, see ﬁgure 1-2, it
was possible for the ﬁrst time to test for superﬂuidity with unequal numbers of |↑⟩and|↓⟩fermions [16], thus opening a door to the new and exciting ﬁeld of imbalanced Fermi
14

gases. Exploring the phase diagram of imbalanced Fermi mixtures as a function ofinteraction strength, density imbalance and temperature was the major task we had
set out to accomplish and this would keep us busy for the next ﬁve years, still withmany questions left to answer today.
Up to this point, much of the information obtained from ultracold atomic Fermi
gases relied on two experimental techniques: 1.) The rapid ﬁeld ramp technique[12], “projecting” fermion pairs onto molecules and 2.) the density distributions wereobtained after time-of-ﬂight, meaning the atoms were released from the trapping
potential before imaging.
Employing these techniques was necessary to observe resonance superﬂuidity but
it was not entirely clear how they would aﬀect the physics quantitatively and qualitatively.

Figure 1-3: Direct observation of superﬂuidity in the unitary limit. Shown are radiallyaveraged density proﬁles for the majority component (blue) and minority component(red). The bimodal feature in the minority density proﬁle clearly shows the onset ofthe formation of a superﬂuid core.

We subsequently circumvented the “projection method” by preparing a number
imbalanced Fermi gas which enabled us to directly observe the superﬂuid phase transition [17]: If the core of the atomic sample is in the superﬂuid phase at equal density,
it is surrounded by unpaired fermions at unequal density. Therefore, the number imbalanced Fermi gas shows a bimodal density distribution, see ﬁgure 1-3, similar to a
BEC surrounded by a thermal cloud.
Beginning around 2006/2007 we have started concentrating our eﬀorts on quantitative studies of the BEC-BCS crossover using in-situ density distributions and radiofrequency spectroscopy.
Employing phase-contrast imaging allowed us to overcome the “problem” of thehigh optical density of the trapped atoms and gave us immediate access to the density
diﬀerence between the two spin states. This way, we were able to demonstrate in-situ
15

Figure 1-4: Phase separation in a strongly interacting Fermi gas. As the trap depthof the potential, and therefore the temperature, is lowered, excess majority atoms
are expelled out of the center of the Fermi mixture. The superﬂuid core of thesample is characterized by equal densities of majority and minority components. It
is surrounded by a density imbalanced Fermi gas in the normal phase.

the phase separation of a number imbalanced Fermi mixture into a superﬂuid coresurrounded by a polarized normal gas [18], see ﬁgure 1-4.
This, however, gave us access only to the density diﬀerence and not the individual
density of each spin state. With the upgrade to a new camera, which allowed us
to take two consecutive images separated by only a few tens of µs, we were able toimage not only the density diﬀerence n↑ − n↓ but a diﬀerent linear combination of
the densities a n↑ − b n↓ with a second phase contrast image1 . We were thus able todetermine both the majority and the minority density, see ﬁgure 1-5. Using the local
density approximation and three-dimensional density reconstruction, we were able tomap out the phase diagram of the unitary Fermi gas [19] and the zero temperaturephase diagram of imbalanced Fermi gases for various interaction strengths in the
BEC-BCS crossover. The results will be presented in parts in chapter 2.
At the same time we performed experiments on radiofrequency spectroscopy in
BEC1. Our ﬁrst publication on radiofrequency spectroscopy of highly imbalanced
Fermi gases in the normal state in 2007 raised the question of the nature of thisnormal state: We have observed a spectral gap in the normal phase even at the lowest temperatures achievable in the experiment. Initially, we have interpreted this as1

In our previous setup this had not been possible due to the heating of the cloud as a result of theﬁrst phase contrast image. The typical detuning of 40MHz is not suﬃcient to suppress absorptionof the ﬁrst imaging pulse. With a suﬃciently short delay time between the images, the cloud canbe imaged before heating distorts the density proﬁles.

16

Figure 1-5: In-situ three-dimensional density proﬁles in the unitary limit at low temperature T /TF ≃ 0.03. The plot on the left shows the radially averaged column
density proﬁles of majority (red) and minority (blue). Also shown are the column
density diﬀerence proﬁle (solid black) and the non-interacting density proﬁle (dashedblack) ﬁt to the wings of the majority density distribution in order to determine
the temperature. The center plot shows the three-dimensional density distributionsof majority and minority. The plot on the right shows the local density imbalance
σ = (n↑ − n↓ )/(n↑ + n↓ ). Rc denotes the radius of the superﬂuid core.

evidence for paired fermions which remain in the normal state even at zero temperature [20]. It became clear after experiments performed a year later that, in fact, thespectral gap is a signature of the normal Fermi liquid. This topic will be the subjectof chapter 3.
Up to this point, experiments on radiofrequency spectroscopy in 6 Li had been
plagued by several complications described below in chapter 1.5.1 and in the PhD
thesis of my long-time colleague Christian Schunck [21]. In the following years weovercame these problems and developed radiofrequency spectroscopy into a powerful tool for quantitative studies. The ﬁrst step was the introduction of tomographic
radiofrequency spectroscopy [22]: Taking an image immediately after the radiofrequency population transfer and reconstructing the three-dimensional density allowed
us to overcome the problem of trap-averaged radiofrequency spectra and to probe
the local excitation spectrum. The next improvement was the introduction of a newstrongly interacting Fermi mixture, comprised of hyperﬁne states |1⟩and |3⟩of 6 Li, seeﬁgure 1-10. This overcame the problem of strong ﬁnal state interactions and furthermore allowed taking spectra of both spin states. This paved the road for quantitative
studies using radiofrequency spectroscopy and therefore the most recent results presented in chapters 3 and 4.
Before proceeding to the presentation of our experimental results, a few concepts andtheir underlying physics will be discussed in the next few sections.
17

1.2

Feshbach Resonances

Feshbach resonances are now a standard tool in in the ﬁeld of ultracold atomic physics
as they are the mechanism allowing to tune the interaction strength between atoms
in diﬀerent states. For an extensive treatment please refer to [23]. The principleof a Feshbach resonance in the case of 6 Li is shown in ﬁgure 1-6: Two free neutral
atoms in the triplet conﬁguration at a relative kinetic energy of E ≃ 0 interacting
through the van der Waals potential are coupled to a quasibound state. The singlet
and triplet states of the interatomic potential are coupled by the hyperﬁne interaction
VHF = aHF (s1 · i1 + s2 · i2 ). Since singlet and triplet state have diﬀerent magnetic
moments the energy of the incoming colliding atoms can be tuned with respect to abound state in the singlet potential. A scattering resonance occurs when the energyof the incoming atoms is resonant with that of the bound state.

Figure 1-6: a) A Feshbach resonance occurs when the energy of two free atoms is
coupled to and resonant with a bound state in the interatomic potential. b) The
resulting energy level structure for two particles in a box

The s-wave scattering length, completely characterizing the interactions between
atoms at low enough temperatures (T < 1mK), can be shown to diverge and change
its sign around a Feshbach resonance. Below the Feshbach resonance, where the~2interatomic potential supports a bound state2 of energy Eb = ma
2 , the excited statesin ﬁgure 1-6b) consist of free atoms interacting repulsively, corresponding to a positive
scattering length a > 0. The metastable ground state consists of molecular dimers
2

The experimentally relevant bound state in the singlet potential is the vibrational state withquantum number n = 38.

18

whose residual interaction depends on four-body physics and can be calculated exactly
and is given by add ≃ 1.18a. Above the Feshbach resonance the interatomic potential
does not support a bound state and colliding atoms simply acquire a phase shift
in the scattering event given by the usual result δ = arctan(−ka) corresponding to
attractive interactions between the atoms.
It is important to note that the bound state in the interatomic potential is only
populated when the magnetic ﬁeld is adiabatically ramped from above the Feshbachresonance to below the Feshbach resonance if three-body interactions are suppressed.
This is the case if the Franck-Condon factor, quantifying the wavefunction overlap
between the incoming free atoms and the bound molecular state, is small enough forthe time scale of three-body relaxation to be longer than other relevant time scalesof the experiment, like the inverse Fermi energy ~/ϵF . Therefore, a sample prepared
below the Feshbach resonance represents a repulsive Fermi gas. This has important
implications for the experiments presented in this thesis and has been used recently
in search for ferromagnetic states in ultracold Fermi gases [24].
It is insightful to compare the above model of a Feshbach resonance to the morefamiliar case of a discrete state coupled to a continuum, e.g. an atom in an excitedelectronic state. A simple model has been presented in [25] based on a discrete state
coupled to an unbounded continuum (−∞ < Econt < ∞) of scattering states. Let
us now remove the unphysical restriction of the absence of a lower bound for thefree wave states. The resulting energy level structure corresponds the above singlechannel model for a Feshbach resonance once the bound state energy Eb crosses thelower bound Ekin = 0 to negative energies.
Before proceeding to the mean-ﬁeld treatment of a many-body system, whose
interactions can be tuned through the strongly interacting regime, let us review a
simple toy model capturing many qualitative features of the many-body treatment[26]: A particle interacting with a δ-potential characterized by the scattering lengtha, into which Pauli blocking eﬀects are incorporated by conﬁning it to a spherical boxof the size of the interparticle spacing, see ﬁgure 1-7.
One of the important features that can be reproduced is the existence of an excitation gap between the ground state and the excited states above the Feshbachresonance, where pairing is a many-body eﬀect. This model also demonstrates that
there is a smooth crossover between the diﬀerent physical regimes in spite of thediverging scattering length around the Feshbach resonance.

19

Figure 1-7: a) Toy model of a Fermi mixture with tunable interactions 1/kF a: Theinteraction strength is characterized by the pseudo-potential V (r) = 4π~2 a/mδreg andPauli blocking is mimicked by choosing a box of the size of the interparticle spacing
R. b) Many qualitative features of the many-body treatment can be reproduced.
Taken from [26]1.3

BEC-BCS Crossover: A Mean-Field Description

The tunability of the interaction strength by simply changing the external magnetic
ﬁeld gives ultracold atomic physics a unique advantage over condensed matter experiments. Let us consider a many-body system comprised of two fermionic species,|↑⟩ and |↓⟩. Without interactions, the momentum distributions are simply given byΘ(k = kF ), with the Fermi momentum kF .
For small positive interactions, corresponding to a < 0, there is no bound state in
the interatomic potential but the presence of the Fermi sea results in Cooper pairing[5], a many-body bound state. It was shown that the ground state of this system is
not a Fermi Liquid, see chapter 3, but a Superﬂuid described by the famous BCSwavefunction, named after the authors of their seminal paper [6], Bardeen, Cooper
and Schrieﬀer. A simple picture for the eﬀect of the presence of the Fermi seas is a
restriction of phase space available for particles above the Fermi surface to scatter
into. The Fermi sea makes this essentially a two-dimensional system3 , in which abound state exists for arbitrarily small interactions.
For a deeply bound molecular state with small residual repulsive interactions,corresponding to a > 0, the system is comprised of composite bosons. The groundstate is a Bose-Einstein condensate (BEC) of dimers and can be described by the3

The naı̈ve picture does not hold for N -dimensional space. The ﬁgure of merit is a constantdensity of states and not the restriction of the phase space to N − 1 dimensions.

20

BEC mean-ﬁeld description.
A mean-ﬁeld description of the entire BEC-BCS crossover regime, including thestrongly interacting regime |a| → ∞, based on [27, 28] can be found in [23]. Here wewill review a few of the results. The generalized BCS-wavefunction takes the same
form as in the limit a → 0− :|ψBCS ⟩ =∏(uk + vk c†k↑ c†k↓ )|0⟩

(1.1)

k

where the uk , vk are variational parameters obeying the normalization condition|uk |2 + |vk |2 = 1 and the c†kσ are the creation operators for a particle in spin state σ.

Figure 1-8: Chemical potential µ (black dashed) and the superﬂuid gap ∆ (red) as
a function of interaction strength in the BEC-BCS crossover regime in a mean-ﬁeld
description based on the seminal work of Nozières Schmitt-Rink [28].

We use the HamiltonianH

V0 ∑ †
ck+ q ↑ c†−k+ q ↓ ck′ + q2 ↓ c−k′ + q2 ↑2
2
V k,k′ ,qk,σ∑ †V0 ∑ † †→ϵk ckσ ckσ +ck↑ c−k↓ ck′ ↓ c−k′ ↑
V
′
k,σ
k,k
=

∑

ϵk c†kσ ckσ +(1.2)

where V0 is a measure of the interaction strength between the particles, V is thevolume of the system. The second row represents the standard BCS approximationin which only terms of q = 0 are retained in the interaction, thereby neglecting ﬁnite
21momentum Cooper pairs4
The energy can be minimized under variation of E − µN with respect to uk , vk .
The results are:

u2k
vk2

(
)
ξk
1
=
1+
2
Ek
(
)
1
ξk
=
1−
2
Ek

(1.3)
(1.4)
(1.5)

√
Figure 1-9: Quasiparticle excitation spectrum Ek = ξk2 + ∆2 for various interaction strengths in a mean-ﬁeld description. A qualitative change can be observed at
1/kF a = 0.553.
Physically, vk2 represents the momentum distribution of particles and u2k represents
the distribution of holes. Here we have deﬁned the kinetic energy ξk = ϵk − µ withrespect to the chemical potential µ and the quasiparticle excitation energy Ek =√
ξk2 + ∆2 , ∆ being the superﬂuid gap. The above equations are identical to the usual
BCS result, the diﬀerence for atomic gases with arbitrary interactions strength beingthat the chemical potential and the superﬂuid gap have to be solved self-consistently
as a function of interaction strength 1/kF a:∫

d3 k 1(2π)3 2Ek∫
d3 k 2n = 2v(2π)3 k

1
−
=
V0

(1.6)
(1.7)

The results for µ and ∆ are plotted in ﬁgure 1-8.
4This is a severe approximation with important results. See discussion below and in chpater 4.1.

22

The resulting quasiparticle excitation spectrum is shown in ﬁgure 1-9
Although the above picture provides valuable qualitative insight, the results havebeen shown to be quantitatively inaccurate: a) In the deep BEC limit the residualinteraction between molecules, characterized by the dimer-dimer scattering length
can be shown to be add = 2a, diﬀering from the exact result [29] add ≃ 0.6a. b) Evenin the BCS limit, the mean-ﬁeld result overestimates of the superﬂuid gap by a factorof ∼ 2.2 - see discussion in 4.1. The reason for both of the above discrepancies is
the reduction of the Hamiltonian 1.2 by setting q = 0. The same inaccuracy can beobserved in the unitary limit: c) The chemical potential in the unitary limit in themean-ﬁeld treatment is µ ≃ 0.59ϵF , while it has been shown theoretically to range
between 0.32ϵF and 0.45ϵF [30, 31, 32, 33, 34, 35, 36, 37] and experimentally to range
between 0.32ϵF and 0.51ϵF [38, 39, 14, 40]. c) The superﬂuid gap is ∆ ≃ 0.69ϵFin the above mean-ﬁeld theory and has been theoretically predicted to be on theorder of 0.45ϵF − 0.54ϵF [41, 42]. An experimental determination of the gap using
momentum-resolved radiofrequency spectroscopy has found ∆ ≃ 0.8ϵF close to thecritical temperature [43]. In chapter 4 we will use radiofrequency spectroscopy of
Fermi systems characterized by a small density imbalance between |↑⟩ and |↓⟩ todetermine the superﬂuid gap to be ∆ ≃ 0.44ϵF [44]. d) Another point of quantitativeinterest is the critical interaction at which the character of the system changes fromfermionic to bosonic. This should be related to the position of the minimum in thequasiparticle dispersion, ﬁgure 1-9. For interactions stronger than 1/kF a = 0.553,
the lowest single-particle excitation energy is at ﬁnite momentum k > 0 and costs anenergy ∆, whereas for weaker interactions the lowest excitation energy is at k = 0 at√
an energy of ∆2 + µ2 . The interaction strength at which this transition happens is
the one at which the chemical potential µ is zero, see deﬁnition for Ek . The criticalinteraction strength determining the change of character from fermionic to bosonicwill be discussed in more detail in chapter 3.
We will discuss the approximations made in the above treatment and its shortcomings in chapter 4, in which we present a determination of the superﬂuid gap in
the unitary regime using radiofrequency spectroscopy.

1.4

Experiment

The experimental setup and production of a strongly interacting degenerate gas of6
Li has been described in detail in [45, 23]. Here, we will only brieﬂy review theprocedure: An atomic beam is created in an oven containing 23 Na, a boson, and 6 Li,
23

a fermion. After a Zeeman slowing stage, ∼ 1010 23 Na atoms and ∼ 109 6 Li atoms
are loaded into a dual species Magneto-Optical Trap (MOT) for a typically one second. After optically pumping 23 Na and 6 Li into their respective stretched states inorder to avoid spin exchange collisions and for their magnetic trapping properties,
the atoms are loaded into a magnetic trap of the Ioﬀe-Pritchard type. Both 23 Na and6
Li are subsequently evaporated within 20s using microwave/radiofrequency radiation. At this point the 6 Li atoms are typically in the quantum degenerate regime ata temperature of T ≃ 0.3TF with typically a few 107 6 Li atoms and no 23 Na atoms
left. The 6 Li atoms are subsequently loaded into an optical dipole trap (ODT) with awaist size of ∼ 70µm and an power of a few Watts, resulting in a typical trap depthof a few ten µK. Next the magnetic ﬁeld is ramped up to around 690G within 2s,
where there is a Feshbach resonance between the hyperﬁne states |1⟩ and |3⟩5 . Here aspin mixture of the states |1⟩ and |3⟩ of variable number imbalance is created using a
Landau-Zener sweep. For the experiments on radiofrequency spectroscopy presented
in this thesis, the |1⟩-|3⟩ mixture is preferred over the |1⟩-|2⟩ mixture because the ﬁnal
state interactions are weak and can easily be accounted for [46]. Next the power of
the ODT is ramped down for evaporative cooling in 4s to a typical trap depth of a fewµK. For the experiments presented in this thesis the typical atom number for highly
imbalanced samples is 5 · 106 at a temperature of T ≃ 0.15TF and for the balanced
or moderately imbalanced samples 1 · 107 at a temperature of T ≃ 0.05TF .

1.5

Radiofrequency Spectroscopy: An Overview

1.5.1

A Little Bit of History

Radiofrequency spectroscopy has been experimentally realized on trapped cold neutral atoms for the ﬁrst time a little over 20 years ago [47], from which the authors wereable to deduce the temperature. Subsequently, forced evaporative cooling of neutral
atoms using radiofrequency has been suggested [48, 49]. This was a crucial step to inorder to increase phase space density [50, 51] and eventually made the realization of
Bose-Einstein condensation possible [7, 8]. Radiofrequency induced transitions havealso been used as an output coupler in order to create the ﬁrst atom Laser [52] andto create spinor condensates [53, 54].
In the years following, radiofrequency spectroscopy has developed into a powerful5

We have realized strongly interacting superﬂuid Fermi gases in other combinations of hyperﬁnestates [46]: There is the much employed Feshbach resonance between states |1⟩ and |2⟩ at 834G, andalso the |2⟩-|3⟩ Feshbach resonance at 810G

24

tool to study single-particle excitations in degenerate quantum gases. The ﬁrst density
dependent radiofrequency spectrum has been observed in Bose-Einstein condensates[55, 56], a clear signature of a mean-ﬁeld shift due to interactions between the atoms.
With the advent of mixtures of degenerate Fermi gases [57, 58, 59, 60, 61, 62]
radiofrequency spectroscopy has been used to study a variety of aspects of this new
system: In the ﬁrst radiofrequency spectroscopy experiments performed at JILA [63]and at MIT [64] the scattering length around a Feshbach resonance has been determined by an observation of the mean-ﬁeld shift and was shown to switch sign aroundthe Feshbach resonance. The authors of [64] also found an absence of the mean-ﬁeld
shift in a two-level system, in spite of the strong interactions between the atoms induced by a Feshbach resonance. This can be shown to be a direct consequence of theasymmetry of the fermionic wavefunction [65]. In the same year, the JILA group has
veriﬁed the creation of molecules consisting of two fermions below the Feshbach resonance [10] using radiofrequency dissociation spectra. The binding energy was shown~2
to closely follow Eb = ma
2 . In 2004, the Innsbruck group [13] observed bimodal radiofrequency spectra throughout the entire BEC-BCS crossover, see chapter 1.3: At
low enough temperatures a second peak appeared in the radiofrequency spectra. Thiswas interpreted as evidence for pairing and superﬂuidity throughout the BEC-BCS
crossover. Three years later, after the experimental realization of number imbalanced
mixtures of Fermi gases [16, 40], it was found that the minority component in ahighly imbalanced normal mixture also shows a gap in the excitation spectrum [20].
This spectral gap was initially interpreted as a signature of pairing of fermions in theimbalanced ground state.
At this time, a few shortcomings of standard radiofrequency spectroscopy became
obvious. Up to this point spectra were obtained by radiofrequency transfer to athird state and subsequent time-of-ﬂight for an accurate measure of the transferredatom number. Quantitative, and even qualitative, interpretation of the spectra wasdiﬃcult due a variety of factors (for a detailed discussion see [21]), which have all
been overcome by our group in the past few years:
1. Problem: Inhomogeneous trapping potential. Since many-body eﬀects dependon the density, the spatially varying density yields a spatially varying gap ∆,mean-ﬁeld shift, interaction strength 1/kF a and normalized temperature T /TF .
Even separate phases can exist within the trap [18].
Solution: We have performed spatially resolved radiofrequency spectroscopy[22]. Immediately after the radiofrequency pulse an absorption image of the25

ﬁnal state is taken, projecting the three-dimensional density distribution onto
a two-dimensional image. The three-dimensional density distribution can beobtained using the Inverse Abel Transform6 [66]. The resulting spectra represent
the response of a homogeneous system in the local density approximation.
2. Problem: Strong ﬁnal state interactions alter radiofrequency spectra qualitatively and quantitatively. If the atoms in the ﬁnal state are non-interacting, orat least weakly interacting, the spectrum shows the typical dissociative shape√thI(ω ∝ ω−ω), see 1.5.2. If the ﬁnal state is also strongly interacting, the lineω2
shape is altered. Furthermore, bound-bound transitions [46, 67] appear.
Solution: We have a created a degenerate fermion mixture using the hyperﬁnestates |1⟩ and |3⟩. State |2⟩ is only weakly interacting with both states |1⟩ and|3⟩. The local spectra can be used to determine the pair size of fermion pairs
in the unitary regime [46, 21].
3. Problem: Hartree energy: Mean-ﬁeld shifts usually neglected in interpretations
can shift radiofrequency spectra. These are diﬃcult to predict theoretically inthe strongly interacting regime.
Solution: Hartree terms can be identiﬁed and quantiﬁed in radiofrequency
spectra of imbalanced mixtures. This is discussed in chapter 4.
4. Problem: Absence of majority spectra: The qualitative and quantitative interpretation of the minority spectra requires comparison to the majority spectra.
Solution: The use of |1⟩-|3⟩ mixtures has the additional advantage of allowing
us to perform spectroscopy on both majority and minority components of the
mixture using state |2⟩ as the ﬁnal state. In previous experiments, obtaining
radiofrequency spectra from both states |1⟩ and |3⟩ would have required a twophoton transition from state |1⟩ to state |3⟩.
Recently, momentum resolved radiofrequency spectroscopy has been performed
at JILA [43] allowing to extract information about the spectral function of stronglyinteracting fermions, see also 3.3.2.
6

The inverse Abel transform is given by f (x, z) = − π1

column density measured in a typical experiment

26

∫∞
x

dn2D (y,z)

dy √ dy2

y −x2, where n2D (y, z) is the1.5.2

A Little Bit of Theory

Radiofrequency is broadly deﬁned as electromagnetic radiation in the frequency range
from 300Hz to 3GHz7 . Since the 6 Li |1⟩-|3⟩ Feshbach resonance is located at a magnetic ﬁeld of 690G and the hyperﬁne constant for 6 Liis aHF ∼ 152MHz, typical transition frequencies in our experiments are around 80MHz, corresponding to a wavelengthof 3 − 4m. As a consequence the momentum of the radiofrequency photon is negligible compared to the recoil of the atom. Another consequence is that radiofrequency
transitions are typically magnetic dipole transitions - they act as a spin-ﬂip operator8 .

Figure 1-10: Energy level structure of 6 Lias a function of magnetic ﬁeld. The relatively
small hyperﬁne constant aHF ≃ 152MHz results in a small splitting between the
ground states with F = 3/2 and F = 1/2 of 3/2 aHF ≃ 228MHz and in entering
the Paschen-Back region at relatively low ﬁeld (state |2⟩ has a maximum at ∼ 27G).
However, the admixture of the upper hyperﬁne levels is still large enough to ensure
eﬃcient radiofrequency transfer, see text.

Looking at the energy level structure in ﬁgure 1-10 of 6 Li, one might be tempted
to assume that the magnetic ﬁeld around 690G would bring 6 Lideep into the PaschenBack regime and that the radiofrequency radiation ﬂips the nuclear spin. However,7

Wikipedia entry: http://en.wikipedia.org/wiki/Radio spectrum
There are, however, electric dipole transitions in the radiofrequency range, e.g. The 2 S1/2 to2
P1/2 transition at a frequency of 1000MHz in hydrogen.
8

27

using perturbation theory it can be shown that the hyperﬁne interaction V = aHF S · I
results in an admixture to the wavefunction9 of the upper manifold ms = 1/2 into
the lower manifold ms = −1/2, containing the experimentally relevant states |1⟩ and|2⟩, on the order of ten percent. Since the magnetic moment of the nucleus is a few
thousand times smaller than that of the electron, the transitions are mostly due toelectronic spin-ﬂip. This raises the question whether, for the same radiofrequency
power, a |1⟩-|2⟩ transition will have a matrix-element similar to the |3⟩-|2⟩ transition.
Using the Breit-Rabi formula [68] it can be shown that the matrix elements, and hence|1⟩→|2⟩|3⟩→|2⟩/ΩR≃
Rabi frequencies ΩR = µ · B/~, are equal to within a few percent (ΩR
0.94). Using fully spin-polarized atomic samples we were able to conﬁrm the close
agreement of the Rabi frequencies experimentally by directly measuring the Rabi
frequency of spin transfer. This point is of importance for both experiments presented
in chapters 3 and 4.

1.5.3

Radiofrequency Spectroscopy on the Microscopic Level

After establishing the nature of radiofrequency spectroscopy as a spin-ﬂip operator,
one can immediately deﬁne the corresponding interaction operator:

Vrf=

2π~Ω2R)∑( ††ck3 ck↑ + ck↑ ck3→

2π~Ω2R∑ †ck3 ck↑

k

(1.9)

k

where ΩR is the Rabi frequency, c†k2 is the creation operator for a free particle inthe ﬁnal state and ck↑ the annihilation operator for a single particle in the initial state.
The second line holds if the ﬁnal state is empty, which is the case in a typical exper9
In ﬁrst order perturbation theory, the admixture to state |1⟩ (good quantum numbers: ms =−1/2, mI = +1) is given by|n(1) ⟩

= aHF

∑ ⟨k (0) |S · I|n(0) ⟩
(0)

k̸=n

(0)

En − Ek|k (0) ⟩⟨ 1 , 1 , 1, 0|S+ I− | 21 , − 12 , 1, 1⟩ 1 1| , , 1, 0⟩= aHF 2 2
2µB B
2 2
(1.8)
where we have used S · I = Sz Iz − S+
I − S− I+ and only kept the relevant states in the second√ −equality. Therefore, the admixture is 2µ2aBHF
B ≃ 0.1

28

iment10 . From equation 1.9 it can be seen that the radiofrequency operator probes
the single-particle excitation spectrum: One particle is removed from its environment
and the energy cost, including interaction and kinetic energy, is provided by the radiofrequency photon: ~ωRF ≡ Ek + ϵk − µ. Using the generalized BCS-wavefunction|ψBCS ⟩, see section 1.3, and Fermi’s Golden Rule one can calculate the radiofrequency
spectrum in linear response [23]:I(ω) =

2π ∑|⟨f |Vrf |ψBCS ⟩|2 δ(~ω − Ef )~ f

(1.10)

which can be written as [23]:√I(ω) ∝

ω − ωth
ω2

√
1+

ωth
µ
+
ω
~ω

(1.11)

with µ the chemical potential of the generalized BCS state and the deﬁnition√
for the threshold frequency ωth ≡ µ2 + ∆2 − µ. The ﬁrst square-root factor is thedominant feature of the spectrum and quite generic, with the square-root representing
the three-dimensional density of states and the 1/ω 2 dependence can be understoodin the limit of high energy: It has been shown recently [69] that the momentum
distribution of any system of fermions interacting through a δ-potential behaves as
C/k 4 for high momenta. C is the called the contact coeﬃcient. In this limit, thekinetic energy dominates the dispersion relation and ω ∝ k 2 . The contact coeﬃcient
will be the subject of chapter 3.7.4.

Transitions from states other than the approximately resonant one, state |3⟩ in the above case,are usually highly suppressed as a typical linewidth of an experimental radiofrequency transitionis on the order of kilohertz, whereas other hyperﬁne states are typically detuned by hundreds of
Megahertz.
1029Chapter 2
Imbalanced Fermi Systems and thePhase Diagram
2.1

BEC-BCS Crossover with Density Imbalance:
A Mean-Field Description (and Beyond)In a mean-ﬁeld description, the results of chapter 1.3, see also 4.3 are readily generalized, see [70, 71] and references therein. For imbalanced Fermi systems n↑ ̸= n↓ thefree energy to be minimized now takes the form ⟨H − µn − h∆n⟩, where H is the
Hamiltonian deﬁned previously in equation 1.2, µ is the systems chemical potential
ﬁxing the total density n = n↑ + n↓ , h is the species-imbalance chemical potential,
ﬁxing a constant density diﬀerence ∆n just as µ ﬁxes the total particle density. By
comparison to electrons in a metal subject to a magnetic ﬁeld it is clear that h plays
the role of an eﬀective Zeeman ﬁeld and ∆n can be regarded as an eﬀective magnetic
moment. From now on we will refer to h as the Zeeman ﬁeld. The dimensionless normalized density diﬀerence σ = ∆n/n is typically used as a parameter of the system.
σ is usually referred to as the polarization.
It can be shown that the excitation spectrum 1-9 takes the form:√
ξk2 + ∆2 − h√E↓ (k) =ξk2 + ∆2 + h
E↑ (k) =

(2.1)
(2.2)

where, as before, ξk = ϵk − µ. Now the coherence factors vk , uk are modiﬁed to:
u2k = 1 − vk2 =E↑ (k) − ξk − hE↑ + E↓30(2.3)

Figure 2-1: Zero temperature mean-ﬁeld phase diagram of fermions in the BEC-BCS
N −N
crossover. Plotted is the phase as a function of number imbalance P = N↑↑ +N↓↓ andinteraction strength 1/kF a (taken from [71]). The grey area is a phase separated state
between a superﬂuid of equal densities and a partially polarized normal phase. SFM
stands for a polarized superﬂuid. Pc1 and Pc2 denote the phase boundaries separating
the diﬀerent phases.

And the number and gap equations 1.7 are:∫d3 k 1 − f (E↑ ) − f (E↓ )(2π)3 E↑ (k) + E↓ (k)∫)
d3 k ( 2
2
n = 2
u
f
(E
)
+
v
f
(−E
)
σ
−σ
k
k
(2π)3

1
−
=
V0

(2.4)
(2.5)

It can be seen that in the limit of equal densities σ = 0 the above equations reduceto the familiar BCS results of chapter 1.3.

2.2

Theoretical Phase Diagrams

With the above equations one can produce the zero-temperature phase diagram in
the BEC-BCS crossover shown in ﬁgure 2-1.
This treatment has been generalized to ﬁnite temperatures [72].
Just as in the balanced case presented in chapter 1.3, the above mean-ﬁeld treatment for the imbalanced case exhibits many of the qualitative features of the phase

31

Figure 2-2: Zero temperature phase diagram obtained by ﬁxed-node Monte Carlo calculations, including interactions in the normal phase (taken from [73] and modiﬁed).
N −N
Plotted is the phase as a function of number imbalance P = N↑↑ +N↓↓ and interaction
strength 1/kF a. The arrows point to the qualitatively very diﬀerent regimes, which
will be presented in the following chapters. The nomenclature is as follows: Polarized
superﬂuid SFP , balanced superﬂuid SF0 , partially polarized normal phase NP P , fullypolarized normal phase NF P . The diﬀerent phases are discussed in chapter 2.3.

diagram. But the mean-ﬁeld treatment fails when put to the test quantitatively. A
discussion of the shortcomings of a mean-ﬁeld treatment have been provided at theend of chapter 1.3 and will be discussed in more detail in chapter 4.1. In essence, thequantitative inaccuracy can be traced back to neglecting interactions in the normal
phase. The authors of [73] perform ﬁxed-node Monte Carlo simulations taking theinteractions in the normal phase into account [74, 75], see ﬁgure 2-2. These normal
state interactions will be the subject of chapter 3, in which we use radiofrequency
spectroscopy to experimentally determine the properties of this Landau Fermi Liquidof a strongly interacting Fermi system.
As is indicated in ﬁgure 2-2, in the following chapters 4 and 5 we will focus on
quantitative studies of the single-particle excitation spectrum using radiofrequency
spectroscopy in two diﬀerent physical regimes: The highly imbalanced normal phaseand the (almost) balanced superﬂuid phase. In chapter 6 we will present another32

quantitative study, in which we use the density distribution in a non-homogeneous
potential to determine the equation of state for ﬁnite temperature at unitarity.

2.3

Discussion of the Theoretical Phase Diagram

T = 0: The phase diagram in ﬁgure 2-2 shows a very rich structure and here we will
discuss the diﬀerent phases based on [71, 73, 76].
The line P = 1 or N↓ = 0 is the fully polarized non-interacting Fermi gas (NFP ).
Here, the system consists of fermions ﬁlled up to momentum kF , resulting in thewell-known equation of state:
3
ENFP = N↑ EF ↑
5

(2.6)

~2 k 2

with the Fermi energy EF ↑ = 2mF .
The limit of a single impurity (P = 1− ) is the line inﬁnitesimally below P =1. This polaronic phase, a phase of dressed quasiparticles interacting with a Fermisea, already yields rich information about the interacting system. In particular, in
chapter 3 we will show that a dilute gas of such Fermi-polarons (more than a single
impurity) is characterized by a binding energy, an eﬀective mass m∗ ̸= m and weakinteractions and essentially follows the physics of the single impurity limit. This
allows to extrapolate into the much more complicated partially polarized normal phase(NPP ). We encounter this phase at ﬁnite imbalance (concentration x ≡ N↓ /N↑ > 0) if
the two-body bound state energy is not too large. In the regime x ≪ 1 the equationof state can be expanded as [74]:(
ENPP = N↑ EF ↑3
3 m 5/3 3 2− Ax +
x + Fx
5
5 m∗
5

)
(2.7)

The ﬁrst term is the energy of the majority Fermi sea, assumed not to be appreciably perturbed by the presence of a few minority particles. The second term represents
the interaction energy of polarons with the majority particles. It is attractive andquite signiﬁcant in the unitary limit: A ≃ 0.6ϵF . The third term represents the energy of a Fermi sea of non-interacting polarons since1 (EF ↓ N↓ )/(EF ↑ N↑ ) = mm∗ x5/3 .
The eﬀective mass renormalization has been shown to be fairly small and on the order of m∗ = 1.17m in the unitary limit [77, 74, 78]. The fourth term represents the1

Since the eﬀective mass might depend on the density, the third term in the parametrization inequation 2.7 might have to be modiﬁed to depend on x7/3 instead of x5/3 .

33

(weak) interactions between polarons and has been calculated to be F = 0.14 usingﬁxed-node Monte Carlo calculations [74].
Once the density imbalance between majority and minority becomes smaller, thesystem enters a phase separated state (SF0 +NPP ) with a balanced superﬂuid separated from a partially polarized normal phase. The unitary limited balanced superﬂuid state is described by the equation of state:µ = ξ EF

(2.8)

where ξ = 1 + β with the Bertsch parameter β. The value of ξ is still subject todiscussion, ranging between 0.32 < ξ < 0.49 [30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
14, 40]. In this regime of equal densities, the existence of the superﬂuid gap expels
all excess particles out of the superﬂuid as their presence would “cost” an energy ∆.
It is obvious that the phase diagram depends strongly on the precise energetics ofthe normal and the superﬂuid phase and they will be the subject of the following
chapters.
In the deep BEC regime, it is clear that the imbalanced ground state consists of a
BEC of strongly bound molecules weakly interacting with excess majority atoms (allminority atoms are bound into molecules). The interactions between fermionic atoms
and the bosonic molecules are characterized by the scattering length aad ∼ 1.18a [79]2
and the typical mean-ﬁeld energy UMF = 4π~maad nd . In this regime, the superﬂuid iscalled partially polarized (SFP ).
If UMF is increased beyond a critical value, the system will phase separate into apartially polarized superﬂuid and a fully polarized normal cloud (SFP +NF P ). In ahomogeneous system the phase separation would be evident. For the experimentally
relevant harmonic trapping potential, this transition would be less obvious as the
majority cloud extends beyond the size of the superﬂuid. Since the superﬂuid density
continuously approaches zero and the transition is of ﬁrst order, it can only be observable through a kink in the trapped three-dimensional majority density distribution
at the interface, a signature very hard to detect experimentally.
For increasing interactions one might enter a small sliver of a phase separated
state between a partially polarized superﬂuid and a partially polarized normal region(SFP +NP P ). This phase does not exist in a mean-ﬁeld description and the authorsof [73] concede that this phase might not be realized.
A further increase in the interaction strength results in a phase separated state
between a balanced superﬂuid and a partially polarized normal gas. The criticaldensity imbalance σc in the normal phase at the interface between the superﬂuid
34

and the normal region was calculated [73] and experimentally conﬁrmed [19] to beσc ≃ 0.36.

2.4

Experimental Phase Diagram

As is intuitively obvious and as can be seen from ﬁgure 2-2, as 1/kF a is increased
towards the BEC side of the Feshbach resonance, more and more minority particles
pair up with majority particles, thereby increasing the critical polarization σc in thenormal phase. At a certain critical interaction strength, all minority particles will bebound in generalized molecules, no matter how high the density imbalance - this is
the polarized superﬂuid regime.
Its experimental signature is immediately accessible through the majority andminority density proﬁles, see ﬁgure 2-3. The top row shows elliptically averaged
column density proﬁles and the column density diﬀerence. The reconstructed threedimensional density proﬁles of majority and minority are shown in the second row.
As the BEC limit is approached, one can indeed clearly see the region of the partially polarized normal region shrinking until all minority particles are paired into a
Bose-Einstein condensate of molecules, phase separated from a fully polarized normal
region. The density imbalance σ = (n↑ − n↓ )/(n↑ + n↓ ) is shown in the third row,
conﬁrming that the critical imbalance indeed approaches unity. This critical imbalance is plotted in ﬁgure 2-4 as a function of interaction strength and compared to theMonte Carlo phase diagram, as well as to the data in the following chapters using a
diﬀerent method: Radiofrequency spectroscopy.

2.5

Phase Diagram at Finite Temperatures

So far we have limited the discussion to the zero-temperature case. Theoretically,making a quantitative prediction at ﬁnite temperature for a density imbalanced system remains a formidable challenge even within the realm of mean-ﬁeld theory as
one must include the eﬀect of thermally excited molecules and quasiparticle excitations (see also chapter 4), which might be strongly interacting in the unitary regime.
Again, one must resort to Monte Carlo calculations2 and experiments [19, 80] to test
the accuracy of theories. In fact, even the balanced two-component Fermi mixture at2

While there have been various Monte Carlo calculations for the imbalanced case at zero temperature and the balanced case at ﬁnite temperature, to date the treatment of imbalanced Fermi gases
at ﬁnite temperature remains elusive.

35

Figure 2-3: Density proﬁles of imbalanced Fermi mixtures in a harmonic trap. The
top row (a)(e) shows the averaged column-density proﬁles for various magnetic ﬁelds.
Green: Majority, blue: Minority, black: Column density diﬀerence. The black dottedline is a zero-temperature Thomas-Fermi distribution ﬁt to the majority wing (r >R↓ ). The middle row (f)(j) and the bottom row (k)(o) show the reconstructed threedimensional density distributions and the spin polarizations obtained from the proﬁlesin the top row. The critical polarizations σc at the phase boundary r = Rc are
indicated by the right-pointing arrows. The values for R↑ (in µm), Rc /R↑ , and R↓ / =R↑ were respectively: for (a), (f ), (k), 381, 0.33, 0.33; for (b), (g), (l), 380, 0.33, 0.33;
for (c), (h), (m), 362, 0.35, 0.59; for (d), (i), (n), 371, 0.44, 0.72; for (e), (j), (o), 367,
0.41, 0.76. T /TF 0 ≃ 0.05, and TF 0 ≃ 1.0µK (see the text for deﬁnitions).

ﬁnite temperatures in the unitary regime is still under current theoretical and experimental investigation. We will present an outlook on our recent experimental eﬀorts
in this regard in chapter 5.
For completeness, we will brieﬂy present our results on the ﬁnite temperaturephase diagram in the unitary limit. For more details refer to [19] and Christian
36

P

1.0

0.8

0.6

0.4

0.2

0.0
0.6

0.4

0.2

0.0

-0.2

1 / kF a

Figure 2-4: Experimental phase diagram approaching zero temperature. Criticalinteraction strength determined by several methods: Solid green triangles: Density
distribution. Solid red circles: Radiofrequency spectroscopy of polarons, chapter 3.
Open blue squares: Radiofrequency spectroscopy of moderately imbalanced Fermi
mixtures based on ﬁgure 4-9 in chapter 4. For comparison, the ﬁrst order phasetransition line from Monte Carlo calculations is added as solid black line, see ﬁgure
2-2.

Schunck’s thesis [21].
We have used an analysis of the density proﬁles of majority and minority components and recorded the critical density imbalance in the normal (σc ) and in the
superﬂuid (σc ) region, this time as a function of the normalized local temperature
T /TF . The resulting ﬁnite temperature phase diagram is shown in ﬁgure 2-5.
The fact that at ﬁnite temperature the superﬂuid supports ﬁnite polarizations σ
in the form of thermally excited quasiparticles will be used in chapter 4 to determine
the superﬂuid gap ∆ and the Hartree energy U .
Furthermore, ﬁnite temperatures make the superﬂuid less robust against imbalance, for instance the critical polarization is reduced from σc ≃ 0.36 at zero temperature to σc ≃ 0.2 at T /TF ≃ 0.07, see ﬁgure 2-5. This might explain the discrepancy
between the Monte Carlo calculations and our data in ﬁgure 2-4. For an accurate37

Figure 2-5: Experimental phase diagram at ﬁnite temperature in the unitary limit.
The critical polarizations σc (black solid circles and square) and σc (grey solid circles) are displayed along the local T /TF at the phase boundary. The yellow area
(σs < σ < σc ) represents a thermodynamically unstable region, leading to the phaseseparation, see chapter 2.3. We identify the tricritical point, separating a ﬁrst orderphase transition (T /TF < 0.07) from a second order phase transition (T /TF > 0.07).
The linear ﬁt to the σc values is shown as a guide to the eye for the normal-tosuperﬂuid phase transition line. Each data point consists of ﬁve independent measurements and error bars indicate the standard deviation.

comparison, a theoretical phase diagram is needed for ﬁnite temperatures.

38

Chapter 3
The Fermi-Polaron
3.1

N+1 = The Impurity Problem

A seemingly simple problem: What is the energy of a single |↓⟩ particle (fermion orboson) interacting with an environment of |↑⟩ fermions, composing a Fermi sea, which
we will denote as |F S⟩↑ ? Apart from the universal interest the impurity problem has
as discussed below, we have seen in the previous chapter that the interactions in
the normal state have substantial impact on the phase diagram of imbalanced Fermi
mixtures. This becomes obvious when a quantitative comparison is made between
the mean-ﬁeld phase diagram, neglecting normal phase interactions, and the Monte
Carlo calculations including these interactions, see ﬁgures 2-1 and 2-2.
In general, no real physical system is free of imperfections and impurities. Modeling the ions in a metal as having a perfect crystalline structure and treating theelectrons as free waves in Bloch bands is surprisingly successful in describing many
properties at high temperature. However, at low temperatures, on the order of theinteraction energy between impurity and environment, new phenomena appear. A
famous example is the Kondo eﬀect, where immobile magnetic impurities decrease
the mobility of conduction electrons, therefore giving rise to an enhanced resistancein metals below the Kondo temperature [81]. Without the presence of an impurity,
the resistance decreases quadratically with the temperature, the usual signature ofPauli blocking.
A diﬀerent example for an impurity problem is an electron moving in a crystal
lattice creating a localized polarization, or phonon wavepackets, by displacing theions in a crystal from their respective equilibrium positions. The electron dressed byphonons forms the lattice polaron [82]. The diﬀerence to the above examples and theexperimental system presented later in this chapter is that the fermion is interacting
39

with a bosonic environment, the lattice phonons.
The phonons, or lattice polarization, created by one electron can, in turn, interact
with another electron giving rise to the small attractive interaction responsible for
Cooper pairing [5] and are therefore an essential ingredient for the formation of thesuperﬂuid phase. This immediately raises the question of the connection between
interacting polarons (the limit of T → 0 and σ → 1 in 2-5) and the normal stateof a balanced Fermi gas above the superﬂuid transition temperature TC (the limitof σ → 0 and T > Tc in 2-5). If superﬂuidity can be regarded as condensation ofinteracting polarons, a good understanding of the normal polaronic phase might prove
essential.
As interacting Fermi systems have been successfully described within the framework of Landau’s Fermi Liquid Theory, we will introduce the basic concepts of quasiparticles and their properties below, largely based on [83].

3.2Another Angle on the Impurity Problem

Above, the argument is made that the a single fermion interacting with a Fermi sea
constitutes a quasiparticle which can be described in terms of Fermi liquid theory.
One can look at this problem from the other end: Consider two fermions in a box
at almost zero kinetic energy, whose interaction potential is close to supporting abound state and that this bound state can be tuned above or below the energy of
the incoming particles. This is a typical realization of a strongly interacting system
around a Feshbach resonance and can be solved exactly [23]. Their energy is shown
ﬁgure 3-1a: There exists a bound state in the interatomic potential below the Feshbachresonance (dashed line), above the Feshbach resonance the particles scatter oﬀ eachother and acquire a phase shift related to the scattering length a.
How is the energy of the particles modiﬁed if one were to add particles of one
kind to the system, one by one, building up a majority Fermi sea? The result is also
shown in ﬁgure 3-1a as the solid black line. It can be seen that the energy changes
substantially: At unitarity where we expect the binding energy to be zero, it is in facta sizeable fraction of the majority Fermi energy. Away from the Feshbach resonancethe modiﬁcation reduces and eventually approaches the mean-ﬁeld limit. The energy
diﬀerence between two-body physics and many-body physics is shown in ﬁgure 3-1b.
Another important result is that, even when the interatomic potential supports abound state, no two-body bound state is formed. The particle remains a fermionic(quasi-) particle instead of forming a composite boson or molecule. Physically it
40

b)

F

2

e / ygrene ytirupmIe / ) am/1 - ygrene ytirupmI(

0.0

F

a)

0.5

1.0

0.6

0.4

1.5

0.2

2.0

2.5

0.0

1

0

-1

1 / kF a

1

0-1

1 / kF a

Figure 3-1: Two-body physics vs many-body physics: a) Dashed line: Binding energy~2
as a function of interaction strength kF1 a according to the simple model of two
ma2
particles interacting through a scattering resonance, see Feshbach resonance 1.2. Solidline: Energy from a variational many-body calculation, see equation 3.23. b) Energy
diﬀerence between the two-body and many-body calculation.

is clear, though, that once the two-body binding energy reaches a critical value, abosonic molecule will form regardless of the presence of a Fermi sea. We expect this~2
critical interaction strength to be reached when the two-body binding energy ma
2 ison the order of the Fermi energy of the Fermi sea. It is important to note, see also
ﬁgure 3-11 and 3-1 that the energy of this bosonic quasiparticle is not described by~2
but will also be modiﬁed due to the presence of the Fermi sea. We will discuss
ma2
this “Molaron” below.

3.3

Landau’s Fermi Liquid Theory

How is it possible that the naı̈ve model for metals as free electrons moving in a lattice
structure is so successful in describing the properties of solids when the electrons
are, in fact, interacting through the Coulomb force, which is, although screened, stillrather strong? The explanation and theoretical framework has been provided by L.D.
Landau more than 50 years ago [82]. At the core of Fermi Liquid Theory is the notionof a quasiparticle, characterized by its eﬀective mass m∗ , its quasiparticle residue Z
and the interaction parameters Fs and Fa .
In a non-interacting Fermi system free particles of wavevector k are the eigenstates.
This, together with Pauli’s exclusion principle, completely characterizes the groundand excited states: The ground state is just a ﬁlled Fermi sea - at zero temperature the41occupation probability is one for k ≤ kF and it is zero for k > kF . A particle added
above the Fermi momentum kF or a hole added below kF constitutes an elementary
excitation of the system. These excitations are also eigenstates of the system andtherefore have an inﬁnite lifetime.
Consider now a system of fermions in which the interactions are gradually, i.e.
adiabatically, switched on. The system will remain in its ground state but now withan interaction term in the Hamiltonian. Free particles are no longer eigenstates ofthe system as particles will scatter and “dissolve” into the ground state by way ofparticle-hole excitations in the Fermi sea. Similarly, if interactions are switched onstarting with a system in its ground state plus an elementary excitation at k > kF ,this excitation will be an approximate eigenstate of the real interacting system if
its lifetime is long enough that it qualiﬁes as an eigenstate. As the Pauli principle
together with energy conservation restrict the available phase space for decay, one cansee that the lifetime of an added particle of energy ξk = ϵk − µ, measured relative to
the Fermi energy, has a scattering probability, or lifetime, proportional to 1/τ (ξ) ∝ ξ 2 .
Hence, this excitation can be treated as an approximate eigenstate if its energy isclose to the Fermi energy ξk → 0. This elementary excitation is called a quasiparticleof momentum k to the real ground state. Similarly, an elementary excitation ofk < kF is called a quasihole.

3.3.1

Quasiparticles

Quasiparticles are fermions since the system adiabatically evolves from a system of
non-interacting fermions. In the ground state the real system can be represented as isshown as the solid black line in ﬁgure 3-2 - there are no quasiparticles present above
kF and no quasiholes below kF . It is important to stress that ﬁgure 3-2 does notshow the distribution of quasiparticles, as they are only well-deﬁned entities close to
the Fermi energy. In other words, quasiparticles are elementary excitations and giveno information about the ground state of the system. Also shown is the eﬀect ofinteractions on the distribution of the bare particles.
Just as in the non-interacting case, the excitation of the system can be characterized by the deviation from the ground state:δn(k) = n(k) − n0 (k)(3.1)∑
and for small densities of quasiparticles Nexc = k |δn(k)| ≪ N and energies near
the Fermi energy, the energy of the system can be expanded as:
42

nk

1.0

ground state:
no quasiparticles above k ,no quasiholes below kFFbare particles
0.5

Z = z

k

F

0.0
0.0

0.5

1.0

1.5

k / kF

Figure 3-2: Cartoon picture of the ground state momentum distribution (solid blackline) and bare particle momentum distribution (solid red line).

E[δn(k)] = E0 +∑ϵk δn(k) +k1 ∑f (k, k ′ )δn(k)δn(k ′ )2V k,k′(3.2)

where E0 is the (interacting) ground state energy, ϵk is the energy of non-interacting
2 2
quasiparticles ϵk =: ~2mk∗ , with the deﬁnition of the eﬀective mass m∗ and f (k, k ′ ) contains the interactions between quasiparticles.
The eﬀective mass m∗ will be diﬀerent from the bare mass m due to the interactions
between the bare particles. It is insightful to think of quasiparticles as bare particles
being “dressed” by their surroundings - a quasiparticle is a superposition of diﬀerent
unperturbed states. In this picture it is easy to see why the eﬀective mass of aquasiparticle diﬀers from that of a free particle: It is dragging along a cloud of dressing
particles. A quantity related to the eﬀective mass is the quasiparticle residue Z, whichquantiﬁes how much “bare particle” is contained in a quasiparticle at momentum k.
Z also gives the size of the “jump” in the bare particle distribution at k = kF called
the Migdal discontinuity, see also ﬁgure 3-2. We will get back to this point in thenext section on the microscopic description of interacting Fermi systems.
The function f (k, k ′ ) in equation 3.2 is diﬃcult and tedious to compute. However, it is generally clear that the interaction between quasiparticles is important in
the description of collective excitations, such as sound waves, spin waves, etc. Morespeciﬁcally, f (k, k ′ ) can be decomposed into a symmetric part f s (k, k ′ ), independent43

of spin, and an antisymmetric part f a (k, k ′ ), and then expanded in Legendre polynomials. The resulting coeﬃcients Fls,a can be connected to parameters measurable inexperiments, such as the eﬀective mass m∗ related to F1s , the speed of sound or thecompressibility ∂nrelated to F0s and the paramagnetic spin susceptibility χp related∂µ
to F0a .

3.3.2

Fermi Liquid Theory on a Microscopic Level

We begin with the deﬁnition of the single particle Green function G(xt, x′ t′ ). It
contains all the information about one particle or one hole. It is deﬁned as:
G(xt, x′ t′ ) := i⟨ϕ0 |T {ψ(x, t)ψ † (x′ , t′ )}|ϕ0 ⟩(3.3)

where |ϕ0 ⟩ is the true ground state wavefunction of the system and generally unknown, T {...} is the time ordering operator, and ψ(x, t), ψ † (x, t) are the annihilationand creation at operator, respectively. The interpretation of formula 3.3 is immediately clear: G(xt, x′ t′ ) is the probability amplitude of ﬁnding a particle at x at timet, when a particle has been added at x′ at time t′ .
For simplicity, let us assume that the system is translationally invariant and thatthe time evolution is independent of the starting time t′ . The Fourier transform isthen G(k, t). G(k, 0− ) is proportional to the momentum distribution n(k):G(k, 0− ) = −in(k)(3.4)

A hole is added at t′ = 0: ak (0)|ϕ0 ⟩. If one were to measure at an inﬁnitesimal
time step later, one would have have measured the probability of being able to createa bare hole, which is proportional to the momentum distribution n(k). Note that theadded particle is a bare particle and therefore not an eigenstate. Eventually, for longenough times the particle will scatter and the according excitation will damped as
expressed by the fact that limt→∞ G(k, t) → 0.
At this point it is convenient to introduce the spectral function A(k, ω):A+ (k, ω) =∑|⟨ϕn |a†k |ϕ0 ⟩|2 δ(ω − ϵn )(3.5)|⟨ϕn |ak |ϕ0 ⟩|2 δ(ω − ϵn )

(3.6)

n

A− (k, ω) =∑n

A(k, ω) = A+ (k, ω) + A− (k, ω)

44

(3.7)

Similarly to the physical interpretation of equation 3.3, A+/− is the probability
that adding/removing a particle with momentum k to the system will be associatedwith, in other words “costs”, the energy ~ω. For free particles the energies ϵn are just2 k2
the ϵk = ~2m
and A(k, ω) is a sum of discrete δ-functions as the fermionic excitations
are eigenstates. Including interactions, an added particle is not an eigenstate any more
but a superposition of the eigenstates of the interacting system. However, following
the argument above, excitations close to the Fermi energy constitute quasiparticles
and one can expect A(k, ω) still to exhibit maxima at the new resonance frequencies
ϵk spread out over a width Γk , determining the inverse lifetime of the quasiparticle.
More explicitly, with the deﬁnition of the spectral function and the Green function
above:−iµt−iG(k, t) = e∫ ∞A+ (k, ω)e−iωt dω

(3.8)

0

with the chemical potential µ. This can be rewritten deforming the contour of
integration and choosing α ≫ 1/t:
∫ iα−iG(k, t)eiµt≈

A+ (k, ω)e−iωt dω − 2πi∑0ζj e−iωj t

(3.9)

j

where ζj are the residues of A+ (k, ω). One of the poles of A(k, ω) will give adominant contribution. The according frequency is ω = ξk − iΓk . The other poles
and the integral along the imaginary axis give contributions of various frequencies
and very quickly decohere. Deﬁning zk := 2πıζk , the time evolution of an added bareparticle is:iG(k, t) = zk e−iξk t−Γk t + iGinc (k, t)(3.10)

Therefore, with A(k, ω) = −2 ImG(k, ω) for k → kF the spectral function can be
written as:
A(k, ω) = Zδ(ω − ξk ) + Ainc (k, ω)(3.11)

From the equations above it is clear that Z = zkF can be interpreted as theprobability of free propagation of a quasiparticle. More generally, it can be shownthat zk measures how much bare particle is contained in a quasiparticle.
Inserting a complete set of eigenstates into 3.3 and using the above deﬁnitions
3.7 for the spectral function and performing the Fourier transform, one arrives at the
Lehmann-representation for the Green function:
45

(

∫ ∞
G(k, ω) =dω
0′

A+ (k, ω)A− (k, ω)− ′′+ω − ω + µ − i0
ω + ω − µ − i0+)(3.12)

which we will need in the next chapter when making the connection between the
self-energy Σ(k, ω) amd the spectral function.

3.3.3

Calculating m∗ and Z

In order to explicitly calculate the properties of an interacting system described bya Hamiltonian of the form H = H0 + HI , we not only need the single particle Greenfunction but the two-particle Green function, which is deﬁned similar to the oneparticle Green function in equation 3.3:
K(x1 , x2 , x3 , x4 ) := ⟨ϕ0 |T {ψ(x1 )ψ(x2 )ψ † (x3 )ψ † (x4 )}|ϕ0 ⟩(3.13)

With this deﬁnition, the Schrödinger equation can be expressed in terms of theGreen functions:(∂~ 2 ∇2
−i
∂t
2m

)
′

∫

G(x, x ) −d3 x′′ V (|x| − |x′′ |)K(x′′ , x, x′′ , x′ ) = iδ(x − x′ ) (3.14)

After Fourier-transforming this equation can be expressed as:
)
( 2 2
~k
− ω − Σ(k, ω) G(k, ω) = 12m(3.15)The quantity Σ(k, ω) is called the self-energy and contains all the eﬀects of interactions in the system. It is in essence the correction to the free-particle kinetic
energy ϵk by interactions and is therefore closely related to the eﬀective mass m∗ andthe quasiparticle residue Z, see discussion following equation 3.2. Equation 3.15 and3.10 can be combined to give the well-known result for the quasiparticle residue Z:

46

zk =
=

=
=

I
1
G(k, ω)dω
2πi
I
1
dω
( ~2 k 2)
2πiω − 2m − Σ(k, ω)()−1∂Σ(k,ω)I
1 + ∂ω |ω=ϵk
1
dω ′
2πi
ω′
)−1
(
∂Σ(k, ω)
|ω=ϵk
1+
∂ω

(3.16)(3.17)(3.18)(3.19)As mentioned above, one can relate the eﬀective mass to the self-energy Σ(k, ω)from equation 3.15. The pole of the Green function giving the dispersion relation
satisﬁes ω = ϵk + Σ(k, ω). Expanding around the ground state k = 0 and ω = µ + ω ′gives:µ + ω ′ = ϵk + Σ(0, 0) +| {z }∂Σ(ϵk, ω ′ )∂Σ(ϵk, ω ′ ) ′
ϵk +
ω
∂ϵk
∂ω ′

µ

leading to:1 − ∂Σm∗∂ω=∂Σ
m
1 + ∂ϵk(3.20)k
Here we made the substitution ϵk = ~2m
and replaced Σ(0, 0) by µ, which is only
true in the case of a single impurity particle as no momentum states are blocked.
Finally, for completeness let us make the connection between the self-energy1
Σ(k, ω) and the spectral function A(k, ω) by inserting the standard identity ω±iη=1
P ω ∓ iπδ(ω) into the Lehmann representation 3.12 and using equation 3.15:
2 2A(k, ω) = −2 Im[G(k, ω)]−2Im[Σ(k, ω)]=~2 k2(ω − 2m − Re[Σ(k, ω)])2 + (Im[Σ(k, ω)])2(3.21)

The above equations for m∗ , Z and A(k, ω) from the self-energy will become
important in the next section when the self-energy will be calculated for a particular47

limit of a Fermi Liquid, namely a |↓⟩ impurity particle interacting with a |↑⟩ Fermi
sea.
One result of the above equation is that the (bare) particle momentum distributionn(k) can be written as, using 3.4 (T = 0):∫ 0n(k) =−∞dωA− (k, ω)(3.22)

with the energy measured relative to the Fermi energy.
For the special case of a Fermi Liquid of a single particle immersed in a Fermi sea
of majority particles, the according momentum distributions are plotted in ﬁgure 3-3.
nk

1.0
majority

0.9

0.8
Z
Z

1

0.03

0.02

minority

a

C
4

k

0.01

0.00
0.0

0.5

1.0

1.5

2.0

2.5
k / kF

Figure 3-3: Momentum distribution for the polaron system at unitarity. It can be seenthat the minority momentum distribution has a δ-peak of weight Z at k = 0 and acontinuous momentum distribution at k > 0. The majority momentum distribution is
described by a stepfunction only in the thermodynamic limit. For a ﬁnite volume V we
see the particle-hole excitations in the ground state. The discontinuity in the majority
momentum distribution is given by the quasiparticle residue of the majority Z↑ . It
approaches unity in the thermodynamic limit. At large momenta, the momentum
distributions of majority and minority overlap and are characterized by the contact
coeﬃcient C, see chapter 3.7.4.

From the expression for the spectral function in equation 3.11 one can readily see
another interpretation of the quasiparticle residue Z: At T = 0 the distribution of48

the bare particles jumps by Z at k = kF . If k < kF , the δ-function is included witha weight of Z in the above integration as ξk < 0. If k > kF , it is not included. The
coherent part of the spectral function only determines the size of the jump in the
momentum distribution and stems from the fact that excitations change in characterfrom quasiparticles to quasiholes at k = kF . The actual distribution is determinedby the complicated integral over the incoherent part of the spectral function, seeequation 3.11, and therefore not necessarily inﬂection symmetric around kF . We will
see this below in the speciﬁc example of an impurity particle interacting with a Fermi
sea.

3.4

A Special Case: A |↓⟩ Swimming in a |↑⟩ Fermi
sea

In this section we will use some of the results from the previous section and apply them
to a particular limit of a Fermi Liquid: A |↓⟩ impurity particle strongly interactingwith a Fermi sea of |↑⟩ particles. Of particular interest are the value of the selfenergy Σ(0, 0) ≡ µ and the Fermi liquid parameters m∗ and Z as they are directly
accessible in our experiments. The general cooking recipe to approach this problem
would consist of the following steps:• get vertex function Γ [84] describing the full interaction based on the Hamiltonian of the system• T-matrix approximation for Γ: The impurity particle interacts repeatedly with
only a single particle in the Fermi sea of |↑⟩ at a time. In other words, upona scattering event the impurity particle creates only a single particle-hole pair.
It can do so any number of times and hence perturbation theory can be carried out to arbitrarily high orders, giving the well-known ladder series (ladderapproximation).
• get self-energy Σ(k, ω) from Γ [84]• extract physical quantities from Σ(k, ω)In order to simplify the above calculation, in addition the non-selfconsistent approximation is often made of an undisturbed majority momentum distribution thatcan still be described by a step-function - it does not exhibit the characteristic C/k 449

behavior, see chapter 3.7.4 and ﬁgure 3-3. In the Chevy Ansatz, see below, the momentum distribution of the majority retains the high momentum tail for ﬁnite volume
but approaches a step function in the thermodynamic limit, as one would expect fora single impurity particle interacting with a majority Fermi sea.
If the constraint of an undisturbed majority momentum distribution is released in
the T-matrix approximation, however, the resulting minority energy is E↓ ≃ −0.9ϵF[85]1 compared to the exact result E↓ ≃ −0.615ϵF [87] and experimental result E↓ ≃−0.64ϵF , see below.
A more insightful but mostly equivalent approach has been provided by F. Chevy:
It is a variational calculation based on a trial wavefunction for the system in the veinof the T-matrix approximation, namely the impurity particle only interacting withsingle particle-hole pairs. It allows, however, for a modiﬁed majority momentum
distribution. The Ansatz can be written as:|ψ⟩ = ϕ0 |k′ ⟩↓ |F S⟩↑ +∑ϕqk c†k↑ cq↑ |k′ + q − k⟩↓ |F S⟩↑(3.23)

q<kF
k>kF

The Hamiltonian for this N + 1 body system is:
Ĥ =

∑
k,σ

ϵk c†kσ ckσ +g0 ∑ †
ck+ q ↑ c†−k+ q ↓ ck′ + q2 ↓ c−k′ + q2 ↑2
2
V k,k′ ,q(3.24)

where ϵk = ~2 k 2 /2m, V is the volume of the system and the c†kσ ,ckσ are the usual
creation and annihilation operators for fermions with momentum k and spin σ.
In the standard way the⟨ variationalparameters ϕ0 and ϕqk can then be found⟩by minimizing the energy Ψ|Ĥ|Ψ − E↓ ⟨Ψ|Ψ⟩ under the constraint | ⟨Ψ|Ψ⟩ |2 =∑ F
2|φ0 |2 + k>k
q<kF |φkq | = 1
Minimizing with respect to ϕ0 and ϕqk leads us to the following set of coupledequations:

E↓ ϕ0

g0 ∑
= ϵk′ ϕ0 +
V q<k

(
ϕ0 +

∑

)
ϕqk

k>kF

F

E↓ ϕqk = ϕqk (ϵk − ϵq + ϵk′ +q−k
1

g0 ∑
V q<k

F

(3.25)

(
ϕ0 +

∑

)
ϕqk′′ )

(3.26)

k′′ >kF

This treatment is identical to the 1/N expansion and setting N = 1 at the end of the calculation,
where N − 1 is the number of diﬀerent spin states involved. Hence, the binding energy yields thesame result as in [86].

50

Elimination of ϕ0 and ϕqk ﬁnally yields a self-consistent equation for the energyE↓ :E↓ = ϵk′ +

1 ∑
1
∑
1
1
V q<k g0 + V k>kF ϵk −ϵq +ϵ 1′(3.27)

k +q−k −E↓F≡ ϵk′ + Σ(ϵk′ , E↓ )(3.28)

m1
which can be cast into the more convenient form, using g10 = 4π~
2a − V

E↓
ϵk′
=
−2
ϵF
ϵF

∫ 1
dy
0

y2
1 − 2kπF a − I

(

E↓
,y
ϵF

∑

)

1
k 2ϵk [23](3.29)

with the deﬁnition for the integral I(ϵ, y)(∫ ∞I(ϵ, y) =

dx
1

x
ln
2y

(

2x2 + 2xy − ϵ
2x2 − 2xy − ϵ

)

)
−1

(3.30)

In summary, we get for the self-energy Σ(ϵk′ , E↓ ) = E↓ :
E↓
= −2
ϵF

∫ 1
dy
0

y2
1 − 2kπF a − I

(

E↓
,y
ϵF

)

(3.31)In order to get the quasiparticle residue Z and the eﬀective mass m∗ , one can
combine equations 3.31, 3.28 and 3.29. The results will be discussed in the section of
the experimental results 3.7.2 and 3.7.3.

3.4.1

Radiofrequency Spectroscopy of a Fermi Liquid

From the discussion following equation 3.7, it can readily be seen that the spectral
function has a central meaning in radiofrequency spectroscopy: The radiofrequency
photon “asks” the system for its transition probability given an energy ~ω. Since theradiofrequency photon induces a transition to the ﬁnal state regardless of a particles
initial momentum, the total transition probability will be a sum over all momentum
states weighted by the distribution of particles:I(ω) = 2π~Ω2R∑

A(k, ϵk − µ − ~ω)nF (ϵk − µ − ~ω)(3.32)kwhere ΩR is the Rabi frequency, ω := ωRF − ωHF the energy of radiofrequency
photon relative to the hyperﬁne transition energy and nF (~ω) = 1/(1 + exp(β~ω)).
Using equation 3.11, we can immediately see the structure of the radiofrequency
51

spectrum for a single quasiparticle at k = 0:I(ω) = 2π~Ω2R Zδ(~ω + E↓ ) + Iinc (ω)(3.33)and is given by a sharp peak plus an incoherent background. Using the deﬁnition
3.7 for the spectral function we obtain:
Iinc (ω) = 2π~Ω2R∑|φkq |2 δ(~ω + E↓ − ϵk + ϵq − ϵq−k )(3.34)

q<kF
k>kF

which can be solved analytically, see appendix C.

3.5

Generalization of the Impurity Problem

The treatment of the previous section is, of course, an idealized one. Two important
generalizations will be discussed in this section:• We have only considered a single particle in an otherwise non-interacting Fermisea, a situation which can not be realized in our physical system. Hence, theabove treatment has to be generalized to the N +M -body case. For ﬁnite impurity concentration, eﬀects of the presence of the minority Fermi sea have to betaken into account. Also, residual mediated interactions between the quasiparticles will exist, similar to the case of interacting lattice polarons discussed above.
By construction of Chevy’s wavefunction it does not contain quasiparticle interactions. The quasiparticle interactions should in principle be computablein the T-matrix approximation. Another approach [88] has been a variational
approach of the N + M -body problem generalizing Chevy’s wavefunction. In
our case of an attractive interparticle potential, quasiparticle interactions also
raise the question of the range of applicability of Fermi Liquid Theory at very
low temperatures: At very low temperatures one might encounter the case ofpairing and condensation of polarons.
• If the binding energy of the Feshbach molecular state between the |↓⟩ and an|↑⟩ particle exceeds the Fermi energy of the |↑⟩ Fermi sea, we expect many-bodyeﬀects to be of less importance and we expect the formation of a molecule. At
this point the quasiparticle changes from a fermionic quasiparticle to a bosonicquasiparticle. However, the energy and Fermi Liquid parameters of this quasiparticle, sometimes called “Molaron” reminiscent of the term “Polaron”, are52

still strongly modiﬁed by the presence of the Fermi sea. Again by construction,
Chevy’s wavefunction does not capture this behavior. However, one can deﬁne
a generalized wavefunction taking molecule formation into account [76, 89, 90].

3.5.1

N + M = M (any) Impurities

One step towards extending the variational wavefunction Ansatz has been made in[88]:|ψ⟩ = ϕ0 |F S⟩↓ |F S⟩↑ +∑ϕqkr Θqkr c†k↑ cq↑ c†r+q−k↓ cr↓ |F S⟩↓ |F S⟩↑(3.35)

q<kF
k>kF
r=kF

This Ansatz looks more intimidating than equation 3.23 but has a similarly simple
interpretation: The ﬁrst term describes two non-interacting Fermi seas, the second
a particle-hole excitation in the majority Fermi sea by a minority particle at the
Fermi level. The stepfunction Θqkr = Θ (|r + q − k| − kF ↓ ) ensures the restriction|r + q − k| > kF ↓ to satisfy Pauli exclusion. Therefore, the main diﬀerence between equations 3.23 and 3.35 is the decreased phase space available for the impurity
particles to scatter into. Intuitively, one might therefore expect less dressing of thepolarons by majority particles resulting in a decrease in binding energy. This would
imply an eﬀective repulsive interaction between polarons. This Ansatz neglects scattering of impurity particles inside the Fermi sea and can therefore be regarded as aﬁrst step towards symmetrization of the treatment using a variational wavefunction.
Accordingly, the total energy of the system as a function of impurity concentration,
the zero-temperature equation of state, can be found in [88] and will not be shown
here. The results show good agreement with the Monte Carlo simulations of [73],which are shown in ﬁgure 3-18.

3.5.2

Radiofrequency Spectroscopy for Finite Impurity Concentration

In this section we will consider the eﬀect of ﬁnite impurity concentration on theradiofrequency spectrum. We will neglect the eﬀect of quasiparticle interactions as
this is still a topic of discussion and concentrate on the eﬀect of an eﬀective massm∗ that is diﬀerent from the bare mass m. The cartoon picture is shown in ﬁgure3-4a): As diﬀerent momentum states are occupied for ﬁnite impurity concentrations,the resonance frequency will depend on the momentum k.
53

a)

b)

w

)w(I

FR

wRF

k / kF

Figure 3-4: Eﬀect of the eﬀective mass on the minority radiofrequency spectrum. a)
Diﬀerent momenta will respond at diﬀerent frequencies. b) The result is a continuous
spectrum with a sudden drop, see equation 3.36.

The coherent part of the spectral function is thus Acoh
− (k, ω) = Zδ(~ω+E(k)) withm
2 2∗E(k) = −~ k /2m = − m∗ ϵk relative to the impurity Fermi energy. The coherent
part of the spectrum then becomes

Icoh. (ω) = 2π~Ω2R∑

Acoh− (k, ϵk − E↓ − ~ω)kwhere the sum extends up to the impurity Fermi momentum kF ↓ . With the free,
3D density of states ρ(ϵ), this is∫ EF ↓mdϵ ρ(ϵ)Zδ(ϵ − E↓ − ~ω − ∗ ϵ)
m
0
(
)
Z
~ω + E↓= 2π~Ω2R×m ρ
1 − m∗
1 − mm∗)(m
θ (1 − ∗ )EF ↓ − ~ω − E↓m

Icoh. (ω) =2π~Ω2R(3.36)

This coherent part of the spectrum starts at the polaron ground state energy ~ω =54

|E↓ |, then grows like a square root and jumps to zero when ~ω − |E↓ | = (1 − mm∗ )EF ↓ .
On resonance, where m∗ ≈ 1.2, this occurs at ~ω − |E↓ | = 0.2x2/3 EF ↑ ≈ 0.04EF ↑ forx = 0.1. This is still smaller than the Fourier width of the radiofrequency pulse usedin the experiment of about 0.1ϵF . The size of the jump is given by 2π~Ω2R 1−Zm∗ ρ(EF ↓ )mand reﬂects the impurity Fermi surface in the radiofrequency spectrum. This behaviorof the coherent part of the spectrum was treated in [91] and was discussed recently
in [85]. Therefore, for an arbitrarily high spectral resolution the eﬀective mass should
be revealed both by the maximum and the shape in the radiofrequency spectrum.

3.5.3

Polaron vs Molaron or Fermi Liquid vs Bose Liquid

The variational wavefunction for the polaron describes, by construction, a Fermi Liquid. We expect, however, for a large bound state energy the formation of a composite
boson, a molecule regardless of the presence of a Fermi sea. A rough estimate for thecorresponding critical interaction strength is given by comparing the Fermi energy ϵF~2resulting in (kF1a)c ≃ 0.7. For higher binding
to the two-body binding energy Eb = 2m
energies, corresponding to smaller values of kF1 a , the molecule will be similarly dressedby particle-hole excitations of the majority Fermi sea. Due to the internal structureof the molecule, however, the treatment shows a higher complexity and three-body
physics is needed. The according variational wavefunction in the spirit of equation
3.23 is [76, 89, 90]:





∑
∑


†
†
†
†
†
†
N −1
′qd
ξ
u
u
u
⟩↑
|ψ⟩ = 
ξ
d
u
+
′
′
kk
k
−k k
q−k−k k k q  |F S


k>kF
q<kF

(3.37)

k>kF
k′ >kF

where d†k and u†k are the creation operators for a |↓⟩ particle and a |↑⟩ particle, respectively. The interpretation is similar to equation 3.23: Instead of a free |↓⟩particle,a |↓⟩-|↑⟩ pair with zero center-of-mass momentum, a molecule, is created. The momentum, and spatial, distribution is determined by the function ξk . Note that thesum over the momentum states contained in the molecule is restricted to the volume outside of the majority Fermi sea due to Pauli blocking. The second term inequation 3.37 describes particle-hole excitations in the majority Fermi sea by way of
imparting the momentum q − k onto the molecule. This wavefunction can be shown
to reproduce the weakly interacting limit, in which the energy is given by:

55

Figure 3-5: Energy of an impurity particle in a Fermi sea for various theories takenfrom [76]. The Ansatz 3.37 is shown as red line with full squares. For comparison,also shown are the diagrammatic Monte Carlo results (black/green diamonds) andthe Ansatz 3.23 (blue line).

Em = −~22~2 πaad+n↑ − ϵF2ma2
2/3m(3.38)

where a is the scattering length, aad ≃ 1.18 is the atom-molecule scattering length[79] and n↑ is the atom density. The ﬁrst term is simply the two-body binding
energy, the second term the density dependent mean-ﬁeld shift. The full variational
calculation has been done by the authors of [76, 89, 90] and is shown in Fig.3-5.

3.5.4

Finite Size Eﬀects of the Impurity Problem

As one can see in ﬁgure 3-1, the energy of an impurity particle is signiﬁcantly alteredas it is not only interacting with one but N particles. An interesting question arises
in an intermediate regime: How does the energy evolve for small numbers N ? What
happens for N = 3, N = 10, N = 100? This question might help understand thediscrepancy between [16] and [40] in the determination of the critical density imbalance σc , above which the superﬂuid state is quenched. For better comparison withexperiment, these calculations [88] have been performed for harmonic conﬁnement.
In the spirit of Chevy’s wavefunction, a minority particle in an unperturbed harmonic
potential plus elementary excitations in the majority, one can determine the energy56

of the system [88]2 . The energy is shown in Fig. 3-6. The energies for a few particularvalues are ϵ4 ≃ −0.0289ϵF , ϵ10 ≃ −0.135ϵF , ϵ1000 ≃ −0.51ϵF and ϵ100,000 ≃ −0.59ϵFin the case of an isotropic potential. For an anisotropic potential these mesoscopic
eﬀects become more pronounced, e.g. for an aspect ratio of 35 ϵ100,000 ≃ −0.56ϵF ,10% from its homogeneous value. Since the critical imbalance for the formation ofa superﬂuid phase directly depends on the energy of the normal phase compared tothat of the superﬂuid phase, the above results need to be borne in mind when experimental phase diagrams are compared to the homogeneous case using the local
density approximation. In the case of the experiments performed at BEC1 at MIT
the typical atom number is a few 106 and a typical aspect ratio α < 10 the expecteddiscrepancy is below 1% and therefore below our experimental resolution.

Figure 3-6: Energy η(α, N ) as a function of majority particle number N for various
aspect ratios α, compared to η = 0.607 for the uniform system [94] (horizontal lines).

2

Pairing states beyond the variational Ansatz, such as the formation of trimers [92, 93], arenaturally not included in this treatment and might not provide an accurate measure for the true
ground state.

57

3.6

Applicability of Fermi Liquid Theory to theN+1 Body System

The treatment of Fermi liquid theory in textbooks is usually restricted to:
1. Repulsive interactions, as attractive interactions result in the Cooper-instability[5].
2. High enough temperatures such that the system does not undergo a phase transition into a superﬂuid or magnetic state, mediated by more subtle (attractive)
interaction mechanisms (eg. phonons).
3. Low enough temperatures such that Pauli blocking results in a long enoughquasiparticle lifetime for them to be treated as eigenstates of the system.
In metals this results in a temperature range of a few Kelvin up to thousandsof Kelvin. For He3 the range of validity is typically a few mK up to 1K. In our
experimental system at hand, entering the superﬂuid phase is rendered impossible
due to the number imbalance of |↓⟩ particles and |↑⟩ particles, see [95, 96] and ﬁgure
2-5. In this case, the system is guaranteed to be in the normal phase even at zerotemperature. Typically, our experiments were performed at a normalized temperatureof T /TF ≃ 0.15 and a Fermi energy of ϵF ≈ 25kHz, resulting in a scattering rate of( TTF )2 ϵF ≈ 1kHz much smaller than ϵF , therefore validating a Fermi liquid approach.
It is worth noticing that our experiments constitute the ﬁrst realization of a Fermi
liquid with attractive interactions.

3.7

Experimental Results:
Radiofrequency Spectroscopy of the Polaron

After the above theoretical preparations, let us now turn to the comparison withexperiment.
As can be seen from equation 1.9, radiofrequency spectroscopy probes the singleparticle excitation spectrum and is therefore ideally suited to determine the impurity
binding energy with the level of precision commonly achievable employing radiofrequency spectroscopy. Furthermore, by comparing majority and minority spectra we
can distinguish between the free part in equation 3.23 producing a δ-peak in the
spectrum and the particle-hole excitations producing an incoherent background, see58

also section 3.4.1. This allows us to determine the probability of free propagation of
the minority particle, the quasiparticle residue |ϕ0 |2 = Z. A direct result of this is
the possibility to identify a critical interaction strength, beyond which the minority
forms a molecule upon encounter with a majority particle and hence Z = 0.

3.7.1

Preparation and Parameters

The creation of a strongly interacting degenerate Fermi gas of 6 Li has been describedin section 1.4. For the highly number imbalanced samples on the order of P =N↑ −N↓> 90% in this experiment we have used a two-photon Landau-Zener sweep
N↑ +N↓
from state |1⟩ to |3⟩: Using a frequency splitter in reverse, two radiofrequencies are
fed into an antenna. One of the frequencies was red-detuned by 40kHz from the |1⟩|2⟩ transition. The other frequency was swept over a range of 5kHz, centered aroundthe |2⟩-|3⟩ transition frequency plus 40kHz. This gave a stability of the minority
number characterized by ﬂuctuations smaller than 20%. In comparison, preparationof the system using two subsequent single-photon Landau-Zener sweeps resulted in
ﬂuctuations of the minority number by a factor of 2.
The high number imbalance of P > 90% resulted in less eﬃcient thermalizationof the system than in a balanced system [19] and the temperatures we have achieved
range from T ≃ 0.12TF to T ≃ 0.17 TF .
After evaporation, a radiofrequency pulse around the |1⟩→ |2⟩(|3⟩→ |2⟩) transition
frequency was applied for 500µs, resulting in a spectral resolution of 2kHz or 0.1EF ↑ .
The power of the radiofrequency pulse was adjusted so that the maximum populationtransfer was ∼ 30%. An image was taken immediately after the radiofrequency pulse.
The two-dimensional density distribution was elliptically averaged and the threedimensional density proﬁle was reconstructed using the inverse Abel transform3 [66].
The local response to the radiofrequency excitation was recorded as a function ofdetuning from the bare hyperﬁne transition frequency.
Due to the inhomogeneous trapping proﬁle the minority density varied from amaximum density to zero for almost constant density of the majority atoms. Therefore, we were able to determine the polaron energy and quasiparticle residue not only
approaching the limit of a single impurity but also for higher impurity concentrations.
Typical radiofrequency spectra for low impurity concentration are shown in ﬁgure
3-7 along with the spectra of the majority component. The bulk of the majority
3

The inverse Abel transform is given by f (x, z) = − π1

density.

59∫∞
x

dn2D (y,z)

dy √ dy2

y −x2, where n2D is the column

a)

b)

c)

d)

.u.a / refsnart mota
0

2

4

6

0

1

2

3

4

5

6

0

rf offset /

1

e

2

3

4

0

1

2

3

4

F

Figure 3-7: Radiofrequency spectroscopy of polarons. Shown are spatially resolved,
3D reconstructed radiofrequency spectra of the majority (blue, state |1⟩) and impurity (red, state |3⟩) component in a highly imbalanced spin-mixture. a) Molecular
limit, b), c) Emergence of the polaron, a distinct peak exclusively in the minority
component. d) At unitarity, the polaron peak is the dominant feature in the impurity
spectrum, which becomes even more pronounced for 1/kF a < 0 (not shown). For the
spectra shown as dashed lines in d) the roles of states |1⟩ and |3⟩ are exchanged. The
local impurity concentration was x = 5(2)% for all spectra, the interaction strengths
1/kF a were a) 0.76(2) b) 0.43(1) c) 0.20(1), d) 0 (Unitarity).

spectrum is found at zero oﬀset, corresponding to the free (Zeeman plus hyperﬁne)energy splitting between states |1⟩ and |2⟩. However, interactions between impurity
and majority particles lead to a spectral contribution that is shifted: The radiofrequency photon must supply additional energy to transfer a particle out of its attractive environment into the ﬁnal, non-interacting state [23]. In ﬁgure 3-7a, impurity
and majority spectra above zero oﬀset exactly overlap, signalling two-body molecularpairing. The steep threshold gives the binding energy, the high-frequency wings arise
from molecule dissociation into fragments with non-zero momentum [23, 10].
As the attractive interaction is reduced, however, a narrow peak appears in the
impurity spectrum that is not matched by the response of the majority particles(ﬁgure 3-7b,c,d). This narrow peak, emerging from a broad incoherent background,signals the formation of the Fermi polaron. The narrow width and long lifetime are
expected: At zero temperature the zero momentum polaron has no phase space fordecay and is stable. At ﬁnite kinetic energy or ﬁnite temperature T it may decay into
particle-hole excitations [87], but phase space restrictions due to the spin up Fermi
sea and conservation laws imply a decay rate ∝ (T /TF )2 ∼ 1% in units of the Fermi
energy. Indeed, the width of the polaron peak is consistent with a delta function
within the experimental resolution. The full width at half maximum (FWHM) is
shown in ﬁgure 3-8. On the molecular side the width of this fully incoherent disso60

ciation spectrum is a measure of the molecule size [46], but in the unitary limit andtowards the BCS side of the Feshbach resonance the width of this peak approaches
the experimental resolution of the radiofrequency pulse.

1.0

e / MHWF

murtceps ytirupmi fo F

1.5

0.5

0.0
0.8

0.6

0.4

0.2

0.0

-0.2

1 / kF a

Figure 3-8: FWHM of spectra of the type shown in Fig.3-7 as a function of interaction
strength. The dashed line shows the Fourier limit of a 500µs pulse.

In general, one might expect the width of the coherent polaron peak to increase
with increasing impurity concentration as the polarons might be rendered unstableby interactions and have shorter lifetimes. Figure 3-9 shows a series of spectra forvarious impurity concentrations. Therefore, if the broadening eﬀect is present, it ismasked by our ﬁnite experimental resolution, see ﬁgure 3-8.
The incoherent background is perfectly matched by the radiofrequency spectrumof the majority particles. This is expected at high radiofrequency energies ~ω ≫ ϵFthat are probing high momenta k ≫ kF and thus distances short compared to theinterparticle spacing. Here, an impurity particle will interact with only one majority
particle, leading to overlapping spectra. We will return to this point below in chapter
3.7.4, in which we will give an estimate of the contact coeﬃcient C.

61

x = 0.05

x = 0.07

x = 0.08

x = 0.10

x = 0.12

x = 0.13

x = 0.15

x = 0.16

0

1

2

3

(rf offset - m) / eF

Figure 3-9: Normalized polaron spectra as a function of radiofrequency for various
impurity concentrations x = n↓ /n↑ .

3.7.2

Results I:
Chemical Potential µ↓ of the N + 1-body System

To measure the polaron energy E↓ = µ↓ , we determine the peak position of the impurity spectrum as a function of the local interaction strength 1/kF a. The data for an

62

0

1

2

3

(rf offset - m) / eF

Figure 3-10: The averaged spectrum (black line) overlaps well with the individual
spectra from ﬁgure 3-9 (red lines).

impurity concentration of x = 5% are shown in ﬁgure 3-11, along with the variational
upper bound given by the wavefunction equation 3.23 and the diagrammatic Monte
Carlo calculation of [87]. As ﬁnal state interactions are weak, they can be included asa simple repulsive mean ﬁeld shift 4π~2 afe n↑ /m, with afe the scattering length between
the ﬁnal state and the majority atoms [65]. Polaron energies have been predicted via
the variational Ansatz in equation 3.23 [78], the T-matrix approach [97, 91, 98, 99],
the 1/N expansion [86], ﬁxed node Monte Carlo [74, 73] and diagrammatic Monte
Carlo [87]. With the exception of the 1/N expansion, these all agree with each otherand with the present experiment to within a few percent. In particular, in the unitary
limit where 1/kF a = 0 we ﬁnd a polaron energy of E↓ = −0.64(7)ϵF (−0.72(9)ϵF )when state |3⟩ (|1⟩) serves as the impurity. This agrees well with the diagrammatic
MC calculation, −0.615ϵF [100], and the analytical result −0.6156(2)ϵF [98]. Analysisof experimental density proﬁles yields a value of −0.58(5)ϵF [101].
The relatively large value for E↓ directly implies that the normal state, modeled
as a Fermi sea of weakly interacting polarons, is favored over the superﬂuid state
up to a critical concentration (44%), much higher than that predicted by mean ﬁeld
theories (4%) [70], see ﬁgures 2-1 and 2-2. These neglect interactions in the normal
63

0.5

1.0
F

e / noitisop kaep

1.5

2.0

2.5

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

1 / kF a

Figure 3-11: Peak position of the impurity spectrum as a measure of the polaron energy E↓ for various interaction strengths in the limit of low concentration x = 5(2)%(solid circles). Open circle: Reversed roles of impurity and environment. Dotted
line: polaron energy from variational Ansatz equation 3.23 [78], the solid line including weak ﬁnal state interactions. Dashed line: Energy of a bare, isolated moleculein vacuum. Blue dash-dotted line: Mean ﬁeld limit for the energy of an impurity
atom. Solid (open) diamonds: Diagrammatic Monte Carlo energy of the polaron(molecule) [87].

state and therefore imply a polaron binding energy of zero.

3.7.3

Results II: Quasiparticle Residue Z of the Polaron

The spectral weight of the polaron peak directly gives the quasiparticle residue Z.
Experimentally, we determine the area under the impurity peak that is not matchedby the majority’s response and divide by the total area under the impurity spectrum,see ﬁgure 3-12.
Figure 3-13 presents Z as a function of interaction strength and impurity concentration x. Figure 3-14 shows Z for an impurity concentration of x = 5%. As
expected, Z approaches 100% for weak attractive interaction kF a → 0− , where thebare impurity only rarely recoils oﬀ majority atoms. As the mean free path shortens
and the bare impurity starts to surround itself with environment atoms Z decreases.

64atom transfer / a.u.
0

1

2
3
rf offset / ε F

4

5Figure 3-12: Determination of the quasiparticle residue Z. Impurity spectrum(red), majority spectrum (blue) and spectral response of non-interacting atoms (black
dashed), folded over from negative radiofrequency oﬀsets.

On resonance, we ﬁnd Z = 0.39(9) for x = 5%, with only a weak dependence on x
(ﬁgure 3-13). Theoretical values for Z vary: Ansatz Eq.(3.23) predicts Z = 0.78 for asingle impurity, while Ref. [86] predicts Z = 0.47 (0.30) for vanishing (5%) impurity
concentration. Our procedure might yield a lower bound on the actual value of Z, as
the incoherent part of the impurity spectrum might be depleted around threshold. We
have also determined an upper bound to the quasiparticle residue Z, which has been
extracted under the assumption that the incoherent part of the impurity spectrum iscompletely depleted, see ﬁgure 3-14.
Eventually, for strong attraction between the impurity and the majority particles,
Z vanishes and we observe complete overlap of the impurity and environment spectra.
This signals the formation of a two-body bound state between the impurity andexactly one majority atom. For a spin down concentration of x = 5% we determine thecritical interaction strength where the polaron peak vanishes to be 1/(kF a)c = 0.76(2).
This is in good agreement with the independently determined critical interaction
1/kF a = 0.74(4) beyond which one ﬁnds a superﬂuid even for the smallest impurity
concentration [80]. This is a multicritical point [102, 73, 87] where a Fermi liquidof weakly interacting polarons undergoes a phase transition into a Bose liquid ofmolecular impurities. Fixed-node Monte Carlo calculations place this transition at avalue of 1/kF a = 0.73 for x → 0 [73]. Our 1/(kF a)c is lower than the value 0.90(2)from diagrammatic MC [87] for a single impurity. The Ansatz in equation 3.23 does65

1 / kF a
0.8

0.6

0.4

0.2

0.0

0.0

-0.2

Z eudiser elcitrapisauq

0.8

0.6

x noitartnecnoc ytirupmi

0.4

0.1

0.2

0.0

0.2

0.3

Figure 3-13: Quasiparticle residue Z as a function of interaction strength and impurity
concentration. The color coding indicates the magnitude of Z and is an interpolationof the data points shown in the graph. Open circles: Data points consistent withzero (Z < 0.03), solid circles: Z > 0.03, the solid line marking the onset of Z. Thisdeﬁnes a line of critical concentration xc as a function of interaction strength. Bluecross: Critical interaction strength for the Fermi liquid - molecular BEC transition
for x → 0 [73].

not predict a transition, as it does not test for the formation of molecules. In ﬁgure
3-13, the color coding reveals where molecular behavior is observed (yellow), andwhere the spectra show polaronic behavior (red to black). It can be seen that thecritical interaction strength for the formation of molecules depends only weakly on
the impurity concentration x.
It is an interesting question what the eﬀect of the ﬁnite temperature of T /TF ≃0.15 is when we determine the critical interaction strength: If the polaron and themolaron state were coupled, the polaron state would exist as a thermal excitationof the molaron state for (1/kF a)c > 0.76, and vice versa. The result would be an
overestimate of the critical interaction strength as there is no complete overlap if

66

0.8

Z eudiser elcitrapisauq

pxe

0.6

0.4

0.2

0.0
0.8

0.6

0.4

0.2

0.0

-0.2

1 / kF a

Figure 3-14: Quasiparticle residue Z as a function of interaction strength in the limit
of low impurity concentration x = 5(2)%. Solid black circles: State |3⟩ impurity in|1⟩majority Fermi sea. Open circle: Reversed roles, state |1⟩impurity in a |3⟩majority
Fermi sea. Solid grey circles: Upper limit of the quasiparticle residue determined by
a Gaussian ﬁt to the polaron peak normalized by the spectral weight of the minority.
The red line is the quasiparticle residue as calculated from the variational Ansatz inequation 3.23.

part of the minority is polaronic and therefore its wavefunction delocalized. A simple
estimate by much one would overestimate the critical interaction strength would
consist of a comparison of our temperature to the energy diﬀerence between the twostates, see the diamond data points from diagrammatic Monte Carlo calculations
in ﬁgure 3-11, and would shift the critical interaction strength down by ∼ 0.2 to(1/kF a)c ≃ 0.56, which is not consistent with theoretical and other experimental
predictions, see previous paragraph.
However, the thermal population of the excited polaron state might evolve on avery slow time scale if the population requires breaking the molaron before formationof a polaron. This picture is similar to an endothermic chemical reaction with alarge activation energy. In this case our temperature would have to be compared tothe binding energy of the molaron which is several ϵF , see ﬁgure 3-11. Hence, the67thermal population of the polaron state would be negligible and our determinationof the critical interaction strength (1/kF a)c > 0.76 unaﬀected by temperature.
Another caveat with respect to the determination of the critical interaction strength
is the ﬁnite impurity concentration. We have shown experimentally that there is only
a weak dependence on the impurity concentration. However, the phase transition
from the phase separated polarized superﬂuid phase to the fully polarized normal
phase is expected to be ﬁrst order, see chapter 2.3 and ﬁgure 2-2. The minority density would therefore jump from a ﬁnite value to zero, making the true molaron phaseinaccessible for our experiments4 .
The phase diagram revisited The black line of critical concentration xc as afunction of interaction strength has been added to ﬁgure 2-4 and validates the extrapolation of our previous study of the phase diagram based on trapped density
distributions.

3.7.4

Results III: The Contact Coeﬃcient C

After concentrating on the coherent quasiparticle peak, let us now investigate the incoherent background spectrum. At the end of chapter 3.7.1 we have given a physicalinterpretation of why the majority and minority spectra overlap at high radiofrequency: High energy corresponds to the large momentum components and therefore
to short distances, where we expect one minority atom to interact with only one
majority atom.
It can be generally shown [103] for a Fermi system interacting through a zero-range
potential that the momentum distribution behaves like nk = kC4 , with the so-called
contact coeﬃcient C [104, 69]. This distribution at large momenta is independent of
the phase or temperature5 and is completely determined by the contact coeﬃcient C,
which is the same for both spin states, see also ﬁgure 3-3. Tan [104, 69] has derived
an “adiabatic sweep theorem” which relates the contact coeﬃcient C to the derivativeof the energy of the system to the interaction strength:
~2 C
du
=−
4π m
d(1/a)(3.39)

with the energy density u = E/V . The fact that the energy of the ground state
4This argument is true only for zero temperature. In the experimentally relevant case of ﬁnite
temperature, a polarized superﬂuid probably exists even in the limit P → 1, see also ﬁgure 2-5
5
The value of C does depend on the state of the system, of course.

68

(and the excited states) is related to C highlights the importance of the contact
coeﬃcient. The role of the contact coeﬃcient can be further elucidated by calculating
the derivative of the energy density in equation 3.39. Let us consider the general formof the Hamiltonian with contact interactions characterized by the coupling constant
g:
H=

∑ ~2
σ

2m

∇ψσ† · ∇ψσ + g ψ1† ψ2† ψ2 ψ1(3.40)

If we plug this Hamiltonian into the adiabatic sweep theorem in equation 3.39, wearrive at the following relation for the contact coeﬃcient C:
~2
du
C =
4πm
d(−1/a)du=d(−1/g)
du
= g2
dg
⟨
⟩
dH
2
= g
dg
= g 2 ⟨ψ1† ψ2† ψ2 ψ1 ⟩(3.41)

where, in the second-to-last equation, we have used the Feynman-Hellman theorem[105, 106, 107, 108]. From the above equation it is clear that C can be interpreted
as as the probability that an |↑⟩ and a |↓⟩ particle are close to each other, which has
been formally shown by the authors of [109], hence the name contact coeﬃcient. In
general it can be shown, see [110] and references therein, that C is related to the(2)short distance behavior of the pair correlation function G↑↓ (r):(2)

G↑↓ (r → 0) ≃C 1
(4π)2 r2

(3.42)

From this one can calculate the total number of pairs of particles of opposite spin
which are separated by a distance smaller than rp :∫(2)

Npair (rp ) =
r<rp

C
=
rp
4π
69

d3 rG↑↓ (r)(3.43)

This way, one can connect the contact coeﬃcient to the probability of Feshbach
molecules to be in the closed channel [111]. This probability has been determined by
the Rice group using photoassociation spectroscopy [112].
With relation 3.39 one can immediately determine the contact coeﬃcient for the Fermi
polaron knowing the energy as a function interaction strength, see ﬁgure 3-11. As it
involves a derivative of experimental data, one might get a more accurate measure bytaking advantage of the high frequency wings of the polaron radiofrequency spectra,
which in linear response can be calculated using Fermi’s Golden Rule 1.10:
I(ω → ∞) −→ 2π~Ω2R

∑C
k

k4

δ(~ω − 2ϵk + µ)(3.44)

where ΩR is the Rabi frequency for the radiofrequency transfer. A comment onenergy conservation in the formula above: One ϵk has to be “paid” for creating aparticle in the ﬁnal state at momentum k. Since the radiofrequency radiation is
exciting the ground state of the system, see ﬁgure 3-2 for the polaron case, removing
a particle of momentum k out of its environment constitutes an excitation of energy
ϵk − µ, where ϵk is the free particle kinetic energy. This is true only for momentak ≫ kF .
Now the spectrum can be readily calculated. In order to simplify the ﬁnal result, we calculate the normalized spectrum as a function of the reduced frequency~ω → ~ω/ϵF , as they are represented in ﬁgure 3-7. We have also deﬁned the dimensionless quantity s = C kF ↑ kF3 ↓ and we measure the radiofrequency from the chemical
potential µ, see also [113, 85, 114]:
√
I(ω)
2 s
∫
=
3 (~ω)3/2
I(ω) dω(3.45)The generic decay of the spectra as ω −3/2 has already been encountered in thederivation of equal density radiofrequency spectra in the BEC-BCS crossover, seechapter 1.5.3. The normalized experimental spectra multiplied by (~ω)3/2 should
saturate at high momenta at the value of the dimensionless contact coeﬃcient s. The
results for various interaction strengths are shown in ﬁgure 3-15.
In order to increase the signal-to-noise ratio in the wings of the radiofrequency
spectra we have taken advantage of the negligible change in the shape of the radiofrequency spectra as a function of position in the trap, see ﬁgure 3-10, and averaged
over the normalized local minority spectra corresponding to diﬀerent minority concentrations. We were thus able to extract the contact coeﬃcient from the plateau70

1 / kF a = 0.75(4)0.6

1 / kF a = 0.46(3)1 / kF a = 0

)'w(I

0.4

2/3

'w
0.2

0.0
0

1

2

3

4

5

w'

Figure 3-15: Spectra multiplied by ω 3/2 for various interaction strengths. The tails
are expected to decay as C/ω −3/2 with the contact coeﬃcient C. A plateau at highω therefore allows direct access to the contact coeﬃcient C.

regions shown in ﬁgure 3-15 as a function of the interaction strength. The result isshown in ﬁgure 3-16. These values for the contact coeﬃcient can be compared to theresult from the model presented above 3.4. s can be readily calculated by inserting
equation 3.29 into equation 3.39. The result is also shown in ﬁgure 3-16.
Our experimental data appear to be systematically below the theoretical prediction. A possible explanation is the small spectral weight in the high frequency wings,
e.g. at unitarity the signal-to-noise ratio beyond ~ω/ϵF = 3 is too small for a determination of C and we have relied on the plateau ϵF < ~ω < 2.6ϵF for a determinationof C. Figure 3-17 shows the comparison of the minority spectrum based on the Chevy
Ansatz and the asymptotic behavior of equation 3.45.
According to the Chevy Ansatz shown in ﬁgure 3-17 one would not expect theasymptotic behavior below ~ω/ϵF ≃ 4. The spectra in ﬁgure 3-7, however, indicate perfect overlap between majority and minority components already at lower
radiofrequencies on the order of ~ω/ϵF ≃ 1, which one would expect only in the highmomentum limit, see also ﬁgure 3-3.
We do not expect the ﬁnite temperature of T ≃ 0.15TF in the experiment to signiﬁcantly change our extracted value of the contact coeﬃcient as it was shown to be only
weakly dependent on temperature for T < 0.3TF due to the phonon contribution of

71

0.30

0.25

0.20

s
0.15

0.10

0.05

0.00
0.8

0.6

0.4

0.2

0.0

-0.2

-0.4

1 / kF a

Figure 3-16: Dimensionless contact coeﬃcient s extracted from ﬁgure 3-15 as a function of interaction strength. Solid black circles: C extracted from the high frequency
wings of the minority spectra. Solid grey circles: C extracted from the high frequency
wings of the majority spectra. The signal-to-noise ratio does not allow for an extraction of s on the BCS side. Solid red line: prediction according to Ansatz 3.23. Dashed
red line: Contact coeﬃcient in the BEC limit.

T 4 to the free energy [115].
It is interesting to consider the contact coeﬃcient in the weakly interacting limits: In
2athe deep BCS limit the energy is simply given by the mean ﬁeld UMF = 4π~n↑ . Withmequation 3.39 the dimensionless contact coeﬃcient can immediately be determined tobe s = 9π4 2 (kF a)2 . In the deep BEC limit with the energy given by the two-body
2~2
4 1binding energy U = 2 Eb = ma
2 , the dimensionless contact coeﬃcient is s = 3π k a . ItF
is also shown in ﬁgure 3-16. The predicted value for unitarity using the Chevy-Ansatz
is s = 0.072.
In the case of a balanced system at T = 0, in order to ﬁnd the contact coeﬃcient,
one might try to use the mean-ﬁeld result for the energy as a function of the interaction
strength, see chapter 1.3, and use the adiabatic sweep theorem 3.39. The result is∆2
s = 4ϵ
2 . In the BEC limit, one would expect the result to be identical to the impurity
F
scenario√ as the superﬂuid is composed of tightly bound dimers. In the BEC limit
4 1
16 √ϵF∆ = 3πand therefore s = 3π
, as in the impurity case above.
kF a
kF a
72

0.10

Z

-E )
d(w

)w(I

0.05

0.00
0

2

4

6

8

w / e

F

Figure 3-17: The minority spectrum calculated from the variational Ansatz equation
3.23. The incoherent background can be seen to approach the ω −3/2 behavior at highfrequencies. One would expect this for frequencies higher than ~ω/ϵF ≃ 4.

With the general expression for the contact coeﬃcient above, one would expect
s to be exponentially small in the BCS limit. This, however, is an artefact of theshortcomings of the mean-ﬁeld theory, see chapters 1.3 and 4.1. The leading ordercorrection to this result yields s = 9π4 2 (kF a)2 as in the impurity case.
At unitarity, the contact coeﬃcient for a balanced superﬂuid at zero temperature is s ≃ 0.1 [91][113]. This is a larger value than for the polaron system, wheres = 0.072, as the polaron has a delocalized component characterized by the quasiparticle residue Z.
To date, there are few radiofrequency spectroscopy experiments from which one
could determine the contact coeﬃcient. Early experiments using 40 K [116] have studied the short-range correlation between fermion pairs performing experiments in thehigh frequency tails of radiofrequency spectra. In early radiofrequency experiments
using 6 Li [13, 20, 22], the high frequency tail was suppressed due to ﬁnal state interactions. It has been shown [117] that the high frequency tail of the spectrum in fact
decays as ω −5/2 instead of ω −3/2 . This behavior is also reﬂected in the average clock
shift [118, 91]

73

∫

∫

ω̄ =

dω ωI(ω) / dω I(ω))
1
1 1 d(u)−=g gf n d(1/g)()
14ϵ2F 1
= s·
−
n
g gf
(

(3.46)

2

where gi = 4π~m ai , the a and ai are the scattering lengths for the initial and ﬁnal
state, respectively, n is the density and where we have used the adiabatic sweep
theorem 3.39 and C = s · kF4 for the balanced system.
In atomic 6 Li, a system with a weakly interacting ﬁnal state and therefore access tothe contact coeﬃcient has only recently become available [46, 44, 119]. Since the ﬁnal
state interactions are small but non-zero, the ω −5/2 behavior of the high-frequency
tails in the radiofrequency spectrum will show up eventually at high enough energy.
We can estimate this energy by comparing the bound state energy of the initial state
Eb with the bound state energy of the ﬁnal state Eb′ . More precisely, based on theFranck-Condon factor describing the wavefunction overlap, it has been shown [117]that the dissociation spectrum in equation 1.11 must be modiﬁed to include theenergy dependent factor (~ω + Eb − Eb′ )−1 . This factor only alters the shape of the~2
dissociation spectrum if ~ω is on the order of Eb − Eb′ . With Eb′ = ma
2 , the boundstate energy of the ﬁnal state of a |1⟩-|2⟩ (|1⟩-|3⟩) molecule at 690G is on the order of
300kHz > 10ϵF and therefore not relevant for the spectra shown in ﬁgure 3-15.
Recently [120], high-momentum transfer Bragg spectroscopy has been used to accessthe contact coeﬃcient from the structure factor.

3.7.5

Results IV: Eﬀective Mass m∗ and Quasiparticle Interactions

We have so far considered the limit of few impurities. By increasing their density, we
can study the eﬀect of interactions between polarons. In ﬁgure 3-18 we show that thequasiparticle peak position depends only weakly on the impurity concentration in theunitarity limit, see also ﬁgure 3-9. Polarons are thus weakly interacting quasiparticles,
despite the strong interactions between the bare impurity and its environment.
The peak position could be modiﬁed due to the eﬀective mass m∗ of polarons,
larger than the mass of the bare impurity. Transfer of a moving polaron into thefree ﬁnal state then requires additional kinetic energy. This leads to an upshift and74

a broadening on the order of the Fermi energy diﬀerence between initial and ﬁnal()
state, x2/3 ϵF 1 − mm∗ . On resonance, this is 0.04ϵF for x = 0.1. The eﬀect could bepartially masked by the predicted weak repulsion between polarons [73] that would
downshift the resonance frequency by −0.02ϵF for x = 0.1.

e / .sop kaep

F

0.7

0.8

0.9
0.00

0.05

0.10

0.15

impurity concentration x = n¯ / n­Figure 3-18: Peak position at unitarity (1/kF a = 0) as a function of impurity concentration (solid circles). The line shows the expected peak position (see equation
42.7), ~ωp /ϵF = A + (1 − mm∗ )x2/3 − 56 F x + 3π
kF afe , using the Monte Carlo value∗
A = 0.615 [87], the analytic result m = 1.2 [98], the weak repulsion between polarons with F = 0.14 [73] and weak ﬁnal state interactions with scattering length
afe .

3.8

Experimental Results: Collective Oscillations

As discussed at the end of chapter 3.5.2 and in previous chapter, the eﬀect of theeﬀective mass m∗ on the radiofrequency spectrum is below our experimental resolution
and one has to resort to diﬀerent methods of determining m∗ . Since the eﬀective mass
is a kinetic property of the system, collective excitations are a natural choice to try
to experimentally access the eﬀective mass of the polaron.
If the conﬁning potential is harmonic V (r) = m2 ωr2 r2 , in the local density approximation the eﬀective potential experienced by the minority particle is modiﬁed dueto the attraction from the majority particles:

W (r) = V (r) − AϵF ↑ (r)= (1 − A)V (r)(3.47)(3.48)

with the local majority Fermi energy ϵF ↑ (r) = µ − V (r) and A ≡ E↓ /ϵF ≃ −0.6.
The attractive van der Waals interaction increases the depth of the potential seen by75

the minority atom. In addition, the eﬀective mass of the quasiparticle is modiﬁed dueto the dressing with majority particles. The eﬀective potential can be written as:m∗ 2
ωr (1 + A)r2
(3.49)
2m ′2 2≡ω r(3.50)
2√
with the new oscillation frequency ω ′ = ωr mm∗ (1 + A). At unitarity where one
expects A ≃ 0.6 and m∗ ≃ 1.17, the resulting eﬀective dipole oscillation frequency
would deviate from the bare oscillation frequency ω by about 20% and should observable in experiments. There are several ways to excite this dipole oscillation:
W (r) =

3.8.1

Optical Potential, I

One might displace the majority cloud using a sheet of an attractive or repulsive
optical potential to excite a relative motion between majority and minority. This isa relatively “violent” excitation and typically distorts the density distribution andresult in a non-harmonic potential experienced by the minority atoms.

3.8.2

Optical Potential, II

One can circumvent the problem of exciting a relative motion between the two spinstates if the minority cloud is created already with relative momentum. This canbe done using Raman spectroscopy: Two laser beams far-detuned from the atomic
transition (∼ 1GHz) intersect each other at the position of the atoms. The resonancecondition is now momentum dependent: Only if the detuning δ between the Laser
2 ∆k 2(∆k being the diﬀerence of thebeams is chosen such that ~δ = ~ωHF + E↓ + ~ 2m
wavevectors of the lasers), atoms will be transferred into state |2⟩. Therefore, minority
atoms are created in state |2⟩in momentum state ∆k. The results are shown for the
weakly interacting limit in Fig.3-19. Even for weak interactions of kF a ≃ 0.05 strong
damping can be observed. This might point to too a high temperature resulting
in a decay of the polaron into particle-hole excitations. However, eﬃcient cooling is
notoriously diﬃcult in this scheme as there is only a short time window for evaporative
cooling after the creation of a minority cloud with momentum.

76

Figure 3-19: Dipole oscillations of in the weakly interacting limit.

3.8.3

Magnetic Gradient, I

The external magnetic ﬁeld can be ramped to a lower value where the magnetic
moments between states |1⟩ and |3⟩ are diﬀerent enough, see ﬁgure 1-10, to initiate
a relative motion between majority and minority. This method does not distort thedensity distribution and allows for evaporative cooling before the dipole excitation.
This method has been used recently to completely separate two atomic clouds and to
study the subsequent spin transport in strongly interacting Fermi systems [121]. The
result for a small relative displacement is shown in ﬁgure 3-20: The oscillation can beseen to be overdamped, as the center-of-mass relaxation time is an order of magnitude
higher than the oscillation frequency. This, again, might be due to the temperatureof the sample being too high. We have observed that ramping the magnetic ﬁeld
over too wide a range tends to heat up the clouds such that it is diﬃcult to achievetemperatures below T /TF ≃ 0.2 even for equal spin mixtures.
There is an additional complication in this and the previous scheme: The majority
and minority atoms are on the repulsive branch of the Feshbach resonance, see ﬁgure 16, and must decay into the metastable ground state ﬁrst. This also presents a feature:
This system can be regarded as an excited state of the polaron and it is interesting
how this new state relates to the weakly interacting limits: On the molecular side,
the excited state should indeed consist of two free atoms. The energy is shifted by
the mean-ﬁeld repulsion, see equation 3.38, of the molecules. Similarly, the energy
should be mean-ﬁeld shifted above the Feshbach resonance.

77

a)

c)

120

m / ecnereffid ssam-fo-retneC

m

0 ms

100

t

80

0

200

400

600

800

= 375 ms60

b)1000 ms

40

20

0
0

200

400

Position /

600

800

mm

0

200

400

600

800

1000

Relaxation time / ms

Figure 3-20: Overdamped spin dipole oscillation of an impurity particle stronglyinteracting with a Fermi sea. a) Initial condition: Minority cloud ∼ 130µm displaced
from majority center-of-mass. b) After 1000ms the minority cloud has migrated tothe center. c) The center-of-mass diﬀerence of majority and minority clouds shows aslow relaxation on the time scale of several hundred milliseconds [121].

3.8.4

Magnetic Gradient, II

The problem of ramping the magnetic ﬁeld over too wide a range and the complication mentioned above of excited polarons can be overcome by associating moleculesby ramping the magnetic ﬁeld below the Feshbach resonance and applying a magnetic ﬁeld gradient. In the deep molecular limit all the minority atoms are bound inmolecules in the singlet state, with zero magnetic moment and therefore not sensitive
to the magnetic ﬁeld gradient to ﬁrst order. Dissociating the molecules by ramping
above the Feshbach resonance yields polarons with a ﬁnite momentum. A complication in this scheme is that in order for the magnetic moment of the molecules tosigniﬁcantly diﬀer from that of the free atoms, the molecules are so strongly boundthat three-body collision are no longer Pauli-blocked. This causes loss of atoms andheating of the cloud. By completion of this thesis, we have not yet found a viablewindow of parameters.

3.8.5

Magnetic Gradient, III

Another method has been used recently to experimentally determine the eﬀective
mass m∗ : Exciting a quadrupole oscillation by suddenly changing the magnetic ﬁeld
and its curvature [77]. The eﬀect is twofold: The sudden change in the conﬁnement,78

Figure 3-21: Aspect ratio of the minority cloud as a function of time after excitationof the quadrupole oscillation mode. The excitation was created by suddenly changing
the scattering length

due to a change in curvature of the magnetic ﬁeld, excites in an in-phase oscillation and the sudden change in the scattering length between majority and minority
excites out-of-phase oscillations. The in-phase excitation is overdamped and the minority cloud size slowly relaxes whereas the out-of-phase excitation shows oscillatory
behavior and allows to extract the eﬀective mass. From the frequency of collective
oscillations, Nascimbène et al. were thus able to extrapolate to the limit of highimbalance with a value for the eﬀective mass of m∗ = 1.17(10)m. Recently, ourgroup has also realized these quadrupole oscillations, see ﬁgure 3-21. This method
overcomes all problems mentioned above: High temperature, excited polaronic state,
three-body relaxation.

3.8.6

Size Measurement

As is indicated in equation 3.50, the conﬁning potential seen by the minority atoms is
aﬀected by the presence of the majority atoms. The attractive interaction quenches
the minority cloud volume - the minority cloud is sucked into the center of the trap.
For a given particle number N , in an isotropic system at T = 0 the minority radius
is determined by ϵF ≡ ~ω(6N )3 = m2 ω 2 R2 . As a result:

79

R↓
=
R↓0

√

mω
m∗ ω ′√ √m m∗ 1=m∗ m 1 + A
)1/4
(
m 1
=
m∗ 1 + A(3.51)

where R↓ is the interacting minority Fermi radius and R↓0 is the non-interacting
minority Fermi radius for a given atom number, the bare/eﬀective mass is m/m∗ andthe bare/eﬀective dipole oscillation frequency is ω/ω ′ (see previous chapter).
As can be seen in equation 3.51 the size is not very sensitive to A, m∗ and therefore
not well suited for an accurate measurement.

80

Chapter 4
Quasiparticle Spectroscopy
4.1

Quantitative Studies of the BEC-BCS Crossover

We have already learned in chapter 1.3 that mean-ﬁeld theories predict the correct
qualitative feature but fail when put to the test of quantitative comparison. Let us
brieﬂy discuss the shortcomings of the mean-ﬁeld treatment in chapter 1.3: The only
simpliﬁcation1 that has been made consisted of neglecting interactions between pairsof non-zero momentum by setting q = 0 in 1.2. This, however, has wide-ranging
results: As q = 0, the magnitude and the phase of the gap are spatially homogeneousand therefore do not allow for density ﬂuctuations. It has been shown in the BCS
limit [122] that density ﬂuctuations induce an interaction between the fermions in
the medium, similar to the van-der Waals interaction between neutral atoms, inwhich the ﬂuctuating dipole moment of one atom can induce a dipole moment ina nearby atom. The eﬀect can be calculated to modify the interaction coeﬃcientin the BCS Hamiltonian in equation 1.2. Since V0 < 0 in
V0,eﬀ = V0 + V02 N (0) 1+2ln(2)3the BCS regime, the eﬀective interaction strength is reduced. This lowers the value
for the superﬂuid gap to ∆/EF = (2/e)7/3 exp(−π/2kF a). Compared to the BCS
value of ∆/EF = 8/e2 exp(−π/2kF a) the value is reduced by a factor of ∼ 2.2 and is
known as the Gorkov Melik-Barkhudarov reduction [123] of BCS theory2 . While thevalidity of the Gorkov Melik-Barkhudarov theory was originally limited to the BCS
limit, when blindly applied to the unitary limit it yields a value for the superﬂuid gapof ∆ ≃ 0.49ϵF , surprisingly close to the recent theoretical and experimental values
1

We have also assumed a variational wavefunction, see equation 1.1. It can be shown, however,that this is the ground state wavefunction of the BCS Hamiltonian.
2
In neutral superﬂuids the Gorkov Melik-Barkhudarov reduction of the superﬂuid gap can besigniﬁcant [122], in charged superﬂuids the frequency at which the correction becomes important ispushed up to the Plasma frequency due to the strong Coulomb interaction [124].

81

[30]. Recently, the Gorkov Melik-Barkhudarov theory has been extended throughout
the BEC-BCS crossover [125].
Also, in close connection to the previous point, interactions in the normal phase
are not captured by equation 1.2. This has important consequences for the discussionof imbalanced Fermi gases in chapter 2 as mean-ﬁeld theories overestimate the energyof the normal state. As a result, the superﬂuid state becomes energetically morefavorable relative to the normal state and the system remains superﬂuid even forhigh density imbalances.
In general, it remains a diﬃcult challenge to theorists to produce a quantitatively
accurate model due to the absence of a small parameter in the unitary regime. The
above mentioned Gorkov Melik-Barkhudarov calculation presents the second termin an expansion series that does not necessarily converge. In the unitary regime one
often has to resort to Monte Carlo techniques which however suﬀer from the infamous
sign-problem or convergence problems.
This highlights the importance of accurate quantitative experiments. In this chapter we will present a method to determine the superﬂuid gap using radiofrequency
spectroscopy. To this end, we have identiﬁed and quantiﬁed the Hartree self-energy,
which plays a crucial role in the quantitative interpretation of radiofrequency spectra.
Finally, in this chapter we will make the connection between the normal polaronic
phase for high imbalance and the superﬂuid phase for zero imbalance.

4.2

Extracting the Superﬂuid Gap from Radiofrequency Spectra

In this chapter we present a quantitative analysis of the spectral peaks in the superﬂuid phase for small density imbalance. This will allow us to determine the superﬂuid
gap ∆ at very low temperatures. Earlier work [13, 22] tried to determine the gap fromthe onset of the pair dissociation spectrum. However, the radiofrequency spectra are
not only sensitive to ﬁnal state interactions, see discussion in chapter 1.5.1, they arealso shifted by Hartree energies, as we show below. Furthermore, radiofrequency spectroscopy can excite all fermions, even deep in the Fermi sea, see eg. [23]. Therefore,
the onset of the pair dissociation spectrum occurs for atoms with momentum k = 0∆2and, in the BCS limit depends quadratically on the gap parameter (ωth = 2ϵ). TheF
excitation gap can be directly observed if quasiparticles near the dispersion minimum
are selectively excited, as in tunneling experiments.

82

Since ﬁnal state interactions and Hartree terms create line shifts, two peaks areneeded for the analysis, the dissociation peak and the quasiparticle peak in our case.
In essence, it is the separation between the peaks in spectra like ﬁg. 4-4b, whichallows us to determine ∆.

4.3

BCS Theory: Quasiparticles

We have encountered the excitation spectrum of the generalized BCS state in chapter
1.3, see ﬁgure 1-9 and have called it the quasiparticle excitation spectrum. Let us
elucidate the role of quasiparticles by diagonalizing the Hamiltonian 1.2 instead ofusing the variational wavefunction. We will start by simplifying the interaction termand ﬁnd the “best” two particle operator from the four particle operator (known asincomplete Wick contraction):∑∑c†k↑ c†−k↓ ck′ ↓ c−k′ ↑ →k,k′⟨c†−k↓ ck′ ↓ ⟩ c†k↑ ck′ ↑ + ⟨c†k↑ c−k′ ↑ ⟩ c†−k↓ ck′ ↓

k,k′

−

∑

⟨c†k↑ ck′ ↓ ⟩ c†−k↓ c−k′ ↑ − ⟨c†−k↓ c−k′ ↑ ⟩ c†k↑ ck′ ↓

k,k′

+

∑

⟨c†k↑ c†−k↓ ⟩ ck′ ↓ c−k′ ↑ + ⟨ck′ ↓ c−k′ ↑ ⟩ c†k↑ c†−k↓

k,k′

∑

≡

U (c†k↑ ck↑ + c†k↓ ck↓ )

∑(
k

+

∆ c†k↑ c†−k↓ + ∆∗ c−k↓ ck↑

)
(4.1)

k

∑ † †
with U = ⟨c†k↑ ck↑ ⟩ = ⟨c†k↓ ck↓ ⟩ and ∆ =k′ ⟨ck′ ↑ c−k′ ↓ ⟩. The ﬁrst term is theHartree energy, usually absorbed into the chemical potential in condensed mattersystems. The second is the signature of the superﬂuid phase. The Fock term iszero in our case as the energies of our states |1⟩ and |3⟩ are not degenerate. Byneglecting the quartic terms in the above contraction we have explicitly neglected
ﬂuctuations and quasiparticle interactions and we have entered the realm of the meanﬁeld description3 . Now the Hamiltonian is quadratic and the standard Bogoliubov
3
Actually, density ﬂuctuations have already been neglected by choosing the simpliﬁed Hamiltonian 1.2, see discussion above 4.1.

83

transform can be applied to diagonalize the Hamiltonian:γk↑ ≡ uk ck↑ − vk c†−k↓

(4.2)

†
γk↓

(4.3)

≡

uk c†−k↓ + vk ck↑†
Where the uk , vk must be chosen such that the γk↑ , γ−k↓obey the usual fermionic22commutation rules, which results in the previous uk + vk = 1. Now the Hamiltonian∑takes the form (with U = k Uk ):H − µN + U N =∑†Ek γkσ
γkσ

(4.4)

k,σ
†
It is indeed diagonal in the new basis {γk↑ , γ−k↓}, where the γ † are quasiparticle√
creation operators with the eigenenergy Ek = ∆2 + (ϵk − (µ − U ))2 . It should be
noted that the superﬂuid ground state is the quasiparticle vacuum: γk↑ |BCS⟩ =γk↓ |BCS⟩ = 0 and that a single excess particle creates a quasiparticle:

uk c†k↑ |BCS⟩ =()
uk c†k↑ − vk c−k↓ |BCS⟩

†
= γk↑
|BCS⟩

(4.5)
(4.6)

For very high momenta k ≫ kF it can be seen from equation 1.5 that vk ≪ 1
and the quasiparticle is essentially a free particle, unaﬀected by the pairing gap, as is
expected. For k ≪ kF the opposite is true and the quasiparticle is essentially a freehole. At k ≃ kF , the quasiparticle is a coherent superposition of a particle and a hole.
At k = kF , the excitation energy has a minimum and assumes the value ∆. This canbe understood from a diﬀerent angle: If a quasiparticle, or an excess fermion, occupies
momentum state k, this momentum state is not available for Cooper pairing, raising
the energy of the system by ∆. For the same reason excess fermions in an imbalanced
fermion system in an inhomogeneous potential are expelled out of the superﬂuid in
the trap center. This leads to phase separation into a balanced superﬂuid and apolarized normal phase, up to a critical polarization, see chapter 2.

4.3.1

Radiofrequency Spectroscopy of Quasiparticles

Let us revisit chapter 1.5.3 and consider the case of a single quasiparticle present in
a BCS superﬂuid in a homogeneous potential. In this case the BCS spectrum shouldbe essentially unaltered except for deviations on the order of 1/N . The quasiparticle
spectrum can be calculated easily. Energy conservation now requires ~ωRF ≡ −Ek +84

.u.a / esnopser lartceps-D-D +|U|

-0.5

0.0

0.5

rf offset /

1.0

1.5

eF

Figure 4-1: Radiofrequency spectrum with a quasiparticle at k = kF . The RF spectrum consists of a quasiparticle peak at a negative frequency −∆ and the pair dissociation spectrum at positive frequencies (red dotted line), see equation 4.7 and 1.11.
The Hartree contribution U acts as a momentum independent eﬀective attraction
and hence shifts the entire spectrum into the positive direction (solid blue line), seeequation 4.9.

ϵk − µ: The quasiparticle is immersed in the repulsive environment of the superﬂuid,
hence is its energy −Ek − µ measured relative to the Fermi energy. In addition afree particle is created in the (non-interacting) ﬁnal state with energy ϵk . The lowest
energy quasiparticle has a momentum of kF and its radiofrequency spectrum is givenby:
IQP (ω) ∝ δ(~ω − (−∆))(4.7)

Therefore, in the limit of small quasiparticle population, the spectral peak should
directly reveal the superﬂuid gap ∆, see ﬁgure 4-1
Moreover, as quasiparticles are fermions, a ﬁnite but small population of quasiparticles should, in principle, allow to reconstruct the dispersion relation Ek aroundthe minimum at kF for a given temperature4 .
4For a given temperature T , the quasiparticle spectrum is given by (see equation 1.10)∫
IQP (ω) =dk k 2 nQPk δ(ω − ωk )k=f ′ (ω)f 2 (ω)
1 + expβ (f 2 (ω)−ω−µ+h)where nQP
is the Fermi-Dirac distribution of the fermionic quasiparticles, f (−1) (k) ≡ ωk is thek85Closing the Gap:
Tunneling vs Radiofrequency Spectroscopy
It is insightful to compare radiofrequency spectroscopy to tunneling experiments used
to study superconducting systems: In radiofrequency spectroscopy, by ﬂipping thespin of an atom it has been essentially removed from the system by transferring
it into a non-interacting state, therefore creating a quasiparticle, see equation 4.6.
In tunneling experiments, this corresponds to the transfer of an electron from thesuperconducting region into the normal conductor at a suﬃciently high voltage of ∆/e,where e is the electron charge. As opposed to tunneling experiments the minimum
energy required for the radiofrequency transition process is not ∆, but rather ∆2 /2EFin the BCS limit, see chapter 4.2 above5 . Therefore, radiofrequency spectroscopy does
not measure the gap ∆ directly but a more complicated quantity in the BEC-BCS
crossover.
The opposite process of injecting a particle into the superﬂuid can not be done
easily in radiofrequency experiments since the ﬁnal state is empty. However, in tunneling experiments an electron tunnels into the superconductor when a voltage of−∆/e is applied, again creating a quasiparticle. Performing radiofrequency spectroscopy of quasiparticles at low concentrations, however, allows us to close this gap
between tunneling and radiofrequency spectroscopy: We extract the gap not by creating a quasiparticle at kF (Stokes process), but by injecting a particle at kF andsubsequently removing it at the cost of −∆ (Anti-Stokes process).

4.3.2“Cold Infusion” of Quasiparticles or:
The Importance of Being Imbalanced

As can be seen from the phase diagram in ﬁgure 2-5 and from the fact that the superﬂuid ground state is the quasiparticle vacuum, no quasiparticles will be present at
T = 0. The quasiparticle population can be interpreted as a measure for the temperature. One might therefore be tempted to increase the temperature until a spectral
signature of quasipartcles, see equation 4.7, can be observed. In a recent publicationof our group [46] low temperature (T /TF ≃ 0.1) radiofrequency spectra did not show
inverse function of f and satisﬁes energy conservation f (−1) (k) = −Ek + ϵk − (µ − h), with the
Zeeman ﬁeld h, see chapter 2.1. This deﬁnes a highly non-linear diﬀerential equation for f (ω)
from which the quasiparticle dispersion Ek can, in principle, be deduced using the experimental
spectrum IQP (ω). In practice, however, even a numerical solution is diﬃcult to ﬁnd for the current
experimental signal-to-noise.
5
The reason is that the electrons of the metal in the normal state occupy all energy levels up to
the Fermi energy, in cold atom experiments the ﬁnal state is typically empty.

86

.u.a / esnopser lartceps

a)

b)

0

2

4

c)

0

2

4

d)

0

RF offset /

2

4

0

2

4

eF­

Figure 4-2: Local radiofrequency spectra of an equal spin mixture for various normalized local temperatures T /TF ↑ (solid black line). a) T /TF ↑ ≃ 0.20, b) T /TF ↑ ≃ 0.22,c) T /TF ↑ ≃ 0.34, d) T /TF ↑ ≃ 0.55. No local double peak spectrum can be resolvedin the radiofrequency spectrum. For comparison, the double peak spectrum of animbalanced mixture with T /TF ↑ ≃ 0.06 is added (red dashed line).

any signature of the presence of quasiparticles. Using BCS theory, we can give an
estimate of the temperature for which we expect a sizeable quasiparticle population:
For a thermal population of quasiparticles the temperature has to be on the orderof the gap ∆.
√ In BCS theory, the temperature dependence of the gap is given by∆(T )= 1.74 1 − TTc , with ∆0 the superﬂuid gap at zero temperature. At unitarity,∆0∆0 /Tc ≃ 3 (both for the mean-ﬁeld treatment and Monte Carlo calculations). Therefore, one would expect to populate quasiparticles only very close to the transition
temperature T /Tc ≃ 0.95, when the gap is only one third of its low-temperature
value. The experimental results in ﬁgure 4-2 show a decrease of the spectral gap withincreasing temperature, but no local double peak structures could be resolved for any
temperature.
Our solution to this problem was the creation of a density imbalanced Fermi
mixture at the lowest temperature we can achieve in the experiment T /TF < 0.1.
The associated Zeeman ﬁeld h results in a splitting of the dispersion relation into two
branches see equation 2.2 - the majority dispersion is given by: E(k) → E(k) − h,see also ﬁgure 4-3. In this case the temperature needed for a sizeable quasiparticle
population is decreased from ∆ to ∆ − h. In our experiment, ∆ − h < 0.1ϵF and
we have therefore created “cold” quasiparticles. Of course, the Zeeman ﬁeld h mustbe smaller than the critical ﬁeld hc1 /µ ≃ 0.96 [74], beyond which the normal state
becomes energetically favorable over the superﬂuid state. This is called the Clogston√Chandrasekhar limit [95, 96, 16] and in the BCS limit is given by ∆/ 2.

87

Ek

1.0

0.5

m+ h

D
0.0
0.0

0.5

1.0

k / kF

Figure 4-3: Creation of “cold” quasiparticles. The population imbalance thermallygenerates quasiparticles even at low temperatures on the order of ∆ − h, where h is
the Zeeman ﬁeld characterizing the number imbalance P . µ↑ is deﬁned as µ + h, seeequation 2.2.

4.3.3

Experimental Results and a Surprise in the Lab

Radiofrequency spectra for an imbalanced system are shown in ﬁgure 4-4. The inhomogeneous trapping proﬁle allows us to probe the single-particle excitation spectrum
in diﬀerent physical regimes within a single sample by recording the local response of
three-dimensional reconstructed density proﬁles in the ﬁnal state immediately after
application of the radiofrequency pulse. The spectra in ﬁgure 4-4 reveal a gradualchange in the nature of the pairing correlations. The balanced superﬂuid is characterized by identical spectral responses of majority and minority particles and has been
the subject of previous studies, see chapter 1.5.1 and the review articles [23, 126, 127]
and references therein. In the polarized superﬂuid region [19, 70] (and references
therein) the minority spectrum perfectly matches the pairing peak of the majority
spectrum, locally coexisting with the quasiparticle spectral contribution, resulting ina local double peak structure of the majority spectrum, see ﬁgures 4-4b and 4.3.1.
As the local imbalance is further increased beyond the superﬂuid to normal (SFN) transition [19], see ﬁgure 4-4d, the majority spectrum no longer shows a local
double-peak structure. This is consistent with theoretical work [99, 128] attributing
the double peak structure in the normal phase in previously reported radiofrequency
spectra [13, 20] to the inhomogeneous density distribution. For increasing spin polar88

BCD.u.a / esnopser lartceps

A

A

B

0

1

2

3

C

0

1

2

3

RF offset /

D

0

1

2

3

4

-1

0

1

2

3

4

eF

Figure 4-4: Tomographically reconstructed radiofrequency spectra for various regionsof the atomic sample at unitarity. a) Balanced superﬂuid, b) polarized superﬂuid, c)moderately polarized transition region and d) highly polarized normal region. The
panel on the left shows a phase contrast image of the atomic cloud before radiofrequency excitation. The positions of the spectra a) to d) are marked in the phase
contrast image and by the arrows in ﬁgure 4-5. Red: Majority spectrum, blue: Minority spectrum. Local polarizations σloc and local temperature T /TF , respectively:a) -0.04(2), 0.05(1), b) 0.03(1), 0.06(1)c) 0.19(1), 0.06(2), d) 0.64(4), 0.10(2). The
negative value in a) implies that the local polarization as inferred from phase-contrast
imaging underestimates σloc by up to 0.05.

ization the majority and minority pairing peaks lose spectral overlap. This missing
overlap can be interpreted as indication that the minority atoms are no longer boundin pairs, each of them interacting with more than one majority atom. This is thepolaronic phase which has been investigated in chapter 3. At unitarity, within our
experimental resolution, the overlap starts to decrease at the superﬂuid-to-normalinterface, see ﬁgure 4-4c.
Even when the spectral overlap decreases, there is still equal response to the radiofrequency excitation in the high frequency tails, see ﬁgure 4-4c and ﬁgure 4-4d.
These tails correspond to large momentum components in the interparticle wave function and hence address the short range physics. The magnitude and functional formof the high-frequency tail are characterized by the contact coeﬃcient C. According
to the adiabatic sweep theorem, discussed in chapter 3.7.4, the contact coeﬃcient C
carries much information about the many-body state of the system. From the wingsof the polaron spectra in chapter 3 we were able to estimate the contact coeﬃcient. It
is predicted to be ≈ 10% larger in the superﬂuid state as the polaron is characterized
by a delocalized component in the wavefunction, see equation 3.23. Unfortunately, inthis experiment we did not explore high enough frequencies to determine the contact
coeﬃcient in the balanced superﬂuid the spectra multiplied by ~ω 3/2 did not show a89

plateau as they do in the polaron spectra, see ﬁgure 3-15.
As can be seen in ﬁgures 4-4 and 4-5 these physically very diﬀerent regimes aresmoothly connected across the critical density imbalance [22], see above.
A two-dimensional summary of all our results in the unitary limit is shown inﬁgure 4-5.
Minority

Majority

3

1.0

N

SF

N
0.5

2

­F

e /tesffo FR

0.0

1

1.0

s

0

col

0.5

0.0
-1
-1.0

-0.5

0.0

0.5

1.0

r / R­

Figure 4-5: Spatially resolved RF spectra of an imbalanced Fermi gas at unitarity.
a) The right half shows the majority spectra as a function of position in the trap
expressed in terms of the majority Fermi radius R↑ , the left half displays the minority
spectra. The superﬂuid to normal transition region is marked by the gray dashedvertical lines. The local polarization σloc is given by the dashed red line. The error
bars are the standard deviation of the mean value. The arrows indicate the positionof the four spectra shown in ﬁgure 4-4. The image is a bilinear interpolation of 2500
data points, each plotted data point in the image is the average of three measured
data points. The spatial resolution of the image is 0.045·R↑

In ﬁgure 4-6a the position of the peaks of majority and minority spectra are plotted
normalized by ϵF ↑ as a function of position in the trap in the unitary limit. The peak
positions are proportional to the local Fermi energy inside the superﬂuid region within
our experimental resolution. In the polarized superﬂuid phase the spectra show local
double peaks. The position of the two peaks in the limit of small polarization isdepicted in ﬁgure 4-6b for various interaction strengths.
90

a)

SF

1.0

N

b)
2.0

0.8
F

e / tfihs kaepFe / noitisop kaep

1.5

0.6

1.0

0.4

0.5
0.2

0.0
0.0
0.0

0.2

0.4

0.6

0.8

1.0

-0.2

0.0

0.2

0.4

0.6

1 / kF ar / R­

Figure 4-6: a) Normalized peak positions of pairing peaks and quasiparticle peak at
unitarity as a function of position in the trap. The superﬂuid-to-normal boundary(cusp in column density diﬀerence, see appendix D, is marked by the dashed vertical lines. The arrow indicates the limit of low quasiparticle population relevant forb). Majority: blue open squares (pairing peak) and solid black circles (quasiparticlepeak), Minority: solid red triangles. b) Pairing peak and quasiparticle peak positions as a function of the local interaction strength 1/kF a in the limit of small local
imbalance (see arrow in a). Pairing peak: Solid circles, quasiparticles peaks: Opencircles.

One striking and initially unexpected feature of the data presented above is the
position of the quasiparticle peak at positive frequencies (relative to the atomic transition frequency) - instead of being repelled out of the superﬂuid they seem to be
attracted by it! This picture is, of course, wrong and is caused by the presence ofHartree terms, resulting in an overall shift of the systems energy and the radiofrequency spectrum, see below in chapter 4.3.4. In the weakly interacting limits, theHartree term reduces to a simple mean ﬁeld shift. In the strongly interacting regime
one has to resort to Monte Carlo calculations [129, 74, 42] for a numerical value of U.

4.3.4

Hartree: Cause and Eﬀect

We have seen the origin of the Hartree energy U in chapter 4.3. Let us now show whateﬀect they have on the radiofrequency spectrum. The Hartree energy is momentum
independent and can be shown to simply result in an overall shift of the spectrum.
91The quasiparticle peak is located at ~ωQP = −Ek + ϵk − µ, see chapter 4.3.1, suchthat the minimum is now at ϵk = µ − U . It follows √that the minimum excitation√energy now occurs not at k =2m(µ−U )such that:~2mµbut rather at k =~

IQP (ω) ∝ δ(~ω − (−∆ + U ))(4.8)

It is straightforward to show that with the modiﬁed dispersion the dissociation
spectrum retains its functional form (equation 1.11) with a frequency axis shifted byU:√√′′ − ω′
ωωthµth
I(ω ′ ) ∝
1
+
+
(4.9)
′2
′
ω
ω
~ω ′
′
= ωth +U and µ′ = µ−U . The maximum of the dissociation
where ω ′ = ω +U , ωthspectrum occurs at
~ωmax

4
=
3
≃

(√

15
µ′2 + ∆2 − µ′
16

)
−U

4
ωth − U3

(4.10)(4.11)

where µ′ = µ − U and ωth is the dissociation threshold (which is at momentum
k = 0).
Determination of the Superﬂuid Gap and the Hartree Energy
We determined the superﬂuid gap ∆ and the Hartree energy U from the peak positionsin the limit of small density imbalance (σ ≃ 0.03). At unitarity with the chemical
potential µ = 0.42 ϵF ↑ , we obtained:∆unit = 0.44(3) ϵF↑andUunit = −0.43(3) ϵF↑(4.12)

in excellent agreement with the predicted values ∆t = 0.45 ϵF ↑ and Ut = −0.43 ϵF ↑from Quantum Monte Carlo calculations [41]. Our determined values for ∆ and U
values suggest the minimum of the quasiparticle dispersion curve to occur at kmin ≃0.9kF . Table 4.1 shows the gap and Hartree energy for various interaction strengths.
Away from unitarity we relied on Quantum Monte Carlo calculations for the chemical
92

Table 4.1: Superﬂuid gap ∆, Hartree term U and ﬁnal state interaction Eﬁnal in termsof the Fermi energy ϵF ↑ for various interaction strengths 1/kF a.
1/kF a
-0.25
0
0.38
0.68

∆
0.22
0.44
0.7
0.99

U
-0.22
-0.43
-0.59
-0.87

Eﬁnal
0.22
0.16
0.14
0.12

potential µ [33].
For an accurate quantitative comparison ﬁnal state interactions, also listed intable 4.1, had to be taken into account. The eﬀect of ﬁnal state interactions is an
2overall mean ﬁeld shift of Eﬁnal = 4π~mafe n. This shift aﬀects both the quasiparticle
peak and the pairing peak equally.

4.3.5

Why Should That Work?!

Above we have presented the extraction of the superﬂuid gap ∆ and the Hartree
energy U based on two assumptions: a) The quasiparticle dispersion relation is given√by the mean-ﬁeld result Ek = ∆2 + (ϵk − (µ − U ))2 , see 4.3.4, and b) the chemical
potential is given by the Monte Carlo result µ = 0.42ϵF [33]. This immediately raises
two questions:• The above treatment is obviously non-self-consistent as we use the dispersion
from mean-ﬁeld theory (which yields µ = 0.59) but the chemical potential fromMonte Carlo calculations µ = 0.42. We have, however, stressed before thatmean-ﬁeld theory correctly produces qualitative features but fails for quantita√
tive comparison, 2.2. The formula Ek = ∆2 + (ϵk − (µ − U ))2 can therefore
be regarded as a valid parametrization of the quasiparticle dispersion relation
with the free parameters ∆ and U .
• The exact value for the chemical potential is still under discussion and ranges
from 0.32ϵF to 0.49ϵF [30, 31, 32, 33, 34, 35, 38, 39, 14, 40, 36, 37] at zero temperature . Moreover, this value is temperature dependent. How much does thisuncertainty aﬀect the extraction of ∆ and U ? Figure 4-7 shows the extracted
superﬂuid gap and Hartree energy as a function of the value used for the chemical potential. Obviously, the dependence is weak and the variation with µ is
93within the error bar. The reason for the robustness is that the value for the superﬂuid gap is essentially determined by the distance between the quasiparticle
and the dissociation peak and does therefore not rely on the precise knowledgeof the chemical potential. Since the quasiparticle peak is located at −∆ − U ,the stability of −∆ makes U stable also.

0.4

0.2

e / U ,D

F

0.0

-0.2

-0.4

0.3

0.4

0.5

m / eF

Figure 4-7: Dependence of the extracted values for ∆ (red line) and U (blue line) on
the assumed value for the chemical potential µ

In addition, it has to be mentioned that our determination of ∆ and U is naturally
not for zero temperature but for the ﬁnite local normalized temperature of T /TF ≃0.06. However, there is indication from ﬁnite temperature Quantum Monte Carlocalculations [42] that ∆ is only weakly temperature dependent for T /TF < 0.15 andthat U is only weakly temperature dependent for T /TF < 0.3.
Comment: Resolution / Experimental broadening For an accurate determination of the peak positions, Fourier broadening of the spectra must be taken into
account: The peak of an asymmetric distribution is shifted upon convolution with thepulse duration. For comparison with the theoretical spectrum, we have modeled theradiofrequency pulse of T = 200µs length as a square pulse. This results in a FWHM
1of the radiofrequency spectral power of ∆ν = 2 · 2π
1.39 T2 ≃ 4.4kHz. The theoretical
spectrum consists of the dissociation term including the Hartree energy, equation 4.9
and the quasiparticle peak modeled as a narrow (FWHM = 1 kHz) Lorentzian. This94

spectrum was convolved with the Fourier transform of a square pulse f (ω) ∝sin2 ωT2

2 .
( ωT2 )
Figure 4-8 shows that the theoretical spectrum reproduces our data quite well. The
deviation in 4-8b) might be attributed to additional broadening mechanisms like ﬁnite
quasiparticle lifetime, ﬁnite temperature and atomic diﬀusion during the duration of
the radiofrequency pulse. The convolution causes a shift of 0.05 ϵF ↑ in the spectral
peak position of the dissociation spectrum and has been accounted for in the determination of ∆ and U.

spectral response / a.u.

a)

0.0

0.5

1.0
1.5
2.0
RF offset / ε F↑

2.5

3.0

0.0

0.5

1.0
1.5
2.0
RF offset / ε F↑

2.5

3.0

spectral response / a.u.

b)

Figure 4-8: Comparison of experimental (red) and theoretical (black) line shapes
for spectra a) and b) in ﬁgure 4-4. The theoretical curve is obtained from a BCSLeggett mean ﬁeld description including the Hartree term and a convolution with theexperimental resolution of ≃ 4.4kHz (the amplitude of the spectra have been left as
free ﬁtting parameters). The values for ∆ and U as calculated from the peak positions
lead to a reasonable agreement with the data.

95

4.4

Connecting the Polaron Phase to the Superﬂuid Phase

So far we have focused on the balanced phase. However, the inhomogeneous trapping
proﬁle presents a cut through the phase diagram and we realize diﬀerent physicalregimes: Apart from the balanced superﬂuid core, we have realized a polarized superﬂuid phase for various interaction strengths (bear in mind that the experiment is
not performed at zero temperature). We have identiﬁed the polarized normal region
with the limiting cases of the polaronic phase and the fully spin polarized regime ofan ideal Fermi gas.
One ﬁgure of merit that can be identiﬁed to distinguish the diﬀerent phases is
the spectral overlap, see ﬁgure 4-4. In the previous chapter 3 we have presented thequasiparticle residue Z of the polaron as a measure of the dressing of the minority
particles with its surroundings. We will generalize this concept to characterize thepairing also in the moderately polarized normal region and the superﬂuid region and,for lack of a better word, continue to refer to it as the quasiparticle residue.
The quasiparticle residue has been extracted in the same way as for the polaronic
phase, see ﬁgure 3-12 and is plotted in ﬁgure 4-9 as a function of the local polarization
σ for various interaction strengths.
The two physical limits are smoothly connected. The quasiparticle residue does
not change appreciably until the polarization approaches its respective critical value,suggesting that the theoretical description presented in chapter 3.4 remains accurateeven away from the highly polarized limit.
Determination of the Critical Imbalance Using the Quasiparticle Residue
In chapter 2.4 we have presented a method to determine the phase boundary separating the normal phase and the phase separated superﬂuid phase based on the density
distribution in an inhomogeneous potential. Here we will show that we can achieve
very similar results by considering the local polarization σ at which the quasiparticle residue Z assume a ﬁnite value. The according data points based on the data
in ﬁgure 4-9 are added to ﬁgure 2-4 and can be seen to agree well with the density
proﬁle analysis. Both of these methods yield a phase boundary at lower polarization
than the zero-temperature Monte Carlo calculations, see ﬁgure 2-2. The discrepancy
might be explained by the ﬁnite temperature in the experiment T /TF ≃ 0.1 and acomparison should be made with ﬁnite temperature calculations, see also chapter 2.5.

96

Z

0.8

0.6

0.4

0.2

0.0
0.0

0.2

0.4

0.6

s

0.8

1.0

loc

Figure 4-9: Generalized quasiparticle residue Z as a function of the local polarization(n −n )
σloc = (n↑↑ +n↓↓ ) for various interaction strengths. Solid black circles: Unitary limit.
Solid blue diamonds: −0.36 < 1/kF a < −0.24 (BCS side). Solid red triangles: 0.34 <1/kF a < 0.43 (BEC side). The open symbols represent the limit of the polaronic phasefor comparable interactions strengths from the more accurate measurements of thepolaron energy and the quasiparticle residue, see chapter 3. The lines are a guide to
the eye and were used as a rough estimate for the critical polarizations σrmloc shownin ﬁgure 2-4.

97

Chapter 5
The Equation of State of aStrongly Interacting Fermi Gas at
Finite Temperature
5.1

Introduction: Equation of State of a Balanced
Fermi Mixture

As discussed in the previous chapters, reliable quantitative theories remain elusive inthe strongly interacting regime due to the absence of a small parameter when the gas
parameter na3 diverges. However, a signiﬁcant simpliﬁcation arises in the unitary
regime: As the scattering length diverges, it is no longer a relevant quantity for thesystem and all physical quantities show universal behavior, independent of the natureof the interactions. At zero temperature, the only energy scale left in the system is the~2 k2~2(6π 2 n)2/3 , where n is the density
chemical potential or Fermi energy ϵF = 2mF = 2mof a single component. In the universal regime energies scale as ϵF , times scale as
1/ϵF , velocities scale as kF and so forth. This is called the “universality hypothesis”[130]. At ﬁnite temperature, two parameters completely describe the system, thetemperature T and the chemical potential µ.
Therefore, the phase space density can be written as:
nλ3 = f (βµ)
(5.1)
√
2π~2
with the thermal wavelength λ = mkand β = 1/kB T describes any stronglyBT98interacting system1 employing the universal function f (βµ). Equation 5.1 represents
the equation of state relating the density n to the temperature T and the chemical
potential µ. Starting from this equation many thermodynamic quantities can bederived. For example the pressure can be obtained using the Gibbs-Duham relation
dP = n dµ + s dT , which for constant temperature yields:∫ µ0
p(µ0 ) =

n dµ

(5.2)

−∞

In this chapter we will present an experimental study towards a high precision determination of the Equation of State of a unitary Fermi gas at arbitrary temperature.

5.2

Inhomogeneous Potential

In the local density approximation the system can be described as locally homoge∑
neous with a spatially dependent chemical potential µloc = µ − V (r) = µ − i m2 ωi2 ri2 .
This provides the density distribution in a harmonic trap, given the temperature
T and the chemical potential µ (for simplicity, let us assume an isotropic potential
ωi = ω):1 (m 2 2)
n(r) = 3 f βµ − β ω r
(5.3)
λ
2
Equivalently, if the temperature, the chemical potential and the potential proﬁleare known, one can directly determine the equation of state f (x) by inversion of theabove relation:(√f (x) = λ3 n2(µ − kB T x)
mω 2

)
(5.4)

This looks more complicated than it is: Given the density distribution n(r), one
simply has to rescale the x-axis by ﬁnding the local x = βµloc at position r andmultiply the y-axis by λ3 .
Using equation 5.2, similar relations hold for the local pressure of the unitary Fermi
gas:
1

It has to be borne in mind, however, that this is only true for zero-range interactions, see also
chapter 3.7.4. In cold atom experiments the eﬀective range reﬀ is on the order of the range of thevan der Waals interaction r0 , which for alkali atoms is on the order of 50 − 100 a0 and for 6 Li it isr0 = 63 a0 and therefore much smaller than any other length scale.

99

kB T (
m 2 2)
F
βµ
−
β
ω r
(5.5)
λ3
2
with the universal function F (βµ) being related to the function f (βµ) deﬁned∫x
above: F (x) = −∞ d(x′ )f (x′ ). The above equation yields the formula for F :p(r) =

λ3
F (x) =
p
kB T

(√

)
2
(µ − kB T x)
mω 2

(5.6)

where the pressure follows from equation 5.22 .

5.3

Temperature Measurement at Unitarity

In principle, the above determination of the equation of state is straightforward andposes no problem. Experimentally, the main challenge consists of a reliable determination of the temperature and the chemical potential. Thermometry of stronglyinteracting ultracold Fermi gases is a notoriously diﬃcult problem in experiments.
Diﬀerent approaches have been taken to solve this problem: In imbalanced Fermi
gases the shape of the non-interacting wings of the majority component have beenused to determine the temperature [16, 23]. This method is obviously not applicableto the case of a balanced Fermi mixture. In balanced gases, the temperature has
been determined by thermal contact with a weakly interacting bosonic bath whose
temperature can be well characterized [131]. Another approach relied on ﬁtting a
non-interacting density proﬁle to the strongly interacting proﬁles and rescaling of theresulting temperature. This quantity is a measure of the temperature but yields thetrue temperature of the system only under a series of assumptions [14]. Yet anothermethod of temperature measurement relies on determination of the temperature in
the weakly interacting limit by ﬁtting a non-interacting density proﬁle before adiabatically bringing the mixture into the unitary regime, see for instance [12]. Again, this
only yields a quantity characterizing the temperature but not the true temperature,
which changes during an adiabatic sweep. Here we will present a diﬀerent approachbased on the virial expansion of the equation of state.
m 2 2
2
For
∫ a harmonic potential, µ = µ0 − 2 ω r . The pressure then can be written as p(r)m 2 ∞dr 2r n(r) and is basically given by the one-dimensional density distribution.
2ω
r

100

=

5.3.1

Virial Expansion

At high temperatures T ≫ µ or βµ ≪ 1, quantum statistics can be neglected andevery gas looks like an ideal gas. The phase space density is simply given by the
Boltzmann distribution eβµ and the equation of state is n(r) = eβµ /λ3 independentof statistics or interactions. This is the ﬁrst term in an expansion in powers of the
fugacity eβµ :()
nλ3 = f (βµ) ≡ eβµ + 2 b2 e2βµ + 3 b3 e3βµ + ...

(5.7)

where the coeﬃcients bi are the virial coeﬃcients. Bear in mind that that thephase space density is small at high temperatures nλ3 ≪ 1, and hence the chemical
potential negative µ < 0. Therefore, the value of the fugacity for which the series
expansion in equation 5.7 converges, is restricted to 0 < eβµ < 1.
Similarly, for the pressure we have the expansion:
p

(
)
λ3
= F (βµ) ≡ eβµ + b2 e2βµ + b3 e3βµ + ...
kB T

(5.8)
√

The virial coeﬃcients for the unitary Fermi gas are b2 = 2 8 2 [132, 133] andb3 = 0.29095295 [134]. Therefore, equation 5.7 provides a robust method to ﬁt thewings of a balanced Fermi gas. Care has to be taken that the ﬁtting range entails only
regions of a suﬃciently negative chemical potential as the virial expansion up to thirdorder is expected to be valid only for βµloc < −0.8. At this point, the virial expansion
deviates from the diagrammatic Monte Carlo calculation [135] by more than 1%. For
the integrated density, yielding the pressure, see equation 5.2, it is expected to bevalid up to βµloc < −0.4. The value of the local βµloc of course depends on the ﬁtyielding the global βµ and βµ has to be determined self-consistently.

5.3.2

Eﬀects of the Trap: Harmonic vs Gauss

All data in cold atom experiments consist of projection images of an atom cloud.
The two-dimensional density distribution n2D (x, z) is given by the integral of thethree-dimensional density n3D (x, y, z) along the imaging axis, here denoted as they-axis.
Harmonic Potential
In the case of a harmonic potential, the temperature determination using the virial
expansion 5.7 is a straightforward Gaussian integral:
101

∫ ∞
n2D (x, z) =−∞dy n3D (x, y, z)(5.9)

1 kB T ∑ √ kβ (µ− m2 (ωx2 x2 +ωz2 z2 ))bk k eλ2 ~ωy k
1 (kB T ) ∑ √ kβ (µ− m2 ωr2 r2 )n1D (r) =bk k eλ2 ~ωy k=(5.10)(5.11)The last equality holds for an elliptically averaged density distribution.
Gaussian Potential
Typically, see chapter 1.4, in order to access a Feshbach resonance to enter the unitary
regime, the atoms are radially trapped in the potential well created by a focussed laser
beam red-detuned from the dominant atomic transition. The laser beam emanatesfrom an optical ﬁber and is well characterized as a TEM00 mode. The potential wellexperienced by the atoms can therefore be written as:(
V (x, y) = V0 1 − e

2−2 x +y2

2)(5.12)

w

where the trap depth V0 /kB is typically on the order of a few µK and in our setup
the waist3 is w = 70.5µm. Along the third axis, the atoms are trapped by a magnetic
curvature4 characterized by a trapping frequency of ωz = 2π · 22.83Hz. As a typicalFermi energy is on the order of 1µK, anharmonicities play a signiﬁcant role for thishigh precision study, see ﬁgure 5-1.
Since we typically operate at trap ﬁlling factors 0.2 < η < 0.5, we can not neglect
the eﬀects of anharmonicity. Therefore, in order to determine the temperature andthe chemical potential from the virial expansion in equation 5.7, one has to do one of
the following:
3

The power of the infrared laser beam was measured before entering the chamber through a penny
window and after leaving the chamber through the opposite penny window. The total power loss
was 13% so that the laser power at the position of the atoms was 93.3%. The waist was determinedusing a beam proﬁler at the position of the image of the ODT focus outside the chamber, close to
the position of the CCD camera chip.
4
In equation 5.12 we have neglected the conﬁnement along the z-direction resulting from the
πw2
optical potential, which is given by V (z) = V0 (1 − 1z2 ) with the Rayleigh range zR = λ 0 . In
1+ 2

z
R

our case with a waist of w0 = 70.5µm and a wavelength of λ = √1064nm, the Rayleigh range is
2V0
zR ≃ 14mm, resulting in a typical trapping frequency of ωz,ODT = mz
2 ≃ 2π · 1Hz. This changes
R

the trapping frequency of the magnetic potential along the z-axis by less than one per mille.

102

1.0

0.8

0

V / )r(V

0.6

0.4

0.2

0.0
-2

-1

0

1

2

r / waist

Figure 5-1: Comparison of the radial trapping potential to the harmonic approximation and the quartic correction.

Method I: Integrate the virial expansion 5.7 along the imaging axis and use thetwo-dimensional ﬁtting function to n2D (x, z):n2D (x, z)λ

3

∫ ∞
=
−∞

dy n3D (x, y, z)

∫ ∞
=

dy
−∞

≡

∑

∑

(5.13)(m

2 2

k bk ek βµ−k β 2 ωz z e−k β V0 1−e−2)x2 +y 22
w

(5.14)km

2 2

k bk ek βµ−k β 2 ωz z Sk (β, x)(5.15)k

While this can be done, it has a few disadvantages. Two-dimensional ﬁts are lessrobust as the ﬁnal ﬁt tends to depend on the initial guess for the ﬁtting parameters.
Also, the integral along the imaging direction Sk (β, x) has to be done numerically ineach step of the ﬁtting procedure, resulting in slow ﬁtting routines.
Method II: Reconstruct the three-dimensional density n3D (x, y, z) from the twodimensional density n2D (x, z) using the inverse Abel transform5 [66]. A potential
disadvantage is the loss of signal-to-noise in the data since the inverse Abel transform5− π1

If a function f (r, z) is cylindrically symmetric, the inverse Abel transform is given by f (x, z) =∫ ∞ dn2Ddy(y,z)
dy √ 2 2 , where n2D (y, z) is the column density.
x
y −x

103

involves a derivative of experimental data. This can be alleviated by performing anappropriate averaging procedure over equipotential lines6 . One has to average all
pixels in an image belonging to the same three-dimensional potential, see ﬁgure 5-2.

xz

Figure 5-2: Equipotential lines for a hybrid potential: Harmonic conﬁnement along
the x-axis, Gaussian conﬁnement along the z-axis.

We can thus “collapse” all pixels in the image into a one-dimensional form, whose
high-energy wings can be ﬁt independently of the actual trapping potential by thesimple ﬁtting function based on the virial expansion:n(V ) =

)
1 (
2
3
σ
+
2
b
σ
+
3
b
σ
loc
2
3
loc
loc
λ3

(5.16)

with the local fugacity σloc ≡ eβµ−βV .
Method III: An improved signal-to-noise ratio can be obtained by an integration
over the density distribution, yielding the pressure p = kλB3T F (βµ), see equations 5.2
and 5.8. One can now employ partial integration to avoid taking the derivative ofexperimental data in the inverse Abel transform and instead take the derivative of
the potential [136]:1p(ρ0 , z) =
π

∫ ∞
ρ0

[ ∂V
∂ρ

dy n2D (y, z) √(y, z)

y 2 − ρ20

∫ y
+

dρ
ρ0

ρ ∂V
(y, z) − y ∂V(ρ, z)∂ρ∂ρ
(y 2 − ρ2 )(3/2)](5.17)For the case of the Gaussian potential from equation 5.12, this equation becomes:
6This can not be done before the reconstruction procedure due to the Gaussian potential.

1041 4V0 (−2 ρ202 )p(ρ0 , z) =
e w
π w2

∫ ∞
0

(√
)(
√ u (√ u ))2
2
du n2Du + ρ0 , z
1−2 2 D2w
w

(5.18)

2 ∫∞2with the deﬁnition D(x) = e−x 0 du eu for the Dawson function and V0 and w
characterizing the optical trapping potential7 .
With the above equation for the two-dimensional pressure, one can now performthe appropriate averaging procedure described above and depicted in ﬁgure 5-1 toarrive at the one-dimensional pressure p(V ), whose high-energy wings can be ﬁt by
the simple ﬁtting function based on the virial expansion:p(V ) =)
kB T (
3
2
σ
+
b
σ
+
b
σ
3
loc
2
loc
loc
λ3

(5.19)

with the local fugacity σloc ≡ eβµ−βV .
We have employed all three methods successfully to extract the temperature andthe chemical potential, with the second and third method being more robust and lesstime consuming. The third method has the advantage of a higher signal-to-noise ratio
and the virial expansion being valid for higher values of βµ, see chapter 5.3.1. In thefollowing chapter we will present the results for a unitary Fermi gas.

5.3.3

Determination of the Temperature

Figure 5-3 shows a typical pressure curve p(V ) as a function of the potential V ,see previous chapter. The zoom-in shows the region in which the virial expansion
equation 5.19 is valid.
The local value of βµloc (z) = βµ − βV depends on the ﬁtted values for βµ and β.
Therefore, the ﬁtting range has to be determined self-consistently. This is done by
performing the ﬁt indiscriminately in each interval {n · ∆z, zlast }, n < zend /∆z. For
each of the (zend − 1) ﬁts we extract βµ and β. The result for βµ as a function ofﬁtting range is shown in ﬁgure 5-4. The equivalent values of the temperature are notshown here but show a similar behavior.
For a perfect signal-to-noise ratio of the data and a perfect characterization of
our experimental parameters, βµ in ﬁgure 5-4b would be a ﬂat line until the local
7

The magnetic saddle potential results in an additional anticonﬁning term which has to be taken
into account in the analysis of experimental data. If this antitrapping potential is characterized by
the “trapping frequency” ωA , one(√ has to add) the following term to the right hand side of equation
∫∞
1
2
2
2
u + ρ0 , z .
5.18: U = − π mωA 0 du n2D105

virial valid
0.02

mb loc <-0.4

0.10

3

) m m / K m Bk ( / p

0.01

0.05

0.00
0.5

1.0

0.00
0.0

0.5

1.0
V /

1.5

2.0

mK

Figure 5-3: Pressure proﬁle vs potential (solid red line). The proﬁle demonstrates
typical signal-to-noise ratio in our data. Shown is also a zoom into the region, in
which the virial ﬁt (solid blue line) is applicable. The ﬁtting region is marked by thedashed vertical line.

βµloc (z) < −0.4, at which point the virial ﬁt deviates from the theoretical calculations
by more than 1%. This can be veriﬁed in ﬁgure 5-4b: For values βµloc (z) > −0.4, theﬁtted value for βµ becomes unreliable. Eventually, as the ﬁtting region is restricted
to low density wings, where βµloc < −1 for the particular data from the proﬁle shownin ﬁgure 5-4a, the signal-to-noise ratio does not allow a reliable extraction of βµ any
more. We therefore extract βµ at the point where βµloc ≃ −0.4. For this particular
proﬁle, βµ = 1.37(3), T = 236(2)nK and µ = 325(7)nK. The error bars represent thestatistical uncertainty of the ﬁt shown in the inset of ﬁgure 5-3 and do not include
potential systematic errors.
With this method we can assign values of βµ and T to each proﬁle and determine
the equation of state from equation 5.6. This can be done for an arbitrary numberof density proﬁles, allowing us to suppress the statistical error and obtain a highprecision determination of the equation of state.

106

a)

0.0

mb

col

-0.5

-1.0

-1.5

-2.0
0.2

b)

0.4

0.6

1.5

mb

1.0
virial valid

mb

loc

<-0.4

0.5

0.2

0.4
V / kB

0.6

mK

Figure 5-4: Determination of βµ. Shown are a) the local value βµloc as a function of
the ﬁtting range, see text, and b) the value of βµ as a function of the ﬁtting range.
The dashed lines show the restriction of the valid ﬁtting range.

5.4

Preparation: Equation of State of the NonInteracting Fermi gas

As the ﬁt to the virial expansion in equation 5.19 requires precise knowledge of theabsolute value of the density at a certain potential V , it is crucial to carefully calibrate
the imaging system. Many factors determine the relation between the observed opticaldensity for absorption and the true atomic density, such as polarization of the imaging
light, linewidth of the imaging laser, oﬀ-resonant light in the imaging path, alignmentof magnetic ﬁelds, magniﬁcation factor of the imaging system, etc. Here we chosea diﬀerent route: We created a system for which we know the equation of state
107

0.15

3

) mm / Km Bk( / p

0.10

0.05

0.00
0.0

0.5

1.0

1.5
V /

2.0

mK

Figure 5-5: Equation of state of a non-interacting Fermi gas for calibration. Solid blue
line: Polylogarithm from equation 5.20. Solid red line: Experimental data, equationof state from a single proﬁle.

and adjusted the prefactor to calibrate imaging. Our calibration system is the noninteracting Fermi gas.
A non-interacting Fermi gas can be easily produced experimentally as there isa zero-crossing of the scattering length a at 534G [137]. An experimental complication arises from the fact that a non-interacting Fermi gas by deﬁnition does notthermalize - the “true” density distribution will therefore never be reached. We can,however, create a weakly interacting Fermi gas, whose density distribution shouldclosely resemble a non-interacting Fermi gas if the mean-ﬁeld interaction energy ismuch smaller than the Fermi energy. In the case discussed here, UMF ≃ ϵF /100, sothat interaction eﬀects should be negligible.
The pressure of a non-interacting Fermi gas can be derived in the local density
approximation by setting µloc (r) = µ − V (r) and carrying out the integration of the
Fermi-Dirac distribution over the momentum components and µ, resulting in [23]:p(V ) = −kB T
Li5/2 (−eβ(µ−V ) )λ3

108(5.20)

∑zk
where Li5/2 (z) = ∞k=1 k5/2 is polylogarithm of order 5/2. Based on this series
expansion, the virial coeﬃcients, see equation 5.19, can be seen immediately: b1 =√√1
1, b2 = − 82 , b3 = 273 , b4 = 32.
Figure 5-5 shows the non-interacting equation of state, see equation 5.20 with βµ,T and a numerical prefactor characterizing the imaging system as ﬁtting parameters.
The experimental data are well ﬁt by the polylogarithm.

5.5

The Equation of State of the Unitary Fermi
Gas I
F(bm) 3.5

3.0

2.5

2.0

1.5
virial valid
1.0

bm < -0.4

0.5

0.0
-4

-3

-2

-1

0

1

bm

Figure 5-6: Equation of state F (βµ) of a unitary Fermi gas. Solid black line: Diagrammatic Monte Carlo calculation [135]. Dashed black line: Third order virial expansion,valid for βµ < −0.4. Solid red line: Experimental equation of state averaged over 10
proﬁles.

After these preparations, let us now turn to the equation of state F (βµ), seeequation 5.6, of a unitary Fermi gas.
The results are shown in ﬁgure 5-6. Also shown in the graph is the result of
diagrammatic Monte Carlo calculations [135].
109

We can thus experimentally produce an equation of state from experimental datawith very high accuracy and can verify state-of-the-art Monte Carlo calculations. At
the time when this thesis was written, Diagrammatic Monte Carlo calculations [135]were available up to βµ = 1.

5.6

The Equation of State of the Unitary Fermi
Gas II (preliminary)

We can now use the equation of state in ﬁgure 5-6 to ﬁt pressure proﬁles at lower
temperature since a wider ﬁtting range is available. This will allow us to obtain theequation of state for larger values of βµ. The result is shown in ﬁgure 5-7, in which
we determine the equation of state up to βµ ≃ 2.6.
F(m b)

12

10

8

6

eos from
4

previous figure

bm < 0.5
2

0
-3

-2

-1

0

1

2

m b

Figure 5-7: Equation of state F (βµ) of a unitary Fermi gas. Solid black line: Equationof state from ﬁgure 5-6 used for ﬁtting. Solid red line: Experimental equation of state
up to βµ ≃ 2.6 averaged over 2 proﬁles. Dashed black line: Equation of state of the
non-interacting Fermi gas shown for comparison.

Future Work According to the universality hypothesis, see chapter 5.1, when approaching the limit of zero temperature (T ≪ µ or βµ ≫ 1), the equation of state110

is given by a Fermi-distribution with a renormalized chemical potential µ = ξ ϵF ,see chapter 4.3.5. Therefore, for a given atom number, and g(βµ) ≡ −Li f (βµ),βµ)
3/2 (−elimβµ→∞ g(βµ) = ξ −3/2 . For the previously mentioned range in chapter 4.3.5 of thechemical potential, this results in 2.7 < ξ −3/2 < 5.5. Therefore, knowledge of theequation of state at low temperature will allow an accurate determination of the universal parameter ξ. From ﬁgure 5-7 it can be seen that g(βµ = 2.5) ≃ 2. Since thephase transition to the superﬂuid state is expected to occur at βµ ≃ 3, it will beinteresting to determine g(βµ) for lower temperatures using the above method.

111

Chapter 6
Conclusion and Outlook
Experimental studies of ultracold atomic Fermi gases have matured over the years.
Testament is given to this by the questions addressed in experiments - until a few
years ago, many of the open questions were of qualitative nature. Nowadays, manyquestions being addressed and waiting to be addressed are of quantitative nature,such as the precise value of the chemical potential, the superﬂuid gap or the contact
coeﬃcient, the critical interaction strength for phase transitions or the eﬀective massof a polaron.
The temperature dependence of these parameters presents a challenging problem fortheory and experiments alike. In fact, even the equation of state of a balanced unitary
Fermi mixture has only recently been addressed in experiments, and a precise andconvincing experimental and theoretical determination of the critical temperature forsuperﬂuidity remains elusive.
However, the results presented in this thesis dealt with the simplest non-trivial realization of an ultracold atomic Fermi gas: A mixture of two fermion species of equalmass in a single potential well characterized by s-wave interactions. But the ﬁeld
is moving forward fast and many new systems have recently been developed including ultracold atoms in lower dimensions and optical lattices, particles with p-wave
or higher order angular momentum interactions, formation of dipolar bosonic andfermionic molecules, three- and four-particle bound states, and Fermi mixtures withunequal masses.
The following sentence can be found at the end of many a thesis in the ﬁeld of ultracold atoms and is still true: Exciting times lie ahead of us...

112

Appendix A
Realization of a Strongly
Interacting Bose-Fermi Mixture
from a Two-Component Fermi Gas
This appendix contains a reprint of Ref. [80]: Yong-il Shin, André Schirotzek, Christian H. Schunck, and Wolfgang Ketterle, Realization of a Strongly Interacting BoseFermi Mixture from a Two-Component Fermi Gas, Phys. Rev. Lett. 101, 070404(2008).

113

PRL 101, 070404 (2008)

PHYSICAL REVIEW LETTERS

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Realization of a Strongly Interacting Bose-Fermi Mixture from a Two-Component Fermi Gas
Yong-il Shin,* André Schirotzek, Christian H. Schunck, and Wolfgang Ketterle
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA(Received 5 May 2008; published 14 August 2008)
We show the emergence of a strongly interacting Bose-Fermi mixture from a two-component Fermi
mixture with population imbalance. By analyzing in situ density profiles of 6 Li atoms in the BCS-BEC
crossover regime, we identify a critical interaction strength, beyond which all minority atoms pair up withmajority atoms and form a Bose condensate. This is the regime where the system can be effectively
described as a boson-fermion mixture. We determine the dimer-fermion and dimer-dimer scattering
lengths and beyond-mean-field contributions. Our study realizes a gedanken experiment of bosons
immersed in a Fermi sea of one of their constituents, revealing the composite nature of the bosons.
DOI: 10.1103/PhysRevLett.101.070404

PACS numbers: 03.75.Ss, 03.75.Hh, 67.60.Fp

Fermions are the fundamental building blocks of ordinary matter, whereas bosons emerge as composite particles. One of the simplest physical systems to study the
emergence of bosonic behavior is a two-component fermion mixture, where the composite boson is a dimer of thetwo different fermions. A dramatic manifestation of bosonic behavior is Bose-Einstein condensation, representing
the low-temperature phase of a gas of bosons. One way toreveal the composite nature of the bosons is to immerse
such a Bose-Einstein condensate (BEC) into a Fermi sea of
one of its constituents. The degeneracy pressure due to thePauli exclusion principle affects the structure of the composite boson, resulting in a zero-temperature quantum
phase transition to a normal state where Bose-Einsteincondensation is quenched.
In this Letter, we observe this transition experimentally.
We identify the regimes where a two-component Fermi gas
can be described as binary mixture of bosons and fermions,and where the composite nature of the boson becomes
essential. The validity of a Bose-Fermi (BF) description
requires that all minority fermions become bound as bosons and form a BEC. We determine the critical value of
1=kF" a for the onset of superfluid behavior in the limit oflarge population imbalance. Here, a is the fermion-fermion
scattering length, and kF" is the Fermi wave number characterizing the depth of the majority Fermi sea. Of course,
for an equal mixture, the zero-temperature ground state isalways a superfluid in the BEC-BCS crossover. It has beenshown previously that a crossover superfluid can bequenched by population imbalance, also called the
Chandrasekhar-Clogston (CC) limit of superfluidity [1,2].
In this work, we determine the critical point where superfluidity can no longer be quenched by population imbalance; i.e., the CC limit becomes 100%.
In the limit of a BF mixture [3], we observe repulsive
interactions between the fermion dimers and unpaired
fermions. They are parameterized by an effective dimerfermion scattering length of abf  1:23 3a. This value isin reasonable agreement with the exact value abf  1:18a
which has been predicted over 50 years ago for the three0031-9007=08=101(7)=070404(4)fermion problem [4], but has never been experimentallyconfirmed. The boson-boson interactions were found to bestronger than the mean-field prediction in agreement with
the Lee-Huang-Yang prediction [5].
The system for this study is a variable spin mixtureof the two lowest hyperfine states j"i and j#i of 6 Li
atoms (corresponding to the jF  1=2; mF  1=2i andjF  1=2; mF  ÿ1=2i states at low magnetic field) inan optical dipole trap as described in Refs. [1,2]. A broad
Feshbach resonance, located at 834 G [6], strongly enhances the interactions between the two spin states. The
final evaporative cooling was performed at 780 G by lowering the trap depth. Subsequently, the magnetic-bias field
B is adjusted to a target value with a ramp speed of
0:4 G=ms, changing the interaction strength adiabatically. At the end of the preparation, our sample was confined in an effective three-dimensional harmonic trap withcylindrical symmetry. The axial (radial) trap frequency was!z =2  22:8 Hz (!r =2  140 Hz).
The phase diagram for the fermion mixture was obtained
from the analysis of in situ density profiles of the majority(spin ") and minority (spin #) components. The profiles
were recorded using a phase-contrast imaging technique[2]. Under the local density approximation (LDA), lownoise column-density profiles were obtained by averaging
the optical signal along equipotential lines (refer to Ref. [2]for a full description of the image processing). For typicalconditions, the temperature of a sample was T=TF0 &0:05, determined from the outer region of the cloud [2],where TF0  1:0 K is the Fermi temperature of the majority component measured as kB TF0  m!2z R2" =2 (kB is
the Boltzmann’s constant, m is the atom mass, and R" is theaxial radius of the majority cloud).
Figure 1 displays density profiles of imbalanced Fermi
mixtures for various magnetic fields, showing how thespatial structure of a trapped sample evolves in the crossover regime. Near resonance, as reported in Ref. [2], three
distinctive spatial regions are identified: (I) a superfluid
core, (II) an intermediate region of a partially polarized
normal (Npp ) phase, and (III) a fully-polarized, outer wing.

070404-1

 2008 The American Physical Society

BEC

B=740 G
Normalized
column density

2.0

777 G

800 G

Resonance

BCS

834 G

851 G

a

b

c

d

e

f

g

h

i

j

FIG. 1 (color online). Density profilesof imbalanced Fermi mixtures in a harmonic trap. The top row (a)–(e) shows
the averaged column-density profiles forvarious magnetic fields [gray (green):majority, dark gray (blue): minority,
black: difference]. The black dotted line
is a zero-temperature Thomas-Fermi distribution fit to the majority wing (r >R# ). The middle row (f)–(j) and the bottom row (k)–(o) show the reconstructed
three-dimensional density distributions
and the spin polarizations obtained fromthe profiles in the top row. The criticalpolarizations c at the phase boundaryr  Rc are indicated by the right arrows.
The values for R" (in m), Rc =R" , andR# =R" were respectively: for (a), (f), (k),381, 0.33, 0.33; for (b), (g), (l), 380, 0.33,
0.33; for (c), (h), (m), 362, 0.35, 0.59; for(d), (i), (n), 371, 0.44, 0.72; for (e), (j),(o), 367, 0.41, 0.76. T=TF0 & 0:05, andTF0  1:0 K (see the text for definitions).

1.5
1.0
0.5

Normalized density

0.0
3.0
2.0
1.0

Spin polarization

0.0
1.0

k

l

m

n

o

0.8
0.6
0.4

σC

0.2
0.0
0

0.5

1 0

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PHYSICAL REVIEW LETTERS

PRL 101, 070404 (2008)

0.5

1 0

0.5

1 0

0.5

1 0

0.5

RC

R

1

Radial position (r/R )account for the observed profiles. Because of the external
trap potential, the local chemical potential varies form zeroat the edge of a cloud to a maximum value in the center.
Therefore, knowledge of the three-dimensional density
profiles of a single cloud is sufficient to obtain the equationof state [2,3,8,9,11].
For a zero-temperature mixture of bosonic dimers withdensity nb  n# and mass mb  2m and unpaired fermions
with density nf  n" ÿ n# and mass mf  m, the energy
density E can be decomposed as E  E bb  E bf  E f ,where E bb nb  and E bf nb ; nf  are the boson-boson and(b) 1.0

0.8

0.9

0.6

0.8

κ

(a) 1.0
Critical polarization

The core radius Rc was determined as the peak position in
the column-density difference profile and the majority(minority) radius R" (R# ) was determined from the fit of
the outer region, r > R" (r > Rc ) of the majority (minority)
column-density profile to a zero-temperature ThomasFermi distribution. The local spin polarization is definedas  r  n" ÿ n# = n"  n# , where n" and n# are thelocal majority and minority density, respectively.
Further on the BEC side, the sample has a more compressed superfluid core, a narrower intermediate normal
region (gray region in Fig. 1) and a higher critical spinpolarization at the phase boundary c   Rc . Eventually, when B < 780 G, there is no noticeable intermediate region, implying that every minority atom pairs up witha majority atom in the superfluid core. In Fig. 2, we determine the critical point for the disappearance of the partially polarized normal phase in two different ways. Figure 2(a) shows the phase diagram for the Npp phase in theplane of interaction strength 1=kF" a and spin polarization
. An extrapolation of the critical line to c  1 yields
1=kF";c a  0:74 4. Another implication of the absence ofan Npp phase is that the size of the minority cloud R# approaches the radius Rc of the superfluid core. This extrapolation is conveniently done using the dimensionless parameter   R2" ÿ R2# = R2" ÿ R2c  [7], resulting in a value of
1=kF";c a  0:71 5. These values are in good agreementwith recent quantum Monte Carlo (QMC) calculations[10].
The critical point marks the onset of the emergence of a
BF mixture from the two-component Fermi system. One
may suspect that near the critical point, the equation ofstate of the BF mixture is complex, but we show now that avery simple equation of state is sufficient to quantitatively

0.4

0.7

0.2

0.6
0.5

0.0
0.6

0.4

0.2

0

-0.2

0.6

0.4

0.2

0

-0.2

Interaction strength (1/kF↑a)FIG. 2 (color online). Emergence of a Bose-Fermi mixture in
the phase diagram for a two-component Fermi gas. (a) The
critical polarization c as a function of the interaction strength
1=kF" a at the phase boundary. The open circle indicates thepreviously measured critical value on resonance, c0  0:36 [2].
(b)   R2" ÿ R2# = R2" ÿ R2c . The solid (red) lines are (a) an
exponential fit and (b) a linear fit to the data points. c  1 and  1 (i.e., R#  Rc ) imply the absence of minority fermions in
the normal phase. Each data point consists of 9 to 23 independent measurements, and the error bars indicate only the statistical uncertainty.

070404-2

dE bb @E bf 1

 m !2 r2 ;dnb
@nb 2 b z

(2)

where f0 and b0 are the global chemical potentials of
fermions and bosons, respectively, referenced to the trap
bottom.
For the determination of the boson-fermion scattering
length abf , we use a mean-field expression for the bosonfermion interaction energy E bf;M  2@2 =mbf abf nb nf
with mbf  mb mf = mb  mf   2=3m. Since f0 
m!2z R2" =2, Eq. (1) gives the relation, abf r  f0 1 ÿ
3@2
r2 =R2"  ÿ n2=3
f = m nb . We obtained a value for abf
by averaging abf r over a mixed region (r < Rc withnb;f > 0:1n0 ). Here, n0 is the reference density defined asn0  f0 = 3=2 . In this analysis, the noninteracting outerwing provides absolute density calibration [9].
The scattering length ratio abf =a turns out to be almost
constant over the whole range, 700 G < B < 780 G, where
we could study BF mixtures [Fig. 3(a)]. For even lower
magnetic fields, severe heating occurred, probably due tomolecular relaxation processes. By averaging a total of 89
measurements, we obtain abf  1:23 3a, close to the exact value abf  1:18a calculated for the three-fermion
problem [4]. Our finding excludes the mean-field prediction abf  8=3a. The detailed behavior above 750 G
requires further investigation.
(a) 1.8(b) 1.5

1.5

1.2

We now turn to the determination of the boson-boson
scattering length abb which parameterizes the boson-boson
mean-field energy E bb;M  2@2 =mb abb n2b . For a given
abf , the effective potential for bosons in the presence offermions is Vb r  m!2z r2  3@2 =mabf nf r. Then,
Eq. (2) gives b0 ÿ 2@2 =mabb nb r  Vb r. By fitting
the data in the core region (0:1R" < r < Rc and nb >0:1n0 ) to this equation with b0 and abb as two freeparameters, we obtained a value for abb . We used the value
abf determined from the corresponding profiles.
The effective mean-field values for abb =a show a strong
increase by a factor of about 2, as the system approaches
the critical point [Fig. 3(b)]. We attribute this behavior tostrong boson-boson interactions causing non-negligible
quantum depletion in the BEC. In this regime, the equationof state has to include beyond-mean-field corrections, withthe leading term given by Lee, Huang, and Yang (LHY) [5]ass
2@2 abb n2b
128 a3bb nb:(3)1
E LHY 15
mb
Inclusion of the LHY correction leads to smaller fitted
values for abb =a, which are now almost constant over thewhole range of magnetic fields with an average value ofabb  0:55 1a. The exact value for weakly bound dimers
is abb  0:6a [13]. For kF;b a  1, the LHY correction is
0:3E bb;M , i.e., a 30% correction to the mean-field approximation. Here, kF;b  62 nb 1=3 . Recently, the LHY cor740 G

B=720 G

761 G

a

b

c

4

d

e

f

2

1/kF,ba=1

h

i

4

2

Normalized density

boson-fermion interaction energies, respectively, and E f 
3=5 n5=3
is the kinetic energy of fermions [ f
2 2=3 26  @ =2mf and @ is the Planck’s constant divided by
2]. Here, we assume that the effective mass of a fermion
in a dilute mixture is the same as its bare mass [12]. Under
the LDA, the densities nb r and nf r in the harmonic trap
should satisfy@E1 bf  mf !2z r2 ;
f0  n2=3
(1)
f
@nf
2
b0 

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PHYSICAL REVIEW LETTERS

PRL 101, 070404 (2008)0

abb / aabf / a

0

1.2

1

0.6

0

0.9
0.3

0

0.0
740

760

780

0.5

10

0.5

10

0.5

1

Radial position (r/R )

0.6
720

g

0.9

720

740

760

780

Magnetic field (G)FIG. 3 (color online). Characterization of a strongly interacting Bose-Fermi mixture. (a) The scattering length for dimerfermion interactions abf and (b) for dimer-dimer interactions abb
in units of the fermion-fermion scattering length a. Black solid
circles were determined using mean-field theory and open (red)circles including the LHY correction for a strongly interacting
Bose gas. The dashed lines indicate the calculated values abf 
1:18a [4] and abb  0:6a [13]. Each data point represents 7 to 17
measurements, and the error bars indicate only the statisticaluncertainty.

FIG. 4 (color online). Observed profiles of strongly interacting
Bose-Fermi mixtures compared to calculated profiles without
any adjustable parameter. (a)–(c) Density profiles of bosonicdimers [dark gray (blue)] and unpaired excess fermions [gray(green)] for various magnetic fields. The numerically obtained
density profiles (d)–(f) for bosons and (g)–(i) for fermion use
abb  0:6a and abf  1:18a (dashed line: mean-field description, solid line: including the LHY correction). The horizontal
dotted lines in (d)–(f) indicate the boson density corresponding
to 1=kF;b a  1. The values for R" (in m) and Rc =R" wererespectively: for (a), 393 and 0.29; for (b), 381 and 0.33; for (c),366 and 0.35.

070404-3

PHYSICAL REVIEW LETTERS

1.2

PRL 101, 070404 (2008)

1.0
0.6

0.8

1/2

(µ b / µ f)

0.5

0.4

0.0

740G

0.8

700 G

1.2

1.6

µf

2.0

2.4

Fermion concentration

1.0

2.8

-1/2

FIG. 5 (color online). Mean-field phase diagram of a dilute b are the chemical potentials for
Bose-Fermi mixture.  f and 2@2
fermions and bosons in units of 2m
62 n f 2=3 and 4@mbabb n b ,
f
a3

m3 m3

a2

m3 m2

bb
b f
b f
bb
 b  9
respectively, where n f  9
2 a6 m m 6 and n
4 a5 m m 5
bf

b

f

bf

b

f

(see Ref. [18]). The thick lines indicate the phase transitions(solid line: first-order, dashed line: second-order). The thin lines
represent typical cuts through the phase diagram realized in our
trapped samples.

rections have been observed via the upshift of collective
oscillation frequencies for a strongly interacting BEC [14].
Our results show that a two-component Fermi mixture
beyond the critical point can be effectively described as astrongly interacting BF mixture. In Fig. 4, we compare ourexperimental data with numerically obtained density profiles [15] without any adjustable parameter, showing excellent agreement. After including the LHY correction,
small discrepancies are visible only at the highest boson
densities exceeding kF;b a  1, where one would expect
unitarity corrections. It is surprising that we did not need
any beyond-mean-field corrections for the boson-fermion
interaction. Such corrections have been calculated for asystem of point bosons and fermions [16]. However, including them into our fit function degraded the quality of
the fit. Recent QMC simulations have shown that theequation of state of a polarized Fermi gas on the BEC
side is remarkably close to E  E LHY  E bf;M  E f withabb  0:6a and abf  1:18a down to 1=kF" a > 0:5 [10,17]
in agreement with our findings. It appears that the beyondmean-field term is offset by other corrections, possibly due
to the composite nature of the bosons. Further studies ofthis rich system could address beyond-mean-field terms,
characterize the breakdown of the BF description close to
the critical point, and look for finite temperature effects.
One motivation for the realization of BF mixtures is toextend studies of 3 He -4 He mixtures. With tunable interactions near Feshbach resonances, cold atom systems canaccess a wider regime of the phase diagram. Predicted phenomena include phase separation and miscibility [11,18],
boson-mediated, effective fermion-fermion coupling[12,19], and novel collective excitations [20,21]. The density profiles in Fig. 4 show a sharper boundary for higher
magnetic fields. This is consistent with Fig. 5 which predicts that in the same magnetic field range, the transition
from full miscibility to phase separation has taken place.

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An interacting BF system has been also realized in
Rb-40 K mixtures [22,23]. The 6 Li system studied here
has the advantages of using a single atomic species andmuch longer lifetimes of several seconds, but cannot accessattractive boson-fermion interactions.
In conclusion, a two-component Fermi gas with population imbalance is a realization of a long-lived stronglyinteracting BF mixture. This is a new BF system withtunable interactions. Furthermore, it offers intriguing possibilities to study the emergence of bosonic behavior froma mixture of fermions.
We thank A. Keshet for a critical reading of the manuscript. This work was supported by NSF, ONR, MURI,
and ARO Grant No. W911NF-07-1-0493 (DARPA OLE
Program).
87*yishin@mit.edu[1] M. W. Zwierlein et al., Science 311, 492 (2006); M. W.
Zwierlein et al., Nature (London) 442, 54 (2006); Y. Shinet al., Phys. Rev. Lett. 97, 030401 (2006).
[2] Y. Shin et al., Nature (London) 451, 689 (2008).
[3] D. E. Sheehy and L. Radzihovsky, Phys. Rev. Lett. 96,
060401 (2006); P. Pieri and G. C. Strinati, ibid. 96, 150404(2006); M. Iskin and C. A. R. Sá de Melo, Phys. Rev. A 77,
013625 (2008).
[4] G. V. Skorniakov and K. A. Ter-Martirosian, Zh. Eksp.
Teor. Fiz. 31, 775 (1956) [Sov. Phys. JETP 4, 648 (1957)].
[5] T. D. Lee, K. Huang, and C. N. Yang, Phys. Rev. 106, 1135
(1957).
[6] M. Bartenstein et al., Phys. Rev. Lett. 94, 103201 (2005).
[7]  is almost independent of the total population imbalancenear the resonance [8,9].
[8] F. Chevy, Phys. Rev. A 74, 063628 (2006).
[9] Y. Shin, Phys. Rev. A 77, 041603(R) (2008).
[10] S. Pilati and S. Giorgini, Phys. Rev. Lett. 100, 030401(2008).
[11] K. Mølmer, Phys. Rev. Lett. 80, 1804 (1998).
[12] J. Bardeen, G. Baym, and D. Pines, Phys. Rev. 156, 207(1967).
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Rev. Lett. 93, 090404 (2004).
[14] A. Altmeyer et al., Phys. Rev. Lett. 98, 040401 (2007).
[15] For a given E, the density profiles were numericallyobtained to be self-consistent with Eqs. (1) and (2).
[16] W. F. Saam, Ann. Phys. (N.Y.) 53, 239 (1969); A. P. Albus
et al., Phys. Rev. A 65, 053607 (2002); L. Viverit and
S. Giorgini, ibid. 66, 063604 (2002).
[17] N. Prokof’ev and B. Svistunov, Phys. Rev. B 77,
020408(R) (2008).
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070404-4

Appendix B
Phase diagram of a two-component
Fermi gas with resonantinteractions
This appendix contains a reprint of Ref. [19]: Yong-il Shin, Christian H. Schunck,
Andre Schirotzek and Wolfgang Ketterle, Phase diagram of a two-component Fermi
gas with resonant interactions, Nature 451, 689 (2008).

118

Vol 451 | 7 February 2008 | doi:10.1038/nature06473

LETTERS
Phase diagram of a two-component Fermi gas withresonant interactions
Yong-il Shin1, Christian H. Schunck1, André Schirotzek1 & Wolfgang Ketterle1

The pairing of fermions lies at the heart of superconductivity andsuperfluidity. The stability of these pairs determines the robustness of the superfluid state, and the quest for superconductors
with high critical temperature equates to a search for systems withstrong pairing mechanisms. Ultracold atomic Fermi gases present
a highly controllable model system for studying strongly interacting fermions1. Tunable interactions (through Feshbach collisional
resonances) and the control of population or mass imbalanceamong the spin components provide unique opportunities toinvestigate the stability of pairing2–4—and possibly to search forexotic forms of superfluidity5,6. A major controversy has surrounded the stability of superfluidity against an imbalance
between the two spin components when the fermions interact
resonantly (that is, at unitarity). Here we present the phase diagram of a spin-polarized Fermi gas of 6Li atoms at unitarity,
experimentally mapping out the superfluid phases versus temperature and density imbalance. Using tomographic techniques, wereveal spatial discontinuities in the spin polarization; this is thesignature of a first-order superfluid-to-normal phase transition,and disappears at a tricritical point where the nature of the phasetransition changes from first-order to second-order. At zero temperature, there is a quantum phase transition from a fully paired
superfluid to a partially polarized normal gas. These observations
and the implementation of an in situ ideal gas thermometer provide quantitative tests of theoretical calculations on the stability ofresonant superfluidity.
When the two spin components resonantly interact, the behaviourof the system becomes independent of the nature of the interactions.
This case of unitarity has become a benchmark for experimental andtheoretical studies over the last few years. However, there is an
ongoing debate about the stability of resonant superfluidity, reflected
in major discrepancies in predicted transition temperatures for thebalanced spin mixture7–9, and an even more dramatic discrepancy
for the critical imbalance of the two spin components, called the
Chandrasekhar–Clogston limit of superfluidity2,3. Recent quantum
Monte Carlo calculations predicted that superfluidity would bequenched by a density imbalance around 40% (ref. 10), whereas
other studies predicted a critical imbalance above 90% (refs 11–16). Our earlier work17–19 suggested the lower limit but other experiments20,21 were interpreted to be consistent with the absence of the
Chandrasekhar–Clogston limit. This huge discrepancy reveals thateven qualitative aspects, such as the role of interactions in the normal
phase, are still controversial. The lack of reliable thermometry forstrongly interacting systems limits the full interpretations of experimental results.
Here we resolve this long-standing debate by presenting the phase
diagram of a spin-polarized Fermi gas at unitarity. We observe thatthe normal-to-superfluid phase transition changes its nature. At low
temperature, the phase transition occurs with a jump in the spin1

polarization as the imbalance increases, which we interpret as afirst-order phase transition. The local spin polarization or local density imbalance is defined as s 5 (n" 2 n#)/(n" 1 n#), where " and #refer to the two spin components with densities n",#. At high temperature, the phase transition is smooth and therefore of second
order. The two regimes are connected by a tricritical point4,22 andwe estimate its position to be (stc, Ttc/TF") < (0.2, 0.07), wherekB TF: ~B2 (6p2 n: )2=3 =2m is the Fermi energy of the majority component of density n" (kB is the Boltzmann constant, " is the Planck
constant divided by 2p and m is the atomic mass of 6Li). Our lowtemperature results confirm a zero-temperature quantum phasetransition at a critical polarization sc0 < 36%.
This work required the introduction of several techniques. A
tomographic reconstruction of local Fermi temperatures and spinpolarization allowed us to obtain the phase diagram for the homogeneous system, no longer affected by the inhomogeneous density ofthe trapped samples. Furthermore, absolute temperatures wereobtained using in situ thermometry applied to the non-interacting
fully polarized Fermi gas in the outer part of the trapped samples,an ideal thermometer with exactly known thermal properties.
Unlike previous work18,23, this is a direct measurement without any
approximations.
Our experiments are carried out in a trapping potential V(r). The
local chemical potential of each spin component is given asm:,; (r)~m:0,;0 {V (r), where m"0,#0 are the global chemical potentials.
When m:0 =m;0 , owing to imbalanced populations, the chemical
potential ratio g(r)~m; =m: varies spatially over the trapped sample
and so, under the local density approximation, the trapped inhomogeneous sample is represented by a line in the phase diagrams of thehomogeneous system. Figure 1 illustrates the spatial structure of astrongly interacting Fermi mixture in a harmonic trap. In the inner
region, where g is closer to unity, a superfluid with zero (or small)spin polarization will form at zero (or low) temperatures, having a
sharp phase boundary against the partially polarized normal gas in
the outer region. The spin polarization shows a discontinuity at theboundary of the superfluid core at r 5 Rc, a signature of the phaseseparation of a superfluid and a normal gas24. The critical polarization sc ~ limz s(r) represents the minimum spin polarization for ar?Rcstable normal gas; ss ~ lim{ s(r) represents the maximum spinr?Rcpolarization for a stable superfluid gas. At higher temperatures, thediscontinuity in the density imbalance disappears. The main result of
this paper is the observation and quantitative analysis of such density
profiles. Because we have no experimental evidence, we are not discussing the exotic partially polarized phases25 which could exist onlyin the transition layer between the superfluid core and the normal
outer region.
We prepared a variable spin mixture of the two lowest hyperfine
states of 6Li atoms, labelled j:i and j;i, at a magnetic field of 833 G. A

Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics, MIT, Cambridge, Massachusetts, 02139, USA.

689
©2008 Nature Publishing GroupLETTERS

NATURE | Vol 451 | 7 February 2008

a

b

0
I

b
Density(arbitrary units)m↑0m↓0 + δm↓

II

III

1

1

0
Spin polarization

1

1

1

0.5

sc

0
ss

0
0

Rc

R↓

z
0

R↓ R↑

x

R↑

Radial position, rFigure 1 | Schematic of spatial structure of a strongly interacting Fermi gas
in a harmonic trap. a, A two-component (spin " and #) Fermi mixture isconfined in an external potential V(r) / r2 with the chemical potential m"0,#0of each spin component (dm# is the shift for the spin # component owing tointeractions). b, Density distributions of the majority component n"(r) (redline) and the minority component n#(r) (blue line). c, Spin polarizations(r) 5 (n" 2 n#)/(n" 1 n#). At zero temperature, the sample has a three-layer
radial structure: (I), the core region (0 # r , Rc) of a fully paired superfluidwith n" 5 n#; (II), the intermediate region (Rc , r , R#) of a partiallypolarized normal gas; and (III), the outer region (R# , r , R") of a fullypolarized normal gas. The critical polarization sc (or ss) is defined as theminimum (or maximum) spin polarization of the normal (or superfluid)
region. The non-interacting case is shown in the insets. The insets have thesame axes as the main figure.

c

0

-1

1

0

c

Column density (arbitrary units)V(r)the majority cloud, and the uncertainty of the final digit is indicatedby parentheses.
We determined temperature from the in situ majority wing profiles. The outer part of the majority component, forming a noninteracting Fermi gas, fulfils the definition of an ideal thermometer,
namely a substance with exactly understood properties in contact
with the target sample. This new in situ method avoids the modification of the ideal gas profile caused by the collision with the inner coreduring ballistic expansion (ref. 18, see Supplementary Information).
The outer part of the averaged column density difference profile(r . R#) was fitted to a finite temperature Fermi–Dirac distribution
in a harmonic trap (Fig. 4) and the relative temperature T 0 :T =TF0
was determined, where kB TF0 ~B2 (6p2 n0 )2=3 =2m is the Fermi energyof the non-interacting Fermi gas, which has the same density distribution in the outer region as the majority cloud (n0 is the central
density of the non-interacting Fermi gas at zero temperature). We
verified that anharmonicity of the trapping potential does not affectthe fitted temperature (see Methods).
The critical lines of the phase diagram of a homogeneous spinpolarized Fermi gas were obtained by determining the local temperature and spin polarization at the phase boundary. The local relative0
temperature Tlocal:T =TF: was derived from the local density n" (Rc)0according to T (Rc )~T =TF0 |(n0 =n: (Rc ))2=3 . Because we observe nojump in the majority density within our resolution, TF" is welldefined at the boundary. The critical polarizations sc and ss were
measured as sc 5 s(Rc) and ss 5 s(Rc 2 0.05R") (this criterion for ss
was more robust than a fitting procedure, but excludes the possibility
that ss will be equal to sc at high temperature. Therefore, the measured ss should be regarded as a lower bound for the polarization ofLinear density (arbitrary units)aPotential energy(arbitrary units)

broad Feshbach resonance at 834 G enhances the interactions
between the two spin states. Our sample was confined in a threedimensional harmonic trap with cylindrical symmetry. The in situ
density distributions of the majority (spin ") and minority (spin #)components were determined using a phase-contrast imaging technique19 (Fig. 2). We obtained the low-noise profiles ñ by averaging
the column density distribution along the equipotential line anddetermined the three-dimensional density profiles n(r) using theinverse Abel transformation of the column densities ñ(r) (seeMethods Summary). Most of our measurements were performed at
a total population imbalance of d < 50%, where d 5 (N" 2 N#)/(N" 1 N#) refers to the total numbers of atoms in the sample, N"
and N# of the spin " and # components, respectively.
Figure 3 displays the radial profiles of the densities n",#(r) and thecorresponding spin polarization s(r) for various temperatures. The
discontinuity in the spin polarization, clearly shown at very low
temperatures, demonstrates the phase separation of the inner superfluid of low polarization and the outer normal gas of high polarization. At low temperature, the core radius Rc is determined as the kink
(and/or peak) position in the column density difference profile. At
high temperature (but still in the superfluid regime), the discontinuity in s(r) disappears. At our lowest temperature, the radii of theminority cloud and the core region were measured as R# 5 0.73(1)R"and Rc 5 0.430(3)R" (at d 5 44(4)%), respectively, and these values
agree with recent theoretical calculations10,25 within the experimental
uncertainties due to the determination of d. Here, R" is the radius of

d

0
–400
6

–200
0
200
Axial position, z (µm)

400

e

3
0
–60

–30
0
30
Radial position, x (µm)

60

Figure 2 | Double in situ phase-contrast imaging of a trapped Fermi
mixture. Two phase-contrast images of one sample were taken using
different probe frequencies of the imaging beam, measuring the density
difference nd1 5 n" 2 n# (a) and the weighted density difference nd2 5 0.76
n" 2 1.43n# (b), respectively. The images show the two-dimensionalÐ~d1,2 (x,z): nd1,2 (r)dy,
distribution of the column density difference, nowing to the line-of-sight integration. The field of view for each image is
150 mm 3 820 mm. c, The distributions of the column density difference ñd1
(black line) and ñd2 (red line) along the central line (the dashed lines in a andb). TheÐprofiles of the integrated linearÐ density difference,d1,z : n~d1 (x,z)dx (d) and n~d1 (x,z)dz (e), show the identical flatd1,x : nn
top feature except scaling. The aspect ratio of the trapping potential was
l 5 6.15, the majority atom number was N" 5 5.9(5) 3 106, the population
imbalance was d 5 44(4)%, and the relative temperature was
T9 5 T/TF0 5 0.03(1) (see text for definitions).

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NATURE | Vol 451 | 7 February 2008

Figure 3 | Density profiles of trapped Fermi
mixtures with imbalanced populations. The top
row (a–d) shows the averaged column density
profiles for various temperatures (red, majority;
blue, minority; black, difference). The majority
radius R" was determined from the outer region(r . R#, where R# is the radius of the minority
cloud) of the majority profiles using a fit to azero-temperature Thomas–Fermi distribution(black dotted lines). The column densities arenormalized by the central value of the fitted
Thomas–Fermi distribution. The middle row(e–h) and the bottom row (i–l) show thereconstructed three-dimensional profiles and thespin polarization profiles s(r) corresponding to
the profiles in a–d. The core radius Rc wasdetermined as the peak (and/or kink) position in
the column density difference (only fora–c), indicated by the vertical dashed lines. The
two spin polarizations sc at r 5 Rc and ss at
r 5 Rc 2 0.05R" are marked by the right and left
arrows, respectively. The values for T9 , sc, Rc/R",
R" (in mm), N", d (in %) and l were, respectively:
for a, e and i, 0.03(1), 0.34, 0.43, 385,
5.9(5) 3 106, 44(4), 6.15; for b, f and j, 0.05(2),0.24, 0.39, 416, 1.0(1) 3 107, 48(4), 6.5; for c, g andk, 0.07(1), 0.21, 0.29, 443, 1.2(2) 3 107, 54(4), 6.5;
for d, h and l, 0.10(1), not determined, notdetermined (sr 5 0 5 0.15 and condensate
fraction 5 2(1)%), 398, 5.3(4) 3 106, 54(4), 7.7.

Normalized column density

1.5

Normalized density

Increasing temperature

1.5

a

b

c

d

e

f

g

h

1

0.5

0

1
0.5
0
Rc 0.5

0

1

0

0.5

1

0

0.5

1

0

k

j

i

0.5

1

l

0.6
0.4

sC

0.2
sS

0
0

0.2

0.4

0.6 0

0.2

0.4
0.6 0
0.2
Radial position, r/R↑

0.4

0.6

the superfluid at the boundary). The discontinuity in the spin polarization profile implies that there is a thermodynamically unstablewindow, ss , s , sc, leading to a first-order superfluid-to-normal
phase transition. As the temperature increases, the unstable region
reduces with decreasing sc and increasing ss. For high temperaturewhen the bimodal feature in the spin polarization profile disappears,
we recorded the condensate fraction as an indicator of superfluidity,
using the rapid field-ramp technique17. As the temperature decreases,
the condensate fraction gradually increases with a finite central
polarization19. Such smooth variations of the density profile andcondensate fraction across the phase transition are characteristic ofa second-order phase transition.
The phase diagram is characterized by three distinct points: thecritical temperature Tc0 for a balanced mixture, the critical polarization sc0 of a normal gas at zero temperature, and the tricritical point(stc, Ttc) at which the nature of the phase transition changes. Owing
to the lack of a predicted functional form for the phase transitionline in the s–T plane, we apply a linear fit to the measured criticalpoints, suggesting Tc0/TF" < 0.15, sc0 < 0.36 and (stc, Ttc/TF") <(0.20, 0.07). The value for sc0 agrees well with the prediction (fromthe quantum Monte Carlo calculation) of 0.39 (ref. 10). The extrapolation of the phase diagram to s 5 0 is tentative, because the in situ
thermometry could not be applied to small population imbalances
owing to the narrowness of the non-interacting outer region.
The Chandrasekhar–Clogston limit reflects the energetic competition between a superfluid state and a partially polarized normal state,
and occurs at a critical value of 2hc for the chemical potential differtheory, which isence dm~m: {m; . In Bardeen–Cooper–Schrieffer
pffiffivalid for weak interactions, hc ~D= 2 (ref. 3). Here, D is the pairinggap. With the assumption of no interactions in a normal gas,quantum Monte Carlo studies predict hc ~1:00(5)D <1:2m at unitarity11, where m~(m: zm; )=2. The condition m;c ~m{hc v0
requires n# 5 0 for a non-interacting normal gas, implying theabsence of a partially polarized normal phase and consequently
sc0 5 100%. Mean-field approaches12–16, which cannot treat theinteractions in the normal phase accurately, also predict a high critical imbalance sc0 . 90%. Strong interactions in the normal phase,

0

0.2

0.4

0.6

however, have been observed through the compressed shape ofthe minority cloud18 and the shift in the radio frequency excitation
spectrum26. The data in Fig. 5 clearly establish a zero-temperature
Chandrasekhar–Clogston limit for sc0 in the range 30% to 40%. By
analysing the in situ density profiles25,27, we obtained hc <0:95m (seeMethods). Since theory clearly predicts m , D at unitarity9,11, we
have hc , D. If hc were larger than D, polarized quasi-particles
would have negative energies and would already form at zero temperature. Therefore, up to our observed value of hc, the fully pairedaAveraged column density difference (arbitrary units)

Spin polarization

0.8

1

0
0

200

400

R↓

b
1

0
0

200

R↓
Radial position (µm)

400

Figure 4 | Temperature determination using in situ density profiles. The
relative temperature T9 5 T/TF0 (see text for definition) was determinedfrom the outer region (r . R#) of the averaged column density difference
profile (black line) fitted to a finite temperature Fermi–Dirac distribution(red line). The radius of the minority cloud R# was determined from a fit of
the wing profile of the minority component (black dashed line) to a zerotemperature Thomas–Fermi distribution (red dashed line). a, T9 5 0.03(1)and d 5 44(4)%. b, T9 5 0.08(1) and d 5 46(4)%.
691©2008 Nature Publishing GroupLETTERS

NATURE | Vol 451 | 7 February 2008

superfluid state is stable, and a polarized superfluid exists only at
finite temperature.
The interface between two immiscible fluids involves a surface
energy, leading to at least a small violation of the local density
approximation. However, the observed sharp interface along anequipotential line and the flat-top structure of the linear density
difference profiles (Fig. 2d and e) imply that corrections to the local
density approximation are smaller than the resolution of our experiment. These observations are inconsistent with the interpretations
given for the experimental results reported in refs 20 and 21, where it
has been shown that highly elongated small samples are deformed bysurface tension28,29. The scaling of those surface effects to our parameters predicted a deviation of the aspect ratio of the superfluid coreof about 15% from the trap aspect ratio29, whereas our data gives anupper bound of 2%. We note that surface tension would add energy
in the phase-separated superfluid regime and would shift the
Chandrasekhar–Clogston limit to smaller values. Refs 20 and 21
concluded that the Chandrasekhar–Clogston limit should be
dc0 . 95%, which is ruled out by our observations. We are not awareof any suggested effect that can reconcile the data of refs 20 and 21
with our phase diagram for a resonant superfluid. To identify thisref. 70.2
ref. 23

NormalTemperature, T/TF↑

ref. 9ref. 80.1

Tricritical point

Superfluid

finite size effect and to understand fully the nature of the normal
state26, more work on imbalanced Fermi gases is needed.
In conclusion, we have established the phase diagram of a homogeneous spin-polarized Fermi gas with resonant interactions in thes–T plane. This includes the identification of a tricritical point at
which the critical lines for first-order and second-order phase transitions meet, and the final confirmation of a zero-temperature
quantum phase transition—the Chandrasekhar–Clogston limit ofsuperfluidity—for a gas at unitarity. So far, predicted exotic superfluid states such as the breached-pair state in a stronger coupling
regime (Bose–Einstein condensate side)13 and the Fulde–Ferrell–
Larkin–Ovchinnikov state in a weaker coupling regime (Bardeen–
Cooper–Schrieffer side)5,6,12,16,30 have not been observed, but thenovel methods of tomography and thermometry will be important
tools in the search for those states.
METHODS SUMMARY
The experimental procedure has been described in our previous publications17–19. A degenerate Fermi gas of 6Li atoms was first prepared in an opticaltrap, using laser cooling and sympathetic cooling with 23Na atoms. A variable
spin mixture of the two lowest hyperfine states j:i and j;i (corresponding to the
jF~1=2, mF ~1=2i and jF~1=2, mF ~{1=2i states at low magnetic field) was
created at a magnetic field B 5 885 G. The final evaporative cooling was achieved
by lowering the trap depth and all measurements were performed at B 5 833 G.
The temperature of the cloud was controlled by the lowest value of the trap depth
in the evaporative cooling process. The oscillation frequency in the axial direction was fz 5 23 Hz. The two transverse oscillation frequencies fr are equal towithin less than 2%. Two phase-contrast images of the same sample were taken
consecutively with different probe frequencies, n1 and n2 (Fig. 2). The timeinterval between the two images was 10 ms, and the pulse duration of each probe
beam was 15 ms. Because the probe beam was off-resonant, no heating effect of
the first pulse was observed in the second image. The trapped sample wasobserved to have an elliptical shell structure of the same aspect ratio l 5 fr/fz
as the trapping potential over our entire temperature range, and we obtained the
low-noise profiles ñ by averaging the column density distribution along theequipotential line defined as l2 x 2 zz 2 ~r 2 for a given radial position r. The
region for averaging was restricted depending on the type of analysis.
Deviations from the trap aspect ratio were only found for the outer thermal
wings. Details of the phase-contrast imaging technique and the data analysis are
given in Methods and Supplementary Information.
Full Methods and any associated references are available in the online version of
the paper at www.nature.com/nature.
Received 19 September; accepted 8 November 2007.
1.

Unstableref. 10

0.0

2.
3.

0.0

0.1

0.2
Spin polarization

0.3

0.4

4.

Figure 5 | The s–T phase diagram for a homogeneous spin-polarized Fermi
gas with resonant interactions. The critical polarizations sc (black solid
circles and square) and ss (grey solid circles) are displayed along the local
T/TF" at the phase boundary. The yellow area (ss , s , sc) represents athermodynamically unstable region, leading to the phase separation. Above
the tricritical point, the phase transition in the centre of the cloud wasobserved by the onset of pair condensation. For this, a cloud wasevaporatively cooled, until it crossed the phase transition on a trajectoryalmost perpendicular to the phase transition line (see Supplementary
Information). The critical spin polarization and temperature were obtainedby interpolating between points without and with small condensates (blacksolid square). The linear fit to the sc values is shown as a guide to the eye forthe normal-to-superfluid phase transition line. Each data point consists offive independent measurements and error bars indicate standard deviation.
The blue open symbols show theoretical predictions for the criticaltemperature of a homogeneous equal mixture7–9 and the critical polarization
at zero temperature10. The blue solid psquareis the measured critical
ffiffitemperature of ref. 23, multiplied by j with j 5 0.42 (ref. 11) to obtain
local T/TF at the centre. Finite temperature correction may increase theeffective value of j.

5.
6.
7.
8.

9.
10.
11.
12.
13.
14.

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21. Partridge, G. B., Li, W., Karmar, R. I., Liao, Y. & Hulet, R. G. Deformation of a trapped
Fermi gas with unequal spin populations. Phys. Rev. Lett. 97, 190407 (2006).
22. Griffiths, R. B. Thermodynamics near the two-fluid critical mixing point in He3He4. Phys. Rev. Lett. 24, 715–717 (1970).
23. Luo, L., Clancy, B., Joseph, J., Kinast, J. & Thomas, J. E. Measurement of theentropy and critical temperature of a strongly interacting Fermi gas. Phys. Rev.
Lett. 98, 080402 (2007).
24. Bedaque, P. F., Caldas, H. & Rupak, G. Phase separation in asymmetrical fermion
superfluids. Phys. Rev. Lett. 91, 247002 (2003).

25. Bulgac, A. & Forbes, M. M. Zero-temperature thermodynamics of asymmetricFermi gases at unitarity. Phys. Rev. A 75, 031605(R) (2007).
26. Schunck, C. H., Shin, Y., Schirotzek, A., Zwierlein, M. W. & Ketterle, W. Pairingwithout superfluidity: the ground state of an imbalanced Fermi mixture. Science
316, 867–870 (2007).
27. Chevy, F. Universal phase diagram of a strongly interacting Fermi gas with
unbalanced spin populations. Phys. Rev. A 74, 063628 (2006).
28. De Silva, T. N. & Mueller, E. J. Surface tension in unitary Fermi gases withpopulation imbalance. Phys. Rev. Lett. 97, 070402 (2006).
29. Haque, M. & Stoof, H. T. C. Trapped fermionic clouds distorted from the trap
shape due to many-body effects. Phys. Rev. Lett. 98, 260406 (2006).
30. Machida, K., Mizushima, T. & Ichioka, M. Generic phase diagram offermion superfluids with population imbalance. Phys. Rev. Lett. 97, 120407(2006).

Supplementary Information is linked to the online version of the paper at
www.nature.com/nature.
Acknowledgements We thank M. W. Zwierlein and A. Keshet for a critical readingof the manuscript. This work was supported by NSF, ONR, MURI and DARPA.
Author Information Reprints and permissions information is available atwww.nature.com/reprints. Correspondence and requests for materials should be
addressed to Y.S. (yishin@mit.edu).

693©2008 Nature Publishing Groupdoi:10.1038/nature06473

METHODS
Phase-contrast imaging. The optical signal in the phase-contrast imaging is
proportional to the net phase shift of the imaging beam passing through a
Fermi mixture, that is, it is proportional to n: =(n{n0: ){n; =(n{n0; ), where n is
the probe frequency of the imaging beam, and n"0 and n#0 are the resonancefrequencies of the optical transition for the states j:i and j;i, respectively. Whenthe probe beam is tuned to the middle of the two transitions, that is, ton~n0 ~(n0: zn0; )=2, the optical signal reflects the density difference nd 5 n" 2 n#.
In our experiment, two phase-contrast images of the same sample were taken
consecutively with different probe frequencies, n1 and n2 (Fig. 2). The two images
record the density difference nd1 5 n" 2 n# and the weighted density differencend2 5 a"n" 2 a#n#. The first probe frequency n1 was determined by zeroing theoptical signal with an equal mixture and a",# was determined by the signal ratio
between the first and the second image for a highly imbalanced Fermi mixture
with jdj . 95% (an almost fully polarized gas). Finally, we obtainedn" 5 (a#nd1 2 nd2)/(a# 2 a") and n# 5 (a"nd1 2 nd2)/(a# 2 a"). The difference
between n1 and n2 was chosen to lie between 8 and 13 MHz.
Data analysis. Low-noise profiles were obtained by averaging the column density distribution of phase-contrast images along elliptical equipotential lines(l2 x 2 zz 2 ~r 2 ). For the measurement of the critical spin polarization, the averaging region was restricted to jxj , 12 mm to preserve the sharp features at thephase boundary. The diffraction limit for our imaging system was about 2 mm.
For the determination of local quantities in the profiles, we averaged over 65 mm
around a given position. For temperature determination, the averaging region
was restricted to an axial sector of 660u to avoid corrections due to transverse
anharmonicities (see below). The relative temperature T9 is determined as
T 0 :T =TF0 ~({6Li3 ({f)){1=3 , where f is the fugacity obtained from the fit?
Pz k =k s is the polylogarithmic function of order s).
(Lis (z):k~1Anharmonicity of the trapping potential. For the determination of temperatures from the spatial in situ profiles it was necessary to address the anharmonicity of the trapping potential. Our trap is generated by a weakly focused (beamwaist w < 125 mm) infrared gaussian laser beam (wavelength 1,064 nm) near thesaddle point of a magnetic potential. The total trapping potential is given as




2r2
m(2pfz )2r2{ zz 2 ,V (r,z)~U0 exp { 2 z2w2where r2 ~x 2 zy 2 . We note that gravity has been compensated by a magnetic
field gradient. The axial confinement comes mainly from the magnetic potential

with oscillation frequency of fz 5 23 Hz. The transverse magnetic potential isanti-trapping and limits the trap depth according to:"!#
2fr2 zfz2
1
f2
ð1Þ
U ~ m(2pfr )2 w 2 1{ z 2 ln
fz2
4
2fr
where fr is the transverse oscillation frequency in the central harmonic region.
When the trap depth is comparable to the Fermi energy of a sample, the transverse anharmonicity will affect the shape of the cloud. Although in our experiments the inner core and the outer cloud had the same aspect ratio as thetrapping potential, anharmonicities were not negligible in the spatial wings used
to determine the temperature.
This issue was addressed by adjusting the angular averaging region(Supplementary Fig. 3). Because the trapping potential is only anharmonic forlarge r, we could reduce the effect by decreasing the angle of the averaging sector
around the axial z-direction. Both the experimental data and an exact simulation
for an ideal Fermi gas show that the fitted temperature remains almost constant
up to a certain angle and then increases when the averaging sector includes moreof the transverse outer region. In our temperature determination, we chose theaveraging sector to be 660u, which was large enough to create low-noise profiles,
but kept the effect of the anharmonicities to below 10%. The one-dimensional fit
to angularly averaged profiles was computationally more efficient than a twodimensional fit to a selected region of the image. In a two-dimensional fit, one
could also include anharmonic terms in the fitting function.
Critical chemical potential ratio gc. In a harmonic trap, the chemical potentialof the majority and minority components are given as m: (r)~m:0 (1{r 2 =R:2 ) andm; (r)~m:0 (g0 {r 2 =R:2 ), respectively. At unitarity, the global chemical potentialof a fully paired superfluid in the core is given as ms0 ~jeF ~jB2 (6p2 ns0 )2=3 =2m
where ns0 5 n",#(r 5 0) is the central density. The thermodynamic equilibrium
requiresms0 ~(m:0 zm;0 )=2,
where
m:0 ~B2 (6p2 n0 )2=3 =2m.
From
ms0 =m:0 ~j(ns0 =n0 )2=3 , we obtain the chemical potential ratio as:g(r)~g0 {r 2 =R:2
1{r 2 =R:2~2j(ns0 =n0 )2=3 {1z1
1{r 2 =R:2

In our coldest sample (d < 44%), the normalized central density and theradius for the phase boundary were measured to be ns0/n0 5 1.72(4) and
Rc/R" 5 0.430(3), respectively, yielding the critical chemical potential ratio
gc 5 g(Rc) < 0.03 with j 5 0.42 (ref. 11). The critical difference is
hc =m~(1{gc )=(1zgc )<0:95.

©2008 Nature Publishing Group[f`1('%('*/&eXkli\'6473

JLGGC<D<EK8IP@E=FID8K@FE
Phase diagram of a two-component Fermi gas withresonant interactions
Yong-il Shin, Christian H. Schunck, André Schirotzek & Wolfgang Ketterle

Supplementary Information
Supplementary Figures

Supplementary Figure 1: (Color online) Trajectory of the center of a cloud in the phase
diagram during the cooling process. Above the tricritical point, the normal-to-superfluid
phase transition was observed by the onset of pair condensation in the evaporative coolingprocess. The local spin polarization and temperature at the center of the cloud was measured(black solid (open) circles with (without) condensate fraction), and the critical point was
obtained by linearly interpolating with the condensate fraction. The dashed-dot line shows a
guide line for the trajectory of the cloud center. The population imbalance of the sample was! 55%. A non-interacting mixture with this imbalance has a spin polarization " ! 30% at
the center at zero temperature.

nnn%eXkli\%Zfd&eXkli\(

[f`1('%('*/&eXkli\'6473

JLGGC<D<EK8IP@E=FID8K@FE

Supplementary Figure 2: (Color online) Expansion of a population-imbalanced Fermi
mixture. The absorption images of (a, d) the majority and (b, e) minority components weretaken along (a, b) the axial z and (d, e) transverse y directions after expansion. (c) The
azimuthally averaged column density profiles of the majority (red) and the minority (blue)
cloud are obtained from (a) and (b), respectively. The excess majority atoms in the outerregion interact with the core during expansion. The contour lines of the outer part of the
majority cloud (color inset) are not elliptical and have the shape of a horse-track. This showsthat the minority cloud pushes the outer majority atoms in the transverse direction, which isalso indicated by the hump of the majority profile at the edge of the minority cloud. The
population imbalance was ! 55%

Supplementary Figure 3: (Color online) The temperature of the cloud was determined forvarious angles " of the averaging sector. For a large angle, the large-x region is included in
the averaged profile, resulting in a broadening of the spatial wings and consequently higher
value of the fitted temperature. The red line shows the results of a simulation using the same
parameters as the experiment (# = f$/fz = 6.15, TF0 = 1 µK and the trap depth U / kB = 2 µK).
nnn%eXkli\%Zfd&eXkli\)

[f`1('%('*/&eXkli\'6473

JLGGC<D<EK8IP@E=FID8K@FE

Supplementary Discussion
Thermometry of ultracold Fermi gases
In our previous work1,2, temperatures have been determined by fitting the spatial wings of the
majority component after expansion. However, we found that one can neglect collisions withthe minority atoms in the core only for large population imbalances. In a simplified picture,
one can regard collisions with the inner core as collisions with a moving wall, which moves
outward radially and inward axially (due to the magnetic trapping potential). This results indifferent average kinetic energies (transversely and axially) of the free majority atoms in theouter region. Figure S2 shows the density distribution of the majority and minority
components after expansion. Although the temperature has been overestimated by only 20%
for typical experimental conditions ( ! 60%) in refs 1,2, we do not regard this technique as
well-calibrated absolute thermometry.
One other concept for thermometry determines temperature as the derivative of entropy
with energy. So far, this concept could be implemented only for balanced fermion mixtureswith certain approximations, and due to the need of determining a derivative, could only be
used to obtain temperatures averaged over a certain range3.
Polarized superfluid at finite temperature
When the two spin components have a chemical potential difference 2h, the BCS-type
superfluid has two branches of quasiparticles with excitation energies(% k # $ ) 2 " !2hwhere % k & 2 k 2 / 2m . At finite temperature, the superfluid is polarized due to the large
thermal population of the lower branch compared to the upper branch. An interesting
situation arises when h becomes larger than ", i.e. the lower branch has negative energy
quasiparticles, implying that even at zero temperature the superfluid state would have a finite
polarization. Our experiments show hc < " at very low temperature, suggesting that apolarized superfluid state exists only at finite temperature. The breached-pair state with hc >" at zero temperature has been predicted in a stronger coupling regime (on the BEC side of
the Feshbach resonance). Since " gradually decreases with higher temperature, it might bepossible to have hc > " at finite temperature, at least in the weakly-interacting BCS limit
where " smoothly approaches zero at a second order phase transition point. One interesting
problem is identifying this gapless region of h > " in the phase diagram for various coupling
regimes.

Supplementary Notes
1. Zwierlein, M. W., Schunck, C. H., Schirotzek, A. & Ketterle, W. Direct Observation of
the superfluid phase transition in ultracold Fermi gases. Nature 442, 54-58 (2006).
2. Shin, Y., Zwierlein, M. W., Schunck, C. H., Schirotzek A. & Ketterle, W. Observation ofphase Separation in a strongly interacting imbalanced Fermi gas. Phys. Rev. Lett. 97,
030401 (2006).
3. Luo, L., Clancy, B., Joseph, J., Kinast, J. & Thomas, J. E. Measurement of the entropy
and critical temperature of a strongly interacting Fermi gas. Phys. Rev. Lett. 98, 080402(2007).
nnn%eXkli\%Zfd&eXkli\*

Appendix C
Observation of Fermi Polarons in a
Tunable Fermi Liquid of Ultracold
Atoms
This appendix contains a reprint of Ref. [119]: Andre Schirotzek, Cheng-Hsun Wu,
Ariel Sommer, and Martin W. Zwierlein, Observation of Fermi Polarons in a TunableFermi Liquid of Ultracold Atoms, Phys. Rev. Lett. 102, 230402 (2009).

128

PRL 102, 230402 (2009)

Selected for a Viewpoint in Physics
PHYSICAL REVIEW LETTERS

week ending
12 JUNE 2009

Observation of Fermi Polarons in a Tunable Fermi Liquid of Ultracold Atoms
André Schirotzek, Cheng-Hsun Wu, Ariel Sommer, and Martin W. Zwierlein
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA(Received 17 February 2009; revised manuscript received 9 April 2009; published 8 June 2009)
We have observed Fermi polarons, dressed spin-down impurities in a spin-up Fermi sea of ultracold
atoms. The polaron manifests itself as a narrow peak in the impurities’ rf spectrum that emerges from a
broad incoherent background. We determine the polaron energy and the quasiparticle residue for various
interaction strengths around a Feshbach resonance. At a critical interaction, we observe the transition frompolaronic to molecular binding. Here, the imbalanced Fermi liquid undergoes a phase transition into a
Bose liquid, coexisting with a Fermi sea.
DOI: 10.1103/PhysRevLett.102.230402

PACS numbers: 05.30.Fk, 03.75.Ss, 32.30.Bv, 67.60.Fp

The fate of a single impurity interacting with its environment determines the low-temperature behavior of many
condensed matter systems. A well-known example is given
by an electron moving in a crystal lattice, displacing
nearby ions and thus creating a localized polarization.
The electron, together with its surrounding cloud of lattice
distortions, phonons, forms the lattice polaron [1]. It is aquasiparticle with an energy and mass that differ from thatof the bare electron. Polarons are central to the understanding of colossal magnetoresistance materials [2], andthey affect the spectral function of cuprates, the parent
material of high-TC superconductors [3]. Another famous
impurity problem is the Kondo effect, where immobile spin
impurities give rise to an enhanced resistance in metals
below the Kondo temperature [4]. In contrast to the electron moving in a phonon bath, a bosonic environment, inthe latter case the impurity interacts with a fermionicenvironment, the Fermi sea of electrons.
Here we study a small concentration of spin-down impurities immersed in a spin-up Fermi sea of ultracold
atoms. This system represents the limiting case of spinimbalanced Fermi gases and has been recognized to holdthe key to the quantitative understanding of the phase
diagram of imbalanced Fermi mixtures [5–16]. Unlike inliquid 3 He, the s-wave interaction potential between the
impurities and the spin-up atoms in this novel spinimbalanced Fermi liquid is attractive. The vicinity of a
Feshbach resonance allows tuning of the interaction
strength at will, characterized by the ratio of the interparticle distance 1=kF to the scattering length a, where kF is
the spin-up Fermi wave vector [17]. Figure 1 depicts the
scenario for a single impurity: For weak attraction(1=kF a ! "1) the impurity propagates freely in the
spin-up medium of density n" ¼ k3F =6(2 [Fig. 1(a)]. It
merely experiences the familiar attractive mean field energy shift E# ¼ 4(@2 an" =m < 0. However, as the attractive
interaction grows, the impurity can undergo momentum
changing collisions with environment atoms, and thus
starts to attract its surroundings. The impurity ‘‘dressed’’with the localized cloud of scattered fermions constitutes
0031-9007=09=102(23)=230402(4)the Fermi polaron [Fig. 1(b)]. Dressing becomes important
once the mean free path 1=n" a2 of the bare impurity inthe medium becomes comparable to the distance 1=kF
between environment particles or when ðkF aÞ2 1.
Collisions then reduce the bare impurity’s probability offree propagation, the quasiparticle residue Z, from unity.
The dressed impurity can instead move freely through theenvironment, with an energy E# shifted away from thesimple mean field result. This polaronic state is stable until,
for strong attraction (1=kF a 1), equivalent to a deep
effective potential well, the spin-down impurity will bindexactly one spin-up atom, thus forming a tightly boundmolecule [Fig. 1(c)]. This molecule is itself a dressed
impurity, albeit a bosonic one [13].
To prepare and observe Fermi polarons, we start with aspin-polarized cloud of 6 Li atoms in the lowest hyperfine
state j1i (spin-up), confined in a cylindrically symmetric
optical trap (125 -m waist, 145 Hz=22:3 Hz radial/axialtrapping frequency) at a magnetic field of 690 G [17]. A
two-photon Landau-Zener sweep transfers a small fraction
into state j3i (spin-down), and further cooling results in acloud containing 2% j3i impurities immersed in a degenerate Fermi gas of 5 * 106 j1i atoms at a temperature T ¼
0:14ð3ÞTF , where TF is the Fermi temperature. A 100 G
wide Feshbach resonance for scattering between thesestates is centered at 690 G. For various fields around the

FIG. 1 (color online). From polarons to molecules. (a) For
weak attraction, an impurity (blue) experiences the mean fieldof the medium (red). (b) For stronger attraction, the impurity
surrounds itself with a localized cloud of environment atoms,
forming a polaron. (c) For strong attraction, molecules of size aform despite Pauli blocking of momenta @k < @kF ! @=a by theenvironment.

230402-1

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PRL 102, 230402 (2009)

1% in units of the Fermi energy. Indeed, the width of thepolaron peak is consistent with a delta function within theexperimental resolution, as calibrated by the spectra offully polarized clouds. The background is perfectly
matched by the rf spectrum of the environment. This is
expected at high rf energies @! ) !F that are probing highmomenta k ) kF and thus distances short compared to theinterparticle spacing. Here, an impurity particle will interact with only one environment particle, leading to overlapping spectra.
Chevy has provided an instructive variational wave
function [5,9] that captures the essential properties of thepolaron, even on a quantitative level [16] when comparedwith Monte Carlo (MC) calculations [6,12,13]:

resonance, we perform rf spectroscopy on the impurity
species j3i and on the environment particles in j1i bytransferring atoms into the empty state j2i, accessible toeither hyperfine state. This state is sufficiently weaklyinteracting with the initial states to allow a direct interpretation of the resulting spectra [18]. As in previous work,
spectra are spatially resolved and tomographically 3D
reconstructed [19] via an inverse Abel transform, and arethus local and free from broadening due to density inhomogeneities. In addition, phase contrast images yield thein situ density distribution n" , n# and thus the local Fermi
energy !F of the environment atoms and the local impuritynconcentration x ¼ n#" . The Rabi frequencies #R for the
impurity and environment rf transitions are measured (onfully polarized samples) to be identical to within 5%.
Figure 2 shows the observed spectra of the spin-down
impurities and that of the spin-up environment at low local
impurity concentration. The bulk of the environment spectrum is found at zero offset, corresponding to the free
(Zeeman plus hyperfine) energy splitting between states
j1i and j2i. However, interactions between impurity andspin-up particles lead to a spectral contribution that is
shifted: The rf photon must supply additional energy totransfer a particle out of its attractive environment into thefinal, noninteracting state [17]. In Fig. 2(a), impurity andenvironment spectra above zero offset exactly overlap,
signaling two-body molecular pairing. The steep thresholdgives the binding energy, the high-frequency wings arise
from molecule dissociation into remnants with nonzero
momentum [17,20,21]. As the attractive interaction is reduced, however, a narrow peak appears in the impurity
spectrum that is not matched by the response of the environment [Figs. 2(b)–2(d)]. This narrow peak, emerging
from a broad incoherent background, signals the formationof the Fermi polaron, a long-lived quasiparticle. The narrow width and long lifetime are expected: At zero temperature the zero momentum polaron has no phase spacefor decay and is stable. At finite kinetic energy or finite
temperature T it may decay into particle-hole excitations[13], but phase space restrictions due to the spin-up Fermi
sea and conservation laws imply a decay rate / ðT=TF Þ2 (’kq cyk" cq" jq , ki# jFSi" (1)jqj<kF <jkj

The first part describes a single impurity with a welldefined wave vector (k# ¼ 0) that is not localized and free
to propagate in the Fermi sea of up spins jFSi" . In thesecond part the impurity particle recoils off environment
particles that are scattered out of the Fermi sea and leaveholes behind. This describes the dressing of the impurity
with particle-hole excitations. The probability of freepropagation is given by the first, unperturbed part, Z ¼
j’0 j2 . According to Fermi’s golden rule [11,15,17,22], thetwo portions of j%i give rise to two distinct features of the
impurity rf spectrum 'ð!Þ (! is the rf offset from the bareatomic transition):'ð!Þ ¼ 2.@#2R Z/ð@! þ E# Þ þ 'inc ð!Þ:(2)

The first part in j%i contributes a coherent narrow quasiparticle peak to the minority spectrum. Its position is adirect measure of the polaron energy E# , its integral gives
the quasiparticle residue Z. The particle-hole excitations inthe secondPpart give rise to a broad, incoherent background'inc ð!Þ / q;k j’qk j2 /ð@! , !q,k , !k þ !q þ E# Þ: The
polaron energy E# is released as the impurity at momentum
q , k is transferred into the final state, leaving behind an

b)

c)

d)

atom transfer / a.u.

a)X

j%i ¼ ’0 j0i# jFSi" þ

0

2

4

6

0

1

2

3

4

5

6

0

1

2

3

4

0

1

2

3

4

rf offset / εF

FIG. 2 (color online). rf spectroscopy on polarons. Shown are spatially resolved, 3D reconstructed rf spectra of the environment(blue, state j1i) and impurity (red, state j3i) component in a highly imbalanced spin-mixture. (a) Molecular limit; (b),(c) Emergence ofthe polaron, a distinct peak exclusively in the minority component. (d) At unitarity, the polaron peak is the dominant feature in the
impurity spectrum, which becomes even more pronounced for 1=kF a < 0 (not shown). For the spectra shown as dashed lines in (d) theroles of states j1i and j3i are exchanged. The local impurity concentration was x ¼ 5ð2Þ% for all spectra, the interaction strengths
1=kF a were (a) 0.76(2), (b) 0.43(1), (c) 0.20(1), and (d) 0 (unitarity).

230402-2

environment particle in k above and a hole within the
Fermi sea at q [22]. These two spectral features are recovered in theoretical rf spectra for a finite number of impurities, i.e., a Fermi liquid [15,23]. For our analysis we do
not rely on a theoretical fit to the spectra.
To measure the polaron energy E# , we determine thepeak position of the impurity spectrum as a function of thelocal interaction parameter 1=kF a. The data for 5% impurity concentration are shown in Fig. 3(a), along with thevariational upper bound given by the wave function Eq. (1)[22] and the diagrammatic MC calculation of [13]. As final
state interactions are weak, they can be included as asimple repulsive mean field shift 4%@2 afe n" =m, with afe
the scattering length between the final state and the environment atoms [24]. Polaron energies have been predicted
via the variational ansatz [5], the T-matrix approach[9,10,15,16], the 1=N expansion [11], fixed-node MC[6,12] and diagrammatic MC calculations [13]. With the
exception of the 1=N expansion, these all agree with eachother and with the present experiment to within a few percent. In particular, in the unitary limit where 1=kF a ¼ 0 we
find a polaron energy of E# ¼ #0:64ð7Þ+F ð#0:72ð9Þ+F Þ
when state j3i (j1i) serves as the impurity [25]. This agrees
well with the diagrammatic MC calculation, #0:615+F[14], and the analytical result #0:6156ð2Þ+F [16]. Analy-

peak position / εF

a)

0.5
1.0

sis of experimental density profiles yields a value of#0:58ð5Þ+F [26].
The relatively large value for E# directly implies that thenormal state, modeled as a Fermi sea of weakly interacting
polarons, is favored over the superfluid state up to a criticalconcentration (44%), much higher than that predicted bymean field theories (4%) [27]. These neglect interactions in
the normal state and therefore imply a polaron binding
energy of zero.
We have so far considered the limit of few impurities. By
increasing their density, we can study the effect of interactions between polarons. In Fig. 3(b) we show that thequasiparticle peak position depends only weakly on the impurity concentration in the unitarity limit. Polarons are thus
weakly interacting quasiparticles, despite the strong interactions between the bare impurity and its environment.
The peak position could be modified due to the effective
mass m( of polarons, larger than the mass of the bareimpurity. Transfer of a moving polaron into the free final
state then requires additional kinetic energy. This leads toan upshift and a broadening on the order of the Fermi
energy difference between initial and final state,x2=3 +F ð1 # mm( Þ. On resonance, this is 0:04+F for x ¼ 0:1.
The effect could be partially masked by the predicted weak
repulsion between polarons [12] that would downshift theresonance frequency by #0:02+F for x ¼ 0:1.
The spectral weight of the polaron peak directly gives
the quasiparticle residue Z, a defining parameter of a Fermi
liquid. Experimentally, we determine the area under the
impurity peak that is not matched by the environment’s

1.5
1 / kF a

2.0
0.8

0.6

0.4

0.2

0.0

-0.2

0.8
0.7

0.4
1 / kF a

impurity concentration x

b)

0.0

0.8
0.9
0.00

0.05
0.10
0.15
impurity concentration x = n↓ / n↑FIG. 3 (color online). Peak position of the impurity spectrum
as a measure of the polaron energy E# . (a) peak position forvarious interaction strengths in the limit of low concentration
x ¼ 5ð2Þ% (solid circles). Open circle: Reversed roles of impurity and environment. Dotted line: polaron energy from variational ansatz Eq. (1) [5], the solid line including weak final stateinteractions. Dashed line: Energy of a bare, isolated molecule invacuum. Blue dash-dotted line: Mean field limit for the energy ofan impurity atom. Solid (open) diamonds: Diagrammatic MCenergy of the polaron (molecule) [13]. (b) Peak position at
unitarity (1=kF a ¼ 0) as a function of impurity concentration(solid circles). The line shows the expected peak position,
4
kF afe , using the MC@!p =+F ¼ A þ ð1 # mm( Þx2=3 # 65 Fx þ 3%
value A ¼ 0:615 [13], the analytic result m( ¼ 1:2 [16], theweak repulsion between polarons with F ¼ 0:14 [12] andweak final state interactions with scattering length afe .

0.8
0.6
0.4

0.1

0.2
0.0

0.2

quasiparticle residue Z

0.0

2.5

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Z

0.5

0.3
0.0
0.5
0.0
1 / kF a

FIG. 4 (color online). Quasiparticle residue Z as a function ofinteraction strength and impurity concentration. The color coding indicates the magnitude of Z and is an interpolation of thedata points shown in the graph. Open circles: Data points
consistent with zero (Z < 0:03), solid circles: Z > 0:03, the solidline marking the onset of Z. Blue cross: Critical interaction
strength for the Fermi liquid—molecular BEC transition for x !
0 [12]. Inset: Z as a function of interaction strength in the limit of
low impurity concentration x ¼ 5ð2Þ%. Open circle: Reversedroles, j1i impurity in j3i environment. The spectrum in the inset
illustrates the determination of Z [22].

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PRL 102, 230402 (2009)PHYSICAL REVIEW LETTERS

response and divide by the total area under the impurity
spectrum (see spectrum in Fig. 4 and [22]). Figure 4
presents Z as a function of interaction strength and impurity concentration x, the inset shows Z for x ¼ 5%. As
expected, Z approaches 100% for weak attractive interaction kF a ! 0" , where the bare impurity only rarely recoils
off environment atoms. As the mean free path shortens andthe bare impurity starts to surround itself with environment
atoms Z decreases. On resonance, we find Z ¼ 0:39ð9Þ for
x ¼ 5%, with only a weak dependence on x (Fig. 4).
Theoretical values for Z vary: ansatz Eq. (1) predicts Z ¼
0:78 for a single impurity, while Ref. [11] predicts Z ¼
0:47 (0.30) for vanishing (5%) impurity concentration. Our
procedure might yield a lower bound on the actual value ofZ, as the incoherent part of the impurity spectrum might bedepleted around threshold. Eventually, for strong attraction
between the impurity and particles of the medium, Z
vanishes and we observe complete overlap of the impurity
and environment spectra. This signals the formation of a
two-body bound state between the impurity and exactly
one environment atom. For a spin-down concentration of
5% we determine the critical interaction strength where thepolaron peak vanishes to be 1=ðkF aÞc ¼ 0:76ð2Þ. This is ingood agreement with the independently determined criticalinteraction 1=kF a ¼ 0:74ð4Þ beyond which one finds asuperfluid even for the smallest impurity concentration[28]. This is a multicritical point [12,13,29] where a
Fermi liquid of weakly interacting polarons undergoes aphase transition into a Bose liquid of molecular impurities.
Fixed-node MC calculations place this transition at a valueof 1=kF a ¼ 0:73 for x ! 0 [12]. Our 1=ðkF aÞc is lower
than the value 0.90(2) from diagrammatic MC [13] for asingle impurity. Ansatz Eq. (1) does not predict a transition, as it does not test for the formation of molecules. In
Fig. 4, the color coding reveals where molecular behavior
is observed (yellow), and where the spectra show polaronic
behavior (red to black). It can be seen that the criticalinteraction strength for the formation of molecules depends
only weakly on the impurity concentration x.
In conclusion, we have observed Fermi polarons in anovel, attractive Fermi liquid of spin-down impurity atoms
immersed in a spin-up Fermi sea. The energy and residueof this quasiparticle was determined and interactions between quasiparticles were found to be weak. Polarons thus
emerge as the quasiparticles of a Landau Fermi liquid
description of this strongly interacting Fermi mixture. To
study first the impurity limit of N þ 1 interacting particles
before dealing with the full N þ M many-body system will
be a fruitful approach for other strongly correlated systems
realized with cold atoms. An intriguing question is how thelimit of a weakly interacting polaron liquid containing few
impurities connects to the physics of a hydrodynamic,balanced Fermi gas containing Cooper pair fluctuations
above the critical temperature for superfluidity. In lightof our findings, fermion pair condensation could be viewed
as condensation of pairs of polarons with opposite spin.
This is also suggested by the large normal state interaction

week ending
12 JUNE 2009

measured by quasiparticle spectroscopy on the superfluid
state [30].
We would like to thank W. Ketterle, M. Randeria, S.
Stringari, B. Svistunov, S. Todadri, and W. Zwerger for
helpful discussions, and Aviv Keshet for the computer
control system. This work was supported by the NSF, an
AFOSR-MURI, and the Alfred P. Sloan Foundation.

[1] L. Landau, Phys. Z. Sowjetunion 3, 644 (1933).
[2] N. Mannella et al., Nature (London) 438, 474 (2005).
[3] P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys.
78, 17 (2006).
[4] J. Kondo, Prog. Theor. Phys. 32, 37 (1964).
[5] F. Chevy, Phys. Rev. A 74, 063628 (2006).
[6] C. Lobo et al., Phys. Rev. Lett. 97, 200403 (2006).
[7] C. Schunck et al., Science 316, 867 (2007).
[8] A. Bulgac and M. M. Forbes, Phys. Rev. A 75, 031605(R)(2007).
[9] R. Combescot et al., Phys. Rev. Lett. 98, 180402 (2007).
[10] M. Punk and W. Zwerger, Phys. Rev. Lett. 99, 170404(2007).
[11] M. Veillette et al., Phys. Rev. A 78, 033614 (2008).
[12] S. Pilati and S. Giorgini, Phys. Rev. Lett. 100, 030401(2008).
[13] N. V. Prokof’ev and B. V. Svistunov, Phys. Rev. B 77,
020408(R) (2008).
[14] N. V. Prokof’ev and B. V. Svistunov, Phys. Rev. B 77,
125101 (2008).
[15] P. Massignan, G. M. Bruun, and H. T. C. Stoof, Phys.
Rev. A 78, 031602(R) (2008).
[16] R. Combescot and S. Giraud, Phys. Rev. Lett. 101, 050404(2008).
[17] W. Ketterle and M. Zwierlein, in Proceedings of theInternational School of Physics ‘‘Enrico Fermi’’, Course
CLXIV, edited by M. Inguscio, W. Ketterle, and C.
Salomon (IOS Press, Amsterdam, 2008), p. 95.
[18] C. H. Schunck et al., Nature (London) 454, 739 (2008).
[19] Y. Shin et al., Phys. Rev. Lett. 99, 090403 (2007).
[20] C. A. Regal et al., Nature (London) 424, 47 (2003).
[21] We do not observe Pauli blocking effects proposed in [15].
[22] See EPAPS Document No. E-PRLTAO-102-040923. For
more information on EPAPS, see http://www.aip.org/
pubservs/epaps.html.
[23] W. Schneider, V. B. Shenoy, and M. Randeria,
arXiv:0903.3006.
[24] The next order correction due to incoherent scattering,where afe is replaced by the full real part of the scattering
amplitude, is of order ðkF afe Þ2 & 5% and thus negligible.
[25] Switching roles of states j1i and j3i tests for the influenceof final state interactions. They contribute Ef ¼ 0:17ð1Þ,F[0:13ð1Þ,F ] to the peak position when state j3i (j1i) serves
as the impurity.
[26] Y. Shin, Phys. Rev. A 77, 041603(R) (2008).
[27] D. E. Sheehy and L. Radzihovsky, Ann. Phys. (N.Y.) 322,
1790 (2007).
[28] Y. Shin et al., Phys. Rev. Lett. 101, 070404 (2008).
[29] S. Sachdev and K. Yang, Phys. Rev. B 73, 174504(2006).
[30] A. Schirotzek et al., Phys. Rev. Lett. 101, 140403 (2008).

230402-4

Supplemental Material: “Observation of Fermi Polarons in a Tunable Fermi Liquid of
Ultracold Atoms”
André Schirotzek, Cheng-Hsun Wu, Ariel Sommer, and Martin W. Zwierlein
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA(Dated: May 1, 2009)
In this supplemental material we state, starting with the variational Ansatz by Chevy, key properties of the polaron, such as the energy E↓ and the quasiparticle residue Z, and we calculate its RF
spectrum using Fermi’s Golden Rule. We connect this approach and its implication for finite impurity concentration to the results of Fermi liquid theory and the T-matrix formalism used in [1–4].
Furthermore, details are provided about the extraction of the quasiparticle residue Z.
Polaron wavefunction, energy and quasiparticle
residue

We start with the hamiltonian for a dilute two component mixture of fermionic atoms interacting via the vander-Waals potential V (r) [5]. Thanks to the diluteness ofthe system, the potential is of short range R comparedto the interparticle distance 1/kF , so kF R ¿ 1. Its
Fourier transform V (k) is thus essentially constant, g0 ,below kF and rolls off to zero at a momentum on the order of 1/R À kF . The many-body Hamiltonian for thesystem is then
Ĥ =

X
k,σ

²k c†kσ ckσ +g0 X †
ck+ q ↑ c†−k+ q ↓ ck0 + q2 ↓ c−k0 + q2 ↑22V0k,k ,q(1)Here, the label σ denotes the spin state ↑,↓, ²k =~2 k 2 /2m, V is the volume of the system and the c†kσ ,ckσ
are the usual creation and annihilation operators forfermions with momentum k and spin σ. The trial wavefunction for the Fermi polaron with zero momentum proposed by F. Chevy in [6] is|Ψi = ϕ0 |0i↓ |F Si↑ +k>kXF

ϕkq c†k↑ cq↑ |q − ki↓ |F Si↑

E↓ =

1 X
f (E↓ , q)
V

(5)

q<kF

These all depend on the function f (E, q) withf −1 (E, q) =

1
1 X
1
+
(6)
g0
V
²k − ²q + ²q−k − E
k>kF

It is a measure of the interaction strength between spin
up and spin down, modified by the presence of the spin up
Fermi sea. As usual, g0 can be replaced by the physicallyobservable scattering length a for collisions
P 1 between spin
1m
up and down via [5] g10 = 4π~
2a − Vk 2²k .
f

−1

¶
µ
π
mkF
−1 +
(7)
(E, q) =2π 2 ~2 2kF a
µ
¶
1 X
1
1
−
V
²k − ²q + ²q−k − E
2²k
k>kF

The integral in above expression is convergent and gives
µ
¶¾
½
mkF
E q
π
f −1 (E, q) =
−
1
+
I
,
(8)
2π 2 ~2 2kF a
EF kF
µ
¶
¶
µ 2
Z ∞
x
2x + 2xy − ²dx
I(², y) =−1ln
2y2x2 − 2xy − ²
1

q<kF

(2)
The energy is then minimized under variation of the parameters ϕ0 and ϕkq , with the constraint of constantPk>k2
2
norm hΨ|Ψi = |ϕ0 | + q<kFF |ϕkq | = 1. That is,DE
the quantity to minimize is Ψ|Ĥ|Ψ − E↓ hΨ|Ψi. The

derivation can be found in [6], here we quote the result forthe particle-hole excitation amplitudes ϕkq , the quasipar2
ticle weight |ϕ0 | = Z, and the energy E↓ due to additionof the down spin impurity:f (E↓ , q)1
ϕkq = ϕ0
V E↓ − ²k + ²q − ²q−k


X
1
1
∂
1
1 −
f (E, q)2 ≡ Z =
∂E V
|ϕ0 |
q<kF

(3)
(4)
E=E↓

An analytic expression for the integral exists but does notprovide additional insight. The equation for E↓ becomes
E↓
= −2
EF

Z 1
0

dy

y2
1 − 2kπF a − I³

E↓
EF , y´(9)This implicit equation for E↓ can be easily solved numerically. The result is shown as the dashed line in
Fig. 3 of the main paper. Clearly, E↓ is negative dueto the attractive interactions with the medium. In the
weakly interacting limit 1/kF a → −∞, we can neglect
the integral in the denominator and immediately obtainE↓ = 23 2kπF a EF = 4π~2 an↑ /m, which is the mean field
result [7].
The approach turns out to be equivalent to a T-Matrix
description, as shown in [7]. In that language, f (E, q)

2
is (up to a constant) the scattering amplitude in themedium (i.e. the vertex) for the scattering process withtotal energy and momentum
E and q of the collidingP
particles. Σ(0, E) ≡ V1 q<kF f (E↓ + E, q) is the selfenergy at zero momentum and frequency E/~. It is realin this approximation. The expression for the quasiparticle residue Z of a single spin down impurity in Eq. 4
is immediately seen to be equivalent to the well-known
relation [3]Z↓−1 =µ¶∂1−ReΣ(kF ↓ , E)
∂E
E=0

(10)

for a spin down quasiparticle on top of a spin down Fermisea, in the limit of vanishing Fermi momentum kF ↓ .

RF spectrum from the variational Ansatz

Fermi’s Golden Rule allows us to directly predict theshape of the impurity RF spectrum. This topic has beenstudied in detail since the early days of RF spectrosocpy,
and for the problem of highly imbalanced Fermi gasesin [1–4], among others. Chevy’s wavefunction offers a
simple way of calculating the RF spectrum of a single
impurity.
P
The RF operator V̂ = ~ΩR k c†k,f ck,↓ + h.c. promotes
the impurity into the free final state |f i (energy Ef ) without momentum transfer [5]. In the experiment, the final
internal state is the second lowest hyperfine state of 6 Li.
Fermi’s Golden Rule for the impurity starting in state|Ψi isE¯2
2π X ¯¯D¯Γ(ω) =¯ f |V̂ |Ψ ¯ δ(~ω − (Ef − E↓ ))~(11)

This leaves us with two components in the RF spectrum:µ2(12)
Γ(ω) = 2π~ΩR Zδ(~ω + E↓ ) +
k>k
XF
q<kF

¶
|ϕkq | δ(~ω + E↓ − ²k + ²q − ²q−k )2

The first part is a delta-peak shifted by the quasiparticle
energy. As E↓ < 0, it is shifted to higher frequencies: The
RF photon has to supply additional energy to transfer the
impurity out of its attractive environment. The weightof this peak is Z, the quasiparticle residue, allowing theexperimental determination of Z by simply integrating
the area under the prominent peak. Such a delta-peakis typically called ”coherent”, as a broadband excitation
around this energy would not dephase over time. The
second part of the spectrum is incoherent, it consists of abroad continuum of frequencies. Broadband excitationsof this continuum would rapidly dephase, over a timescale
given by the inverse width of the continuum.
RF spectrum in Fermi Liquid theory

This structure of the RF spectrum is a generic feature of quasiparticle spectra. In Fermi liquid theory, thepropagator of a quasiparticle is approximated as a poleat energy E(k) > 0 (relative to the ground state energy),
lifetime 1/γ(k) and residue Z plus an incoherent spectrum [3, 8]
GR− (k, ω) =Z
+ GR,inc(k, ω)−~ω + E(k) + i~γ(k)(13)

The spectral function is given by A− (k, ω) =
~γ(k)
1
inc
− π1 ImGR− (k, ω) = Z π (~ω+E(k))2 +~2 γ(k)2 + A− (k, ω)which tends tof

Where ω is the RF offset from the bare atomic transition
frequency between the internal states labeled by ↓ andf . One possible final state is |0i ≡ |0if |F Si↑ , i.e. azero momentum particle in the final state plus a perfectFermi sea of up spins, with energy E|0i = 0 relative to the
Fermi energy EF of the environment. Other possible final
states are |q, ki ≡ |q − kif c†k↑ cq↑ |F Si↑ with q < kF
and k > kF , i.e. a particle with momentum q − k in
the final state and a Fermi sea with a hole at q andan excited environment particle above the Fermi sea at
k. The energy of these states is E|q,ki = ²k − ²q + ²q−krelative to the environment Fermi energy EF . The matrix
elements areD ¯ ¯ E¯ ¯
0 ¯V̂ ¯ Ψ = ~ΩR ϕ0
¯ ¯ E
D
¯ ¯
q, k ¯V̂ ¯ Ψ = ~ΩR ϕkq

A− (k, ω) = Zδ(~ω + E(k)) + Ainc− (k, ω)(14)in the limit of small damping of the quasiparticle.
A− (k, ω) measures the probability that removing a particle with momentum k will cost an energy −~ω. The
RF spectrum in linear response is given by [2]XA− (k, ²k − µ − ~ω)nF (²k − µ − ~ω)Γ(ω) = 2π~Ω2R
k(15)
where µ is the chemical potential of the quasiparticle andnF (x) = 1/(eβx + 1) is the Fermi function that tends toθ(−x) at zero temperature. This is intuitively understood: For a given momentum k, the RF photon withenergy ~ω has to provide the energy ²k − µ (relative tothe initial chemical potential µ) to create a free particle in the final state. The rest, ~ω − ²k + µ, is used
to remove a particle from the initial state (probabilityA− (k, ²k − µ − ~ω)) if there exists such a particle (factor

3
nF (²k − µ − ~ω)). Eq. 15 is equivalent to Fermi’s Golden
Rule Eq. 11 [9]. In the case where the spectral function is dominated by a quasiparticle peak, the spectrum
becomesXδ(²k + E(k) − µ − ~ω)Γ(ω) = 2π~Ω2R Z
k× nF (²k − µ − ~ω) + Γinc (ω)(17)

identical to the prediction via the trial wavefunction.

Polaron spectral function

To connect the single particle and the Fermi liquid description, we calculate the propagator GR− (k, ω) for theremoval of a single spin down impurity from the wavefunction |Ψi. By definition,† iĤt/~iGR
ck↓ |Ψi θ(t)− (k, t) = hΨ| ck↓ e

X2|hf | ck↓ |Ψi| eiEf t/~ θ(t)(19)

The state ck↓ |Ψi is void of any spin down impurity andhas non-vanishing matrix elements only with either theunperturbed spin up Fermi sea, |F Si↑ (if k = 0), or withparticle-hole excitations |q, k0 i = c†k0 ↑ cq↑ |F Si↑ (in the√case k = q − k0 ). These matrix elements are ϕ0 = Z
and ϕk0 q resp., the corresponding energies Ef = 0 and
Ef = ²k0 − ²q relative to EF . So one has:
k0X
>kF

2

δk,q−k0 |ϕk0 q | ei(²k0 −²q )t/~ )θ(t)

q<kF

R,inc
Z
(k, ω) with inFinally, GR− (k, ω) = ~ω+iη δk,0 + G−
finitesimal η > 0. This is just the Fermi liquid formof GR
− but for a single quasiparticle with zero momentum(E(0) = 0 in (13)), as described by |Ψi. The spectral
function isA− (k, ω) = Zδ(~ω)δk,0 +
k0X
>kF

2

δk,q−k0 |ϕk0 q | δ(~ω + ²k0 − ²q )q<kF

With Eq. 15 this exactly gives the RF spectrum of Eq. 12.

k>k
XF

2

|ϕkq | δ(~ω + E↓ − ²k + ²q − ²q−k )

q<kF

=

2πΩ2R Z 1
~ ω2 V 2k>k
XF

f (E↓ , q)2 δ(~ω + E↓ − ²k + ²q − ²q−k )q<kF

The integral over k exists in analytic form:
1 P
k>kF δ(~ω + E↓ − ²k + ²q − ²q−k ) =V~ω+E↓ qmkF
8π 2 ~2 K( 2EF , kF )
 2 2
y −y

 + y − for y− > 1 ,2with K(², y) = y+ −1
for y− < 1 < y+ ,y0
for 1 > y+ .
q2
andy± = ± y2 + y4 + ²(20)

The incoherent spectrum is thenΓinc (ω) = πΩ2R

fiGR
− (k, t) = (Zδk,0 +Γinc (ω) ≡ 2π~Ω2R(18)

Inserting a complete set of eigenstates, this givesiGR− (k, t) =Using Eq. 3, we can write the incoherent part of thespectrum as:(16)

Connecting to our case of a single quasiparticle with µ =E↓ and k = 0, this directly gives
Γ(ω) = 2π~Ω2R Zδ(~ω + E↓ ) + Γinc (ω)

Calculation of the incoherent background

ZEF
~ω 2

Z 1
0

y2 K
dy ³

³

~ω+E↓
2EF , y1 − 2kπF a − I

³

´

E↓
EF , y´´2(21)

One can check that the total spectrum obeys the sumruleZ ∞dωΓ(ω) = 2π~Ω2R(22)−∞

and in particular that the total weight of the incoherent background is proportional to 1 − Z, which is not
obvious from the form in Eq. 21. For RF frequencies
close to threshold ~ω + E↓ ¿ 2EF , the hole momentum q and the particle momentum k must be close toeach other to fulfill energy conservation, i.e. they haveto be close to the Fermi momentum. The double sum
over q and k thus gives a phase space suppression on
the order of (~ω + E↓ )2 , i.e. the spectrum starts like(~ω + E↓ )2 /ω 2 . This is in contrast to the dissociation
spectrum of a molecule of binding energy EB , where thedensity of states√ above threshold gives a spectrum proportional to ~ω + EB /ω 2 . For large RF energies, large
particle momenta k are involved, the suppression due tothe Fermi sea
p becomes negligible and the spectrum behaves like ~ω + E↓ /ω 2 , as for a molecule of binding
energy E↓ . This is natural as for large momenta, we areprobing short-range physics which involves at most two
particles, a spin up environment atom and the impurity.
In particular, at RF energies ~ω À E↓ , we recover theω −3/2 behavior of the RF spectrum that is universal for
short-range interactions.

4

Since polarons are found to be weakly interacting, they
will form a Fermi sea filled up to the impurity Fermi momentum kF ↓ . The fact that the dispersion of polaronsmE( k) = m
∗ ²k differs from that of a free particle due tothe effective mass m∗ 6= m leads to broadening of the RF
spectra. The RF photon has to supply the difference inmkinetic energies (1 − m∗ )²k between the initial and them
2 2
final state, with a maximal shift (1 − m∗ )~ kF ↓ /2m. The
spectral shape is easily obtained: The spectral function
at momentum k will be dominated by polarons that occupy that momentum state. The coherent part of thespectral function is thus Acoh
− (k, ω) = Zδ(~ω + E(k))
mwith E(k) = −~2 k 2 /2m∗ = − m∗ ²k relative to the impurity Fermi energy. The coherent part of the spectrum
then becomesXΓcoh (ω) = 2π~Ω2R
Acoh− (k, ²k − E↓ − ~ω)kwhere the sum extends up to the impurity Fermi momentum kF ↓ . With the free, 3D density of states ρ(²), thisis
Z EF ↓md² ρ(²)Zδ(² − E↓ − ~ω − ∗ ²)
Γcoh (ω) = 2π~Ω2R
m
0
¶
µ
Z
~ω + E↓2= 2π~ΩR
×
ρ
m
m
1− m
1− m
∗
∗
³
´
m
θ (1 − ∗ )EF ↓ − ~ω − E↓
(23)m
This coherent part of the spectrum starts at the polaronground state energy ~ω = |E↓ |, then grows like a squaremroot and jumps to zero when ~ω − |E↓ | = (1 − m∗ )EF ↓ .
∗On resonance, where m ≈ 1.2, this occurs at ~ω−|E↓ | =0.2x2/3 EF ↑ ≈ 0.04EF ↑ for x = 0.1. This is still smaller
than the Fourier width of the RF pulse used in the experiment of about 0.1EF . The size of the jump is given by
2π~Ω2R 1−Zm∗ ρ(EF ↓ ) and reflects the impurity Fermi surm
face in the RF spectrum. This behavior of the coherent
part of the spectrum was found in [1] and was discussed
recently in [4]. It is intriguing that the sharpness of the
Fermi surface and its discontinuity should, at least in
principle, be observable in the RF spectrum.
Determination of Z from experimental spectra

In order to extract the quasiparticle residue Z, we determine the area under the peak that is not matched by
the environment’s response and divide by the total area
under the impurity spectrum (see spectrum in the inset
of Fig. 4 in the main body of the paper). Due to the
Fourier width of the probe pulse, the strong response of

the environment around zero RF offset (the resonance foratom transfer / a.u.

RF Spectrum of a finite concentration of impurities

0

1

2
3
rf offset / ε F

4

5FIG. 1: Determination of the quasiparticle residue Z. Impurity spectrum (red), environment spectrum (blue) and spectral response of non-interacting atoms (black dashed), folded
over from negative RF offsets.

non-interacting atoms) adds some weight to the environment background at the position of the polaron peak. To
remove this effect in the determination of Z, the part ofthe environment’s response at negative frequency offset
is folded towards the positive side (dashed line in Fig. 1)
and subtracted from the environment spectrum. As itturns out, this procedure changes the value for Z by lessthan 5% for all spectra in Fig. 2 of the main paper.

[1] M. Punk and W. Zwerger, Phys. Rev. Lett. 99, 170404(2007).
[2] P. Massignan, G. M. Bruun, and H. T. C. Stoof, Phys.
Rev. A 78, 031602 (2008).
[3] M. Veillette, E. G. Moon, A. Lamacraft, L. Radzihovsky,
S. Sachdev, and D. E. Sheehy, Phys. Rev. A 78, 033614(2008).
[4] W. Schneider, V. B. Shenoy, and M. Randeria, preprint
arXiv:0903.3006 (2009).
[5] W. Ketterle and M. Zwierlein, in Ultracold Fermi Gases,
Proceedings of the International School of Physics ”Enrico Fermi”, Course CLXIV, Varenna, 20 - 30 June 2006,edited by M. Inguscio, W. Ketterle, and C. Salomon (IOS
Press, Amsterdam, 2008).
[6] F. Chevy, Phys. Rev. A 74, 063628 (2006).
[7] R. Combescot, A. Recati, C. Lobo, and F. Chevy, Phys.
Rev. Lett. 98, 180402 (2007).
[8] P. Nozières, Theory of Interacting Fermi Systems, Advanced Book Classics (Addison-Wesley, Reading, MA,
1997).
[9] A. Fetter and J. Walecka, Quantum Theory of ManyParticle Systems (McGraw-Hill, New York, 1971).

Appendix D
Determination of the Superﬂuid
Gap in Atomic Fermi Gases by
Quasiparticle Spectroscopy
This appendix contains a reprint of Ref. [44]: Andre Schirotzek, Yong-il Shin, Christian H. Schunck, and Wolfgang Ketterle , Determination of the Superfluid Gap in
Atomic Fermi Gases by Quasiparticle Spectroscopy, Phys. Rev. Lett. 101, 140403(2009).

137

PRL 101, 140403 (2008)

PHYSICAL REVIEW LETTERS

week ending
3 OCTOBER 2008

Determination of the Superfluid Gap in Atomic Fermi Gases by Quasiparticle Spectroscopy
André Schirotzek, Yong-il Shin, Christian H. Schunck, and Wolfgang Ketterle
Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA(Received 31 July 2008; published 3 October 2008)
We present majority and minority radio frequency spectra of strongly interacting imbalanced Fermi
gases of 6 Li. We observed a smooth evolution in the nature of pairing correlations from pairing in the
superfluid region to polaron binding in the highly polarized normal region. The imbalance induces
quasiparticles in the superfluid region even at very low temperature. This leads to a local bimodal spectral
response, which allowed us to determine the superfluid gap # and the Hartree energy U.
DOI: 10.1103/PhysRevLett.101.140403

PACS numbers: 05.30.Fk, 03.75.Ss, 32.30.Bv, 67.90.+z

Pairing and superfluidity in fermionic systems are intricately related phenomena. In BCS theory [1], describing
conventional superconductors, the emergence of superfluidity is accompanied by the opening of a gap in the
excitation spectrum of the superfluid. This gap can beinterpreted as the minimum energy required to break a
Cooper pair or, equivalently, to create an elementary excitation, a so-called quasiparticle, inside the superfluid.
However, strongly correlated systems show a more complicated behavior. There are gapless fermionic superfluid
systems, e.g., high temperature superconductors [2] or for
superconductors with magnetic impurities [3]. On the other
hand, there are numerous examples of systems with an
excitation gap in the normal state, e.g., a high temperature
superconductor above its superfluid transition temperature
exhibiting a pseudogap [2].
Here, we use radio frequency (rf) spectroscopy to investigate the nature of pairing and the relation between
pairing and superfluidity in a strongly interacting system ofultracold atomic Fermions.
We can spectroscopically distinguish the superfluid andthe polarized normal fluid by introducing excess fermions
into the system. In a superfluid phase described by BCS
theory, the excess particles can be accommodated only as
thermally excited quasiparticles. A local double-peaked
spectrum reflects the coexistence of pairs and unpaired
particles. In the normal phase, at large spin polarization,
the limit of a single minority particle immersed into a
Fermi sea is approached, which can be identified as apolaron [4–7]. Here the system can be described in theframework of Fermi liquid theory and no stable pairs exist.
We find that these different kinds of pairing correlations
are smoothly connected across the critical density imbalance [8], also called the Clogston-Chandrasekhar limit ofsuperfluidity [9,10].
The rf spectrum of a superfluid containing quasiparticles
shows two peaks, which, in the BCS limit, would be split
by #, the superfluid gap. Therefore, rf spectroscopy ofquasiparticles is a direct way to observe the superfluid
gap # in close analogy with tunneling experiments insuperconductors [11]. From the observed spectrum we
0031-9007=08=101(14)=140403(4)can also determine a Hartree term, see Y. Castin in [12],whose inclusion turned out to be crucial.
For this study, we have combined several recently developed experimental techniques: The realization ofsuperfluidity with population imbalance [13] leading tophase separation [8,13,14], tomographic rf spectroscopy[15], in situ phase-contrast imaging with 3D reconstruction of the density distributions [8]. In order to minimize
final state effects [16] we have prepared an imbalanced
mixture of states j1i and j3i of 6 Li (corresponding to
jF ¼ 1=2; mF ¼ 1=2i and jF ¼ 3=2; mF ¼ #3=2i at low
field) in an optical dipole trap at a magnetic field of B ¼
690 G, at which there is a Feshbach scattering resonancebetween the states j1i and j3i [16,17]. Evaporative coolingat B ¼ 730 G is performed by lowering the power of thetrapping light. After equilibration an rf pulse was appliedfor 200 &s selectively driving a hyperfine transition fromstate j1i or j3i to state (j2ijF ¼ 1=2; mF ¼ #1=2i at low
field). Immediately after the rf pulse an absorption image
was taken of the atoms transferred into state j2i.
The spectra were correlated to the local Fermi energy@2'F" ¼ 2m
ð6(n" Þ2=3 of the majority density n" and to then #nlocal polarization *loc ¼ n"" þn## which is a measure of thelocal excess fermion population. As in a previous publication [8] the local densities were measured using phasecontrast imaging and 3D reconstruction using the inverse
Abel transformation.
The rf spectra shown in Fig. 1 reveal a gradual change in
the nature of the pairing correlations. The balanced superfluid is characterized by identical spectral responses ofmajority and minority particles and has been the subjectof previous studies, see Zwierlein, Grimm, and Regal in[12] and references therein. In the polarized superfluid
region [8,18] (and references therein) the minority spectrum perfectly matches the pairing peak of the majority
spectrum, locally coexisting with the quasiparticle spectral
contribution, resulting in a local double-peak structure of
the majority spectrum, see Fig. 1(b). The spectrum suggests that the majority population can be divided into two
distinct parts: One part consisting of pairs forming thesuperfluid, the other part consisting of quasiparticle exci-

140403-1

 2008 The American Physical Society

b)
c)
d)

spectral response / a.u.

a)

a)

b)

0

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PHYSICAL REVIEW LETTERS

PRL 101, 140403 (2008)

1

2

3

c)

0

1

2

3

d)

0

1

2

3

-1

0

1

2

3

4

rf offset / εF↑

FIG. 1 (color online). Tomographically reconstructed rf spectra for various regions of the atomic sample at unitarity. (a) Balanced
superfluid, (b) polarized superfluid, (c) moderately polarized transition region, and (d) highly polarized normal region. The panel on
the left shows a phase-contrast image of the atomic cloud before rf excitation. The positions of the spectra (a) to (d) are marked in thephase-contrast image and by the arrows in Fig. 2. Red: Majority spectrum, blue: Minority spectrum. Local polarizations loc and local
temperature T=TF , respectively: (a) "0:04ð2Þ, 0.05(1); (b) 0.03(1), 0.06(1); (c) 0.19(1), 0.06(2); (d) 0.64(4), 0.10(2). The negative valuein (a) implies that the local polarization as inferred from phase-contrast imaging underestimates loc by up to 0.05.

alone, but shows up in the spectral overlap with the majority spectrum. As a result, the observed spectral gap in
the normal phase should not be interpreted as a signature ofpairing but rather as strong pairing correlations in the formof a polaron as suggested in [24–26]. Figure 2 summarizes
our data in the unitary regime.

Minority

Majority

3

1.0

0.5

rf offset/ εF↑

2

0.0
1

1.0
0

0.5

σloc

tations in the form of excess fermions. Therefore, a naturalinterpretation of the rf spectrum is to identify one peak as a
Stokes process (rf creates a quasiparticle excitation) giving
rise to the dissociation part of the rf spectrum and the otheras an anti-Stokes process (rf destroys a quasiparticle
excitation).
As the local imbalance is further increased beyond thesuperfluid to normal (SF-N) transition [8,19], see Fig. 1(d),the majority spectrum no longer shows a local double-peakstructure. This is consistent with theoretical work [20,21]
attributing the double-peak structure in the normal phase inpreviously reported rf spectra [22,23] to the inhomogeneous density distribution. For increasing spin polarization
the majority and minority pairing peaks lose spectral overlap. We interpret the missing overlap as indication that theminority atoms are no longer bound in pairs, each of theminteracting with more than one majority atom, a situation
we refer to as polaronic binding. We have seen [19] that on
the Bose-Einstein condensation (BEC) side of the
Feshbach resonance the overlap between minority andmajority spectra does not depend strongly on the presenceof excess fermions as is expected in a molecular picture. At
unitarity, within our experimental resolution, the overlap
starts to decrease at the SF-N interface, see Fig. 1(c).
Even when the spectral overlap decreases, there is stillequal response to the rf excitation in the high frequency
tails, see Figs. 1(c) and 1(d). These tails correspond to large
momentum components in the interparticle wave function
and hence address the short range physics. We expect thispart of the spectrum to be insensitive to changes in thebinding at large distances.
The direct comparison between majority and minority
spectra clarifies our previous experimental results on minority rf spectra in the j1i " j2i mixture [23], in which weconcluded that there is strong pairing in the normal phase.
Although the previous results suffered from final stateinteractions [16], we confirm that the change in pairingcorrelations is indiscernible in the minority spectrum

0.0
-1
1.0

0.5

0.0

0.5

1.0

r / R↑

FIG. 2 (color online). Spatially resolved rf spectra of an imbalanced Fermi gas at unitarity. (a) The right half shows the
majority spectra as a function of position in the trap expressed interms of the majority Fermi radius R" , the left half displays theminority spectra. The superfluid to normal transition region ismarked by the gray vertical lines. The local polarization loc isgiven by the short-dashed red line. The error bars are thestandard deviation of the mean value. The arrows indicate theposition of the four spectra shown in Fig. 1. The image is abilinear interpolation of 2500 data points, each plotted data pointin the image is the average of three measured data points. The
spatial resolution of the image is 0:045 & R" .

140403-2

Ek
0.5
↑

0.0
0.0

0.5

1.0 k / k
F

1.0
2.0

peak position / εF↑

1.0ence between majority and minority components ($" >$# ) forces a finite quasiparticle occupation into the superfluid region already at very low temperature [27]. This
allows us to selectively populate quasiparticles at the minimum of the dispersion curve, see Fig. 3(a).
In Fig. 4(a) the position of the peaks of majority andminority spectra are plotted normalized by "F" as a function of position in the trap in the unitary limit [19]. The
peak positions are proportional to the local Fermi energy
inside the superfluid region within our experimental resolution. In the region of superfluidity with finite polarization
the spectra show local double peaks. The position of thetwo peaks in the limit of small polarization is depicted inFig. 4(b) for various interaction strengths.
It was unexpected that the quasiparticles appear at positive frequencies (relative to the atomic transition frequency). This is caused by the presence of Hartree
terms, resulting in an overall shift of the systems energy
and the rf spectrum [19]. In the weakly interacting limits,
the Hartree term reduces to a simple mean-field shift. In thestrongly interacting regime one has to resort to quantum
Monte Carlo (QMC) calculations [28–30] for a numericalvalue of U.
In a mean-field description of the balanced superfluid
starting from the BCS-Leggett ansatz for the BEC-BCS
crossover, see Zwierlein in [12] taking into account the
Hartree term U, the dispersion relation of the quasiparticles
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
can be expressed as Ek ¼ #2 þ ð"k þ U ! $Þ2 , see
2 2
Y. Castin in [12] where "k ¼ @2mk is free particle kinetic
energy and $ is the chemical potential. This mean-fieldformalism gives the analytic expression for the two peak
positions. A quasiparticle at the minimum of the dispersion

0.8

peak shift / εF↑b)spectral response / a.u.

We now turn to a quantitative analysis of the spectral
peaks in the superfluid phase for small density imbalance,
and to the determination of the superfluid gap. Earlier work
[15,22] tried to determine the gap from the onset of the pairdissociation spectrum. However, the rf spectrum is not only
sensitive to final state interactions, it is also shifted by
Hartree energies, as we show here. Furthermore, rf spectroscopy can excite all fermions, even deep in the Fermisea, see M. Zwierlein in [12]. Therefore, the onset of thepair dissociation spectrum occurs for atoms with momentum k ¼ 0 and, in the BCS limit depends quadratically on#2
the gap parameter (!th ¼ 2"). The excitation gap can beFdirectly observed if quasiparticles near the dispersion minimum are selectively excited, as in tunneling experiments.
Our solution is to study not the ground state of a superfluid, but excited states where quasiparticles are present. Ina simple BCS description, quasiparticles are in pure momentum states, but increase the total energy of the systembecause their momentum state is no longer available to theother particles for pairing. Consequently, in an excitation
spectrum, quasiparticles appear at negative frequencies
relative to the bare atomic transition frequency. The lowest
energy quasiparticle appears at frequency !#, see Fig. 3.
Final state interactions and Hartree terms can also createline shifts, and two peaks are needed for analysis, thedissociation peak and the quasiparticle peak in our case.
In essence, it is the separation between the peaks in spectralike Fig. 1(b), which allows us to determine #.
Thermal population of quasiparticles requires a temperature on the order of the excitation gap #. At unitarity,
this temperature can be estimated to be 95% of the criticaltemperature, away from the low temperature limit addressed in this Letter. Indeed, in samples of equal population of the spin states we were not able to spectroscopicallyresolve any local double-peak structures [19]. This problem can be overcome by introducing density imbalance
between the constituents: The chemical potential differa)

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PHYSICAL REVIEW LETTERS

PRL 101, 140403 (2008)

0.6
0.4
0.2

+|U|

1.5

1.0

0.5

0.0
0.0
0.0

0.2

0.4

r / R↑
-0.5

0.0

0.5

rf offset / εF↑

1.0

0.6

0.8

1.0

-0.2

0.0

0.2

0.4

0.6

1 / kF a

1.5

FIG. 3 (color online). Creation and spectroscopy of quasiparticles. (a) Population imbalance thermally generates quasiparticles even at low temperatures comparable to # ! $" . $" is thechemical potential of the majority component. (b) The rf spectrum consists of a quasiparticle peak at negative frequencies andthe pair dissociation spectrum at positive frequencies (dottedline). On resonance, the Hartree contribution U acts as aneffective attraction and hence shifts the entire spectrum into
the positive direction.

FIG. 4 (color online). (a) Normalized peak positions of pairingpeaks and quasiparticle peak at unitarity as a function of positionin the trap. The SF-N boundary (cusp in column density difference [19]) is marked by the dashed vertical lines. The arrow
indicates the limit of low quasiparticle population relevant for(b). Majority: blue open squares (pairing peak) and solid black
circles (quasiparticle peak). Minority: solid red triangles.
(b) Pairing peak and quasiparticle peak positions as a functionof the local interaction strength 1=kF a in the limit of small local
imbalance [see arrow in (a)]. Pairing peak: solid circles; quasiparticles peaks: open circles.

140403-3

PRL 101, 140403 (2008)

PHYSICAL REVIEW LETTERSTABLE I. Superfluid gap %, Hartree term U, and final stateinteraction Efinal in terms of the Fermi energy $F" for various
interaction strengths 1=kF a.
1=kF a%U

Efinal

!0:25
0
0.38
0.68

0.22
0:44
0.7
0.99

!0:22
0:43
!0:59
!0:87

0.22
0:16
0.14
0.12

curve will respond at an rf offset of !rf ¼ !Ekmin ! # þ$kmin ¼ !% ! U, and the maximum of the pair dissociaqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
02tion spectrum occurs at @!max ¼ 43 ð #02 þ 15
16 % ! # Þ !
U ’ 43 !th ! U, where #0 ¼ # ! U and !th is the dissociation threshold (which is at momentum k ¼ 0).
We determined the superfluid gap % and the Hartree
energy U from the peak positions in the limit of small
density imbalance (&loc ’ 0:03). At unitarity with thechemical potential # ¼ 0:42$F" , confirmed in previous
experiments and theory, see [31] and references therein,
we obtained % ¼ 0:44ð3Þ$F" and U ¼ !0:43ð3Þ$F" , in excellent agreement with the predicted values %t ¼ 0:45$F"
and Ut ¼ !0:43$F" from QMC calculations [32]. Our
determined values for % and U values suggest the minimum of the quasiparticle dispersion curve to occur at
kmin ’ 0:9kF . Table I shows the gap and Hartree energy
for various interaction strengths. Away from unitarity werelied on QMC calculations for the chemical potential #[33].
For an accurate quantitative comparison [19] final stateinteractions, also listed in Table I, had to be taken into
account. The effect of final state interactions is an overall
2amean-field shift of Efinal ¼ 4*@
m n. This shift affects boththe quasiparticle peak and the pairing peak equally.
In conclusion, in crossing the superfluid to normalboundary we observed a gradual crossover in the pairing
mechanism by comparing majority and minority spectra.
The majority spectrum shows a local double-peak spectrum in the polarized superfluid region which allowed us todetermine the superfluid gap % and the Hartree terms U.
The spectra in the normal phase are consistent with apolaron picture.
We thank W. Zwerger and M. Zwierlein for stimulating
discussions and A. Sommer for critical reading of themanuscript. This work was supported by the NSF and
ONR, through a MURI program, and under ARO Grant
No. W911NF-07-1-0493 with funds from the DARPA OLE
program.

[1] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev.
108, 1175 (1957).

week ending
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[2] P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys.
78, 17 (2006).
[3] A. A. Abrikosov and L. Gork’ov, JETP 12, 1243 (1961).
[4] F. Chevy, Phys. Rev. A 74, 063628 (2006).
[5] R. Combescot, A. Recati, C. Lobo, and F. Chevy, Phys.
Rev. Lett. 98, 180402 (2007).
[6] N. Prokof’ev and B. Svistunov, Phys. Rev. B 77, 020408(2008).
[7] P. Massignan, G. M. Bruun, and H. T. C. Stoof,
arXiv:0805.3667 [Phys. Rev. A (to be published)].
[8] Y. Shin, C. H. Schunck, A. Schirotzek, and W. Ketterle,
Nature (London) 451, 689 (2008).
[9] A. M. Clogston, Phys. Rev. Lett. 9, 266 (1962).
[10] B. S. Chandrasekhar, Appl. Phys. Lett. 1, 7 (1962).
[11] I. Giaever, Phys. Rev. Lett. 5, 147 (1960).
[12] M. Inguscio, W. Ketterle, and C. Salomon, in Ultracold
Fermi Gases, Proceedings of the International School ofPhysics ‘‘Enrico Fermi’’ Course CLXIV (IOS Press,
Amsterdam, 2008).
[13] M. W. Zwierlein, A. Schirotzek, C. H. Schunck, and
W. Ketterle, Science 311, 492 (2006).
[14] G. B. Partridge et al., Science 311, 503 (2006).
[15] Y. Shin, C. H. Schunck, A. Schirotzek, and W. Ketterle,
Phys. Rev. Lett. 99, 090403 (2007).
[16] C. H. Schunck, Y. Shin, A. Schirotzek, and W. Ketterle,
Nature (London) 454, 739 (2008).
[17] M. Bartenstein et al., Phys. Rev. Lett. 94, 103201 (2005).
[18] D. E. Sheehy and L. Radzihovsky, Ann. Phys. (N.Y.) 322,
1790 (2007).
[19] See EPAPS Document No. E-PRLTAO-101-019841 forauxiliary material. For more information on EPAPS, see
http://www.aip.org/pubservs/epaps.html.
[20] P. Massignan, G. M. Bruun, and H. T. C. Stoof, Phys.
Rev. A 77, 031601 (2008).
[21] E. J. Mueller, arXiv:0711.0182v1 [Phys. Rev. A (to be
published)].
[22] C. Chin et al., Science 305, 1128 (2004).
[23] C. H. Schunck et al., Science 316, 867 (2007).
[24] M. Punk and W. Zwerger, Phys. Rev. Lett. 99, 170404(2007).
[25] M. Veillette et al., arXiv:0803.2517v1 [Phys. Rev. A (to be
published)].
[26] A. Perali, P. Pieri, and G. C. Strinati, Phys. Rev. Lett. 100,010402 (2008).
[27] M. M. Parish, F. M. Marchetti, A. Lamacraft, and B. D.
Simons, Nature Phys. 3, 124 (2007).
[28] J. Carlson and S. Reddy, Phys. Rev. Lett. 95, 060401(2005).
[29] C. Lobo, A. Recati, S. Giorgini, and S. Stringari, Phys.
Rev. Lett. 97, 200403 (2006).
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arXiv:0801.1504v1.
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75, 023610 (2007).
[32] J. Carlson and S. Reddy, Phys. Rev. Lett. 100, 150403(2008).
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Phys. Rev. Lett. 93, 200404 (2004).

140403-4

Auxiliary Material: Determination of the Superfluid Gap in Atomic Fermi Gases by
Quasiparticle Spectroscopy
PACS numbers: 03.75.Ss, 32.30.Bv

Determination of the superfluid boundary

It has been shown previously [1] that at unitarity thedifference column density profiles serve as an indicator
for the SF-N boundary. The discontinuity of the minority
density results in a pronounced ”cusp” in the difference
profile n↑ (r) − n↓ (r), see fig. 1. In the main body of thepaper the phase boundary has been determined by thispeak position. Hence, ”normal” refers to spatial regions
beyond the peak, ”superfluid” refers to spatial regions
inside.
The color coding in the graphs in fig. 1 shows where thespectral overlap (definition see below) between the majority and minority pairing peaks is lost. On resonance(B = 690 G) this position shows excellent agreement withthe position of the cusp in the difference density profileand can therefore serve as an alternative indicator for
the SF-N boundary. This coincidence breaks down awayfrom resonance: On the BCS side (B = 710 G) the spectra are less ”robust” against polarization and spectral
overlap is lost before the column density difference showsa peak. The reverse situation occurs on the BEC side of
the resonance (B = 671 G). Note that on the BEC side
the minority cloud does not extend much further thanthe peak position in the column density difference.

(red) shows a change in curvature as the SF-N boundary is crossed, in contrast to the majority Fermi energy.
Therefore, the ratio of minority peak and majority Fermi
energy shows a sudden increase towards the edges of thecloud. The value of the peak position of ωpol ' 0.9 ²F ↑(where final state interactions of Ef inal ' 0.05 ²F ↑ havebeen taken into account) is higher than the theoreticallycalculated value of ' 0.6 ²F ↑ [2].
Quasiparticles in an equal mixture

In a previous publication [3] low temperature RF spectra did not show any signatures of quasiparticles. We
have attempted to create thermally generated quasiparticles in an equal density mixture for higher temperatures
T /TF ≥ 0.20. The experimental results in fig. 4 show adecrease of the gap parameter, but no local double peakstructures could be resolved for any temperature.
We can estimate the temperature required to populatequasiparticles, assuming that the temperature has to beon the order of the gap
qand that ∆(T ) is given by the BCS)
Trelation ∆(T∆0 = 1.74 1 − Tc [4]. At unitarity, ∆0 /Tc '3. One would expect to populate quasiparticles only very
close to the transition temperature T /Tc ' 0.95, when
the gap is only one third of its low-temperature value.

Quantification of spectral overlap

In order to calculate the spectral overlap of a spectrum, the quasiparticle peak was fitted by a gaussian andsubtracted from the majority spectrum. The overlap isthen defined as one minus the difference of the integrated
spectra normalized by the sum of the integrated spectra,
see fig. 2. As mentioned in the main text and above, on
the BEC side of the Feshbach resonance almost complete
spectral overlap can be observed into the normal region.
On the BCS side the reverse situation occurs.
Minority peak shift for high imbalance

Fig. 4 in the main body of the text shows the peak
positions normalized by the local majority Fermi energy²F ↑ . Fig. 3 shows the bare peak positions as observed in
the experiment.
One unexpected finding in fig. 4a in the main body ofthe text is the sudden increase of the minority peak position for high imbalances. This behavior can be traced
back to the data in fig. 3: The minority peak position

The theoretical dissociation spectrum including the
Hartree term U

Starting from a BCS-Leggett mean-field wavefunction
and applying Fermi’s Golden Rule, the RF spectrum canbe described by (neglecting the Hartree term):√Γ (ω) ∝

ω − ωth
ω2

r

1+

ωth
2µ
+
ω
ω

(1)

p
where ωth = ∆2 + µ2 − µ is the dissociation threshold and µ is the chemical potential.
The corresponding quasiparticle dispersion relation is:p
Ek = ∆2 + (²k − µ)2
(2)
2 2

k
.
²k being the free particle dispersion ²k = h̄2m
Hartree terms U modify the quasiparticle excitation
spectrum [5, 6]:p(3)
Ek = ∆2 + (²k − (µ − U ))2

2
column density difference / a.u.

1.0

1.0

a)

1.0

b)

c)

0.5

0.5

0.5

0.0
0.0

0.5

1.0

0.0

0.0
0.0

0.5

1.0

0.0

r / RTF

0.5

1.0

FIG. 1: (Color online) The difference column density profile n↑ (r) − n↓ (r) (radially averaged). a) BCS side (B = 710G), b)
Unitary limit (B = 690G), c) BEC side (B = 671G). Each profile shown is the average of 10 individual profiles. Blue marks
the region of complete spectral overlap between majority and minority components, red marks the region where there is nocomplete spectral overlap. The black arrows at the bottom indicate the radial size of the minority component.

a)30

εF↑ / kHzoverlap

1.0

0.5

20
10
0
0.0

0.5

1.0

b)
0.0
0.0

20

0.5

FIG. 2: (color online) Overlap of majority and minority pairing peak as a function of position in the trap for various interaction strengths. A power law was fitted to the curves asa guide to the eye. Unitary limit (black circles): 1/kF ↑ a = 0,phase boundary at rc /R↑ ' 0.46; BEC side (red triangles):
1/kF ↑ a = 0.39(1), rc /R↑ ' 0.45; and BCS side (blue diamonds): 1/kF ↑ a = −0.25(1), rc /R↑ ' 0.35

peak position / kHzr / R↑
15

10

5

0
0.0

resulting in an RF spectrum ofrp00ω 0 − ωth
ωth
2 µ0
0
+
1
+
Γ (ω ) ∝
ω 02
ω0
ω0

0.5

1.0

r / RTF(4)0
where ω 0 = ω + U , ωth= ωth + U and µ0 = µ − U .
This demonstrates that the spectrum retains its functional form but the entire spectrum is shifted by U .

FIG. 3: (color online) a) Local majority Fermi energy in kHZ,
the error bars are the standard deviation of the mean value.
b) Peak positions of majority and minority in kHz. Majority (Blue): Pairing peak (higher frequencies, only discernible
in the SF region) and quasiparticle peak (lower frequencies).
Minority (red): The peak can be traced well into the normal
region.

Resolution / Experimental broadening

For comparison with the theoretical spectrum, we
model the RF pulse of T = 200 µs length as a squarepulse, in the frequency domain, resulting in a FWHM of1
the RF spectral power of ∆ν = 2 · 2π
1.39 T2 ' 4.4 kHz.

The theoretical spectrum consists of two parts: 1) A dissociation peak including the Hartree energy, described byeqn. 1 with ∆ and U as given in table I in the main text.
2) A quasiparticle peak modeled as a narrow (FWHM= 1 kHz) Lorentzian with a peak height adjusted so that

spectral response / a.u.

3
!

$!

0

2

4

#!

0

2

4

"!

0

2

4

0

2

4

RF offset / ε F↑

FIG. 4: (Color online) Local RF spectra of an equal spin mixture for various normalized local temperatures T /TF ↑ . a)
T /TF ↑ ' 0.20, b) T /TF ↑ ' 0.22, c) T /TF ↑ ' 0.34, d) T /TF ↑ ' 0.55. No local double peak spectrum can be resolved in the RF
spectrum. For comparison, the double peak spectrum of an imbalanced mixture with T /TF ↑ ' 0.06 is added with a red dashed
line.

a)
spectral response / a.u.

the Fourier transform of a square pulse f (ω) ∝

0.0

0.5

1.0
1.5
2.0
RF offset / ε F↑

2.5

3.0

b)

sin2 ωT
2

2 .
( ωT
2 )
Fig. 5 shows that the theoretical spectrum reproduces
our data quite well. The deviation in 5b) might be attributed to additional broadening mechanisms like finite
quasiparticle lifetime, finite temperature and atomic diffusion during the duration of the RF pulse. The convolution causes a small shift of 0.05 ²F ↑ in the spectral peak
position due to the asymmetry of the theoretical spectrum and has been accounted for in the determination of∆ and U.

spectral response / a.u.

The value given above for the experimental resolution
is confirmed by looking at the blurring of the sharp onsetof pair dissociation. Equation 1 predicts that the threshold and peak position in the strongly interacting regime
differ by less than 10% of the Fermi energy. Adjusting
the experimental resolution to ∼ 4 kHz accounts for theexperimentally observed difference of threshold and peak
position of about 0.3 ²F ↑ .
0.0

0.5

1.0
1.5
2.0
RF offset / ε F↑

2.5

3.0

FIG. 5: (color online) Comparison of experimental (red) andtheoretical (black) line shapes for spectra a) and b) of figure
1 of the main body of the paper. The theoretical curve isobtained from a BCS-Leggett mean field description including
the Hartree term and a convolution with the experimental
resolution of ' 4.4 kHz. The values for ∆ and U as calculated
from the peak positions lead to a reasonable agreement withthe data.

it resembles our data. This spectrum was convolved with

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